What the figures prove

• at x = 0.00 the signed product is d² − r² ×21
• at x = -9.00 the signed product is d² − r² ×20
• the distance between points 1 and 2 is the fraction 3/5 ×10
• point 1 of the nine is the same distance from the centre ×9
• the chord for inscribed angle 10° is 2R sin 10° ×9
• the 4-by-4 count checks pair by pair ×4
• the chord at 0° splits into pieces whose product is r² − d² ×4
• the half-chord over the split 2 + 8 has square 16 ×4
• the apex at 120° sees AB at the same angle ×3
• the secant at 180° has near × far = 144 ×3
• a corner of a piece is α+60° ×2
• a corner of a piece is β+60° ×2
• a corner of a piece is γ+60° ×2
• each third at A is 26° ×2
• each third at B is 18° ×2
• each third at C is 16° ×2
• the assembled triangle's angle at A is 78° ×2
• the assembled triangle's angle at B is 54° ×2
• the assembled triangle's angle at C is 48° ×2
• 1/MX − 1/MY is the same for every pair of chords
• 325 = 5²·13 is a sum of two squares in 24 ordered, signed ways
• a corner of a piece is 60°
• a corner of a piece is α
• a corner of a piece is β
• a corner of a piece is γ
• a hexagon has fourteen triangulations
• a hexagon on a circle has a constant total
• a hexagon well off its circle does not
• A lies on the circle
• a triangle whose incentre is on the Euler line has two equal sides
• a trisector and the far side are not parallel
• AB = cos α
• abc/4K is the radius of the circle through the corners
• across every random butterfly the two distances agree
• AD = cos β
• all six feet are the same distance from the midpoint of P and P*
• among isosceles triangles the margin is smallest at the equilateral one
• and each is as long as the least total distance
• and each product of two still fixes a corner of the trisectors' triangle
• and its incentre is the inner circle's centre
• and its second and third
• and its y coordinate
• and not after three
• and only the flat one is equilateral
• and PA·PB = PC·PD
• and so do the other two
• and so does the third
• and so is the nine-point centre with two of them
• and that distance is half the radius of the circle through the corners
• and the angle at C in the diameter's triangle is right
• and the centroid sits twice as far from the orthocentre as from the circumcentre
• and the equilateral triangle is the smallest margin of all
• and the incentre stays off the line on every scalene triangle in it
• and the same construction on the other trisectors is not
• and the same on the other side
• and the tangent meets the radius at a right angle
• and the three pieces make up the side
• and their radii differ
• and touching each escribed one
• as the point nears the circle its conjugate runs away
• at Euler's distance the rotation number is one third
• B and D lie on the circle with diameter AC
• B lies on the circle
• BC = sin α
• BD = sin(α + β): a chord of a unit-diameter circle is the sine of the angle it subtends
• CD = sin β
• climbing from random sets of four comes close to the tetrahedron, and for five and six points the climbs stall above the chain
• conjugating twice returns the point
• cos of an odd multiple is invariant and of an even multiple is not
• each junction is 120°
• each pair is two incidences: each point on the other's circle
• each pair of Malfatti circles touches
• each side over the sine of the opposite angle is the diameter
• each trisector cuts its angle in exact thirds
• eighteen of the twenty-seven are equilateral
• equal products put D on the circle through A, B and C
• equilateral exactly when the three choices do not add to 2 mod 3
• every angle of the triangle is below 120°
• every apex on the arc gives the same angle
• every chain of four tangents closes
• every chain of three tangents closes
• every chord through M has the same product of pieces
• every conjugate lies on one conic
• every prime in the best squared distance leaves remainder 1 on division by 4
• every side of every equilateral one makes (B − C)/3 with the base, give or take 60°
• every side subtends 120° at the point
• every triangle in the sweep has its nine-point circle touching its inscribed one
• every triangle in the sweep puts its nine points on one circle
• every triangulation of a heptagon has four flips
• every triangulation of the cyclic hexagon has the same total
• every trisector triangle in the sweep is equilateral
• for every triangle the signed distances add to R + r
• for the Fermat point the path is straight
• greedy beats Malfatti on every triangle of the grid
• in both coordinates
• in the corrected coordinate every step advances by the same fraction of a turn
• in the plane both facts hold
• including PQ itself
• inscribed angles on the same chord are equal
• junctions in the hexagon's shortest network
• Newton's method solves Malfatti's three side equations
• no random set goes below √3/2
• OI² = R(R − 2r) on every triangle
• on the ellipse the wings cross on opposite sides
• on the ellipse the wings cut the chord equally
• on the hyperbolic plane the three sides are not equal
• on the parabola the wings cross on opposite sides
• on the parabola the wings cut the chord equally
• on the sphere the spread grows with the excess while the triangle is small
• on the sphere the three cubed rotations still compose to nothing
• on the sphere the three sides are not equal
• one angle, not several
• one segment per pair
• opposite angles add to a straight angle
• opposite angles of a cyclic quadrilateral add to π
• past five points the ratio falls steadily, towards about 0.784
• Ptolemy: the diagonals' product is the sum of the products of opposite sides
• pushed off the circle, the totals spread
• random points in a cube are far from the worst case: averages above nine-tenths, minima above the tetrahedron's
• rotating A by 60° about B gives the apex C′
• so BC / sin A is the diameter
• so do the angles at C and B
• so the circumradius is at least twice the inradius
• the angle at D stands on the same arc BC as the angle at A
• the angle between the chord and the tangent equals the inscribed angle
• the angle on a diameter is a right angle
• the angles add to 180°
• the angles at A and C stand on the same arc BD
• the angles at A and D stand on the same arc and are equal
• the angles at D and B stand on the same arc AC
• the apex is on the major arc
• the apex lies on the circle
• the broken path C′P′PC has the length of the total distance from P
• the centre's angle is twice the apex's
• the centroid has a larger total than the Fermat point
• the centroid's partner is the symmedian point
• the chain does not close after three steps
• the chain's corners fall in proportion to one over the tangent length
• the chord is a diameter
• the circumcentre, centroid and orthocentre are collinear
• the circumcentre, the centroid and the orthocentre lie on one line
• the collinearity holds across the sweep
• the conic passes through O, H and the symmedian point
• the construction and the minimisation find the same point
• the corner C has an angle of at least 120°
• the crossing of the construction lines is no longer the minimum
• the distance satisfies Fuss's relation
• the exterior angle is twice the base angle
• the Fermat views take no options
• the greedy circles cover more of the triangle than Malfatti's
• the grid is between 3 and 14 points on a side
• the hexagon with its centre attains √3/2
• the hexagon's shortest network has length 3√3
• the incentre is its own partner
• the incentre is not on that line, which the determinant reports rather than being told
• the incentres make a right angle at every corner
• the inner circle is less than half the outer
• the inner triangle's first two sides are equal
• the isosceles family passes through nought at the equilateral shape
• the iteration finds a network no longer than the spanning tree
• the least total is at the wide corner
• the lines from A and B are not parallel
• the lines from B and C are not parallel
• the lines from C and A are not parallel
• the lines from corners to opposite apexes are equal
• the marked lengths are realised in many directions
• the meeting point has weights a·sec⁴(A/4) on the corners
• the named quantities split as the angle criterion predicts
• the network with two 120° junctions has length 1 + √3
• the nine-point centre is the midpoint of that line
• the nine-point circle touches the escribed opposite A circle from outside
• the nine-point circle touches the escribed opposite B circle from outside
• the nine-point circle touches the escribed opposite C circle from outside
• the nine-point circle touches the inscribed circle from inside
• the option is one this view reads
• the orthocentre's partner is the circumcentre
• the pair count by direction agrees with checking every pair
• the pocket beside the second circle never holds the largest third circle
• the point is inside the circle
• the quadrilateral's angles add to 360°
• the regular simplex's ratio falls with every dimension, from √3/2
• the regular tetrahedron saves 18.7%, and a chain of twelve points along the helix saves more
• the rotation number never increases as the inner circle moves out
• the sample includes obtuse triangles, where one distance counts negative
• the search never goes below √3/2
• the shape sweep is between 6 and 60 steps on a side
• the short-side pairing wins and the diagonal pairing is worst
• the side is 8R sin α sin β sin γ
• the side opposite A makes (B − C)/3 with BC
• the splitters at A and B are not parallel
• the splitters at B and C are not parallel
• the splitters at C and A are not parallel
• the spread is rounding error exactly for the invariant functions
• the spread shrinks in proportion to the excess
• the Steiner views take no options
• the sweep covers a decent share of the shape space
• the sweep found enough non-degenerate triangles to mean something
• the sweep found enough triangles
• the sweep found enough triangles to mean something
• the sweep runs over between 20 and 2000 triangles
• the tangent length squared is the same number
• the tangent ray on the far side of the chord makes the inscribed angle with it
• the third circle goes deeper exactly when k(α)² beats k(β)
• the third line passes through the same point
• the three angles are the angles of a triangle
• the three angles are those of a triangle
• the three angles are those of a triangle and none is a sliver
• the three lines meet at one point on every triangle of the sweep
• the three reflected lines are parallel
• the three reflected lines meet where the weights say, a²/x : b²/y : c²/z
• the three unit pulls cancel at the Fermat point
• the triangle has three corners and some area
• the triangles are similar, so PA/PD = PC/PB
• the two circles are not concentric
• the two diagonals give the same total
• the two halves are the whole angle
• the two sides agree at every angle
• the two sides of a piece are not parallel
• the two triangulations agree on Σ cos 3x and disagree on Σ cos 2x
• the two triangulations' inradii add to the same total
• the view is one the family draws
• the weights for the centroid give its x coordinate
• the weights for the circumcentre give its x coordinate
• the weights for the incentre give its x coordinate
• the weights for the nine-point centre give its x coordinate
• the weights for the orthocentre give its x coordinate
• the wings cross on opposite sides of M
• the wings cut the chord at equal distances from its midpoint
• three roads meet at 120° at every junction
• triangle OPA is isosceles
• triangle OPB is isosceles
• triangulations of a heptagon
• two touching circles on one line touch it 2√(r r′) apart
• unequal products put D off it
• up to seven points no branching pattern beats the one that follows the chain
• which passes through the three corners
• while MX − MY itself varies
• with the circles concentric every step turns by the same angle
• Σ cos is constant over the flip graph

• the two curves cross at a right angle at (0.543, 0.919) ×28
• the point found by straightedge along line 1 lies on the conic through the five ×18
• the ellipse with semi-axis 1.55 has the same foci ×9
• the hyperbola with semi-axis 0.42 has the same foci ×8
• line 1 is tangent to the ellipse ×6
• the solved conic passes through point 1 ×5
• every point of the 0.55 curve keeps the ratio ×3
• the foci are between 0.3 and 2.5 from the centre ×2
• a circle needs a cut that produces one
• a ellipse needs a cut that produces one
• a hyperbola needs a cut that produces one
• a parabola needs a cut that produces one
• all but a few of the lines give a finite construction
• and each point is as far from the focus as from the directrix
• and its sixth point lies on the conic
• and on its hyperbola
• and on the hyperbola
• and reaches it going forwards, not backwards
• and so they are for twenty-four other pairs
• and the image of Y on the one through Y
• and the other corner along the other hyperbola
• and the two distances differ by the same amount wherever it strikes
• and the two legs together are the same length wherever it bounces
• and travels towards it rather than away from it
• between 3 and 60 rays are traced
• between 6 and 60 lines are constructed
• between one and four curves
• each corner lies on its ellipse
• each curve is a name and six finite coefficients
• each ratio is between nothing and 2.4
• each vertex lies on the ellipse
• every ellipse's semi-axis is longer than the focal distance
• every hyperbola's semi-axis is shorter than the focal distance
• every pair of curves was checked at its crossing
• every parallel ray reflects to the focus
• five points in general position determine one conic
• five points, each within four units of the centre
• for any two points of the first ellipse, the distances across the stretch are equal
• in the ellipse every segment touches the same confocal conic
• in the ellipse the caustic parameter is constant to rounding
• moved off the ellipse, some corner decisively breaks the alignment
• neighbouring tangents meet at a finite corner
• no three of the five points are nearly collinear
• sides one-two and four-five meet at a finite point
• sides three-four and six-one meet at a finite point
• sides two-three and five-six meet at a finite point
• six distinct points of contact, given by their angles in degrees
• six distinct points, given by their angles in degrees
• some paths in the bent table lock to a quarter-turn per bounce: an island chain of period four
• the bent table has a band of starts locked at a quarter-turn
• the bent table's quantity moves, more for the larger bend
• the construction fits a drawable window
• the crossed distances across the stretch are equal
• the crossing lies on the ellipse
• the curve named a hyperbola really is one
• the curve named a parabola really is one
• the curve named an ellipse really is one
• the cutting plane is tipped by less than a quarter turn
• the discriminant and the count of crossings with a large circle agree that a hyperbola is a hyperbola
• the discriminant and the count of crossings with a large circle agree that a parabola is a parabola
• the discriminant and the count of crossings with a large circle agree that an ellipse is an ellipse
• the drawn construction is finite
• the drawn line's direction is between nought and a half turn
• the ellipse has a longer axis and a shorter one
• the ellipse passes through a quarter-turn at one start; the bent table holds it across a range
• the ellipse's semi-axes are between 0.3 and 4
• the hexagon's corners fall near enough to draw
• the hyperbola has both semi-axes positive
• the image of X lies on the confocal hyperbola through X
• the incoming ray has a length to normalise
• the island's width grows like the square root of the bend
• the mirror is an ellipse, a parabola or a hyperbola
• the normal to the curve has a length to normalise
• the order visits each of the six points once
• the outgoing ray has a length to normalise
• the point is on the upper half of the curve
• the radius to the first focus has a length to normalise
• the radius to the second has a length to normalise
• the ratio is between nothing and 2.4
• the reflected ray lies on the line through the near focus
• the reflected ray passes through the other focus
• the stretch carries a corner of the first ellipse along its hyperbola to the second
• the tangent direction has a length to normalise
• the tangent makes the same angle with both focal radii
• the three construction points lie on the Pascal line by construction
• the three diagonals pass through one point
• the three meeting points fall near enough to draw
• the three meeting points of opposite sides lie on one line
• the three values give three different kinds of curve
• the two diagonals of the curved quadrilateral are equal
• the view is one the family draws
• the window is between 1 and 8 wide
• tilted off the ellipse, some tangent decisively breaks the concurrency
• two clearly different points on the inner ellipse
• two ellipses, the second larger, both longer than the focal distance
• two hyperbolas, the second wider, both shorter than the focal distance
• with an outer ellipse of the same shape but different foci, the crossed distances differ
• with the second ellipse and hyperbola on different foci the diagonals differ

• the smallest square has side gcd(34, 13) ×4
• the squares tile the 34 by 13 rectangle ×4
• a ≥ b ≥ 1 are whole numbers, a at most 90
• a ≥ b ≥ 1, a at most 12
• a and b are read by the tiling and the two game views
• a count
• a quartic with signs ++−++
• and no choice of signs at all does better than rounding to the nearest
• and with the rule that the first run longer than one decides
• at is a short list of whole points
• at is read only by the sturmtable view
• coeffs is read only by the descchain, descproof views
• consecutive Fibonacci positions alternate between win and loss
• coords is plain or corner
• coords is read only by the fewcurves view
• d is a whole number from 3 to 60
• d is read only by the fewcount view
• deg is read only by the descgap view
• each solution satisfies both equations
• every bar fits the drawing
• every degree in the list has a band of five
• every interval the count isolates holds exactly one real root found separately
• every polynomial found obeys the rule
• exactly one of n and F − n can be paid
• five sign changes along the curve
• for this polynomial the bound is attained
• Haas's c = 1.1 lies in the band at d = 54
• N is from 12 to 60
• N is read only by the gamegrid view
• never more than five, as Li, Rojas and Wang proved
• no polynomial has more positive roots than sign changes
• p(−x) is never above p(x) for positive x
• poly is read only by the sturm, sturmtable, isolate views
• range is an interval [lo, hi]
• range is read only by the sturm view
• Rolle: one more root at most than the next member
• rounding to the nearest is never slower than rounding down
• system is read only by the fewcurves, fewtrace views
• the coins share no factor and fit the picture
• the count drops by exactly one across each real root
• the count reaches five
• the drawn verdict is the searched one
• the drop from −∞ to +∞ is the number of distinct real roots
• the halving isolates every real root
• the leftover strip is still there after every pass
• the only pairs never found are the two Grabiner proved impossible
• the option is one this view reads
• the pair is two whole numbers, the first at least the second
• the polynomial is one of the named examples
• the positive roots found on the graph are the ones Sturm's chain counts
• the ratio of the averages falls towards its limit
• the rule's excess is never negative and always even
• the search agrees with the golden-ratio rule
• the slopes are 12 ln 2/π² and 12 ln φ/π² per unit of ln N
• the slowest pairs for rounding to the nearest are built from Pell numbers
• the squares peeled at each pass are the continued fraction of the ratio
• the system has five positive solutions
• the system is one of the two named
• the two general bounds for n = 2, k = 3
• the verdict follows the side of φ the ratio falls on
• the view is one the family draws
• three coins with no common factor
• trials is read only by the descgap view
• two positive roots force two negative ones
• upTo is from 5 to 11
• upTo is read only by the gamefib view
• V(lo) − V(hi) equals the number of distinct real roots found separately
• whole coefficients, highest first, leading positive and constant non-zero
• with every root real the rule is exact

• the polynomial for power 1 gives the exact sum at n = 1 ×150
• 0 and 3: triangular triples match odd-square triples ×121
• the 1-th 5-gonal number is 1 dots, none repeated ×9
• Fermat's bound of 3 is reached for 3-gonal numbers ×6
• for power 2 the remainder approaches m/12 ×4
• no number up to 20000 needs more than 4 4-gonal numbers (Fermat, Cauchy) ×4
• the first 6 odd numbers sum to 6² ×4
• a range from 60 to 400 ×2
• a number from 1 to 300
• a range from a thousand to fifty thousand
• a range from ten thousand to two million
• a triangular index from 1 to 7
• an even index from 10 to 40
• and it is the centre
• below a million exactly 241 numbers need five tetrahedral numbers, the last 343,867
• between 3 and 12 steps
• each of the 54 represents every number up to 300, as the fifteen theorem promises
• every even number missed is 4^a(16b + 6), a family no form of this shape can reach
• every number an escalator misses first is one of 1, 2, 3, 5, 6, 7, 10, 14, 15
• every number is a sum of three triangular numbers
• every odd Bernoulli number past the first is zero
• exactly 54 diagonal forms pass the fifteen test
• Faulhaber's polynomial gives the staircase's total
• four tetrahedral numbers give many more representations than three
• n is two triangular numbers exactly when 4n + 1 is a sum of two squares
• no number below ten million needs more than seven octahedral numbers
• only the centre cell is left over
• polygons with three to eight sides
• powers from 2 to 7
• powers up to between 3 and 8
• Ramanujan's fifty-fifth form, x² + 2y² + 5z² + 5w², misses 15
• share × √(log x) changes little over two decades
• the kind is gnomon or one of triangular, galileo, staircase, faulhaber, correction, bernoulli, eight, trisum, twotri, twotridensity, threetri, polydots, polycount, polytable, escalator, ramanujan54, lopsided
• the L-shaped shells fill the square
• the leading coefficient is 1/(m+1), the integral's
• the next is 1/2, half the last step
• the odd numbers missed up to the range end at 2719, eighteen of them
• the power is between 1 and 6
• the share keeps drifting down
• the signs alternate, positive at 2, 6, 10, …
• the size matches 2(2k)!/(2π)^(2k)
• the staircase stands above the curve by about half its last step
• the three triangles add to the number
• to two million the most needed is 3, 5 and 8
• two to six shells
• up to 20000, only 9, 21, 31, 43, 55 and 89 need five pentagonal numbers

• generation 0: the counting rule agrees with the drawn subdivision on thin halves ×7
• … and one thick half
• … and two thick halves
• −2cos 144° = φ
• √2's takes it out twice, after the first step
• 1 to 7 generations
• 1 to 8 cuts
• 2cos 72° = 1/φ
• 2cos 72° is a root of w² + w − 1
• 4 to 14 subtraction steps
• 8/5's stops, because a common unit exists
• a thick half becomes one thin half …
• a thick half has a 108° apex
• a thin half becomes one thin half …
• a thin half has a 36° apex
• and as long as the piece of the side it cuts off
• and it climbs towards φ from below at every cut
• and on thick ones
• and so is 2cos 144°
• diagonal minus side is the smaller pentagon's diagonal
• diagonal over side is φ in every pentagon
• each evaluation shrinks the bracket by 1/φ
• each inner pentagon is smaller by φ²
• each of the first nine Markov values is attained by a repeating pattern of 1s and 2s
• each step shrinks the pair by φ
• each subtraction leaves something smaller than what it subtracted
• every Markov number below 10^30 is the largest member of exactly one triple
• every triple generated satisfies x² + y² + z² = 3xyz
• every value below 3 is a Markov number's value
• four to twelve evaluations
• golden section beats paired bisection, which beats thirds, after forty evaluations
• in the largest disc the ratio of thick to thin is within 0.05 of φ
• leg over base is φ
• one square per Fibonacci number
• one to five levels
• the 1s and 2s come in pairs, read round the block
• the bisector is as long as the base
• the chord to the second neighbour over the chord to the first is φ
• the convergents alternate about φ
• the convergents' constants approach the Markov value from above
• the diagonal's end piece is s/φ
• the final bracket still contains the peak
• the first two values are √5 and √8
• the kept point lands on the new bracket's c point exactly when c² + c = 1
• the last convergent is close to φ
• the long side is the next Fibonacci number
• the longest row settled by n evaluations is one less than the Fibonacci number two places on
• the number of convergents plotted is between 2 and 20
• the number of whirling squares is between 1 and 12
• the pair at every step is in the ratio φ
• the pieces fill their parent exactly
• the ratio has converged to φ
• the rectangle is nearly golden
• the search stalls near the square root of the double-precision epsilon, about 10⁻⁸
• the short side is the one before it
• the thick half built here has its 108° apex
• the values increase with the Markov number and stay below 3
• the view is one the family draws
• the whirling squares tile their rectangle
• φ's subtraction always takes the smaller out exactly once

• the circles round vertex 0 subtend exactly a full turn at it ×7
• ring 1: and the circle it surrounds from outside ×2
• ring 1: and touches the containing circle from inside ×2
• ring 1: each circle touches the next ×2
• a circle missing the centre inverts to a circle
• a circle orthogonal to the mirror is carried to itself
• a circle through the centre inverts to a line
• a point inside goes outside and a point outside comes in
• a point on the mirror circle is left exactly where it is
• and a line missing the centre inverts to a circle
• and a point of the circle lands on it
• and conjugates the imaginary part
• and every edge of the triangulation is a tangency of two circles
• and every point of the straight line lands on its circle
• and its curvature is a whole number
• and one for each of the eight ways of choosing inside or outside
• and so do their images, because a circle's image is a circle
• and the two short gaps overshoot
• and touches each of them on the page
• between two and four points are inverted, none of them at the centre
• both images are circles here
• centre and all
• circles whose vertices are not joined do not overlap
• each drawn circle really touches each given one
• each image gap is the original gap divided by the two distances from the centre
• each new circle satisfies Descartes's relation with its three parents
• every curvature in the packing is a whole number
• every pair of the seed circles touches
• every point of the circle lands on its image
• every point of the circle through the centre lands on its image
• every walk round an interior vertex closes on the circles already placed
• four points on one circle have a real cross-ratio
• four points, in order round the circle and none of them nearly coincident
• in both parts
• neither boundary passes through the inversion centre
• no chain circle passes through the inversion centre
• no point being inverted sits at a mirror's centre
• no point being inverted sits at the centre
• off the circle the identity becomes a strict inequality
• off the circle the three images are not collinear
• on a circle the products satisfy Ptolemy's identity exactly
• one inversion leaves the real part of the cross-ratio alone
• so the two short image gaps add to the long one
• the angle between the two curves is the same after the map as before
• the chord's end sits on the mirror
• the circle's image is a circle
• the concentric ring closes exactly when the radius ratio is the sine
• the constructed point is the one inside the mirror
• the drawn ring is genuinely lopsided rather than a disguised concentric one
• the first mirror has a sensible radius
• the four points are not concyclic, so their cross-ratio has a real imaginary part to conjugate
• the fourth point is pushed off the circle by less than half a radius
• the gasket is drawn to between one and six generations
• the image of the centre is not the centre of the image
• the image of the crossing point is on both image circles
• the inner circle has a sensible radius
• the inversion centre is inside the inner circle and off its centre
• the line inverted does not pass through the centre
• the mirror circle has a sensible radius
• the ring has between three and twelve circles
• the ring round the centre closes up
• the second mirror is offset by a sensible amount
• the seed satisfies Descartes's relation
• the tangent construction lands on the same point the formula gives
• the three centres are not collinear
• the three given circles lie outside one another
• the three images fall on one straight line
• the two circles really cross
• the two distances from the centre multiply to the square of the radius
• the view is one the family draws
• three boundary radii, repeated round the outside
• three circles in general position admit exactly eight tangent circles
• three given circles, each with a sensible radius
• two inversions put the cross-ratio back exactly
• which passes through the centre of the mirror

• the 3-gon obeys Bonnesen's inequality ×10
• 2 pieces against a wall do exactly as well as a 4-gon does in the open ×7
• the 3-gon encloses less than a circle of the same perimeter ×7
• the 3-gon is drawn at the same perimeter as the rest ×7
• regular 3-gon does not beat the circle ×5
• the arc in the 60° corner is 300 long ×4
• 6 reflected copies of the 60° sector make a whole disc ×3
• the 2-piece fence is 300 long ×3
• the 2-piece fence's share of the half-disc is the 4-gon's quotient ×3
• 2:1 rectangle does not beat the circle
• a pentagon symmetrised has more than five corners
• a quadrilateral with perpendicular unit diagonals has area one half and diameter one
• a quotient from 0.9 to just below 1
• a regular polygon of 3 to 12 sides
• a Reuleaux polygon has an odd number of corners
• a Reuleaux triangle, pentagon or heptagon, cut into one to four pieces per arc
• a rhombus is higher the further its angle is from a right angle
• a triangle symmetrised about a side's perpendicular is a triangle
• a unit cube has six units of face
• a unit square cell needs two units of wall of its own
• after 60 it is nearly a disc
• an L does not beat the circle
• and every one holds less than the half-disc
• and holds twice the area
• and is still short of the circle's
• and it is the circle
• and its area is a quarter of the perimeter, squared
• and never lengthens the boundary
• and shortens the boundary
• and the area is strictly larger
• and the best angle between the pieces is a right angle
• and the circle's quotient is exactly one
• and the ring's square like the inverse fourth power
• and the sector holds the fence squared over twice the angle
• between 10 and 80 cells
• between 50 and 400 random shapes
• between one and four fences of 2 to 12 pieces
• each 15-gon has diameter one and the largest perimeter
• each arc has radius one and turns through π/m about the opposite corner
• each more regular sides comes closer to the disc and never reaches it
• each round of symmetrisation brings the shape nearer round
• each structure improves on the one before
• even the best rectangle falls short of the circle
• every extra side encloses strictly more
• every nudged 5-gon is higher than the regular one
• every random convex shape obeys Bonnesen's inequality
• every shape with that quotient has the same allowance
• every side subtends π/n at a point one unit away
• every step keeps the area
• exactly one shape in the list reaches one, and it is the circle
• fences of up to between 4 and 12 pieces
• five steps bring the triangle close to equilateral
• for odd n the optimum is the regular polygon
• four corner angles from 30 to 180 degrees
• Graham's hexagon beats the regular one by more than 0.025
• hexagons need the least wall, triangles the most
• more pieces hold more
• more sides always need less wall
• no cell beats the regular polygon with its number of sides
• no triangle on the grid is lower than the equilateral one
• nothing of diameter one has more area than the disc
• reflecting the dent leaves the perimeter exactly as it was
• Reuleaux triangle does not beat the circle
• so the fence holds less than its length squared over 2π
• so the quotient rises
• so their average wall is at least the hexagon's
• symmetrisation keeps the area
• the 12-gon's circumradius is found
• the 12-gon's inradius is found
• the 12-gon's ring is no wider than its deficit allows
• the averaged pair is never longer
• the best split is an even one
• the bump is tuned to hold 0.99 of the circle's area
• the bump's ring is no wider than its deficit allows
• the cells average six sides
• the cells cover the square exactly once
• the circle does not beat the circle
• the circle of that perimeter encloses more than any of the polygons drawn
• the computed value for a rectangle matches π²(r + 1/r)
• the deficit falls like the inverse square of the number of sides
• the dent is between nothing and 160 units deep
• the diameter graphs have 5, at least 6 and at least 8 edges
• the doubled curve is twice the fence
• the doubled curve obeys the closed-curve inequality
• the ellipse is tuned to hold 0.99 of the circle's area
• the ellipse's circumradius is found
• the ellipse's inradius is found
• the ellipse's ring is no wider than its deficit allows
• the equilateral triangle's exact value, 4π²/√3
• the excess falls as 4ζ(3)/n³ of the disc's value
• the excess grows as the square of the nudge, the mark of a minimum rather than a corner
• the hexagon's circumradius is found
• the hexagon's inradius is found
• the hexagon's ring is no wider than its deficit allows
• the largest rectangle in the sweep is the square
• the lowest grid triangle is the one nearest equilateral
• the lowest mode keeps one sign throughout
• the optimisation reproduces the published optimum
• the option is one this view reads
• the path is Hammersley's or the optimised one
• the polygon cut from a Reuleaux polygon has diameter one and perimeter 2n sin(π/2n)
• the polygons drawn have between 3 and 60 sides
• the raster agrees with the formula for each radius
• the regular pentagon is lowest in both measures
• the regular polygons' shortfall from the disc falls like 1/n²
• the reuleaux's circumradius is found
• the reuleaux's inradius is found
• the reuleaux's ring is no wider than its deficit allows
• the search finds the quadrilateral with perimeter 2 + 4 sin 15°
• the shared perimeter is between 60 and 900 units
• the sides of the cells add to exactly six per cell
• the square's circumradius is found
• the square's exact value, 2π²
• the square's inradius is found
• the square's ring is no wider than its deficit allows
• the stadium is tuned to hold 0.99 of the circle's area
• the stadium's circumradius is found
• the stadium's inradius is a root of the quadratic
• the stadium's inradius is found
• the stadium's ring is no wider than its deficit allows
• the sweep takes between 20 and 2000 samples
• the triangle's circumradius is found
• the triangle's inradius is found
• the triangle's ring is no wider than its deficit allows
• the view is one the family draws
• turning directions approach a disc
• two fixed directions stop short, at a shape both leave alone
• two pentagons with almost the same quotient differ clearly in eigenvalue
• two to four shapes
• which holds an eighth of the fence squared — half of a square
• whose two sides are equal
• πρ² − Lρ + A is not positive at the 12-gon's two radii
• πρ² − Lρ + A is not positive at the bump's two radii
• πρ² − Lρ + A is not positive at the ellipse's two radii
• πρ² − Lρ + A is not positive at the hexagon's two radii
• πρ² − Lρ + A is not positive at the reuleaux's two radii
• πρ² − Lρ + A is not positive at the square's two radii
• πρ² − Lρ + A is not positive at the stadium's two radii
• πρ² − Lρ + A is not positive at the triangle's two radii

• the even construction on 6 points has n/2 ×8
• Melchior's inequality holds for 8 random points of a 4 × 4 grid (1) ×3
• a line meets a cubic in at most three points
• a line with three points on it yields a strictly smaller distance
• a polygon of 3 to 10 sides
• a regular m-gon's chords point in exactly m directions
• and Csima and Sawyer's
• between four and twelve points on the curve
• Böröczky's set has exactly n/2 ordinary lines
• de Bruijn and Erdős: at least as many lines as points
• each of the q² slanted lines of the plane over Z/q has q points
• each pair of points lies on one connecting line only
• every construction respects Kelly and Moser's bound
• every family has at least as many lines as points
• every line through P either vanishes (if ordinary) or survives without P
• every point on a connecting line gives a dual line through its crossing
• every set with exactly as many lines as points is a near-pencil
• every triple is either collinear or clearly not — no decision rests on rounding
• every zero-sum triple of whole numbers is a row
• Melchior's inequality holds for a triangle, its midpoints and its centroid
• Melchior's inequality holds for five in a line and one off it
• Melchior's inequality holds for four points in general position
• Melchior's inequality holds for the nine-point grid
• Melchior's inequality holds for three points
• Melchior's inequality holds on every configuration
• no configuration beats one row per three pairs
• no configuration found beats the proved bound
• no configuration searched avoids ordinary lines altogether
• no point of the grid is on a line with every other point by itself
• no set searched has fewer lines than points
• no two dual lines are parallel after the turn
• on two lines the busiest point is on about half
• removing an end of an ordinary line removes that line entirely
• Roberts's floor holds on the drawn arrangement
• some connecting line holds exactly two of the points
• some point is off some connecting line
• some set of this size is not all in one line
• the block fits the picture for k from 2 to 6
• the busiest point is on one more line than the larger side has points
• the census runs to between four and eight points
• the closest point-and-line pair uses a line with exactly two points
• the connecting lines between them account for every pair of points
• the cyclic group on the cubic meets Sylvester's count at every n
• the distance the argument produces is strictly the smaller one
• the drawn witch has the counted rows
• the five-by-five grid has 140 connecting lines
• the grid searched is three or four across
• the grid searched is three or four points across
• the highlighted triple adds to zero modulo n
• the highlighted triple is three point indices
• the option is one this view reads
• the ordinary line chosen exists
• the point set is one this family knows
• the points are drawn between 12 and 60 units apart
• the search finds no configuration with every point on fewer than n/2 lines
• the search runs to between four and seven points
• the set has an ordinary line to remove a point from
• the set is not all in one line
• the smaller set, not in one line, already has as many lines as points
• the three-point lines are the zero-sum triples of the cyclic group
• the view is one the family draws
• the window runs from −k to k, k between 2 and 5
• the witch never beats the known maximum
• three points of the witch are in line exactly when their indices add to zero modulo n
• three points of y = x³ + ax are in line exactly when their x add to zero
• with seven points on each of two lines the busiest point lies on eight lines

• the order-4 axes carry 9 turns between them ×7
• a turn of order 3 sits on an axis of order 3 ×5
• and each such axis carries a turn of order 4 ×4
• 3 triangles at a vertex leave a gap ×3
• 5-cell: and the edge count agrees with that degree ×3
• 5-cell: every vertex is the same distance from the centre ×3
• 5-cell: every vertex meets the same number of edges ×3
• 2ᵏ times the A determinant is k + 1
• 2ᵏ times the E determinant is 9 − k
• 3 hexagons at a vertex cannot leave a gap
• 3 pentagons at a vertex leave a gap
• 3 squares at a vertex leave a gap
• 4 pentagons at a vertex cannot leave a gap
• 4 squares at a vertex cannot leave a gap
• 6 triangles at a vertex cannot leave a gap
• a cube can pass through a hole in a cube of the same size scaled down by 3√2/4
• a label above six never extends past two mirrors
• a star polygon has between 5 and 13 points
• along this arc the factor peaks near 1.04, short of the 3√2/4 a tunnel off the arc reaches
• and 32 − 90 + 60 = 2, the Euler number of any sphere
• and at the depth found every edge is the same length
• and every one of them is the same triangle
• and sends every corner to a corner
• and steps by at least 2, less than half of p
• and steps by between 2 and 6
• and the angle at each point is π − 2πq/p
• and the group has that many mirrors
• as a surface it has thirty-two corners
• at several numbers of charges most starts miss the lowest valley
• below sixteen charges every start finds the same arrangement; from sixteen on, several valleys appear
• compact hyperbolic diagrams stop after five mirrors
• cuboctahedron: and lies on a circle, so it is regular
• cuboctahedron: and the walk round a corner visits every face at it once
• cuboctahedron: each face has equal sides
• cuboctahedron: Euler's formula still holds
• cuboctahedron: every edge is the same length
• cuboctahedron: every vertex is surrounded by the same faces in the same order
• each candidate is a turn rather than a reflection
• each polytope is one this family builds: 5-cell, tesseract, 16-cell, 24-cell
• each solid is one this family cuts: truncated-tetrahedron, truncated-cube, truncated-octahedron, cuboctahedron, icosidodecahedron, truncated-icosahedron
• every diagram found has a name
• every edge of the solid has the same length
• every other valley is above the lowest by less than one per cent
• every Platonic solid can pass through a hole in a copy of itself
• every shadow's area lies between 1 and √3
• every turn has an axis to be about
• exactly five vertex figures close up
• face on, the square shadow holds a square of the same size and no larger
• five charges settle as a triangular bipyramid, below the best square pyramid
• for these numbers of charges every one of the eight random starts settles into the same arrangement
• from random starts a local search sometimes finds a passage and sometimes gets stuck below factor 1
• icosidodecahedron: and lies on a circle, so it is regular
• icosidodecahedron: and the walk round a corner visits every face at it once
• icosidodecahedron: each face has equal sides
• icosidodecahedron: Euler's formula still holds
• icosidodecahedron: every edge is the same length
• icosidodecahedron: every vertex is surrounded by the same faces in the same order
• no neighbour lies straight along the corner's own axis
• order of the group A₃
• order of the group A₄
• order of the group B₃
• order of the group B₄
• order of the group D₄
• order of the group F₄
• order of the group H₃
• order of the group H₄
• read as twelve pentagrams meeting five at a point, the number is −6
• sixteen charges have two valleys, at 92.911655 and 92.920354
• tesseract: and the edge count agrees with that degree
• tesseract: every vertex is the same distance from the centre
• tesseract: every vertex meets the same number of edges
• the axes account for every turn, the one that does nothing included
• the corner polygon's edge starts shorter than the remaining edge and ends longer
• the correction to N²/2, divided by N^(3/2), stays between −0.6 and −0.4
• the cube and the dodecahedron are not
• the cube and the octahedron reach the same factor
• the diagrams on one to eight mirrors number 1, 4, 3, 5, 3, 4, 4, 4
• the energy falls steadily as the twist grows from a cube to the antiprism
• the faces at a corner close into a ring
• the group is finite
• the hole is cut through a hexagonal shadow, seen along a direction near a long diagonal
• the kaleidoscope is tetrahedral, octahedral or icosahedral
• the number of distinct turns is twice the number of edges
• the option is one this view reads
• the path visits every point exactly once
• the path winds about the centre q times
• the point is off at least one mirror
• the projected solid has a measurable extent
• the search comes within a tenth of a degree of every proved value
• the search reaches the published minimum energy
• the shadow of a unit cube has area 1 face on and √3 along a long diagonal
• the shortfalls from six neighbours add to exactly twelve, as Euler's formula for the sphere requires
• the solid is one of the five
• the solids made of triangles solve both problems
• the spiky surface is sixty triangles
• the star has between 5 and 13 points
• the step and the number of points share no factor, so the path closes once
• the tetrahedron, octahedron and icosahedron are minima
• the three mirrors generate the icosahedral group of order 120
• the three mirrors generate the octahedral group of order 48
• the three mirrors generate the tetrahedral group of order 24
• the traced solid has Euler characteristic two
• the turn in four dimensions is at most half a circle
• the view is one the family draws
• triangles where the determinant's sign matches the angle sum
• truncated-cube: and lies on a circle, so it is regular
• truncated-cube: and the walk round a corner visits every face at it once
• truncated-cube: each face has equal sides
• truncated-cube: Euler's formula still holds
• truncated-cube: every edge is the same length
• truncated-cube: every vertex is surrounded by the same faces in the same order
• truncated-icosahedron: and lies on a circle, so it is regular
• truncated-icosahedron: and the walk round a corner visits every face at it once
• truncated-icosahedron: each face has equal sides
• truncated-icosahedron: Euler's formula still holds
• truncated-icosahedron: every edge is the same length
• truncated-icosahedron: every vertex is surrounded by the same faces in the same order
• truncated-octahedron: and lies on a circle, so it is regular
• truncated-octahedron: and the walk round a corner visits every face at it once
• truncated-octahedron: each face has equal sides
• truncated-octahedron: Euler's formula still holds
• truncated-octahedron: every edge is the same length
• truncated-octahedron: every vertex is surrounded by the same faces in the same order
• truncated-tetrahedron: and lies on a circle, so it is regular
• truncated-tetrahedron: and the walk round a corner visits every face at it once
• truncated-tetrahedron: each face has equal sides
• truncated-tetrahedron: Euler's formula still holds
• truncated-tetrahedron: every edge is the same length
• truncated-tetrahedron: every vertex is surrounded by the same faces in the same order
• turning one square through 45° gives the minimum for eight charges
• twelve charges have the icosahedron's known energy
• two hundred starts for fifty-five charges find at least five valleys, the lowest at 1287.772721
• untwisted, the best height gives exactly the cube

• 157 is among the congruent numbers whose triangles are too large for the search
• 361 congruent square-free numbers up to 1,000 by Tunnell's criterion
• 44, 117, 240 has three whole face diagonals and no whole space diagonal
• a larger exponent gives a ball that contains the smaller one
• a marked third side is inside the range drawn
• a triple whose squares add gives a right angle
• a triple whose squares do not add gives an angle that is not right
• and below one it is not, which is why p < 1 gives no distance at all
• and every one the search found is in the tree
• and half the rectangle — same base, apex on the parallel through the altitude
• and has not appeared before in the tree
• and is primitive
• at nine dimensions the inner sphere reaches the cube's faces
• at ten it is outside them
• consecutive knots are one unit apart
• doubling (12, 36) gives (25/4, −35/8) on the curve
• each fixed ball drawn behind is between 1 and 12
• each is a right triangle with the stated area
• each leg at the corner is a positive length
• each leg is between nothing and a quarter turn
• each leg of the triangle is a positive length
• each side is a positive length
• each side of the box is a positive length
• every box from Saunderson's formula is an Euler brick
• every node of the tree is a right triangle in whole numbers
• every number the search finds a triangle for passes Tunnell's test
• every point drawn is at distance one in this norm
• every primitive brick has an edge divisible by this number
• every square-free number of the form 8k + 5, 6 or 7 up to 1,000 passes Tunnell's test
• four triangles and the tilted square fill the left square
• four triangles and the two upright squares fill the right square
• no perfect cuboid has every edge below 300,000
• no primitive triple has area a square, twice a square or three times a square
• so the square and the rectangle are equal
• the altitude cuts the big square into exactly these two rectangles
• the angle at C is a right angle
• the angle at the corner is a right angle
• the ball meets the diagonal at 2^(-1/p) in each coordinate
• the box has three positive sides
• the constructed triangle's hypotenuse is the root of the sum of the squares
• the corner is square exactly when the squares add
• the dimensions drawn run from 1 to between 4 and 40
• the doubled point gives the triangle (7/10, 120/7, 1201/70) of area 6
• the drawn third side is c
• the eight triangles are all the same
• the exponent is between 0.4 and 12 — below 1 it is drawn as a warning
• the exponent of the norm is between 0.4 and 12
• the first right face is half the product of its legs
• the fixed balls behind it are between 1 and 12
• the floor diagonal meets the vertical edge at a right angle
• the given triangle and the constructed right triangle are congruent
• the hyperbolic excess in c² is a²b²/3 at small size
• the inner circle touches each corner circle
• the largest triangle drawn has legs under a quarter turn
• the legs are positive lengths
• the long leg is b
• the point (12, 36) comes from the triangle (3, 4, 5)
• the proof is one the family draws
• the rope carries one knot per unit of its length
• the rope's three sides are whole numbers of knots, each under sixty
• the second right triangle has the floor diagonal and the height as its legs
• the sensitivity at the right angle is c/ab radians per unit
• the short leg is a
• the slanted face's area squared is the sum of the three right faces' areas squared
• the smallest box with a whole space diagonal and two whole face diagonals is 104, 153, 672
• the smallest primitive brick by its longest edge is 44, 117, 240
• the space diagonal squared is the sum of the three squares
• the sphere shortens the hypotenuse and the hyperbolic plane lengthens it
• the spherical hypotenuse is shorter than the flat one
• the spherical hypotenuse satisfies cos c = cos a cos b
• the spherical shortfall in c² is a²b²/3 at small size
• the square on the hypotenuse is c²
• the square on the leg has the leg's area squared
• the three faces at the corner meet at right angles
• the three lengths close into a triangle
• the tilted square equals the two upright squares
• the tilted square is c²
• the tree finds exactly as many primitive triples under 200 as the search does
• the tree is checked against a search up to a hypotenuse between 50 and 2000
• the tree is drawn to between 1 and 3 generations
• the tree of triples is drawn to between 1 and 3 generations
• the triangle is half the square — same base, apex on the opposite side
• the triple 3, 4, 5 gives Halcke's brick
• the two floor edges meet at a right angle
• the two triangles are the same triangle turned
• with p at least one, the midpoint of any two points of the ball is inside it

• the 3-sided curve is built at the common width ×4
• the 3-sided Reuleaux curve has one width at every angle ×4
• the integral of h round a 3-sided curve is π times the width ×3
• a Reuleaux polygon needs an odd number of sides
• all the shapes have one width
• and is convex
• and it holds for the solid of revolution, measured
• and it is the same in every direction
• and its radius of curvature stays positive, so it is convex
• and lands between the ball and the conjectured minimum
• and the circle's is a quarter of π times it
• and the drawn boundary really is that long
• and the drawn points confirm it, direction by direction
• and the largest is the edge times (√6 − 1)/√2, across two opposite curved edges
• and the Reuleaux triangle the least
• and the smooth family gets only part of the way to the triangle
• and the support function itself is not flat, so the flat sum says something
• and the sweep runs past the point where convexity fails
• and they do so in every direction, not only the one drawn
• at the width the function was given
• between one and four smooth amplitudes, each at most twelve
• between thirty and a hundred and sixty samples per angle
• between twenty and two hundred samples
• between two and five odd side counts, each at most eleven
• Blaschke's relation holds for the ball, which is the check that it is stated right
• convexity fails exactly where h + h″ first vanishes
• every arc bulges away from the centre
• every harmonic in the support function is odd
• every shape in the table has the same width
• ground and plank stay one width apart
• so the solid is not of constant width
• the area falls the whole way, so the minimum is on the constraint boundary
• the boundary is sampled between five hundred and twenty thousand times
• the built curve has constant width
• the circle encloses the most
• the circle is convex
• the four corners are equally spaced
• the generating curve has the stated width
• the harmonic is odd, between three and seven
• the least is about nine tenths of the most
• the plotted sum is flat at the width
• the smallest width is the edge itself
• the smooth curve has the common width
• the spun triangle encloses less than the ball
• the triangle's area is (π − √3)/2 times the square of the width
• the two support distances in opposite directions add to the width
• the view is one the family draws
• the width does not depend on how far it has rolled
• the width is positive
• the width is the arc radius
• though it misses by only a few per cent

• the recursion and the gamma function agree on the 0-ball ×21
• and the circumscribed value is 6 tan(π/6) ×7
• and the inscribed value is 6 sin(π/6) ×7
• the 6-gons bracket pi ×7
• a thin band well inside the sphere
• an even number of sides for each half-polygon
• and along every ray the inner chord is exactly half the outer one while the areas are a quarter and three quarters
• and by ninety-six sides the bracket sits inside Archimedes' own two fractions
• and in ten it reaches out past them
• and its surface is largest in seven
• and more with every doubling
• and the excess shrinks as the sides double
• and the surface peaks exactly two dimensions later
• and the whole ball is two of those
• as a smooth function of the dimension, the volume peaks a little past five
• at every height the dome's disc and the ring have the same area
• between 200 and 4000 points
• between three and seven doublings are tabulated
• cos²θ is a root of the remaining factor
• dimensions up to between 10 and 30
• each band is a pair of heights between −1 and 1, lower first
• each cell of the map has the area of the cell of the sphere it shows
• each doubling adds about a quarter of what the last one added
• each doubling tightens the bracket from both sides
• each slice density integrates to one
• heights chosen evenly put the share the cap's area predicts in the cap
• in nine dimensions the middle ball just reaches the faces of the cube
• latitudes chosen evenly put the share of latitude in it instead
• Liu Hui's estimate stays below the circle — Huygens's inequality
• one doubling fewer, the upper end misses Zu Chongzhi's bound
• one label for each band
• only even powers of cos θ occur
• read Liu Hui's or Snell's way, the 96-gon gains at least three digits on Archimedes' bracket
• Snell's combination (2 sin + tan)/3 is always above π
• Snell's rule gains four digits per tenfold sides
• so the hemisphere's volume is the cylinder's less the cone's
• square roots reach exactly the five
• the areas are integrated at between four hundred and twenty thousand slices
• the drawn polygon has between three and twenty-four sides
• the errors fall like n⁻², n⁻⁴ and n⁻⁶
• the even-dimensional volumes add up to e to the π
• the graticule is every 10, 15 or 30 degrees
• the historical bounds bracket π and Liu Hui's value lies below it
• the inscribed stack of cones has less surface than the sphere
• the lune is the inner triangle less the outer one
• the lune on the leg has exactly half the triangle's area, which is 1/2 of 1
• the lune's area equals the difference of two triangles
• the ratio of successive slivers tends to a quarter
• the resolvent cubic has whole coefficients after scaling
• the rings account for the whole disc
• the sectors are equal
• the segment exceeds four-thirds of its triangle — Huygens's lemma
• the share within the outer tenth is one less nine tenths to the power n
• the shear leans by between a tenth and one and a fifth of the height
• the sheared stack has the same area
• the side count is a whole number from 4 to 24
• the slice is taken strictly between the base and the top
• the sphere's band has the area of the cylinder's band of the same height
• the stack's surface is 4π cos(π/2k), a closed form Archimedes could bound
• the triangle of base 2πr and height r has the disc's area
• the two regions together are the unit square
• the two sectors are equal, so π cancels
• the two solids are compared at between a hundred and four thousand heights
• the unit ball is largest in five dimensions
• the view is one the family draws
• the whole sphere has four times the area of its great circle
• two to five dimensions
• Zu Chongzhi's polygons give 3.1415926 < π < 3.1415927

• the centre of cell 0 is nearest to site 0 ×20
• a large λ loses the corner arms
• a maximal disc contains no boundary point in its interior
• a planar triangulation of n points has at most 3n − 6 edges
• a point is under the chord exactly when it is inside the interval
• a site has a cell exactly when its lifted point is a corner of the lower hull
• a site inside the circle is nearer to both ends than the ends are to each other
• a spanning tree has one edge fewer than it has sites
• a λ-medial axis barely moves while the wobble is under a tenth of λ
• about 1.1 per cent are triangles, as Calka computed
• about 29.5 per cent of cells are hexagons
• an axis point on a corner arm exists
• an axis point on a trunk exists
• an axis point on a wobble branch exists
• an intermediate λ balances spurious branches against lost ones
• and at least enough to be connected
• and jumps once the wobble passes about a fifth of λ
• and so the refuted edge is not in the shortest tree after all
• at least one pair of circles actually crosses
• at least one site has no cell, which is what the picture is about
• below it the axis is one piece
• between 2 and 60 sites, each a point in or near the unit square
• by the largest count drawn, the pairs outnumber the triangulation's edges several times over
• cells with more sides are larger on average
• each graph in the chain has no more edges than the next
• each panel is on the side of the threshold the figure claims
• every Delaunay triangle spans a face of the lower hull
• every edge of the minimum spanning tree is an edge of the triangulation
• every Gabriel edge is a Delaunay edge
• every Lloyd step lowers the total squared distance to the nearest site
• every lower face projects to a Delaunay triangle
• every nearest neighbour edge is a shortest tree edge
• every relative neighbourhood edge is a Gabriel edge
• every shortest tree edge is a relative neighbourhood edge
• every site keeps a cell
• every upper face's circumcircle contains every other site
• exactly one of the two triangulations is the Delaunay one
• more than half of the whole fall happens in the first two rounds
• most interior cells settle at six sides, by more than double the next count
• no point beats the centroid
• no triangle's circumcircle holds another site
• on the torus the cells have exactly six sides on average
• the axis comes apart once λ passes the neck's half-width
• the axis has been found
• the bisector is where the two distances tie
• the cells account for the whole box, with nothing double-covered
• the cells of a pattern on the torus have exactly 6N sides in total
• the cells still account for the whole box
• the cells tile the torus exactly
• the chain is not a chain of equalities
• the circle on a tree edge as diameter holds no other site
• the circumcircle verdict and the lifted-hull verdict agree
• the cost falls at every round
• the edge chosen to be refuted really does have a site inside its circle
• the farthest-point triangulation uses only sites on the convex hull
• the gap between chord and parabola is −(x − a)(x − b)
• the highlighted cell is one of the sites
• the interior Voronoi corners of boundary samples lie along the medial axis
• the iteration has actually moved, rather than starting at its own fixed point
• the lift finds as many triangles as the circumcircle test does
• the mean area is exactly one cell's share
• the middle cell is absent at the small weight and present at the large one
• the middle of each cell has that cell's site nearest in the power distance
• the moment about the site is the moment about the centroid plus area times the gap squared
• the more sides a cell has, the fewer its neighbours have on average
• the settled cells are far closer to equal in area than the scatter's
• the settled diagram holds a larger share of six-sided interior cells
• the shape is one this family knows
• the tested point is inside the interval, which is what makes the pair illegal
• the triangulation is not empty
• the trunk's nearest points are spread widely, the wobble branch's are bunched
• the variance of the area is about 0.280
• the whole axis jumps far at the smallest wobble
• the whole axis wanders far from the clean rectangle's, the λ-axis stays close
• the λ that cuts the tooth's branch grows in proportion to the tooth
• where two circles cross, the two power distances agree

• the product to 1 vanishes at 0 ×17
• the product to 1 vanishes at -1 ×14
• the 1:1 figure crosses itself 2pq − p − q times ×11
• at phase 1.178 the 1:2 figure has the generic crossing count ×7
• the chords of the regular 3-gon from one corner multiply to 3 ×6
• both vanish at 0 ×4
• both vanish at -3 ×3
• the doubled arc from 0.25 of a quarter ends where Fagnano's formula says ×3
• the gap after 10 terms lies between 1/(N + 1) and 1/N ×3
• 1 + 1/2² + 1/3² + … is π²/6
• 1 + 1/2⁴ + 1/3⁴ + … is π⁴/90
• 1 + 1/2⁶ + 1/3⁶ + … is π⁶/945
• 5 and 17 pass and 7 fails
• a coprime pair up to five
• a generic Lissajous figure crosses itself 2pq − p − q times
• a million terms and the tail 1/N give π²/6
• a step up to 1 and two to six sectors
• and they stay on the hyperbola
• between three and seven offsets
• cos 1° has degree 48 over the rationals
• cosh and sinh are the even and odd halves of e^t
• each sine is a root of 4x³ − 3x + sin 3°
• each step keeps sine and cosine on the circle
• each step multiplies the error by about 4 sin² 1°
• every slice between neighbouring points has the same area
• every station lies on the figure eight
• exactly the multiples of three degrees are constructible
• more factors bring the product closer at x = ½
• sl blows up beside each marked pole
• sl repeats along (1 + i)ϖ, imaginary part
• sl repeats along (1 + i)ϖ, real part
• sl repeats along 2ϖ
• sl vanishes at each marked zero
• the catenary's parameter is between 0.2 and 2
• the circle's parameter is between 0 and 2π
• the circle's quarter arc is π/2
• the circular sector's area is half its parameter
• the correct digits roughly double each step
• the cubic's discriminant has the sign that forces complex cube roots
• the drawn arrangement is a zigzag
• the drawn list has the zigzag number of members
• the figure eight, walked as chords, is 2ϖ long
• the figure eight's quarter arc is π over twice the arithmetic–geometric mean of 1 and √2
• the first station is one equal share of arc from the centre
• the folding phases really fold
• the four families fail at 4, 8, never, never
• the gap after N factors is about π/(8N)
• the grid runs to between 2 and 5
• the hyperbola's parameter is between 0 and 2.2
• the hyperbolic sector's area is half its parameter
• the iteration and the Taylor series agree to fifty-five decimal places
• the length is between 3 and 6
• the middle root, twice the real part of a cube root, is sin 1°
• the number inside Cardano's cube root has modulus one eighth
• the offset is generic — the curve does not fold back on itself
• the parabola and the catenary really differ
• the point is on the circle
• the point is on the hyperbola
• the point's height is the wave's height there
• the product's x² coefficient is the Taylor series'
• the product's x⁴ coefficient is the Taylor series'
• the product's x⁶ coefficient is the Taylor series'
• the range is between 1 and 3.2
• the ratio is between one and three
• the relative error is within the next pole's 3^-(n+1)
• the row shows both a folded and an open figure
• the second is not odd
• the steps pass 1 at the seventh and 2 at the fifty-sixth
• the three roots add up to nought
• the time spent in each cell is close to the product of two arcsine densities
• the two coordinates are a point of the unit circle
• the two halvings bracket sin 1°
• the view is one of sector, squeeze, waves, catenary, lissajous, lissgrid, lissphase, lissdense, lemwalk, lemlength, lemdouble, lemagm, lemcomplex, lemdivide, prodpartial, prodsame, polychords, basel, wallis, prodcoef
• twenty thousand factors give sin(π/2) to four places
• two whole frequencies up to six with no common factor
• Wallis's partial products climb towards π/2 from below
• x + y is e to the parameter
• π divided by the mean is ϖ

• so sin 2π1x and sin 2π2x differ by at least 1 somewhere ×90
• the mean square of sin 2π1x − sin 2π2x over [0, 1] is 1 ×90
• the grid never beats the stated modulus of sin(2πkx) at k = 1 ×40
• the grid never beats the stated modulus of sin(2πx + k) at k = 1 ×40
• the grid never beats the stated modulus of xᵏ at k = 1 ×40
• x − x²/2 + x³/3 − …: 1 terms stay within the bound ×30
• x − x³/3 + x⁵/5 − …: 1 terms stay within the bound ×30
• members 7 and 13 share an arc, so they are within 0.765 everywhere ×28
• the n = 3 approximant is 1/1 at 0/1 ×28
• at 0/2 every member is 1 ×10
• the sum of 1 terms is within 0.5000 of the function everywhere on [0, 1], x = 1 included ×10
• a member later than the last one picked lies in the arc at stage 0 ×9
• the two members agree at node 0/8 ×9
• 1 − 2x + 3x² − 4x³ + … at x = 1/2 is the function it is said to sum to
• 1 − x + x² − x³ + … at x = 1/2 is the function it is said to sum to
• 1 − x + x² − x³ + …: the largest gap on [0, 1] never drops below 1/2
• 2 arctan t − t solves y′ = cos(t + y)
• 4 to 12 nodes
• a member fits inside the tube exactly when the convergence is uniform
• a well-chosen polynomial of the same degree does better than the iteration
• all four kinds of fraction occur
• and between the nodes they are nearly 2 apart
• and comes within two hundredths of it
• and N is the first such member
• at 2π/3 the increment shrinks like a square root
• at an irrational point the approximants settle towards 0
• at the golden point the averaged exponent lies near three-quarters, well clear of one-half and one
• at x = 1 the partial sums stay away from the function's value there
• at π the remainder after the slope shrinks like h to the three-halves
• between 16 and 400 members
• between 4 and 10 halvings
• between the multiples of 1/m! the members fall to 0
• between two and seven steps, each from 1 to 11
• between two and six levels of cutting
• between two and six member numbers, each a whole number up to 400
• each column of the picture settles on the limit
• each member is within 1/√n of |x|, worst at 0
• each point is passed by the bump and left behind
• eps is read only by the covering view among these
• every level covers each point exactly once
• every point settles on the limit
• every shifted wave has the same modulus at every k
• every step has slope between −1 and 1
• finer polygons sit closer to the solution
• from N on, the members are inside the band off the strip
• near the ends the differentiated series runs off to minus infinity
• no point of the interval beats the stated largest gap
• once a point is within the tolerance it stays within it
• one half holds at least half the members
• one member is within the tolerance on the whole interval
• one to four members are drawn, from inside the table
• one to three approximants, each with n from 2 to 40
• one to three sequences this family draws
• one to three values of m, each from 1 to 4
• R vanishes at π, where every term sin(n²π) is zero
• refining the grid finds a larger gap, because the largest is reached nowhere
• some point is still far from the limit
• steps is read only by the climbing view
• term counts up to between 8 and 60
• the antiderivative is at most 2/n³ and reaches it
• the bars run to between 20 and 80
• the circle of phases is cut into 4 to 16 arcs
• the differentiated series at 0 alternates between 2 and 0
• the fullest arc holds at least its share
• the gap to the previous pick is within the previous arc's bound
• the Gauss sum vanishes exactly when p and q are both odd
• the grid finds the largest gap, because it is reached at a point
• the height of each member is 1/n
• the intervals have total length below ε
• the largest gap for x / n falls with n
• the largest gap for xⁿ does not fall with n
• the largest gap for xⁿ never drops below where it started
• the largest gap is below a constant over √n
• the largest gap shrinks as n grows
• the largest gap shrinks exactly when the convergence is uniform
• the measured distance between partial sums respects the tail bound
• the measured square-root coefficient matches the Gauss sum on both sides
• the members are given in increasing order
• the members considered number 500 to 20000
• the members fall at every point
• the members run to between 10 and 60
• the partial sums at x = 1 do not settle
• the points jumping by at least 1/5 are the fractions with q ≤ 5
• the power family's modulus rises with k
• the removed length is the geometric sum
• the search for a member inside the tube runs to between 20 and 2000
• the second frequency is one more than a multiple of the node count, so the two agree at every node
• the sequence is one this family draws
• the series is one this family draws
• the series sums to x inside the interval
• the sine family's modulus reaches 2
• the stated gap between two members is the one on the grid
• the steepest slope grows with every term
• the steepest slope is at most π times the sum of (ab)ᵏ
• the steepest slope of each member is n
• the stretch δ is between 0.02 and 0.2
• the sup distance is plotted for between 6 and 80 values of n
• the symmetric difference quotient settles on −1/2
• the table of gaps runs to between 4 and 12
• the threshold is 1/k for k from 2 to 12
• the time runs to between 1 and 6
• the tolerance is between 0 and a half
• the tolerance is between 0.02 and 0.3
• the tube has a half-width between 0.02 and 0.6
• the upper sums of a function with finitely many spikes fall like 1/N
• the view is one the family draws
• the weight on partial sums equal to 1 is 1/(1 + x)
• the weights add to one, the unshown tail included
• the worst point moves in towards 0 as the steps go on
• two or three series
• two to five levels
• two to five step counts, increasing, up to 256
• two to six term counts, increasing, up to 60
• two values of x between 0.3 and 0.995, increasing
• which is the function the series sums to
• x − x²/2 + x³/3 − … at x = 1/2 is the function it is said to sum to
• x − x²/2 + x³/3 − …: the largest gap on [0, 1] falls away
• x − x³/3 + x⁵/5 − … at x = 1/2 is the function it is said to sum to
• x − x³/3 + x⁵/5 − …: the largest gap on [0, 1] falls away

• A has three to eight corners
• A's corners are all on its own hull, so the set is convex
• and a function that is not convex fails the same test, so the test has content
• and all but a handful of borderline cases are one or the other
• and every corner of the second strictly on the other
• and never more than three
• and positive against the target, which is the refutation
• and the other has more than one
• and the second is outside it
• and with every three assumed, the whole family has a common point
• B has three to eight corners
• B's corners are all on its own hull, so the set is convex
• between ninety and thirty-six hundred directions swept
• between one and six targets
• between six and twenty points in the scatter
• between twenty and four thousand random quadruples
• between two and eight values are averaged
• between two and seven points are averaged
• between two and six touch points, all inside the interval
• both ends of the chord are inside the interval drawn
• Efron's identity: the mean count equals n times the mean share missed by n − 1 points
• every corner of the first set is strictly on one side
• every point is inside the interval drawn
• every point of the hull lies in the hull of three of the points
• every three of the family have a common point
• every two of the family have a common point
• every value is positive and inside the interval drawn
• four points in a disc are in convex position with chance 1 − 35/(12π²)
• no line in any of the 720 directions separates the two sets
• no vector is both a non-negative combination and refuted by a direction
• one positive weight per point
• simulation agrees with Valtr's formulas
• so a refuting direction exists for the second
• some chord of this function dips below it
• the average of the values is at least the value at the average
• the average of the values is at most the value at the average
• the chord runs left to right
• the chord stays above the curve at every sampled point
• the chord stays below the curve at every sampled point
• the conjugate of x² is p²/4, found by search
• the convex function has exactly one local minimum on the interval
• the crescent's hull swallows its own mouth, so the crescent is not convex
• the curve stays above the tangent everywhere, not only nearby
• the direction is non-positive against every generator
• the disc's count follows 3.383 n^(1/3)
• the disc's hull corners are spread evenly round it
• the disc's hull has the most corners
• the first target is inside the cone
• the function drawn is a convex one
• the function is one this family draws
• the gap the line leaves is exactly the distance between the sets
• the geometric mean is at most the arithmetic mean
• the great majority of quadruples were non-degenerate
• the mean of the logarithms is at most the logarithm of the mean
• the mean of the logarithms names the geometric mean
• the number of layers grows like n^(2/3)
• the scatter has a two-dimensional hull
• the search examined some pair of ends
• the search runs over between 20 and 200 candidate ends
• the small set lies outside the crescent
• the square's count follows (8/3)(ln n + γ − ln 2)
• the square's hull corners crowd into its own corners
• the transform is drawn for a convex function
• the triangle's count follows 2(ln n + γ)
• the two generators are independent
• the two means agree exactly when every value is the same
• the two parts' hulls share the point the dependence produces
• the two sets are disjoint
• the view is one the family draws
• the weakened hypothesis is on or off
• the weights add to one
• transforming twice returns the original function
• two functions, side by side
• two generating vectors
• Valtr's formulas give Sylvester's 25/36 and 2/3 at four points
• while the sweep does find a wall when one exists, so it is not simply blind
• with only the pairs assumed, the whole family can have nothing in common

• so over 0.7702 exactly the factor is 2 ×4
• a base that is not e does not have slope equal to height
• and at base e the slope is the height
• and is what the closed form gives over that interval
• and the gap left is smaller than the last term added
• and the limit of the staircase is e itself
• and the place where that area is exactly 1 is e
• and the ratio falls towards 1 every step
• both factors are above 1 and their product fits
• both intervals lie inside the integrated range
• each solution starts at a positive height of at most 4
• every partial sum is below e
• more splits is closer to e
• no number of splits reaches e
• so by the end it is under one per cent
• so the two areas add to the area of the product
• the approximation is always an underestimate here
• the area is taken from 1 up to somewhere between 1 and 12
• the area under 1/x from 1 is the logarithm
• the base is a positive number of at most 20
• the curve integrated from the equation is the exponential
• the factor is the same wherever along the curve the interval starts
• the gap closes like one twelfth of 1/n, measured at the last row
• the growth rate is between a twentieth and three in size
• the interval over which the height changes by that factor is inside the window drawn
• the region from a to ab has the same area as the region from 1 to b
• the region runs from 1 to somewhere between 1 and 12
• the table runs to between 4 and 20
• the table shown gets e right to three decimals
• the tangent drawn is the curve's own slope
• the two factors are both above 1 and their product is at most 12
• the view is one the family draws
• the year is split between 2 and 6 ways, each a whole number up to a million

• the 0-th line of the period-2 train sits on the envelope ×94
• harmonic 1 has decayed by exp(−1²t) at t = 0.005 ×18
• sine 1 against sine 2 integrates to nothing ×12
• the partial sum of Q₂₅₆ at 0 up to frequency 0 ×7
• the 1-term curve is sampled fast enough to be the function rather than an alias ×5
• the partial sum of Q₂ at 0 up to frequency 0 ×5
• sine 1 against sine 1 integrates to π ×4
• the lines of the period-2 train are 1/T apart ×4
• harmonic 1 of the heat profile has decayed by exp(−m²π²t) ×3
• the projection onto sine 1 is the harmonic this family draws ×3
• between 6 and 40 harmonics ×2
• the projection onto sine 2 cancels to nothing ×2
• 113 frequencies need 169 samples on a square grid and 124 on the best lattice
• a longer period puts the lines closer together
• a missing harmonic has a node at the pluck
• a ramp width up to 0.2
• a time from 0 to 2
• and every Lₙ past ten is within 0.02 of the logarithm
• and it is never negative
• and its finest oscillation is more than three pixels wide
• and so is its absolute area
• and the slopes grow faster than the amplitudes shrink, which is the whole condition
• and the terms it leaves out are below a twentieth of what the window shows
• at every tick the two sines agree
• at three turns a second the eight-spoked wheel stands still, just below it turns backwards
• between 5 and 121 harmonics carry the profile
• between two and eight times, each from 0 to 4
• between two and five times, each from 0 to 2
• between two and four harmonics, each between 1 and 12
• between two and four periods, each positive
• between two and six times, none of them negative
• by the last frame the profile is a single sine to within two percent
• each block's peak stands higher than the last
• each window is between a hundred-thousandth and two wide
• each window is drawn from between 200 and 4000 samples
• each window is sampled fast enough for the terms it draws
• every term count is a whole number between 1 and 200
• every winning lattice is within nine degrees and five per cent of hexagonal, and within a degree and a half from W = 10 on
• halfway through, the shape is the pluck inverted and mirrored
• in three dimensions the best lattice needs well under two-thirds of the cubic grid's samples
• its signed area is 1
• L₁ is 1/3 + 2√3/π
• Lₙ − (4/π²) ln N has settled to its limit
• Lₙ rises with N
• magnifying ten times does not flatten the curve ten times
• more terms is closer, away from the jump
• N from 2 to 40
• one to three kernel sizes from 1 to 60
• one to three pluck positions, each away from the ends
• reconstruction fails below twice the top frequency and succeeds above it
• the amplitudes shrink
• the apparent turn is within half a spoke gap
• the area of |D₁₂| by quadrature matches Fejér's closed form
• the area of |D₄| by quadrature matches Fejér's closed form
• the averaged kernel is the mean of the first N + 1 Dirichlet kernels
• the best lattice always beats the square grid and settles near 2√3/π samples a frequency
• the best lattice's 124 samples give the pattern back exactly; a square grid of 121 does not
• the cardinal series rebuilds the signal between the samples
• the closed-form b₁ is the projection of the tent onto sin(1πx)
• the closed-form b₂ is the projection of the tent onto sin(2πx)
• the closed-form b₅ is the projection of the tent onto sin(5πx)
• the coefficients of the sawtooth wave fall like 1/m^1
• the coefficients of the square wave fall like 1/m^1
• the coefficients of the triangle wave fall like 1/m^2
• the construction is the one with no tangents
• the cubic grid needs 2/√3, √2, 2, 16 and 2²⁴ times as many samples in dimensions 2, 3, 4, 8 and 24
• the curve drawn is the sum of the harmonics
• the deepest zoom still has something to look at
• the fine detail fades faster than the coarse shape
• the folded frequency lies between nought and half the rate
• the frequencies multiply by an odd whole number
• the function never leaves [−1, 1]
• the motion repeats after a time of 2
• the partial sum at 0 comes within a tenth of Lₙ and never exceeds it
• the pluck is at least a twentieth from either end
• the pulse fits inside every period drawn
• the range runs to between 100 and 100,000
• the rings outrun the grid inside the square
• the signed area of Dₙ is 1
• the spectrum has something in it
• the steepest chord gets steeper as the spacing shrinks
• the string keeps a slope at least half the pluck's steepest at every time
• the travelling halves and 400 terms of the series agree to within the series' own tail
• the view is one the family draws
• the wave is one the family draws
• the Wilbraham–Gibbs integral
• two to four blocks — the fourth sits at 2⁶⁴

• 1/1 + … + 1/2 is not a whole number ×190
• one number in 1..2 carries the most factors of 2 ×190
• the density is positive at 0.00 ×56
• run 3 of the q = 0.5 series is under its own bound ×24
• 1 golden signs land on F(4) − 1 points ×20
• the histogram bar at -2.47 matches the density ×19
• the histogram bar at 0.07 matches the density ×19
• the vector for λ^1 is λ^1 ×19
• the denominator of H(1) carries 2 to the power ⌊log₂ 1⌋ ×16
• H(2) is not a whole number ×15
• term 2 is under the envelope ×14
• the first 1 terms add to the closed form ×14
• φ's powers close in on whole numbers by a factor 1/φ at 3 ×14
• piece 1 is 0.50 of what was left ×13
• and p³ does not, for p = 5 ×9
• p² divides the numerator of H(4) ×9
• stage 0 has two to the 0 intervals ×8
• and the upper bound on block 0 is the k-th power of 2^(1−p) ×7
• and their lengths add to (2/3) to the 0 ×7
• block 0 sits between its own two bounds ×7
• block 1 reaches a half ×4
• H(4) lies strictly between 2 and 3 ×4
• and that bound is 1/(k log 2)^0.5 ×3
• between 50 and 2000 terms ×2
• the blocks run to 30 to 100 digits ×2
• the partial sum differs from 1/(1 − 2) by exactly 2 to the power of its number of terms ×2
• the tail of 1/n^1.1 past 100000 is bounded by an integral ×2
• the transform at λ = 0.65 dies away along its own powers ×2
• 2 is first passed at n = 4
• 20 to 80 terms
• 3 at n = 11
• a partial sum agrees with the limit in at least as many digits as p divides the next term
• a point chosen by the measure has local exponent near the information dimension
• a ratio with even numerator has odd denominator
• a table small enough to read
• after Aitken's process the shortfall is still about 1/n, halved
• Aitken's process recovers the limit exactly from any three terms of a sequence with geometric error
• and all three agree at p = ½
• and between them they are all 32 strings of 5 bits
• and by the last drawn stage it is already tiny
• and by this stage they already cover most of the interval
• and ends at one
• and it closes on it
• and it is still coming down towards the limit
• and it is the largest power of 2 not above n
• and lists none of them twice
• and no address uses the digit one
• and points of different classes are different points
• and the denominator of the sum carries exactly that many
• and the numerator is odd
• and the remainder is the ratio to the power of the number of pieces
• and they add to what the series says
• and two of the series have ratios climbing to one, one of which converges and one of which does not
• and what has gone is the geometric series it looks like
• and what is left is exactly the tail
• at 1/3 the count is the Cantor set's log 2 / log 3
• at 4,000 digits the exact share is within 0.2% of the local limit
• at one half every bar has height a quarter
• at one half it is exactly one
• at one half the distribution function is a straight line
• at p = 0.4 it is large and growing with n
• at p = 1 the variance left after the first tenth is tiny
• at radius 10⁻⁵ the three densities are near 1, 0 and 1/2
• at the critical exponent the ratio is the same at every scale
• at x = 0 every sign is plus and the value is H(digits)
• at x = 1/3 the digits alternate and so do the signs
• below one half most of the interval is empty (λ = 1/3)
• below ten million the blocks agree with adding the terms one by one
• Bertrand: a prime lies between 20/2 and 20
• between 12 and 40 rationals are covered
• between 2,000 and 40,000 samples
• between 3 and 20 random runs
• between 3 and 7 ternary digits are listed
• between 3 and 9 stages are drawn
• between 8 and 16 binary digits
• between 8 and 30 partial sums are listed
• between eight and twenty-two terms are drawn
• between four and twelve blocks are drawn
• between ten and forty terms are drawn
• between three and nine translates are drawn
• between three and six exponents between 0.3 and 3 are compared
• between three and six exponents between nought and three are compared
• between three and ten stages are built
• between two and five series the family knows are compared
• between two and four values of λ
• between two and six classes are drawn
• by 4 it is more than ten times thinner than the normal curve
• consecutive partial sums differ by the next term
• each address names the left end of its own interval
• each average is closer to π/4 than the farther of the two it averages
• each block leans further out than the one below it
• each class is sampled at between 6 and 30 points
• each further power of the expansion gains at least two digits
• each of the three sides receives a third of the walks, as the symmetry requires
• each rational is inside its own interval
• each round of averaging more than halves the largest error
• every matrix drawn has its eigenvalues inside the unit circle
• every partial sum drawn is under the head plus the geometric tail's total
• every partial sum falls short of the limit
• every partial sum sequence reaches (I − A)⁻¹
• every power outside the squares stays out of reach
• every random run has settled down over its second half
• every repeated-digit string leaves a larger sum than every other string
• every such sum has an even denominator
• every triangle is isosceles with its two longest sides equal
• every walk drawn reaches the boundary
• every whole number from 2 up is crossed
• every window of width 0.02 meets the filled set in some length and none lies inside it
• exactly one number up to 12 carries the highest power of 2
• five to forty thousand sample points
• for the four-digit strings the sum is within a fifth of one per cent of ln 10 times the wait
• forty terms of Euler's transformed series reach twelve places
• forty terms of Leibniz's series still leave an error above five thousandths
• from 1 to 9, a larger forbidden digit leaves a larger sum
• from 2 to 9 a larger first digit falls less short
• from inside the notches gather more than the tips; from outside the tips gather more than the notches
• from levels 2 to 5 the walks' entropy grows by about ln 3 a level, dimension near one
• from one half up the whole interval is used (λ = 1/√2)
• from one half up the whole interval is used (λ = 1/2)
• from one half up the whole interval is used (λ = 1/φ)
• Hölder exponent ≤ information dimension ≤ dimension of the support
• inside the circle the partial sums settle on the inverse
• it starts at nought
• just below 1 every sign is minus
• leaving out 1 removes the most and leaving out 0 the least
• leaving out 9 reproduces the value Baillie published in 1979
• leaving out 99 gives within a tenth of 110 ln 10
• Legendre's formula matches the exact factorial
• n runs to between 10 and 30
• no candidate length puts the union between one and three
• no point is in two classes at once
• no term is used twice
• no two addresses read the same in binary
• no two seeds is a small rational away from another, which would be a typo
• odd primes below 100,000 dividing their left factorial
• on Leibniz's series the error falls like 1/n before Aitken and like 1/n³ after
• on the circle they stay bounded without settling
• one to four balances are counted
• one to four Pisot numbers this family knows
• one to three of the roots 2, 3 and 4
• only one number up to 20 is a multiple of 19
• outside it they grow without bound
• over the last factor of ten in length, one and two balances keep growing while three and four barely move
• Richardson's extrapolation from 160 terms reaches ten places
• so 19 divides the denominator of H(20) exactly once
• so its total is finite
• so the translates of the selection do not overlap
• strings led by 1 fall furthest short and strings led by 0 sit above
• ten to sixty random points
• the 2-adic digits times the denominator give the numerator, to 24 places
• the alternating run is heading for log 2
• the answer is caught between an even and an odd partial sum
• the balanced block sums times √j approach ln 10/√(0.4π)
• the bars carry all the probability
• the base is 2, 3, 5 or 7
• the block has 10 to 60 digits
• the block recursion agrees with term-by-term addition below ten million
• the block sums fall at the rate set by the automaton's largest eigenvalue
• the block's sums over every balance add up to about ln 10
• the bound on each block is exactly a half
• the bounding series is geometric exactly when p is above one
• the comparison holds a series from each side of the threshold
• the comparison holds a series that converges and one that does not
• the computed value agrees with the published decimals of Catalan's constant
• the covered rationals leave no wide gap, so the covering meets every part of the line
• the covering budget is between a hundredth and a half
• the crossings of 2 to between 4 and 11 are drawn
• the denominator of H(12) carries exactly 2^3
• the density at 0 misses 1/4 by a few millionths
• the density at 2 is 1/8 to double precision
• the density falls away from the centre
• the diagonal matrix's powers only shrink
• the distribution function climbs to one
• the doubly balanced block sums times j approach ln 10/(0.4π)
• the drawn lean is the sum of the drawn steps
• the drawn running total is the partial sum
• the entropy per sign has settled for the golden ratio
• the entropy per sign has settled for the pentanacci
• the entropy per sign has settled for the tetranacci
• the entropy per sign has settled for the tribonacci
• the enumerated bars match the closed form at λ = 1/√2
• the enumerated bars match the closed form at λ = 1/∛2
• the envelope dominates the series from some term onwards
• the error is smaller than the last term used
• the expected number of solutions below 100,000 is about 2.2
• the exponent is between a half and three
• the family is one of even, primes, squares or powers
• the first middle removed is between a third and an eighth
• the fit stays on the drawing
• the forbidden string is one to four decimal digits
• the function is constant across each removed interval
• the function never decreases
• the function stays inside the envelope
• the gap to ln n has settled on γ
• the geometric series' ratio is the same at every term
• the golden dimension is Alexander and Zagier's 0.99571
• the golden ratio's distribution has dimension below one
• the golden transform holds level at 2πφⁿ
• the halves share bars at λ = 0.6
• the halves share no bar at λ = 0.4
• the largest gap between the sum and ln 10 times the wait shrinks as the string lengthens
• the last certain convergent has a denominator of at least 495 digits
• the last digit of the limit is the left factorial modulo p
• the last term is small
• the left factorial residue agrees with an exact sum
• the length left is the product of what each stage keeps
• the lengths add to less than the budget
• the list holds the rationals it was asked for
• the longest surviving interval shrinks at every stage
• the middle-thirds set of the same depth has far less left
• the n-th root of the size of Aⁿ never falls below the spectral radius
• the neglected tail is under an eighth of a bar
• the neglected tail is under half of a bar
• the no-9 blocks add up to Kempner's 22.92067661926
• the option is one this view reads
• the other two grow a great deal before they shrink
• the overlap is the set's length at t = 0 and positive for every shift up to one half
• the p = 1 row is the harmonic sum, which is a logarithm plus Euler's constant
• the partial sums of Σ 1/k² fall short by about 1/n
• the patterns are different
• the patterns disagree only in a leading block
• the pentanacci's distribution has dimension below one
• the pieces and what is left fill the square exactly
• the Pisot values sit just below the line
• the plain series has four digits, Euler's more than fifteen and Ramanujan's more than thirty-four
• the plotted partial sums are the series
• the prime is 2, 3 or 5
• the prime-digit-sum blocks stay within a seventh of ln 10/ln(4.5j)
• the ratio is strictly between nothing and one
• the ratio of successive terms is n/(2n + 1)
• the ratio tends to one half
• the rearranged total settles near the number it was aimed at
• the removed intervals so far have the length the construction gives
• the residues pass a chi-square test for evenness with nineteen degrees of freedom
• the run is 1 to between 4 and 30
• the sample variance is near Σ 1/n² = π²/6
• the series is one the family knows
• the share of pieces needed to hold nine tenths of the walks falls level by level
• the share of points with density between 0.1 and 0.9 falls at every smaller radius
• the share of the set in the dyadic interval about a point ends near 1 if the point is in the set and near 0 if not
• the shares run to ten thousand or a hundred thousand digits
• the signs are the ones the series has
• the smallest interval is still wide enough to be represented
• the special number is found by powers of two or by a prime
• the staircase is flat over most of its interval
• the sums run over numbers of up to 40 to 200 digits
• the sums run to between a thousand and a million terms
• the sums run to between ten thousand and ten million terms
• the surviving length is bounded away from nothing
• the table runs to between 6 and 20
• the tail is drawn to between 3 and 4.5
• the target is visibly away from the sum the same terms give in order
• the term times 2ⁿ falls like √π/(2√n)
• the tetranacci's distribution has dimension below one
• the three points are in, out of and on the edge of the set
• the total is not small
• the transformed terms shrink by factors near one half and one quarter
• the triangle has side between 8 and 24
• the tribonacci's distribution has dimension below one
• the two parts add to G to seven places after twelve terms
• the two patterns land on the same point exactly
• the union is no longer than the sum of the pieces
• the view is one the family draws
• the Wilson primes below thirty thousand are 5, 13 and 563
• the worst window share of the alternating cells matches the formula
• there is one address for each surviving interval
• this view is for a series that does converge
• three powers already give twelve digits
• three to ten primes between 3 and 61
• two different translates never carry the same point of a class twice
• two or three values of λ
• two sign patterns of equal length
• two to four exponents in (0, 1.5]
• two to four values of λ
• up to seven powers keep the small system well conditioned
• weighted by the walks, a piece's share falls by about the same factor as its length
• what is left and what has gone add to the whole interval
• with a smaller exponent the ratio falls, with a larger one it grows without bound
• Wynn's table reaches twelve places from seventeen terms
• Wynn's table, given seventeen terms of this series, is still off in the second decimal place
• λ = 1/θ satisfies its relation for the golden ratio
• λ = 1/θ satisfies its relation for the pentanacci
• λ = 1/θ satisfies its relation for the tetranacci
• λ = 1/θ satisfies its relation for the tribonacci
• λ is between 0.2 and 0.9

• the integral gives 0! exactly ×11
• the area under x^0e^(−x) is 0! ×8
• at 1 steps the length is still 2 ×7
• the shadow in the direction (1, 2) keeps its full length at stage 0 ×7
• the area under 1/x^0.9 from 1 to 10 matches its closed form ×6
• the integral of sin to the power 2 is (2 − 1)/2 times the one for power 0 ×5
• the area under 1/x^1.25 beyond 1 is 1/(p − 1) ×4
• the horn to 10 holds π(1 − 1/T) ×4
• and it equals the far-end area of the reflected exponent 1.75 ×3
• on block 1 the integrand gains exactly 1 ×3
• the area under 1/√x from 1 to 10 matches its closed form ×3
• the area under 1/x from 1 to 10 matches its closed form ×3
• the area under 1/x² from 1 to 10 matches its closed form ×3
• the staircase of 2 steps is exactly 2 long ×3
• between 256 and 8192 samples ×2
• the area under 1/x^0.25 between 0 and 1 is 1/(1 − p) ×2
• √π lies inside every pair
• ∫ dx/(x ln x) from e^e to 10^50 is ln ln x between the ends
• ∫ from 1 to ∞ of x^(−3/2) is 2
• ∮ x dy and the half-sum ½∮(x dy − y dx) give the same area
• ½∮(x dy − y dx) is π(b² + a²/2) for the limaçon
• a circle's perimeter is 2π and a flattened ellipse's tends to 4a
• a fine enough polygon reaches the curve's length
• a known increasing curve over a real stretch
• adding corners never shortens the inscribed polygon
• after 20,000 lines the estimate is within its own error band of the true length
• and (−½)! = √π
• and are within 1/N of it after N factors
• and by this stage it is already small
• and every one of them falls short of the curve's own length
• and is nearly nothing by the end
• and its length is nowhere near the curve's
• and so does the polynomial
• and the fat one keeps a length in the limit
• and the finest is strictly longer than the coarsest, so the family is climbing
• at least one triangle counts negative, so cancellation is visible
• between 20 and 20,000 lines
• between 6 and 40 strips are drawn
• between three and five corner counts, each up to 64
• between two and six divisions
• both parts only ever increase
• Crofton's formula: the mean count of crossings is half the wall's length
• doubling the corners divides the gap by four
• each area is n times the one before
• each curve has the same length
• each curve's mean crossing count is 2L/(πd), whatever its shape
• each divergent integrand ends above the convergent ones at its own level
• each inscribed polygon is shorter than the ellipse, and doubling the corners shortens the gap
• each integral is still growing at the far end
• each node is a root of the Legendre polynomial
• each panel is at least as close as the one before it
• each stage is met by fewer lines than the one before
• each staircase runs closer to the curve than the last
• every horizontal line crosses the curve going up as often as coming down
• every integrand in the chain still has a finite integral to infinity
• every parametrisation traces the same point set
• every weight is positive
• every wiggle eventually breaks log-convexity
• for a flat ellipse, P − 4a is (2b²/a)(ln(4a/b) − ½) to within a hundredth of the bracket
• for an ellipse the error falls like 1/n²
• for an entire integrand Gauss needs fewer samples
• for nearly circular ellipses the error of each formula falls like a power of h: the first power for Kepler, the average and Euler, the third and fifth for Ramanujan's
• for the wavy curve too the error falls like 1/n², with a larger constant
• Gauss reaches rounding within eighty samples
• Gauss's eighty points beat Simpson's by more than five orders of magnitude
• halving the wiggle doubles how long it survives
• in a general direction the shadow shrinks
• lines is read only by the crofton view
• log of the wiggled curve bends the wrong way over stretches of every unit interval
• log Γ bends one way everywhere
• middle quarter, halving: no cell lies wholly inside the set, so the lower sum is nought
• middle quarter, halving: refining the partition never raises the upper sum
• middle quarter, halving: the upper sum never falls below the set's own length
• middle thirds: no cell lies wholly inside the set, so the lower sum is nought
• middle thirds: refining the partition never raises the upper sum
• middle thirds: the upper sum never falls below the set's own length
• more bars means less error
• no rule reaches a millionth on the corner within two hundred samples
• no search result beats the known wall by more than the allowance for leaks
• no width, no area
• on a function with a kink the two rules end up within about a decade of each other
• on the open interval doubling the samples quarters the error
• out and back covers the image a whole number of times
• out, back, out again covers the image a whole number of times
• refining a partition never lowers the variation
• seed is read only by the crofton, favbuffon view
• sin p is the fraction's error p − qπ, to first order
• sixteen samples of the periodic integrand reach rounding
• so in the limit the middle-thirds set has no length at all
• so the first upper sum is on its way down to nothing
• stage is read only by the cantorlen view
• straight through, accelerating covers the image a whole number of times
• straight through, once covers the image a whole number of times
• the accumulation only ever climbs, because f is positive
• the area between the two shrinks every time the steps are halved
• the area left of the curve up to b
• the area read from the boundary matches the cells counted inside
• the area under 1/√x between 0 and 1 is 1/(1 − p)
• the area under the curve up to a
• the arithmetic–geometric mean and the trapezoid rule agree to twelve places
• the arithmetic–geometric mean settles to the last digit within seven steps
• the boundary integral equals the area weighted by winding number
• the Cantor, noodle and Koch views draw their own curves
• the counts increase
• the curve is one of quarter, parabola, wiggle
• the curve lies inside the disc the lines are thrown at
• the directions (2, 1) and (1, 2) keep the full shadow
• the drawing keeps a few points per column rather than every term
• the drawn partition has between 24 and 240 cells
• the error shrinks as points are added
• the error shrinks by the factor a + √(a² − 1) per sample
• the fat construction's first middle is between a third and an eighth
• the finest one is within a twentieth of the curve everywhere
• the finest partition is a power of two between 64 and 16384
• the four corners project to 0, 3, 6 and 9 quarters along x + 2y
• the function is one of root, square, smooth
• the function is one the figure knows
• the function moves, so the decomposition is not trivial
• the functions are among root, square, smooth
• the gap shrinks like √π/(2n)
• the inner loop is wound round twice
• the inscribed polygon's length, measured, matches the closed form
• the interval runs from nought to at most 2
• the limaçon has an inner loop: 0 < b < a
• the maps differ in length or in speed, so the panels are comparing something
• the maps run at different speeds
• the middle-thirds approximation's length is two-thirds to the power of the stage
• the number of nodes is a whole number from 2 to 20
• the option is one this view reads
• the panels really are in increasing order of rectangles
• the parametrisations are among once, back, thrice, slow
• the partial products rise towards π/2 from below
• the polynomial's ratio does not
• the printed total is the area of the bars drawn
• the quoted exact area is the area under this curve
• the rules are exact up to degrees 1, 3, 5 and 11
• the search found a wall
• the series needs a handful of terms for a mild ellipse and hundreds for a flat one
• the set is approximated at between 5 and 14 stages
• the share of random lines that meet the stage agrees with the shadows' average
• the share of the straight line, w, is between 0 and 1
• the shoelace sum equals the area counted inside
• the signed area is within 1/T of π/2
• the signed triangles add up to the shoelace sum
• the slope of the accumulated area is the height of the curve
• the slowest divergent drawn is still above the fastest-converging drawn at the far end
• the speed integral and the polygon agree for out and back
• the speed integral and the polygon agree for out, back, out again
• the speed integral and the polygon agree for straight through, accelerating
• the speed integral and the polygon agree for straight through, once
• the squares' shadows along x + 2y tile the interval end to end
• the stage drawn is 1 to 7
• the stage-10 polygon is within 0.02 of the limit
• the stage-k Koch curve has length 1.8 × (4/3)^k
• the staircase is built against quarter or diagonal
• the staircase of 1 step is exactly 2 long
• the step counts are between 2 and 5 whole numbers up to 512
• the stub must reach the diagonal and not stop short of it
• the sum to ten million is 30.3145
• the sums run to between ten thousand and twenty million terms
• the sweep runs over between three and eight partition sizes
• the sweep runs up to between 4 and 16384 steps, or an interval end for the variation view
• the term at n = 355 is about 24.6
• the third block ends at a number of a few hundred digits
• the too-well-approximated number's sum races away
• the tracer arm's length times the wheel's roll is the area enclosed
• the tracer is within reach of the arms
• the two areas together are at least ab
• the two regions fill the big rectangle less the small one
• the unsigned area exceeds (2/π) ln T
• the variation of t sin(1/t) climbs by as much at the finest refinement as at the coarsest
• the variation of t² climbs less and less, so it settles
• the variation of t² sin(1/t) climbs less and less, so it settles
• the view is one the family draws
• the volume is below π and the surface above 2π ln T
• the weights add up to the length of the interval
• the wiggle even keeps the recurrence
• the wiggle keeps every factorial
• the wiggle's ratio to Γ repeats with period one
• their difference is the function
• two to four exponents between 0 and 3
• w is read only by the cantorlen view
• while the second has already settled on the set's own length
• with equality when b = f(a)
• x runs up to 10 to a power from 20 to 300
• Γ lies between the two products
• Σ 1/(n (ln n)²) has almost stopped growing
• Σ 1/(n ln n) − ln ln N has settled to a constant

• 100 to 100,000 steps
• a point where the slope formula breaks down in both directions is a genuine crossing
• an optimal schedule for C(c + r, c) steps runs no step more than r times
• and of slope minus one
• and the difference quotient closes on u'v + uv'
• and the gap as a fraction of the window falls with it
• and the rough curve's do not settle on anything
• arbitrarily short secants of slope one
• at a regular point the partial derivative in y is away from zero
• at the minimum the slope of the original curve is exactly the level
• beside a vertical tangent one side meets the curve twice and the other not at all
• between 1 and 4 mirrored pairs are marked
• between 1 and 4 pairs are marked
• between 2 and 4 windows are drawn
• between 4 and 14 spacings are measured
• both ends are higher than the minimum
• both sides grow, so the picture is four pieces and not a signed sum
• both sides of the rectangle are positive lengths
• by the last halving the corner is under a twentieth of the increment
• each magnification is between 2 and 20
• each secant is closer than the last
• each shorter secant is closer to the slope the equation gives
• each step magnifies by between 2 and 20
• every secant from the left has slope −1
• every secant from the right has slope 1
• inside the window round a regular point each vertical line meets the curve once
• nested dual numbers agree with a fourth-order difference of the exact slope
• on every interval about nought the slope runs nearly from −1 to 1
• the binomial schedule beats repeated halving with the same memory
• the bisection really inverts the function
• the central difference bottoms out near its two-thirds power
• the complex step at h = 10⁻²⁰⁰ and the dual number are correct to rounding
• the corner's share of the increment falls at every halving
• the curve is one the figure knows
• the curve, solved for y near the point, has the slope minus the ratio of the two partial derivatives
• the derivative is drawn for the power 2 or 3
• the derivative's formula matches its difference quotients
• the first window is at most 4 wide
• the forward difference bottoms out near the square root of the rounding unit
• the four pieces fill the grown rectangle
• the function increases across the window it is drawn in
• the function is increasing where it is marked, so it has an inverse there
• the function is one the family knows
• the function is one the figure knows
• the gap to the tangent falls by the square of the magnification
• the increment is the two strips and the corner
• the last line's slope is the derivative to rounding
• the last secant is near the tangent
• the level c is reached again and again as x approaches nought
• the level is strictly between −1 and 1
• the level lies between the slopes at the two ends
• the marked point is inside the window
• the marked point is not one where the tangent is vertical
• the marked point lies on the curve
• the minimum is inside the interval, not at an end
• the number whose root is taken is positive and at most 100
• the optimal schedule runs no step more than r times
• the option is one this view reads
• the pair is one the figure knows
• the power is 1, 2 or 3
• the rectangle is drawn inside the domain of both functions
• the recursion agrees with Griewank's formula
• the reverse sweep costs less than five evaluations, the same at every size
• the reverse sweep's partial derivative matches a forward sweep
• the schedule built costs what the recursion says
• the secant slope from the origin is h^(p−1) sin(1/h)
• the secant slope is squeezed by the envelope
• the secants are drawn at a list of 2 to 24 positive offsets
• the secants run one way along the curve or the other
• the slope carried through the iteration converges to 1/(2√a)
• the smooth curve's chords close on its tangent
• the step is a positive length under 1.6
• the step the rectangle grows by is a positive length under 1.6
• the total is r·l − C(c + r, c + 1)
• the two sides disagree, so there is no limit
• the two slopes at a matched pair multiply to one
• the view is one the family draws
• two to five slot counts from 1 to 10
• two to four slots and a chain of four to twelve steps
• two to six slots and ten to two hundred steps

• the [2/2] approximant reproduces coefficient 0 of the series exactly ×144
• with 2 terms the sum is within the first omitted term at x = 0.013 ×120
• with 1 terms at x = 0.05 the error is below the first term left out ×90
• the polynomial passes through node 1 ×21
• the size of coefficient 1 of 1/(1 − 2x) is what the exact value says ×20
• the size of coefficient 1 of 1/√(1 − 4x) is what the exact value says ×20
• the size of coefficient 1 of Euler's series is what the exact value says ×20
• Laguerre node 1 of 4 is a root ×12
• pole 1 of [3/4] is minus one over a Gauss–Laguerre node ×12
• pole 1 of [4/4] is minus one over a Gauss node ×12
• the [2/2] approximant exists with a denominator starting at one ×12
• the optimal nodes for degree 10 equalise every local maximum ×11
• degree 2: every reference point sits at the largest error ×8
• degree 2: the error alternates in sign ×8
• degree 2: the reference has n + 2 points ×8
• the degree-1 sum matches to order 1 at the expansion point ×7
• the degree-2 sum about 1.2 matches the function to order 2 there ×6
• as x shrinks, the point settles at 1/(n+2) for degree 1 ×4
• the integral Euler's series belongs to agrees two ways at x = 0.05 ×4
• the [2/2] denominator does not vanish on the drawn interval ×3
• the best error at x = 0.05 has the size √(2π/x)·e^(−1/x) ×3
• the best number of terms at x = 0.05 is within two of where the terms are smallest ×3
• the degree-2 sum about -1.4 matches the function to order 2 there ×3
• well past the best at x = 0.05, every term added makes the error larger ×3
• at x = 0.5, 4 terms lose to 2 ×2
• between 3 and 41 nodes ×2
• between one and three orders, each between 1 and 12 ×2
• close to nought, 4 terms beat 2 ×2
• the degree is between 1 and 10 ×2
• a degree between 1 and 20
• a function that is already a quotient of polynomials is reproduced exactly
• a function with known poles
• a higher-order approximant is at least as close at the probe point
• Aberth's iteration settles on all the zeros
• an amplification factor is never below one, since the nodes reproduce themselves
• and by the largest count drawn it is orders of magnitude worse
• and by the last coefficient they are within ten per cent of it
• and by the last they are past k/e, so no radius but nought is possible
• and every approximant is closer than the last
• and interpolation at Chebyshev points is within a factor of two of it
• and it dwarfs anything in the middle of the interval
• and its roots are the Chebyshev points
• and never more than a small factor worse
• and the error shrinks by about ρ per degree
• and the terms are smallest near 1/x
• at 160 terms the scaled logarithm follows the limiting curve away from the tip and from 1
• between 6 and 40 terms
• between one and five degrees, each between 1 and 8
• between one and five series
• between one and four degrees
• between one and four singularities, each a point of the plane
• between one and four term counts, each between 1 and 20
• between one and six centres
• between one and three values of x, each between 0.04 and 1
• between three and fourteen node counts, each between 3 and 25
• Chebyshev nodes keep the error small across the whole interval
• Chebyshev's constant follows its formula
• degree 3 for eˣ: every reference point sits at the largest error
• degree 3 for eˣ: the error alternates in sign
• degree 3 for eˣ: the reference has n + 2 points
• degree 4 for 1/(1.2 − x): every reference point sits at the largest error
• degree 4 for 1/(1.2 − x): the error alternates in sign
• degree 4 for 1/(1.2 − x): the reference has n + 2 points
• degree 6 for |x|: every reference point sits at the largest error
• degree 6 for |x|: the error alternates in sign
• degree 6 for |x|: the reference has n + 2 points
• degrees up to between 6 and 20
• each computed zero makes the partial sum vanish
• each drawn pole is a root of the denominator
• each longer partial sum is closer everywhere on the interval
• each new centre lies inside the previous disc, so its coefficients are known there
• each node is the projection of its angle
• equally spaced nodes never amplify less than Chebyshev ones
• evenly spaced roots give a larger maximum
• every pole lies on the cut, at or beyond −1
• every pole lies on the negative real axis
• every term count is a whole number between 1 and 40
• every zero satisfies the balance to within a tenth
• inside the new interval more terms help
• inside the radius, more terms help
• its foci are ±1, the ends of the interval
• ln(1+x) is centred where it is defined
• ln(1+x) is expanded inside its own radius
• most zeros of the long partial sum lie within a tenth of the unit circle
• no centre sits on a singularity
• no division by nought
• no rational has a denominator of nought
• no singularity lies strictly inside a disc
• one to three functions with known poles
• optimal, then extended Chebyshev, then Chebyshev
• outside it they hurt
• outside it, more terms hurt
• past the radius every Taylor error is larger than the last
• past the radius, the Taylor sum of higher degree is the worse one
• the angles of the zeros are close to evenly spread
• the best is no worse than either of the others
• the best polynomial in the comparison: every reference point sits at the largest error
• the best polynomial in the comparison: the error alternates in sign
• the best polynomial in the comparison: the reference has n + 2 points
• the bound is a bound
• the centre is inside the drawn domain
• the Chebyshev end gaps are several times smaller than the middle one
• the coefficients decay at the rate the ellipse through the pole predicts
• the coefficients fall like the power of k the smoothness predicts
• the comparison uses a function with a known Taylor series
• the cosine coefficient of f(cos θ) is the Chebyshev coefficient
• the cut-off series is never better than the best
• the drawn domain is an interval
• the ellipse passes through the nearest pole
• the equally spaced constant grows with every step
• the error in the curve is at most the Lebesgue constant times the error in the data
• the error never exceeds the bound Lagrange's form puts on it
• the evenly spaced nodes have one gap throughout
• the exchange for √(1 + x): every reference point sits at the largest error
• the exchange for √(1 + x): the error alternates in sign
• the exchange for √(1 + x): the reference has n + 2 points
• the function has a closed form to compare against
• the function has a closed form to draw
• the function is one the figure knows
• the function is one the figures know
• the function is one the minimax figures know
• the function is one whose derivatives the figure knows
• the interval drawn is inside the function's radius
• the interval drawn is not empty
• the interval is between 0.4 and 8 wide
• the largest order is between 3 and 12
• the last coefficient is between 8 and 200
• the levelled error never exceeds the true maximum
• the levelled error only rises
• the mode draws the functions whose derivative is monotone
• the mode of the taylor family is one of lebfun, lebconst, lebbumps, lebshift, lebnoise, sums, radius, remainder, xi, centre, discs, interp, lebesgue, nodes, pade, poles, paderate, root, asymptotic, asymsums, minimax, minimaxcmp, remez, monic, chebcoef, bernstein, chebsmooth, cosine, chebsums, chebvsbest
• the nodes are equally spaced or Chebyshev
• the numerator's degree is a whole number close to the denominator's
• the optimal nodes lie within two hundredths of the stretched Chebyshev nodes
• the partial sum is negative at −0.6n for odd n
• the point is one where the function is defined
• the point moves steadily in one direction as x grows
• the pole picture is drawn for ln(1 + x), 1/(1 + x²), eˣ or Euler's series
• the polynomial has the degree it claims
• the probe point is inside the domain
• the radius is the distance to the nearest singularity
• the real zero sits outside the tip by about ½ ln(2πn) divided by n times the curve's steepness
• the roots for 1/(1 − 2x) close on one over the distance to the singularity
• the roots for 1/√(1 − 4x) close on one over the distance to the singularity
• the roots for Euler's series grow at every step
• the series is one of log, runge, exp, geom2, central, lacunary, euler
• the shifts are symmetric about the middle
• the smallest gap is at one of the ends
• the smallest monic polynomial is the Chebyshev polynomial scaled
• the span is between 0.2 and 1.5
• the starting reference is not a symmetric one that levels the error to nought
• the tail of the series is the difference between the function and its partial sum
• the Taylor degree is between 1 and 40
• the Taylor polynomial agrees at the centre
• the unknown point lies strictly between nought and x
• the window reaches past −1 and past 0.5
• two to four degrees up to 60
• with equally spaced nodes the worst error is out near the ends
• x^5 against lower degrees: every reference point sits at the largest error
• x^5 against lower degrees: the error alternates in sign
• x^5 against lower degrees: the reference has n + 2 points

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