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 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
• 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 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
• AD = cos β
• and its second and third
• and its y coordinate
• and PA·PB = PC·PD
• and so do the other two
• 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 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 their radii differ
• and touching each escribed one
• 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
• each side over the sine of the opposite angle is the diameter
• 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 apex on the arc gives the same angle
• every side of every equilateral one makes (B − C)/3 with the base, give or take 60°
• 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 trisector triangle in the sweep is equilateral
• one angle, not several
• opposite angles add to a straight angle
• Ptolemy: the diagonals' product is the sum of the products of opposite sides
• so BC / sin A is the diameter
• so do the angles at C and B
• 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 D stand on the same arc and are equal
• the apex is on the major arc
• the apex lies on the circle
• the centre's angle is twice the apex's
• 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 exterior angle is twice the base angle
• the incentre is not on that line, which the determinant reports rather than being told
• the inner triangle's first two sides are equal
• 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 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 point is inside the circle
• the quadrilateral's angles add to 360°
• the shape sweep is between 6 and 60 steps on a side
• 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 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 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 triangle has three corners and some area
• the triangles are similar, so PA/PD = PC/PB
• the two circles are not concentric
• the two halves are the whole angle
• the two sides agree at every angle
• the two sides of a piece are not parallel
• 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
• triangle OPA is isosceles
• triangle OPB is isosceles
• unequal products put D off it

• 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
• 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
• 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'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 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
• 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
• consecutive Fibonacci positions alternate between win and loss
• every interval the count isolates holds exactly one real root found separately
• N is from 12 to 60
• N is read only by the gamegrid view
• poly is read only by the sturm, sturmtable, isolate views
• range is an interval [lo, hi]
• range is read only by the sturm view
• the count drops by exactly one across each real root
• 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 polynomial is one of the named examples
• the search agrees with the golden-ratio rule
• the squares peeled at each pass are the continued fraction of the ratio
• the verdict follows the side of φ the ratio falls on
• the view is one the family draws
• 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

• the polynomial for power 1 gives the exact sum at n = 1 ×150
• 0 and 3: triangular triples match odd-square triples ×121
• for power 2 the remainder approaches m/12 ×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 ten thousand to two million
• a triangular index from 1 to 7
• an even index from 10 to 40
• and it is the centre
• between 3 and 12 steps
• every number is a sum of three triangular numbers
• every odd Bernoulli number past the first is zero
• Faulhaber's polynomial gives the staircase's total
• n is two triangular numbers exactly when 4n + 1 is a sum of two squares
• only the centre cell is left over
• powers from 2 to 7
• powers up to between 3 and 8
• 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
• 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 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

• 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 inner pentagon is smaller by φ²
• each step shrinks the pair by φ
• each subtraction leaves something smaller than what it subtracted
• 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 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 diagonal's end piece is s/φ
• the last convergent is close to φ
• the long side is the next Fibonacci number
• 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 short side is the one before it
• the thick half built here has its 108° apex
• 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 quotient from 0.9 to just below 1
• 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 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 random convex shape obeys Bonnesen's inequality
• every shape with that quotient has the same allowance
• 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
• four corner angles from 30 to 180 degrees
• 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
• 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 deficit falls like the inverse square of the number of sides
• the dent is between nothing and 160 units deep
• 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 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 polygons drawn have between 3 and 60 sides
• the reuleaux's circumradius is found
• the reuleaux's inradius is found
• the reuleaux's ring is no wider than its deficit allows
• 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 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 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 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 searched avoids ordinary lines altogether
• no set searched has fewer lines than points
• no two dual lines are parallel after the turn
• removing an end of an ordinary line removes that line entirely
• 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 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 ordinary line chosen exists
• the point set is one this family knows
• the points are drawn between 12 and 60 units apart
• 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

• 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
• 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 star polygon has between 5 and 13 points
• 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
• as a surface it has thirty-two corners
• 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 turn has an axis to be about
• exactly five vertex figures close up
• 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
• read as twelve pentagrams meeting five at a point, the number is −6
• 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 faces at a corner close into a ring
• the number of distinct turns is twice the number of edges
• the path visits every point exactly once
• the path winds about the centre q times
• the projected solid has a measurable extent
• the solid is one of the five
• 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 turn in four dimensions is at most half a circle
• the view is one the family draws
• 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

• 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
• each fixed ball drawn behind is between 1 and 12
• 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 node of the tree is a right triangle in whole numbers
• every point drawn is at distance one in this norm
• four triangles and the tilted square fill the left square
• four triangles and the two upright squares fill the right 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 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 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 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 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

• 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
• 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 the whole ball is two of those
• at every height the dome's disc and the ring have the same area
• between three and seven doublings are tabulated
• each doubling tightens the bracket from both sides
• so the hemisphere's volume is the cylinder's less the cone's
• the areas are integrated at between four hundred and twenty thousand slices
• the drawn polygon has between three and twenty-four sides
• the rings account for the whole disc
• the shear leans by between a tenth and one and a fifth of the height
• the sheared stack has the same area
• the slice is taken strictly between the base and the top
• the triangle of base 2πr and height r has the disc's area
• the two regions together are the unit square
• the two solids are compared at between a hundred and four thousand heights
• the view is one the family draws

• the centre of cell 0 is nearest to site 0 ×20
• 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
• and at least enough to be connected
• 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
• 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
• 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
• the bisector is where the two distances tie
• the cells account for the whole box, with nothing double-covered
• the cells still account for the whole box
• 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 iteration has actually moved, rather than starting at its own fixed point
• the lift finds as many triangles as the circumcircle test does
• 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 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 tested point is inside the interval, which is what makes the pair illegal
• the triangulation is not empty
• where two circles cross, the two power distances agree

• 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
• a coprime pair up to five
• a generic Lissajous figure crosses itself 2pq − p − q times
• a step up to 1 and two to six sectors
• and they stay on the hyperbola
• between three and seven offsets
• cosh and sinh are the even and odd halves of e^t
• every slice between neighbouring points has the same area
• the catenary's parameter is between 0.2 and 2
• the circle's parameter is between 0 and 2π
• the circular sector's area is half its parameter
• the folding phases really fold
• 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 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 range is between 1 and 3.2
• the ratio is between one and three
• the row shows both a folded and an open figure
• 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 view is one of sector, squeeze, waves, catenary, lissajous, lissgrid, lissphase, lissdense
• two whole frequencies up to six with no common factor
• x + y is e to the parameter

• 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
• and between the nodes they are nearly 2 apart
• and comes within two hundredths of it
• and N is the first such member
• at an irrational point the approximants settle towards 0
• at x = 1 the partial sums stay away from the function's value there
• between 16 and 400 members
• between 4 and 10 halvings
• between the multiples of 1/m! the members fall to 0
• 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 point is passed by the bump and left behind
• every level covers each point exactly once
• 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
• no point of the interval beats the stated largest gap
• one half holds at least half the members
• 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
• refining the grid finds a larger gap, because the largest is reached nowhere
• term counts up to between 8 and 60
• the bars run to between 20 and 80
• the circle of phases is cut into 4 to 16 arcs
• the fullest arc holds at least its share
• the gap to the previous pick is within the previous arc's bound
• the grid finds the largest gap, because it is reached at a point
• 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 shrinks exactly when the convergence is uniform
• the members are given in increasing order
• the members considered number 500 to 20000
• 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 sine family's modulus reaches 2
• the stated gap between two members is the one on the grid
• the stretch δ is between 0.02 and 0.2
• the sup distance is plotted for between 6 and 80 values of n
• 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 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
• 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
• 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
• 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
• 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 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 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 transform is drawn for a convex function
• 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
• 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
• 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
• 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
• halfway through, the shape is the pluck inverted and mirrored
• 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
• the amplitudes shrink
• 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 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 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 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 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 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
• 3 at n = 11
• a ratio with even numerator has odd denominator
• 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 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 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)
• 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 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 block leans further out than the one below it
• each class is sampled at between 6 and 30 points
• each rational is inside its own interval
• 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 random run has settled down over its second half
• every such sum has an even denominator
• every triangle is isosceles with its two longest sides equal
• every whole number from 2 up is crossed
• exactly one number up to 12 carries the highest power of 2
• 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)
• inside the circle the partial sums settle on the inverse
• it starts at nought
• just below 1 every sign is minus
• 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
• on the circle they stay bounded without settling
• 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
• so 19 divides the denominator of H(20) exactly once
• so its total is finite
• so the translates of the selection do not overlap
• 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 bars carry all the probability
• 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 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 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 exponent is between a half and three
• the first middle removed is between a third and an eighth
• the function is constant across each removed interval
• the function never decreases
• 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 last term is small
• 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 other two grow a great deal before they shrink
• the p = 1 row is the harmonic sum, which is a logarithm plus Euler's constant
• 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 plotted partial sums are the series
• the prime is 2, 3 or 5
• the ratio is strictly between nothing and one
• the rearranged total settles near the number it was aimed at
• the removed intervals so far have the length the construction gives
• 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 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 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 tetranacci's distribution has dimension below one
• the total is not small
• the triangle has side between 8 and 24
• the tribonacci's distribution has dimension below one
• 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
• there is one address for each surviving interval
• this view is for a series that does converge
• 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
• what is left and what has gone add to the whole interval
• 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 area under x^0e^(−x) is 0! ×8
• at 1 steps the length is still 2 ×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
• 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
• 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 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 the fat one keeps a length in the limit
• and the finest is strictly longer than the coarsest, so the family is climbing
• between 20 and 20,000 lines
• between three and five corner counts, each up to 64
• both parts only ever increase
• 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 panel is at least as close as the one before it
• each staircase runs closer to the curve than the last
• every parametrisation traces the same point set
• lines is read only by the crofton view
• 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 width, no area
• 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 view
• 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 under 1/√x between 0 and 1 is 1/(1 − p)
• the area under the curve up to a
• 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 drawn partition has between 24 and 240 cells
• 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 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 inscribed polygon's length, measured, matches the closed form
• the interval runs from nought to at most 2
• 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 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 printed total is the area of the bars drawn
• the quoted exact area is the area under this curve
• the set is approximated at between 5 and 14 stages
• the share of the straight line, w, is between 0 and 1
• the signed area is within 1/T of π/2
• the slope of the accumulated area is the height of the curve
• 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 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 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 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
• 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)

• 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
• 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 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 step magnifies by between 2 and 20
• every secant from the left has slope −1
• every secant from the right has slope 1
• the bisection really inverts the function
• the corner's share of the increment falls at every halving
• the curve is one the figure knows
• the first window is at most 4 wide
• 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 secant is near the tangent
• the marked point is inside the window
• the pair is one the figure knows
• the rectangle is drawn inside the domain of both functions
• the secants are drawn at a list of 2 to 24 positive offsets
• 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 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

• 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 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
• a degree between 1 and 20
• a function that is already a quotient of polynomials is reproduced exactly
• a higher-order approximant is at least as close at the probe point
• 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 it dwarfs anything in the middle of the interval
• and the terms are smallest near 1/x
• 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
• each drawn pole is a root of the denominator
• 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
• 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
• inside the new interval more terms help
• inside the radius, more terms help
• ln(1+x) is centred where it is defined
• ln(1+x) is expanded inside its own radius
• no centre sits on a singularity
• no division by nought
• no rational has a denominator of nought
• no singularity lies strictly inside a disc
• 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 bound is a bound
• the centre is inside the drawn domain
• the Chebyshev end gaps are several times smaller than the middle one
• the drawn domain is an interval
• the equally spaced constant grows with every step
• the error never exceeds the bound Lagrange's form puts on it
• the evenly spaced nodes have one gap throughout
• 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 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 mode draws the functions whose derivative is monotone
• the mode of the taylor family is one of sums, radius, remainder, xi, centre, discs, interp, lebesgue, nodes, pade, poles, paderate, root, asymptotic, asymsums
• the nodes are equally spaced or Chebyshev
• the numerator's degree is a whole number close to the denominator's
• 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 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 smallest gap is at one of the ends
• the span is between 0.2 and 1.5
• the tail of the series is the difference between the function and its partial sum
• the Taylor degree is between 1 and 40
• the unknown point lies strictly between nought and x
• the window reaches past −1 and past 0.5
• with equally spaced nodes the worst error is out near the ends

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