The ledger

What the figures prove

This collection claims that a picture can be an argument rather than an illustration of one. That is a strong claim, so the generators check it while they draw — and this page is every claim they check, collected by listening to the assertions during this build. Nothing on it was typed in.

• A lies on the circle
• and so do the other two
• and the same on the other side
• B lies on the circle
• every apex on the arc gives the same angle
• one angle, not several
• opposite angles add to a straight angle
• the angle on a diameter is a right angle
• 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 exterior angle is twice the base angle
• the quadrilateral's angles add to 360°
• the two halves are the whole angle
• triangle OPA is isosceles
• triangle OPB is isosceles

• 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

• the smallest square has side gcd(34, 13) ×4
• the squares tile the 34 by 13 rectangle ×4
• the leftover strip is still there after every pass
• the squares peeled at each pass are the continued fraction of the ratio

• the first 6 odd numbers sum to 6² ×4
• the L-shaped shells fill the square

• one square per Fibonacci number
• the convergents alternate about φ
• the last convergent is close to φ
• the long side is the next Fibonacci number
• the rectangle is nearly golden
• the short side is the one before it
• the whirling squares tile their rectangle

• 3 triangles at a vertex leave a gap ×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
• exactly five vertex figures close up

• and half the rectangle — same base, apex on the parallel through the altitude
• 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 eight triangles are all the same
• the long leg is b
• the short leg is a
• the square on the hypotenuse is c²
• the square on the leg has the leg's area squared
• the tilted square equals the two upright squares
• the tilted square is c²
• the triangle is half the square — same base, apex on the opposite side
• the two triangles are the same triangle turned

• the 3-sided Reuleaux curve has one width at every angle ×3
• a Reuleaux polygon needs an odd number of sides
• all the shapes have one width
• every arc bulges away from the centre
• ground and plank stay one width apart
• the width does not depend on how far it has rolled
• the width is the arc radius

• the rings account for the whole disc
• the triangle of base 2πr and height r has the disc's area

• the centre of cell 0 is nearest to site 0 ×20
• every site keeps a cell
• 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 triangulation is not empty

• the point is on the circle
• the point's height is the wave's height there
• the two coordinates are a point of the unit circle

• a base that is not e does not have slope equal to height
• and at base e the slope is the height
• and the gap left is smaller than the last term added
• and the limit of the staircase is e itself
• every partial sum is below e
• more splits is closer to e
• no number of splits reaches e
• the table shown gets e right to three decimals
• the tangent drawn is the curve's own slope

• every term count is a whole number between 1 and 200
• more terms is closer, away from the jump
• the curve drawn is the sum of the harmonics

• block 1 reaches a half ×4
• consecutive partial sums differ by the next term
• each block leans further out than the one below it
• no term is used twice
• the answer is caught between an even and an odd partial sum
• the bound on each block is exactly a half
• the drawn lean is the sum of the drawn steps
• the drawn running total is the partial sum
• the error is smaller than the last term used
• the gap to ln n has settled on γ
• the last term is small
• the plotted partial sums are the series
• the rearranged total settles near the number it was aimed at
• the signs are the ones the series has
• the target is visibly away from the sum the same terms give in order
• the total is not small

• each panel is at least as close as the one before it
• more bars means less error
• no width, no area
• the accumulation only ever climbs, because f is positive
• the function is one the figure knows
• the panels really are in increasing order of rectangles
• the printed total is the area of the bars drawn
• the quoted exact area is the area under this curve
• the slope of the accumulated area is the height of the curve

• each secant is closer than the last
• every secant from the left has slope −1
• every secant from the right has slope 1
• the function is one the figure knows
• the last secant is near the tangent
• the two sides disagree, so there is no limit

• the degree-1 sum matches to order 1 at the expansion point ×4
• every term count is a whole number between 1 and 40
• inside the radius, more terms help
• outside it, more terms hurt
• the function is one the figure knows

• and the reassembled rectangle has the same area
• the four tiles are the square
• the larger length is larger
• the length really solves the equation
• the missing corner is (b/2)²
• the negative root solves it too
• the picture is about positive lengths
• the positive root is the one a length can be
• the square tile is x²
• the strips are bx however they are cut
• the three pieces and the corner make a square of side x + b/2
• the two pieces are what is left of a² after b² is removed
• which is completing the square, as arithmetic
• which is the identity

• root 0 raised to the 5 is 1 ×17
• and has no imaginary part left
• and the angles add
• every root is on the unit circle
• in both coordinates
• the lengths multiply
• the roots sum to zero

• a real 2×2 map has two eigenvalues counted with sign
• Av is λv in the first coordinate
• Av is λv in the second

• a·b is |a||b| cos θ
• a·b is the shadow's length times |b|
• the dropped line meets b at a right angle
• the sign of a·b follows the angle
• the vector projected onto is not zero

• the walk reaches m × n / gcd cells on a 3 by 5 grid ×3
• a dial needs at least three positions
• a row is a permutation exactly when its multiplier is coprime to the modulus
• every non-zero row is a permutation exactly when the modulus is prime
• every residue has a place on the dial
• every row of the addition table is a permutation
• it fills the whole grid exactly when the moduli are coprime
• multiplying `order` times returns to 1
• the arithmetic and the walk agree
• the drawn entry is the arithmetic
• the drawn walk passes the top as often as the division says
• the first modulus is between 2 and 12
• the multiplier is not zero
• the orbit visits each residue once before closing
• the order divides m − 1, which is Fermat's little theorem
• the period is the least common multiple
• the second modulus is between 2 and 12
• the walk is between one and 144 steps
• the walk runs at least one full period

• a vertex meets five others
• and in fact at least two, which is Goodman's bound
• EVERY colouring of six people contains a monochromatic triangle
• five edges in the pentagon
• K5 has ten edges
• K6 has fifteen edges
• no triangle has all three edges the same colour
• the three ends make three pairs, and any one of them closes a trio
• this colouring has a monochromatic triangle
• two colours over five edges forces three of one

• a closed circuit needs every degree even
• all four landmasses have odd degree
• an even degree pairs every arrival with a departure
• an odd degree leaves exactly one edge unpaired
• an open walk needs exactly two odd degrees
• every vertex drawn is a vertex used
• Königsberg has seven bridges
• landmass E has degree 3
• landmass I has degree 5
• landmass N has degree 3
• landmass S has degree 3
• no Euler walk exists
• the degrees add to twice the number of edges
• the even panel really shows an even degree
• the odd panel really shows an odd degree

• an odd rim needs four and an even rim three
• four colours are enough
• no two neighbouring regions share a colour
• the map can be coloured with at most five

• a marked square holds a prime, and an unmarked one does not
• and 41² is where it fails
• consecutive integers are neighbours on the spiral
• Euler's polynomial is prime for forty values in a row and then is not
• no two integers land on the same square
• the first polynomial really is the prime-richer one over the range drawn
• the sieve starts where the primes do
• there are primes to compare against

• row 0, entry 0 of Pascal's triangle ×45
• the row count is a whole number between 1 and 64
• the subdivision depth is a whole number between 1 and 8

• each case really does overflow its boxes
• every item is in a box
• more items than boxes forces a box with two
• the fullest box holds ⌈items/holes⌉
• there are at least as many things as boxes
• there is at least one box

• C(0) two ways ×9
• a 6-gon has C(4) triangulations ×4
• each cut uses n−3 diagonals

• and its twelve edges
• every edge of the cube is shared by two faces
• every edge of the dodecahedron is shared by two faces
• every edge of the icosahedron is shared by two faces
• every edge of the octahedron is shared by two faces
• every edge of the tetrahedron is shared by two faces
• the flattened cube keeps its eight corners
• the flattened cube still gives 2
• the solid is one figure-kit knows
• V − E + F is 2 for the cube
• V − E + F is 2 for the dodecahedron
• V − E + F is 2 for the icosahedron
• V − E + F is 2 for the octahedron
• V − E + F is 2 for the tetrahedron

• and has no direction to choose there
• and it lies inside the disc
• and sits on it
• at the left end the map moves the point right or not at all
• at the right end it moves it left or not at all
• dipole has the index it claims
• every arrow is tangent to the sphere
• every orbit runs into the fixed point
• saddle has the index it claims
• source has the index it claims
• the field is one the figure knows
• the field vanishes at the pole
• the fixed point is unique
• the map is one the figure knows
• the map keeps the interval inside itself
• the map sends the disc into itself
• the point found really is fixed
• the solved point is fixed, first coordinate
• the solved point is fixed, second

• a diagram with k crossings is cut into k arcs
• at every crossing the three arcs are all alike or all different
• the colouring really uses more than one colour
• the figure-eight knot has the crossings it claims
• the knot is one the figure knows
• the table is ordered by crossing number
• the trefoil has the crossings it claims
• the trefoil is three arcs
• the unknot has the crossings it claims
• the unknot is a single arc
• there are three moves, and Reidemeister proved there are no others

• carrying the surface round once brings it back on the other face
• the band is drawn at a whole number of half-twists
• the gluing arrow matches the twist the figure is about

• and one lap leaves one loop while two laps leave two
• the cut is inside the band
• the scissors take one lap down the middle and two off-centre

• the image of 0 lies on the circle ×14
• the ray to 0 passes through the pole ×14
• the image of -2 lies on the circle ×13
• the ray to -2 passes through the pole ×13
• the image is on the ray from the pole through the point
• the image lands on the plane

• and at three doors it is two thirds
• everybody has it, so everybody positive has it
• nobody has it, so nobody who tests positive has it
• one of the two strategies wins, and only one
• staying wins as often as the first pick was right
• switching gets better with more doors
• switching wins two thirds against a host who knows, and half against one who does not
• the answer rises with the base rate
• the four cells fill the square
• the game needs at least three doors
• the ignorant host's story throws two of the six worlds away
• the ignorant-host variant is only drawn at three doors, where switching has one meaning
• the shaded fraction is Bayes' theorem
• the two strategies exhaust the possibilities

• 22 is still under a half
• 23 is over it
• one person shares with nobody
• the half-way point is 23 people
• the pair count is k choose 2

• at least one needle is dropped
• every needle claimed was drawn
• so a needle crosses with probability 2/π
• the area under half a sine wave is 1
• the box has area π/2
• the estimate is in a plausible range for pi
• the printed estimate is 2Ln / dc

• a path bounces once per row
• and are centred under the funnel
• and ends in a bin the board has
• the exact bin shares add to one
• the path ends over the bin its right-turns name
• the row count is a whole number between 1 and 24

• a walk in space escapes, and the simulation sees it
• a walk on a line comes back, and the horizon barely hides it
• a walk on a plane comes back too, but more slowly
• an endpoint has the parity of the step count
• and their spread is the square root of the number of steps
• each step moves one place along one axis
• every step is drawn
• every step is one place, either way
• every walk is counted once
• some walk goes outside the envelope, because √n is typical and not maximal
• the position has the parity of the step count
• the quoted constant is Pólya's
• the spread of the endpoints is of the order of the square root of the steps
• the walk comes back to where it started at least once
• the walks are centred on where they started

• |qα − p| comes out below 1/N
• and every convergent from the first onward breaks a record
• every record-breaking fraction is a convergent
• neighbouring convergents differ by a determinant of one
• so p/q is within one over qN of the number itself
• the constant is one this figure knows
• the largest denominator is a whole number between 8 and 200
• the number of boxes is a whole number between 3 and 14
• there are record-breaking denominators to mark
• two of the N + 1 points share a box, as they must

• the quotients rebuild 34/13 ×3
• the number of terms is a whole number between 2 and 12 ×2
• √2 has a convergent above √5, as Hurwitz says every irrational must
• and the last of them has closed on √5
• between two and five constants
• e has a convergent above √5, as Hurwitz says every irrational must
• each convergent is closer than the one before
• neighbouring convergents differ by a determinant of one
• the constant is one this figure knows
• the convergents alternate above and below the value
• the denominator is a whole number between 1 and 10000000
• the golden ratio's convergents never leave the neighbourhood of √5
• the numerator is a whole number between 1 and 10000000
• the tower's last convergent is a number
• there are enough convergents to draw
• π has a convergent above √5, as Hurwitz says every irrational must
• π has a convergent that beats √5 by two orders of magnitude
• φ has a convergent above √5, as Hurwitz says every irrational must

• the side of the big square is a whole number between 2 and 200 ×2
• the side of the small squares is a whole number between 1 and 200 ×2
• and the discrepancy never changes size
• and the new pair is strictly smaller
• every step is strictly smaller than the one before
• the descent carries the discrepancy to minus itself
• the descent runs out of room before the discrepancy runs out
• the descent takes at least three steps
• the pair is one the descent can start from
• the two small squares overlap and still reach the corners
• the two small squares, less their overlap and plus the corners, are the big one

• 6 is called what it is ×7
• the bar for 6 is its divisors laid end to end ×7
• the lattice draws every divisor of 30 exactly once ×4
• 496 is perfect ×2
• the number is a whole number between 4 and 100000 ×2
• and so do the norms of two and three
• between one and eight whole numbers above one
• both first splits divide the number
• both trees end in the same multiset of primes
• every divisor including the number itself adds to twice the number
• no divisor is drawn twice
• no element of this ring has norm two or three, so neither factor can split further
• the cells add up to the divisor sum
• the divisor count is the product of one more than each exponent
• the divisor sum factorises as the two row totals multiplied
• the exponent is a whole number between 2 and 7
• the first tree's leaves multiply back to the number
• the lattice half-width is a whole number between 2 and 5
• the Mersenne number for this exponent is prime
• the number has a factor tree to draw
• the number has at least three prime factors, so its trees can differ
• the number has at most three distinct primes
• the number is a product of exactly two prime powers
• the powers of two below 2^k add up to the Mersenne number
• the rectangle holds every divisor
• the second tree's leaves multiply back to the number
• the two conjugate factors have norms multiplying to 36
• the two trees start differently

• the partitions of 8 into odd parts and into distinct parts come out equal ×2
• and it is the largest one that fits
• every listed partition adds to the number it partitions
• the conjugate is a partition of the same number
• the number being partitioned is a whole number between 3 and 12
• the parts are a descending list of whole numbers
• the square, the arm and the leg account for every dot
• there is a Durfee square to draw
• turning the diagram over twice gives it back

• 3, 4, 5 is a Pythagorean triple ×10
• 5 = 2² + 1² ×5
• the points on the circle of radius √25 are 4(d₁ − d₃) ×5
• a prime is a sum of two squares exactly when it is one more than a multiple of four
• and the point it came from is on the unit circle
• every marked point really is on the circle
• no two slopes give the same triple
• the largest prime to test is a whole number between 10 and 200
• the lattice half-width is a whole number between 1 and 16
• the number is a whole number between 1 and 200
• the slopes are proper positive fractions

• the circles on 0/1 and 1/7 touch ×62
• 0/1 and 1/7 are Farey neighbours ×24
• 1/1 arrives already in lowest terms ×15
• the sequence has one entry per coprime pair up to 7 ×2
• circles that are not neighbours stay clear of each other
• no fraction is produced twice
• reading the drawing left to right reads the fractions in increasing order
• the depth is a whole number between 2 and 5
• the drawing scale is a whole number between 200 and 400
• the order is a whole number between 2 and 12
• the tree is full to its stated depth

• the length is a whole number between 3 and 7 ×2
• the number of colours is a whole number between 2 and 4 ×2
• and exactly a of them are single
• and the number of non-constant classes is (a^p − a)/p
• every class is a single string or a full ring of p
• every other class holds exactly p strings
• every string of the given length is drawn
• p divides a^p − a, which is what the count just showed
• the classes account for every string
• the classes of size one are exactly the constant strings
• the figure draws every string, so there is a ceiling on how many
• the length is prime
• the length is prime — the whole argument needs it to be

• (p | q) comes out of the count above it
• (q | p) comes out of the count below the line
• 3 folds gives the same answer as Euler's criterion
• and every one of them lands in the bottom half
• and their product is minus one to the power of the whole rectangle
• both are odd primes and they are different
• every point is counted exactly once, on one side or the other
• no lattice point lies on the diagonal, because the primes are coprime
• the first prime is a whole number between 3 and 31
• the folded values are all different
• the modulus is a whole number between 5 and 31
• the modulus is an odd prime
• the multiplier is a whole number between 2 and 30
• the multiplier is not a multiple of the modulus
• the points below the line are the sum of the floors of kq/p
• the second prime is a whole number between 3 and 31

• 211 leaves remainder 1 on division by 2 ×10
• the limit is a whole number between 10 and 400 ×2
• the staircase ends at π(1000) ×2
• a square left unstruck holds a prime, and a struck one does not
• every prime factor of the constructed number is outside the list
• one gap between each consecutive pair
• the column count is a whole number between 4 and 25
• the count runs above x / ln x over this range
• the gaps add up to the distance from 2 to the last prime
• the list is between two and six primes
• the pass number is between 0 and 8
• the squares left standing below 100 are the primes

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