Generator

Pascal's triangle, 8 rows

A generator in the discrete library, called 43 times across 13 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

pascal is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

Pascal's triangle, 8 rows

Pascal's triangle modulo 4, where one digit at a time is not enough

C(13, 4) modulo 4, read in windows of 2 base-2 digits

Every fifth row of Pascal's triangle repeats the triangle modulo 125

A polygon of 4 points averaged down to its curve at t = 0.40

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

Granville's rule gives row 0, entry 0 modulo 4 once its 2s are removed ×906
Lucas' theorem gives row 0, entry 0 modulo 2 ×906
the exponent of 2 in row 0, entry 0 is the number of carries ×756
the row adds to one at x = 0 ×401
row 0, entry 0 is 3 modulo 4 exactly when an odd number of pairs of ones are split ×243
row 0, entry 0 of Pascal's triangle ×91
the additive rule built C(32, 0) ×61
row 0, entry 0 is the binomial coefficient ×45
2 divides the coefficient at least 1 times ×10
and no more than 0 times ×6
degree 2 lies above the convex function ×6
degree 2: averaging down the triangle and the closed form agree ×6
column 0 of the sum is n's digit ×4
the polynomial for k = 0 peaks at k/n ×4
the routes to the cut at height 0 are counted both ways ×4
the weight on P0 is C(3, 0) t^0 (1 − t)^3 ×4
between 3 and 8 rows ×3
degree 2: the gap for x² is x(1 − x)/2 ×3
C(5n, 5k) − C(n, k) is a multiple of 5^3 for every n up to 24 ×2
Granville's product, with its sign, is C(13, 4) with its 2s removed, modulo 4 ×2
n is between 2 and 1000 ×2
the digit product is wrong modulo 4 somewhere in 32 rows ×2
the exponent of 2 in the coefficient is the number of carries ×2
the run down diagonal 2 adds to the entry at row 7, place 3 ×2
a higher degree has a smaller largest gap ×1
a p-free factorial is a unit modulo the prime power ×1
and encloses 1/4 ×1
and ends at the last ×1
and the two are not simply equal ×1
and up ×1
at least 8 rows ×1
at most twenty routes are drawn ×1
between 1 and 6 steps up ×1
between 1 and 8 steps across ×1
between three and six control points, each inside the unit square ×1
classifying routes by where they cross the diagonal accounts for all of them ×1
every control point's label sits inside the frame ×1
every drawn entry was checked ×1
every drawn pair agrees modulo 5^3, as the theorem promises ×1
every drawn pair agrees modulo p, which is Lucas' theorem ×1
every entry of the drawn triangle was checked both ways ×1
every point of the curve lies inside the control points' hull ×1
four to ten increasing degrees, up to 1024 ×1
k is strictly between nought and n ×1
one to four functions from square, sine, kink, runge ×1
one to three increasing degrees, up to 256 ×1
row 6 adds to two to the power 6 ×1
row 6 with alternating signs adds to nought ×1
shallow diagonal 8 adds to a Fibonacci number ×1
t is strictly between nought and one ×1
t, if given, is strictly between nought and one ×1
the addition has as many columns as n has digits ×1
the averaged point is the weighted sum, across ×1
the base is a prime between 2 and 11 ×1
the base is a prime up to 7 ×1
the curve starts at the first control point ×1
the cut is a diagonal strictly inside the grid ×1
the diagonal is one the triangle holds ×1
the function is one of square, sine, kink, runge ×1
the hull test accepts the hull's own centroid ×1
the identity is one the mode draws ×1
the kink converges like 1/√n ×1
the mode of the pascal family is one of numbers, parity, mod, sierpinski, sums, paths, carries, power, granville, scaled, casteljau, basis, bernstein, rate ×1
the modulus is p to a power from 1 to 4 ×1
the modulus is p, p² or p³ ×1
the picture is modulo a prime squared, whose remainders have two base-p digits to colour ×1
the prime is 2, 3 or 5 ×1
the prime is one of 2, 3, 5 and 7 ×1
the routes counted by the additive rule agree with the coefficient ×1
the row count is a whole number between 1 and 64 ×1
the row is between 1 and 8 ×1
the row is one the triangle holds ×1
the row's double is inside the triangle, or the total cannot be pointed at ×1
the run starts inside the triangle and leaves room for its total ×1
the shallow diagonal is one the triangle holds ×1
the sine converges like 1/n, however smooth it is ×1
the squares of row 4 add to the middle entry of row 8 ×1
the subdivision depth is a whole number between 1 and 8 ×1
the weights add to one ×1
x² converges like 1/n ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Discrete

A polynomial that counts

Hang a counting sequence on the powers of a variable and the two ways of combining choices — this and that, this or that — become multiplication and addition, so a recursion turns into an equation and the equation can be solved.

Discrete

A remainder read two digits at a time

Lucas' theorem reads a binomial coefficient's remainder on division by a prime off its digits one at a time. On division by the prime's square the same reading is wrong at four odd entries in ten. What replaces it still reads digits — in overlapping pairs, with the prime taken out first and a sign that the carries decide.

Number

Always one before the double

A density says what happens on average and permits long empty stretches. This says something a density cannot — that the stretch from any number to twice it contains a prime, at every scale, without exception.

Discrete

Averaging down the triangle

Change one word in the rule that builds Pascal's triangle — take a share of each entry above instead of adding them — and the triangle stops counting and starts averaging. The same rule then draws smooth curves from polygons and approximates every continuous function by polynomials, at a rate that no amount of smoothness can improve.

Probability

A bell curve assembled out of coin flips

Drop six hundred balls through a board of pegs, each bouncing left or right at random, and they pile up in a shape that can be predicted precisely. Nothing coordinated them.

Geometry

Circles that are diamonds and squares

The theorem hands over a formula for distance. Take the formula as a definition, change the exponent in it, and the set of points one unit from the origin stops being round — while remaining, in every sense that matters, a circle.

Discrete

Every entry counts the routes to it

Turn Pascal's triangle forty-five degrees and it becomes a grid of street corners, with each entry counting the ways of walking there. Identities between the entries then become statements about routes, and the statements are proved by cutting the routes in one place.

Geometry

Every square is a stack of odd numbers

Add up the odd numbers in order and the running totals are 1, 4, 9, 16, 25. This is not a coincidence, and the reason fits in a single picture.

Discrete

One sequence, counting everything

The number of ways to cut a polygon into triangles is 1, 2, 5, 14, 42. So is the number of ways to bracket a product, the number of binary trees, and the number of paths that never cross a diagonal. They are the same count, and the reason is one picture.

Discrete

Pascal's triangle, in two colours

Shade the odd numbers in Pascal's triangle and a fractal appears. Nothing was designed to produce it, and the same shape arrives independently from a completely different construction.

Discrete

The carries decide the divisibility

How many times a prime divides a binomial coefficient is not a fact about the coefficient at all. It is a count of the carries that happen when two numbers are added in that prime's base, which is a question about column addition and has nothing to do with choosing anything.

Probability

The path folded at its first touch

Counting the walks that touch a line looks like a question about a walk's whole history. Fold each one where it first touches, and it becomes a question about where walks end up — which is a binomial coefficient, and is already known.

Discrete

The run that lands one place along

Add up a run of entries down one of Pascal's diagonals and the total is another entry of the triangle — one row further down and one place along. The same triangle holds four more sums of that kind, and each is a different question answered by the same additive rule.

The whole library · What the figures prove