pascal
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.
At its defaults
show: "parity"
show: "sierpinski"
show: "mod"
What it checks while it draws
Collected by running the family and listening to lib/verify.js, not written
here. The count is how many separate times this build put that claim to the test.
- row 0, entry 0 of Pascal's triangle ×45
- the row count is a whole number between 1 and 64 ×1
- the subdivision depth is a whole number between 1 and 8 ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
GeometryEvery 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.
DiscreteOne 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.
DiscretePascal'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.