euler-solid
euler-solid 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: "flatten"
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.
- and its twelve edges ×1
- every edge of the cube is shared by two faces ×1
- every edge of the dodecahedron is shared by two faces ×1
- every edge of the icosahedron is shared by two faces ×1
- every edge of the octahedron is shared by two faces ×1
- every edge of the tetrahedron is shared by two faces ×1
- the flattened cube keeps its eight corners ×1
- the flattened cube still gives 2 ×1
- the solid is one figure-kit knows ×1
- V − E + F is 2 for the cube ×1
- V − E + F is 2 for the dodecahedron ×1
- V − E + F is 2 for the icosahedron ×1
- V − E + F is 2 for the octahedron ×1
- V − E + F is 2 for the tetrahedron ×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.
Every corner pays for itself
Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself.
TopologyNothing on a sphere can be combed flat
Point an arrow along the surface at every place on a sphere, continuously, and somewhere an arrow has to vanish. On a doughnut it can be done. The difference between the two is a number that was already known from counting corners.