Generator

dissect

A generator in the computation library, called 6 times across 1 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

dissect 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

A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

`show: "polygon"`

A 5-sided polygon reduced to 3 triangles. A convex polygon cut by diagonals from one corner into a fan of triangles, with a list of the pieces each stage of the standard chain of dissections costs.

`show: "steps"`

A 9 by 4 rectangle and a 6 by 6 square, from one staircase cut. A rectangle cut by a staircase into two pieces, beside the square the same two pieces make when one of them is slid by a single step.

`show: "dehn"`

The tetrahedron's dihedral angle is not a rational part of a turn. A cube and a regular tetrahedron with their dihedral angles named, above a table of the whole-number numerators of the cosines of every multiple of the tetrahedron's angle, none of which is divisible by three.

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.

a polygon of 5 sides falls into 3 triangles ×1
and exactly one step up ×1
and the same two tile the square ×1
each numerator over its power of three is the cosine of that multiple ×1
no numerator in the sequence is a multiple of three ×1
the apex sits over the base, between its two ends ×1
the cube's dihedral angle is a quarter turn exactly ×1
the fan of triangles tiles the polygon ×1
the number of multiples tested is a whole number between 6 and 30 ×1
the number of risers in the staircase is a whole number between 1 and 5 ×1
the number of sides is a whole number between 3 and 8 ×1
the polygon's radius is between 0.5 and 1.5 ×1
the rectangle and the square have the same area ×1
the rectangle has the triangle's area ×1
the same three pieces tile the rectangle ×1
the side of the square is a whole number between 2 and 24 ×1
the slide is exactly one step across ×1
the square's side is divisible by n(n+1) so every corner lands on a whole number ×1
the three pieces tile the triangle ×1
the triangle is between 0.4 and 1.6 times as tall as its base is long ×1
the triangles' areas add up to the polygon's ×1
the two pieces tile the rectangle ×1
the view is one of steps, polygon, dehn ×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.

Computation

Equal area is enough, and equal volume is not

Any two polygons of the same area can be cut into each other with finitely many straight cuts. The same sentence with area replaced by volume and polygon by polyhedron is false, and what blocks it is an angle.

The whole library · What the figures prove