Generator

One perimeter of 300, spent five ways

A generator in the geometry library, called 5 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.

iso 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

One perimeter of 300, spent five ways. Regular polygons all of the same perimeter, drawn to scale beside the circle of that perimeter, with the area each encloses and the ratio 4πA/L².

`show: "quotient"`

9 shapes ranked by how much they enclose. A bar per shape giving 4πA/L², the area it encloses as a fraction of what a circle of the same perimeter would; the circle is the only one at one.

`show: "rectangles"`

`show: "dent"`

A dent, reflected: same fence, more field. A shape with a dent beside the same shape with the dent reflected outward; the boundary is the same length in both and the second encloses more.

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.

the 3-gon encloses less than a circle of the same perimeter ×7
the 3-gon is drawn at the same perimeter as the rest ×7
regular 3-gon does not beat the circle ×5
2:1 rectangle does not beat the circle ×1
an L does not beat the circle ×1
and is still short of the circle's ×1
and it is the circle ×1
and its area is a quarter of the perimeter, squared ×1
and the area is strictly larger ×1
and the circle's quotient is exactly one ×1
even the best rectangle falls short of the circle ×1
every extra side encloses strictly more ×1
exactly one shape in the list reaches one, and it is the circle ×1
reflecting the dent leaves the perimeter exactly as it was ×1
Reuleaux triangle does not beat the circle ×1
so the quotient rises ×1
the circle does not beat the circle ×1
the circle of that perimeter encloses more than any of the polygons drawn ×1
the dent is between nothing and 160 units deep ×1
the largest rectangle in the sweep is the square ×1
the polygons drawn have between 3 and 60 sides ×1
the shared perimeter is between 60 and 900 units ×1
the sweep takes between 20 and 2000 samples ×1
the view is one the family draws ×1
whose two sides are equal ×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.

Geometry

The most area a fence can hold

One length of boundary, and the question of what shape to bend it into. The answer is a circle, everybody knows it, and the argument that convinced the nineteenth century turned out to prove something slightly different.

The whole library · What the figures prove