Matpic
mathematics, in pictures
Generator

iso

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 waysRegular polygons all of the same perimeter, drawn to scale beside the circle of that perimeter, with the area each encloses and the ratio 4πA/L².3 sidesarea 4,3304πA/L² = 0.6054 sidesarea 5,6254πA/L² = 0.7856 sidesarea 6,4954πA/L² = 0.90712 sidesarea 6,9984πA/L² = 0.977the circlearea 7,1624πA/L² = 1every shape here has a perimeter of 300; only the area changes4πA/L² rises from 0.605 to 0.977 and reaches 1 only at the circle

show: "quotient"

9 shapes ranked by how much they encloseA 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.the circle1.000regular 8-gon0.948regular 6-gon0.907Reuleaux triangle0.897regular 5-gon0.865regular 4-gon0.7852:1 rectangle0.698regular 3-gon0.605an L0.589the circle's 1every shape is drawn at the same perimeter, so the bar is the area and nothing elsethe ratio has no units, so it ranks shapes rather than sizes: doubling a shape leaves its bar exactly where it was

show: "rectangles"

Every rectangle of perimeter 300The area of a rectangle of fixed perimeter against the share of it spent on one side, peaking at the square and falling away to nothing at both ends.00.10.20.30.40.50.60.70.80.910100020003000400050006000share of the half-perimeter on one sidearea enclosedthe square: 5,6254,2194,219the square is the best rectangle at 5,625, and its 4πA/L² is 0.785the curve is flat at the top, which is why a fence that is roughly square loses almost nothing and a fence that istwice as long as it is wide loses an eighth

show: "dent"

A dent, reflected: same fence, more fieldA 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.dented — area 37,180, 4πA/L² = 0.601reflected — area 51,220, 4πA/L² = 0.828the same 882 of fence, 37,180 of area before and 51,220 afterso no shape with a dent can be the best one, which removes every non-convex shape from the runningwithout settling which convex one wins

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.

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