Generator

3 circles, and the 8 patterns they realise

A generator in the logic library, called 9 times across 3 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.

venn 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

3 circles, and the 8 patterns they realise. Closed curves overlapping in the plane, with each region of the arrangement identified by which curves contain it.

`show: "ellipses"`

Four ellipses, and all sixteen patterns. Closed curves overlapping in the plane, with each region of the arrangement identified by which curves contain it.

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.

3 circles realise every one of the 8 patterns ×2
4 circles cannot realise all 16 patterns ×2
and each of the sixteen is a single connected region ×1
every leaf of the formula is a variable ×1
every letter in the set expression names a drawn curve ×1
no pattern occupies more pieces than the arrangement has ×1
the formula is a non-empty string ×1
the four ellipses realise all sixteen patterns ×1
the number of sets is a whole number between 2 and 5 ×1
the pieces the circles cut the plane into match Euler's count ×1
the whole formula is consumed by the parser ×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.

Logic

Four circles cannot do it

Three overlapping circles cut the plane into exactly the eight regions three sets need. Four circles cut it into fourteen, and sixteen are required — so the diagram everyone draws stops working at four, and the reason is a count.

Logic

Twenty-four out of two hundred and fifty-six

Aristotle's syllogisms are four sentence forms in four arrangements, which makes 256 patterns of argument. Fifteen of them are valid. Nine more become valid if you assume the things being talked about exist, and the gap between those numbers is a two-thousand-year-old disagreement.

Algebra

What is lost at eight

Double the quaternions and division still works, but ab times c and a times bc are no longer the same number. What survives is a weaker law that turns out to be enough for a great deal — and the whole multiplication table fits into a picture of seven lines.

The whole library · What the figures prove