venn
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
show: "ellipses"
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.
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.
LogicTwenty-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.