euler-path
euler-path 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: "graph"
show: "parity"
show: "example"
What it checks while it draws
Collected by running the family and listening to lib/verify.js, not written
here. The count is how many separate times this build put that claim to the test.
- a closed circuit needs every degree even ×1
- all four landmasses have odd degree ×1
- an even degree pairs every arrival with a departure ×1
- an odd degree leaves exactly one edge unpaired ×1
- an open walk needs exactly two odd degrees ×1
- every vertex drawn is a vertex used ×1
- Königsberg has seven bridges ×1
- landmass E has degree 3 ×1
- landmass I has degree 5 ×1
- landmass N has degree 3 ×1
- landmass S has degree 3 ×1
- no Euler walk exists ×1
- the degrees add to twice the number of edges ×1
- the even panel really shows an even degree ×1
- the odd panel really shows an odd degree ×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.
Every corner pays for itself
Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself.
DiscreteSeven bridges, and the invention of throwing things away
Euler solved a puzzle about a Prussian city by deleting the city. What survived the deletion was a new branch of mathematics.