Matpic
mathematics, in pictures
Generator

parity

A generator in the topology library, called 6 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.

parity 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

A point, a ray, and 8 crossingsA closed curve wound into a spiral corridor, with a marked point, a ray from it and every crossing marked; an odd count means the point is inside.outsidethe ray crosses the curve 8 times, so the point is outsidefired in any of 360 directions from the same point, the count changes and its parity does not

show: "rays"

16 rays from one point, and one parityRays fired in 16 directions from a single point, each crossing the curve a different number of times, with every count having the same parity.16 rays from one point count 7, 9 crossings — every one of them oddthe number of crossings is a property of the ray; its parity is a property of the point

show: "shade"

Every sample point, classifiedA grid of sample points over the curve, each marked inside or outside by counting one ray's crossings, with the inside shaded by the same rule.219 of the 416 sample points are inside, which puts the area at 100,235 against the 98,063 thecoordinates givea finer sweep than the one drawn finds exactly two connected pieces, which is the theorem the eyewas going to assume

show: "nonsimple"

A curve that crosses itself, where inside stops being one questionA five-pointed star drawn as a single closed loop that crosses itself five times, with a ray from its centre; the crossing rule and the winding rule disagree about the centre.even: outsidewinds 2×: insidethe centre is crossed an even number of times, which the crossing rule reads as outside,and the curve winds twice round it, which the winding rule reads as doubly insidefor a curve that crosses itself the two rules are different questions, and the theorem isabout the curves where they are not

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.

Topology

Which side of the line is inside

A closed curve with no self-crossings divides the plane into an inside and an outside. Nobody doubts it, almost nobody can prove it, and on a curve wound tightly enough nobody can see which side a given point is on either.

The whole library · What the figures prove