Math & Code & Stuff
The cheating mentioned in the last post is this: Although the image data really is part of the hexadecimal expansion of π, I’ve added a palette to the gif to reinterpret the bytes as convenient colors. If we just interpreted the same exact bytes in a reasonable way, say as grayscale values, it looks like this: Nothing there. The trick is that we can reassign the colors using a palette. And in fact…
The SMBC comic Convert included as the “Votey button” a challenge to extract a Waldo visual from the digits of π.
Last time we looked at well-centered polytopes and I wondered whether a vertex-transitive polytope with well-centered \(j\)-faces would be completely well-centered. (For \(j=0\) or 1 the condition is trivial, since points and edges are always well-centered, so we’d have to require \(j=2\) at least.) This seems to be false. But why would I think this?
A well-centered polytope is one whose vertices all lie on a sphere (also known as being circumscribable, or equiradial) and with the center of that sphere (the circumcenter) within the polytope’s interior.
The hard part of FizzBuzz is the newlines.
There’s an easy way to make antiprisms whose sides are isosceles triangles: take two congruent regular \(n\)-gons, rotate one so its vertices are halfway between the original positions (that is, by 1/(2n)-th of a circle), and lift one up. Unless you pick one particular height, you end up with isosceles triangles on the side. (There’s one height that gives you equilateral triangles.)