# Mathematics

This section of the site deals with a number of interesting topics in mathematics.

Tilings: This page begins a discussion of tilings with a discussion of the 17 wallpaper symmetry groups.

Pentagonal Tilings: This page exhibits a pentagonal tiling similar to that devised by Johannes Kepler.

Polycube Puzzles: This page gives several examples of puzzles whereby several parts, made up of small cubes, are to be assembled into a cube.

The Slide Rule: This page explains how slide rules were used.

Infinity: This page tries to explain some topics that are often confusing to nonmathematicians, primarily the Cantorian theory of transfinite numbers.

Archimedian Solids: This page exhibits the beautiful, but mostly little-known, solids one step up in complexity from the well-known Platonic solids.

The Fourth Dimension: This page exhibits some of the regular polytopes in four dimensions, and attempts to explain the 120-cell polytope whose three-dimensional surface consists of 120 dodecahedra.

Dodecahedral Rotations: This page contains a diagram of the possible rotations of a dodecahedron, showing the symmetry involved (which is also that of the A5 alternating group).

Probability: A small page about probability, featuring the odds at Hazard.

Groups, Rings, and Fields: I define these important mathematical terms here, and give a few interesting examples of groups.

Sphere Packing: On this page, I include some diagrams of the face-centered cubic packing of spheres, as well as packings of circles in two dimensions, and I also include a diagram illustrating the E8 packing of hyperspheres in eight dimensions.

Gödel's Proof and the Halting Problem: This page tries to discuss, very briefly, what is perhaps the most profound of mathematical discoveries.

Two Famous Equations: this page tries to explain two equations by Euler: e^(i*pi)=-1 is explained by means of an outline of trignometry, logarithms, and differential calculus, and The Riemann Zeta Function notes how this function relates the prime numbers to analysis (or rather, why it might do so; only the formula for it is explained, I do not attempt to give an introduction to analytic number theory).

Euler's Constant is described on another page.

Magic Squares: An example of a bimagic square, and the simple rules for constructing magic squares of some orders.