This picture by Tilman Piesk shows the 14 Dyck words of length 8. A Dyck word is a balanced string of left and parentheses. In the picture, a left parenthesis is shown as upward-slanting line segment, and a right parenthesis as a downward-slanting one.
Suppose you have a set with 5 elements. There are 10 ways to choose a 2-element subset. Form a graph with these 10 choices as vertices, and with two vertices connected by an edge precisely when the corresponding subsets are disjoint. You get the graph shown here, called the Petersen graph.
This picture by Tilman Piesk shows the 15 partitions of a 4-element set, ordered by refinement. Finer partitions are connected to coarser ones by lines going down. In the finest partition, on top, each of the 4 elements is in its own subset. In the coarsest one, on bottom, all 4 elements are in the same subset.
The Kepler problem concerns a particle moving under the influence of gravity, like a planet moving around the Sun. Newton showed the orbit of such a particle is an ellipse, assuming it doesn’t fly off to infinity. There are many ways to prove this, but the most illuminating is to reparametrize time and think of the orbit as a circle in 4 dimensions. When the circle is projected down to 3-dimensional space, it becomes an ellipse. The animation in this post, created by Greg Egan, shows how this works.
This image by Greg Egan shows 5 ways to inscribe a regular tetrahedron in a regular dodecahedron. The union of all these is a nonconvex polyhedron called the compound of 5 tetrahedra, first described by Edmund Hess in 1876.
Here Greg Egan has drawn two regular dodecahedra, in red and blue. They share 8 corners—and these are the corners of a cube, shown in green. Adrian Ocneanu calls these twin dodecahedra, and has proved some fascinating results about them.
A spheroid is an ellipsoid with an axis of rotational symmetry. This image created by John Valentine shows a sphere inside a mirrored spheroid, reflected almost endlessly.
A Schwarz triangle is a spherical triangle that can be used to generate a tiling of a branched covering of the sphere by repeatedly reflecting this triangle across its edges. Sometimes we get an actual tiling of the sphere, but in general we get a branched covering, because the same point can lie in the interior of several triangles, and there may be branch points at the corners of the triangles.
This is the small cubicuboctahedron, as drawn by Robert Webb’s Great Stella software. It looks simple enough, but it conceals some interesting mathematics.
This picture drawn by Katherine Stange shows what happens when we apply fractional linear transformations $ z \mapsto \frac{a z + b}{c z + d} $ to the real line sitting in the complex plane, where $a,b,c,d$ are Eisenstein integers that is, complex numbers of the form $ m + n \exp(2 \pi i/3) $ with $m,n$ being integers.
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