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↦az+bcz+dz↦az+bcz+d to the real line sitting in the complex plane, where a,b,c,da,b,c,d are Eisenstein integers that is, complex numbers of the form m+nexp(2πi/3)m+nexp(2πi/3) with m,nm,n being integers.
Two regular pentagons and a regular decagon fit snugly at a point: their interior angles sum to 360°. Despite this, you cannot tile the plane with regular pentagons and decagons. However, there is a branched covering of the plane tiled with pentagons and decagons, which map to regular pentagons and decagons on the plane. Here Greg Egan has drawn a portion of this branched covering.
Two regular pentagons and a regular decagon meet snugly at a vertex: their interior angles sum to 360°. However, they can’t tile the plane. However, they come fairly close, as shown in this picture by Greg Egan.
The moving sofa problem asks: what is the shape of largest area that can be maneuvered through an L-shaped hallway of width 1? This animated image made by Claudio Rocchini shows one attempt to solve this problem.
The icosidodecahedron can be built by truncating either an icosahedron or a dodecahedron. It has 30 vertices. It is a beautiful, highly symmetrical shape. But it’s just a shadow of an even more symmetrical shape with twice as many vertices in twice as many dimensions!
This picture by Toby Hudson shows the densest known packing of the regular pentagon.
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