[freebeard]

Originally Posted by myself at #725

The ancient Greeks thought Universe is a dodecahedron, that wraps around to opposing faces like a side-scrolling 8-bit video game. Being a fan of the Platonic solids (and the Archimedian semi-regulars) it sounds as good as anything to me.

Found this today, via Hacker News

Didicosm
Loops Across Space

Adidicosm is a certain kind of three-dimensional space, one of ten platycosms, the name given by the mathematician John Conway[1] to a space that has a finite volume, no boundary, and a flat geometry: the same geometry locally as ordinary Euclidean space. Four of the ten platycosms are non-orientable (a property for which the Möbius strip is the most famous two-dimensional example) which would lead to serious complications if we lived inside such a space, as it would allow matter to be transformed into its own mirror image just by travelling around a loop. The remaining six, including the didicosm, are potentially candidates for the shape of our own universe, if it happens to be finite.[2] (The didicosm is also referred to as a Hantzsche-Wendt Manifold, named after the mathematicians who discovered it in the 1930s, but we will use the name that Conway gave it much later.)

A didicosm can be formed by taking a rectangular prism, as in the image on the right, and “identifying” its faces, so that when you move within the prism to any point on its boundary, you re-enter it at a different point, determined by the way the faces are matched up with each other.

All this is through the frame of Cartesian Coordinates. They get close to Sytnergetics at the end, finding triangulations inside the cube with integer values.

The image on the right shows one solution found from a triangulation of the didicosm. Here, the blue line represents the loop X4, while the triangles have been coloured to show various integer coefficients: yellow=1, orange=2, magenta=3, red=4.