See my Stellations of the Dodecahedron applet to see how a "Star Polyhedron" is generated in 3 dimensions. In that applet, we see that a dodecahedron is a volume of space that is bounded by 12 planes. When those planes are extended infinitely, they criss-cross through each other, chopping up space into many "chunks". The inner chunks are finite and they are distributed in shells around the core dodecahedron. The dodecahedron has only one kind of chunk in each shell, but other polyhedra (like the icosahedron) can have several different types of chunk in each shell.

The same procedure works in 4 dimensions. A 4-dimensional convex polyhedron (properly called a "polytope" or "polychoron") is a volume of 4-dimensional space that is bounded by a number of hyperplanes. For example, the 4-dimensional polytope known as the "120-cell" is bounded by 120 hyperplanes.

(A hyperplane is a 3-dimensional space that slices through the 4-dimensional space, the same way a 2-dimensional plane can slice through our 3-dimensional space.)

The bounding hyperplanes can be extended infinitely so that they criss-cross through each other, chopping up hyperspace into many 4-dimensional "chunks". Again the inner chunks are finite, and they are distributed in shells around the core polytope.

The HyperStar applet displays those finite chunks, one shell at a time. The inner shells are complete -- each shell completely encases the previous shell. The outermost shells have holes in them.

The applet's shell# represents the number of hyperplanes that you have to pass through to reach the interior of the core. Shell# 0 is the core itself. The first couple of shells contain only one kind of chunk in each, but most of the higher-numbered shells contain a variety of different chunks.

## Wednesday, June 27, 2007

### Hyperspace Star Polytope Explanation

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