Quotes About Polytopes

Ludwig Schlafi (1814-1895) proved that there are six regular four-dimensional polytopes (generalisations of polyhedra): the 5-cell made of tetrahedra, the 8-cell or tesseract made of cubes, the 16-cell made of tetrahedra, the 24-cell made of octahedra, the 120-cell made of dodecahedra, and the 600-cell made of tetrahedra.

Schlafi also proved that in five or more dimensions the only regular polytopes are the simplex, or generalized tetrahedron, the hypercube, or generalized cube, and the orthoplex, or generalized octahedron.

Cantor's discovery that lines, planes, cubes, and polytopes were all equivalent as sets of points goes a long way toward explaining why set theory was such a revolutionary development for math-revolutionary in theory and practice both.