Schläfli–Hess 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogs of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
Regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century, although the full set were not discovered until later.
There are six convex and ten star regular 4-polytopes, giving a total of sixteen.
The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. Schläfli discovered that there are precisely six such figures.
Schläfli also found four of the regular star 4-polytopes; (the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell). He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F − E + V = 2). That excludes cells and vertex figures as {5,5/2}, and {5/2,5}.
Edmund Hess (1843–1903) published the complete list in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder.
The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which form its cells and a dihedral angle constraint
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