Ursatope

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The ursatopes are a family of convex polytopes that generalize the tridiminished icosahedron, and specifically how its vertices fall in three parallel planes. They were discovered by members of the hi.gher.space community and are notable for producing CRF polytopes under certain conditions.

Construction[edit | edit source]

The triangular ursahedron with the three planes highlighted.

An n -ursatope is constructed from a base CRF polytope 𝓟 of rank n  − 1.

The ursatope is constructed by placing polytopes on 3 parallel hyperplanes

  • 𝓟 scaled to have unit edge length on the first hyperplane (red)
  • 𝓟 scaled to have edge length equal to the golden ratio on the medial hyperplane (blue)
  • The rectification of 𝓟 scaled to unit edge length on the final hyperplane (green)

The distances between hyperplanes is adjusted until the vertices from the three layers line up to form regular pentagons.

Then the ursatope is the convex hull of the three polytopes.

Properties[edit | edit source]

It happens to be orbiform in general, provided if its base was such. Accordingly it can be used as a vertex figure for other (convex) uniform polytopes then.

Name[edit | edit source]

Ursatopes derive their name from the tridiminished icosahedron the only 3D ursatope. The prefix uses the Latin root ursa-, meaning "bear", in reference to similarity between the OBSA of the tridiminished icosahedron, "teddi", and the teddy bearteddy bear.

Examples[edit | edit source]

The following lists some examples of CRF ursatopes

2D[edit | edit source]

There is one 2D ursatope, the pentagon, formed from a dyad as the base polytope.

3D[edit | edit source]

While an ursatope can be made for any regular convex polygon, only the triangle produces a Johnson solid.

4D[edit | edit source]

There are 3 known CRF ursachora:

All three are formed with a Platonic solid as their base polytope.

5D[edit | edit source]

There are 4 known CRF 5D ursatopes, the first 3 of which are formed with regular polychora as their base polytope.

A Hexacosichoric ursateron might be added as a dimensionally degenerate case only, because its height within the 5th dimension would become zero.

Higher dimensions[edit | edit source]

In dimensions higher than 5, two families of ursatope are known, which have regular bases: those based on the simplex and the cross polytope. A further dimensional family would be based on the demicube.

External links[edit | edit source]