4D Euclidean space

News Archive

June 2018

This month, we introduce another scaliform polytope (a polytope that is vertex-transitive, albeit not necessarily uniform):

truncated tetrahedral cupoliprism

This is the truncated tetrahedral cupoliprism, or tutcup for short. It is formed from the convex hull of two truncated tetrahedra placed in dual orientation to each other in parallel hyper­planes, and has 6 tetrahedra and 8 triangular cupolae connecting the two truncated tetra­hedra. It is listed as K4.55 among Dr. Klitzing's convex seg­mento­chora.

Go to the truncated tetrahedral cupoliprism page to find out more about this cute little scaliform polytope. As usual, we provide the full coordinates.

May 2018

This month's Polytope of the Month is a modification of the familiar snub 24-cell:

The runcinated snub 24-cell

This is the runcinated snub 24-cell, an interesting polytope that, even though it is not uniform, is nevertheless vertex-transitive and CRF, thus belonging to the class of scaliform polytopes. As its name implies, it is derived from the snub 24-cell by radially expanding the 24 icosahedral cells outward. The remaining gaps are filled in by 24 truncated tetrahedra, 96 triangular cupolae, and 96 triangular prisms.

Check out the runcinated snub 24-cell page to explore the structure of this interesting polytope. As is customary, we provide full coordinates.

Apr 2018

This month we present a beautiful specimen of a CRF polytope that exemplifies the fascinating structure of the Hopf fibration of the 4D sphere. This structure is actually present in all of the regular polychora, but it is hidden within their higher degree of symmetry. In the swirl­prismato­diminished rectified 600-cell, however, this higher symmetry is stripped away, leaving bare the Hopf fibration substructure.

swirlprismatodiminished rectified 600-cell

Also known by its Bowers Acronym spidrox, this polychoron, in spite of being non-uniform because of its square pyramid cells, is nonetheless vertex-transitive and has equal edge lengths, and thus belongs to the class of scaliform polytopes. Its 600 square pyramids form 20 rings that wrap and twist around the 12 rings formed by its 120 pentagonal prisms and 120 pentagonal antiprisms, forming a marvelous 4D structure that corresponds with the discrete partitioning of the Hopf fibration according to the structure of the icosi­dodecahedron.

So head on over to the spidrox page to learn the structure of this beautiful polychoron. As usual, we provide the full coordinates.

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Last updated 27 Jun 2018.

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