4D Euclidean space

News Archive

February 2019

This month, we introduce the augmented cantitruncated 5-cell, an augmentation of the uniform cantitruncated 5-cell by a tetrahedral canticupola (K4.76):

The augmented cantitruncated 5-cell

This particular polychoron is notable because 6 of the triangular prisms of the cantitruncated 5-cell lie on the same hyperplane as the 6 triangular prisms of the augment in gyrated orientation, and therefore merge into gyrobifastigium (J26) cells.

Head over to the augmented cantitruncated 5-cell page to learn more about the structure of this interesting polytope with its unusual gyrobifastigium cells. We provide full coordinates as usual.

January 2019

In generalizing 3D polyhedra to 4D, a question that frequently crops up is how to generalize the 3D antiprisms. The common understanding of a 3D antiprism—two polygons in parallel planes rotated relative to each other and connected by triangles—is difficult to generalize to higher dimensions in a consistent way that is also aesthetically-pleasing.

A different way of understanding a 3D antiprism is that the top and bottom polygons are duals of each other. Since the dual operation applies across all dimensions, and furthermore preserves the symmetries of the starting polytope, this provides a nice basis on which to construct a higher-dimensional definition of an antiprism. Under this analysis, the triangles connecting the polygons may be thought of as line pyramids, which become full-dimensioned pyramids in the higher-dimensional generalization.

The cube

In the case of 4D, we can form a polyhedron antiprism by placing the polyhedron and its dual in parallel hyperplanes, and connecting the vertices of one to the faces of the other with polygonal pyramids, and vice versa. Furthermore, the edges of one can be connected to the other by tetrahedra. The image above shows one such example: the cube antiprism (K4.15), formed by placing a cube and an octahedron in two parallel hyperplanes and connecting them with 6 square pyramids and 20 tetrahedra, 8 of which connect the faces of the octahedron to the vertices of the cube, and 12 of which connect their respective edges.

For more details, go to the cube antiprism page and discover its structure, to learn how higher-dimensional antiprisms may be constructed. We provide full coordinates as usual.

Note: the above definition of a 4D antiprism is not compatible with the structure of the so-called Grand Antiprism, one of the uniform polychora. The grand antiprism has an interesting structure based on the Hopf fibration, which is peculiar to 4D and thus difficult to generalize to other dimensions. Furthermore, the grand antiprism stands alone as the only uniform (and indeed, the only CRF) member of its family. Nevertheless, its rings of antiprisms connected by tetrahedra do seem like a fitting analogue of the 3D antiprisms, so it possibly represents a different way to generalize the family of 3D antiprisms to higher dimensions.

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Last updated 28 Feb 2019.

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