# 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):

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.

**25 Feb 2019:**Added the augmented truncated dodecahedron (J68), one of the Johnson solids. As usual, we provide full algebraic coordinates for it.

**1 Feb 2019:**The Polytope of the Month for February is up!

### 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.

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.

**25 Jan 2019:**An important internal rendering system upgrade took place today. You should not notice anything different about the site, but if you notice any images that have glitches, rendering artifacts, or that otherwise look wrong, please let us know. Thanks, and enjoy the site!

**23 Jan 2019:**Added more Johnson solids:

**21 Jan 2019:**Added the augmented truncated tetrahedron (J65), still another Johnson solid.

**16 Jan 2019:**Added the augmented tridiminished icosahedron (J64), yet another Johnson solid.

**15 Jan 2019:**Added the triaugmented dodecahedron (J61), another Johnson solid.

**10 Jan 2019:**Added the metabiaugmented dodecahedron (J60), another of the Johnson solids.

**9 Jan 2019:**The Polytope of the Month for January is up!