# Bicupolic ring (EntityClass, 16)

### From Higher Dimensions Database

There are twelve **bicupolic rings** in Klitzing's list of segmentochora, two of which turn out to be prisms.

Digonal | Trigonal | Square | Pentagonal | |
---|---|---|---|---|

ortho- | K4.9 | K4.25 | K4.73 | K4.154 |

gyro- | K4.8 | K4.27 | K4.64 | K4.133 |

magna- | K4.12 | K4.51 | K4.105 | K4.165 |

## History

These CRFs were originally included as *wedges* in Klitzing's list.

Keiji independently rediscovered the ortho- and gyro- forms in late 2011, and quickfur rediscovered the magna- form shortly afterwards. Keiji has dubbed these shapes **bicupolic rings** in general, and the specific naming pattern is *n*-gonal *form*bicupolic ring, e.g. *square orthobicupolic ring*. student91 noticed that Klitzing's K4.8 could be constructed as the digonal gyrobicupolic ring.

The ortho- and gyro- forms are constructed as in this post. The magna- forms are constructed as in this post (second-to-last paragraph).

## Structure

Each of these bicupolic rings consists of two n-gonal cupolae, a linking element, an opposite element corresponding with a Stott-expanded n-gon, and a ring of lateral cells that fill up the gaps in between.

- For the orthobicupolic rings, the linking element is an n-gonal prism, and the opposite element is a 2n-gon. The lateral cells consist of alternating tetrahedra and triangular prisms.
- For the gyrobicupolic rings, the linking element is an n-gonal antiprism, and the opposite element is a 2n-gon. The lateral cells consist of a ring of square pyramids in alternating orientations.
- For the magnabicupolic rings, the linking element is just an n-gon, and the opposite element is a 2n-gonal prism. The lateral cells consist of an alternating ring of triangular prisms and square pyramids.

In the digonal bicupolic rings, the digonal prism is simply the square, the digonal antiprism is the tetrahedron, and the 2n-gon is also the square (tetragon). Hence, the digonal cupola is just the triangular prism (the digon and the tetragon in parallel planes joined by a ring of alternating squares and triangles).