Digonal gyrobicupolic ring (EntityTopic, 17)

From Hi.gher. Space

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Revision as of 09:17, 8 February 2014


The digonal gyrobicupolic ring is a special member of the set of bicupolic rings. Unlike the others, only the gyro- form is possible, since ortho- or magna- forms would be degenerate. Its cells are 1 tetrahedron, 4 square pyramids and 2 triangular prisms. Its faces are 1+4 squares and 4+4+4 triangles. It has 2+4+4+8 edges and 4+4 vertices.

Keiji studied it explicitly to try to understand more about the segmentochora.


Construction

It is possible to construct the digonal gyrobicupolic ring from three different pairs of polytopes:

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Petrie polygon
 
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Square pyramid and
opposite triangle highlighted
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Tetrahedron and
opposite square highlighted
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Triangular prism and
opposite digon highlighted


This segmentochoron also arises from a bidiminishing of the 3-pyrotomochoron. First, delete any vertex from the 3-pyrotomochoron. That forms the (mono)diminished 3-pyrotomochoron, better known as the trigonal biantiprismatic ring, or K4.6; its cells are 1 triangular prism, 2 octahedra, 3 square pyramids, and 3 tetrahedra. It can be constructed as trigonal prism || gyrated triangle; if a vertex from the gyrated triangle is deleted, it will create a second triangular prism, thus resulting in this segmentochoron.

Projections

The following projection shows this segmentochoron from a viewpoint analogous to that of the other bicupolic rings, to show how the gyrated digons connect to each other via a tetrahedron (digon antiprism) and to the opposite square face via triangular prisms (digon cupolae).

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