It is also the vertex figure of the nonuniform p-qduoantiprism (if p and q are greater than 2). Despite the fact that p, q = 3 would yield a geometrically identical equivalent to the Johnson solid, it lacks a circumscribed sphere that touches all vertices, except for the case p = 5,q = 5/3, which represents a uniform great duoantiprism.
The name of the gyrobifastigium comes from the Latin fastigium, meaning a sloping roof.[5] In the standard naming convention of the Johnson solids, bi- means two solids connected at their bases, and gyro- means the two halves are twisted with respect to each other.
The gyrobifastigium's place in the list of Johnson solids, immediately before the bicupolas, is explained by viewing it as a digonal gyrobicupola. Just as the other regular cupolas have an alternating sequence of squares and triangles surrounding a single polygon at the top (triangle, square or pentagon), each half of the gyrobifastigium consists of just alternating squares and triangles, connected at the top only by a ridge.
Cartesian coordinates for the gyrobifastigium with regular faces and unit edge lengths may easily be derived from the formula of the height of unit edge length
To calculate formulae for the surface area and volume of a gyrobifastigium with regular faces and with edge length a, one may simply adapt the corresponding formulae for the triangular prism:[7]
Topologically equivalent polyhedra
Schmitt–Conway–Danzer biprism
The Schmitt–Conway–Danzer biprism (also called a SCD prototile[8]) is a polyhedron topologically equivalent to the gyrobifastigium, but with parallelogram and irregular triangle faces instead of squares and equilateral triangles. Like the gyrobifastigium, it can fill space, but only aperiodically or with a screw symmetry, not with a full three-dimensional group of symmetries. Thus, it provides a partial solution to the three-dimensional einstein problem.[9][10]
Dual
Quick Facts Elongated tetragonal disphenoid, Type ...
The dual polyhedron of the gyrobifastigium has 8 faces: 4 isosceles triangles, corresponding to the valence-3 vertices of the gyrobifastigium, and 4 parallelograms corresponding to the valence-4 equatorial vertices.
Alam, S. M. Nazrul; Haas, Zygmunt J. (2006), "Coverage and Connectivity in Three-dimensional Networks", Proceedings of the 12th Annual International Conference on Mobile Computing and Networking (MobiCom '06), New York, NY, USA: ACM, pp.346–357, arXiv:cs/0609069, doi:10.1145/1161089.1161128, ISBN1-59593-286-0, S2CID3205780.
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