# Schmitt–Conway–Danzer biprism

Schmitt–Conway–Danzer biprism | |
---|---|

Rank | 3 |

Elements | |

Faces | 2+2 asymmetric triangles, 2+2 parallelograms |

Edges | 2+2+2+2+2+2+2 |

Vertices | 1+1+2+2+2 |

Measures | |

Central density | 1 |

Abstract & topological properties | |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | K_{2}+×I, order 2 |

Chiral | Yes |

Convex | Yes |

Nature | Tame |

A **Schmitt–Conway–Danzer biprism** or **SCD prototile** refers to a certain kind of convex polyhedron which can tessellate 3-dimensional space but only aperiodically.^{[1]} These can be formed by taking a gyrobifastigium and "making it generic", in the sense that equalities between the sides and angles should be avoided.^{[2]}

## Solution to the einstein problem[edit | edit source]

The Schmitt–Conway–Danzer biprism can be made to tessellate 3-dimensional space and no tiling of just Schmitt–Conway–Danzer biprisms has any translational symmetry. However, some of these tilings have screw symmetry, that is symmetries which are a simultaneous rotation and translation. These tilings are thus aperiodic but not strongly aperiodic. This makes the Schmitt–Conway–Danzer biprism a solution to some versions of the einstein problem, which do not require the strong version of aperiodicity.

The Schmitt–Conway–Danzer biprism along with its mirror image can tile the plane periodically.^{[3]}

## History[edit | edit source]

A non-convex variant was originally discovered by Schmitt in 1998. Schmitt conjectured that it could be made convex while maintaining its properties as a prototile.^{[3]} Conway found such a variant and in 1993 Danzer improved the design further.^{[3]}

## External links[edit | edit source]

- Weisstein, Eric W. "Schmitt-Conway Biprism" at MathWorld.
- Wikipedia contributors. "Gyrobifastigium#Schmitt–Conway–Danzer biprism".

## References[edit | edit source]

- ↑ Senechal (1995:209-210)
- ↑ https://ics.uci.edu/~eppstein/junkyard/biprism.html
- ↑
^{3.0}^{3.1}^{3.2}Senechal (1995:210)

## Bibliography[edit | edit source]

- Danzer, Ludwig (1993). "A single prototile, which tiles space, but neither periodically nor quasiperiodically". Cite journal requires
`|journal=`

(help) - Senechal, Marjorie (1995).
*Quasicrystals and geometry*.