Schmitt–Conway–Danzer biprism

Schmitt–Conway–Danzer biprism
Rank	3
Space	Spherical
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	K2+×I, order 2
Chiral	Yes
Convex	Yes
Nature	Tame

The Schmitt–Conway–Danzer biprism, or SCD prototile, is a convex polyhedron which can tessellate 3-dimensional space but only aperiodically. It is abstractly equivalent to the gyrobifastigium.

Solution to the einstein problem

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.

History

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. Conway found such a variant and in 1993 Danzer improved the design further.

External links

• Weisstein, Eric W. "Schmitt-Conway Biprism" at MathWorld.

Bibliography

• Danzer, Ludwig (1993). "A single prototile, which tiles space, but neither periodically nor quasiperiodically".
• Senechal, Marjorie (1995). Quasicrystals and geometry.