Quaquaversal Tiling

Visualization

@
$dim=6
$subspace=[q,0,2,4]
a=[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1]
b=[1,0,0,0,0,0,0,1/2,0,0,0,-3/2,0,0,1/2,0,3/2,0,0,0,0,1,0,0,0,0,-1/2,0,1/2,0,0,1/2,0,0,0,1/2]
q=[0,0,0,9,0,0,0,0,0,0,6,0,0,0,0,0,0,3,3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0]
h0=1*[1,0,0,0,0,1]
h1=1*[1,1,0,0,0,0]
h2=1*[0,0,0,0,0,1]
h3=a^3*[-1,1,0,0,0,0]
h4=a^3*b^3*[-1,-2,0,0,0,-1]
h5=a^2*b^3*[-1,0,0,0,0,-1]
h6=b^4*[0,-1,0,0,0,0]
h7=b^4*a^2*[-1,1,0,0,0,0]
A=2^-1*(h0|h1|h2|h3|h4|h5|h6|h7)*A

Overview

The quaquaversal tiling is a 3D aperiodic tiling introduced by John Conway and Charles Radin (1998) as the 3-dimensional analog of the pinwheel tiling. Its prototile is a 30-60-90 triangular prism that decomposes into 8 smaller congruent copies scaled by $1/2$.

The defining property is orientational density in SO(3): the set of tile orientations that appear in the tiling is dense in the full rotation group $SO(3)$. No rotational or translational symmetry of the tiling exists.

Algebraic Structure

The construction lives in a 6-dimensional rational space $\mathbb{Q}^6$. Two rotation matrices $a$ (order 4, 90° rotation) and $b$ (order 6, 60° rotation) generate the dense subgroup $G(6,4) \subset SO(3)$:

The rendering projection is given by an explicit $6\times 6$ matrix $q$ that selects the three eigenplanes corresponding to rendering indices 0, 2, 4 (AIFS: $subspace=[q,0,2,4] produces a 3D rendering).

The 8 maps are isometries from $\langle a, b \rangle$ composed with integer translations in $\mathbb{Q}^6$, all contracting by $\frac{1}{2}$:

The IFS equation is:

Properties

• Orientational density: every orientation in $SO(3)$ is approximated arbitrarily closely by a tile orientation. This is the 3D counterpart of the pinwheel’s density in $SO(2)$.
• The symmetry group of tile orientations is $G(6,4)$, generated by a 90° and a 60° rotation in perpendicular planes. Conway and Radin proved $G(6,4)$ is infinite and dense in $SO(3)$.
• The prototile is a solid 30-60-90 triangular prism (not a fractal boundary tile).
• The Hausdorff dimension of the attractor equals 3 (it is a solid 3D region).

References

• Conway, J. H. & Radin, C. (1998). Quaquaversal tilings and rotations. Inventiones Mathematicae, 132(1), 179–188.
• Radin, C. & Sadun, L. (1998). Subgroups of SO(3) associated with tilings. Journal of Algebra, 202(2), 611–633.
• Quaquaversal tiling — Wikipedia

Similar