A quaquaversal tiling is a hierarchical substitution tiling of three-dimensional Euclidean space
introduced by Conway and Radin (1998). It is a nonperiodic
tiling whose tile is a right triangular prism
of depth 1 with base a 30-60-90
triangle with lengths 1, 
, and 2. Equivalently, the triangular
prism has five unit edges, two of measure
,
and two of measure 2 (Radin 2021).
The defining substitution dissects the triangular prism into eight mutually congruent copies, each
similar to the original with scale
factor 
.
The children are obtained from two subdivisions of depth 
by rotating
two pieces of one through 
and two pieces of the other through 
about perpendicular
axes (Conway and Radin 1998). The tile is therefore a rep-tile
with 
.
After 
substitutions, the original triangular prism
is partitioned into 
triangular prisms, each with scale
factor 
and volume 
that of the original.
Conway and Radin (1998) combine each subdivision with a dilation by a factor of 2 and iterate about an appropriate fixed
point. In the limit, the resulting construction covers Euclidean
space. Two copies of the unmarked triangular
prism can instead be joined to form a cuboid of dimensions
,
so the tile also admits periodic tilings. The nonperiodicity
is therefore a property of the substitution construction rather than of the tile
as an aperiodic monotile.
Conway and Radin denote the rotation group generated by rotations through 
and 
about perpendicular
axes by 
(Conway and Radin 1998). The group of tile orientations
is 
(Draco et al. 2000, p. 424). In the limit of large regions, the tile
orientations are uniformly distributed in the special
orthogonal group 
with respect to normalized Haar
measure (Conway and Radin 1998, Radin 1999).
Draco et al. (2000, p. 421) give the number 
of distinct orientations after 
substitutions as
 |
(1)
|
where 
is a positive integer. For example, the 64 tiles
after two substitutions have 24 distinct orientations, and the 512 tiles after three
substitutions have 44. The illustration ranks the orientations by decreasing tile
count. These orientations need not occur equally often in a finite region. In a patch
of 
tiles, just 104 orientations account for about 69% of the tiles (Draco et al.
2000, p. 421).
The name "quaquaversal" was taken by Conway from geology, where it describes structures oriented in all directions (Radin 2021).
