Substitution Tiling

A substitution tiling is a tiling whose finite patches occur in repeated applications of replacement rules to finitely many tile types. A geometric rule expands a representative tile and partitions the result into tiles of the original types with disjoint interiors (Frank 2008).

Tile inflation enlarges a tile by a dilation and subdivides the result into tiles of the original sizes. Applying the subdivision without the enlargement gives tile deflation, which refines the pattern at its original overall scale. The ratio of enlarged to original edge lengths is called the inflation factor. For a common inflation factor , this can be written

(1)

where each is an isometry placing a child tile and counts the children of type in the expanded tile of type . In dimension , the column vector of tile volumes satisfies

(2)

Thus is an eigenvalue of the nonnegative matrix . More general geometric rules use an expanding linear transformation instead of a common scale factor (Frank 2008).

Substitution does not itself imply nonperiodicity. A square subdivided into four equal squares generates a periodic tiling. The quaquaversal tiling instead gives a three-dimensional nonperiodic tiling using a triangular prism that is a rep-tile (Conway and Radin 1998). The Mosseri-Sadoc tiles provide another three-dimensional example with inflation factor equal to the golden ratio.

More general rules allow different contraction factors among children (Smilansky and Solomon 2021).

A two-piece example starts with a right triangle having side lengths in the ratio , where are integers and satisfies

(3)

At each stage, every currently largest triangle is split by the altitude to its hypotenuse, and all smaller triangles are left unchanged. The two children are similar triangles with areas and times that of their parent. The number of distinct sizes present at any one stage satisfies

(4)

where is the greatest common divisor. In particular, when and are relatively prime (E. Pegg, Jr., pers. comm., Oct. 19, 2026). The illustration shows stages 0, 1, 2, 4, and 8 for , (top) and for the Kepler triangle, , (bottom). Colors distinguish sizes within each stage, from largest to smallest in blue, orange, green, and purple.

More generally, fix a finite dissection of one tile into smaller similar copies with relative measures , ..., , where and the are positive integers. The measure is length, area, or volume in the appropriate dimension. Starting with one tile, repeatedly replace every largest tile by this rule. Then

(5)

where and . To prove this, write each relative measure as . The exponents are multiples of , and their span remains at most by mathematical induction. If the current minimum is , retained exponents are at most , new ones are at most , and the new minimum is at least . Thus at most distinct sizes coexist.

A four-piece rule splits a selected triangle by its altitude to the hypotenuse and then splits both children the same way, treating the entire dissection as one replacement step. The relative child areas have exponents , , , and , giving the bound .

These largest-first refinements are two-dimensional counterparts of Kakutani partitions. Smilansky (2020) proves a result in uniform distribution theory that applies to these refinements. Choose one point in the interior of each triangle at every stage. For any subregion of the original triangle whose boundary has area zero, the fraction of selected points in that subregion tends to its area divided by the area of the original triangle.

The following finite dissection example uses six tetrahedra (Pegg 2026), denoted here by , ..., . Writing for the golden ratio, a row specifies edge lengths

(6)

in the vertex-pair order 12, 13, 14, 23, 24, 34. The rows and numbers of children are

type	edge exponents	children
		13
		17
		7
		11
		8
		11

Each of the six tetrahedra is partitioned into smaller similar copies drawn from the same six types. The exploded view separates the pieces to show the dissections, with colors identifying types. The rules use contraction factors with drawn from 1, 2, ..., 7, so all contraction factors are powers of . The children have disjoint interiors and their union is the parent tetrahedron, so the subdivisions can be iterated indefinitely. Whether these rules generate a nonperiodic tiling has not yet been established. These six shapes are distinct from the traditional golden tetrahedra.

The Ammann-Kramer-Neri tiling provides a three-dimensional rhombohedral example with tile deflation reducing lengths by the cube of the golden ratio (Dietl and Eschenburg 2017).