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GeezerVison is showing mathworld.wolfram.com — Substitution Tiling -- from Wolfram MathWorld 
Substitution TilingA 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
where each
Thus 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
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
where
More generally, fix a finite dissection of one tile into smaller similar copies with relative
measures
where
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
The six-piece dissection of the Kepler triangle has 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
in the vertex-pair order 12, 13, 14, 23, 24, 34. The rows and numbers of children are
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 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). See alsoAmmann-Kramer-Neri Tiling, Golden Tetrahedron, Mosseri-Sadoc Tiles, Nonperiodic Tiling, Plastic Constant, Quaquaversal Tiling, Rep-Tile, Substitution System, Supergolden Ratio, Tile Deflation, Tile Inflation, TilingPortions of this entry contributed by Ed Pegg, Jr. Explore with Wolfram|Alpha
ReferencesConway, J. H. and Radin, C. "Quaquaversal Tilings and Rotations." Invent. Math. 132, 179-188, 1998. https://doi.org/10.1007/s002220050221.Dietl, R. M. K. and Eschenburg, J.-H. "The Icosahedral Quasiperiodic Tiling and Its Self-Similarity." J. Geom. 108, 319-354, 2017. https://doi.org/10.1007/s00022-016-0342-2.Frank, N. P. "A Primer of Substitution Tilings of the Euclidean Plane." Expo. Math. 26, 295-326, 2008. https://doi.org/10.1016/j.exmath.2008.02.001.Papadopolos, Z. and Ogievetsky, O. "On Inflation Rules for Mosseri-Sadoc Tilings." Mater. Sci. Eng. A 294-296, 385-388, 2000. https://doi.org/10.1016/S0921-5093(00)01166-7. Pegg, E. Jr. "The 3D Golden 6 Substitution System." Wolfram
Demonstrations Project. 2026. https://demonstrations.wolfram.com/The3DGolden6SubstitutionSystem/. Pegg, E. Jr. Mathematical Games. Episode 46: "Sporadic Similar
Triangles." Oct. 22, 2026. https://www.youtube.com/watch?v=bp7mOio0ef0.Smilansky,
Y. "Uniform Distribution of Kakutani Partitions Generated by Substitution Schemes."
Israel J. Math. 240, 667-710, 2020. https://doi.org/10.1007/s11856-020-2075-z.Smilansky,
Y. and Solomon, Y. "Multiscale Substitution Tilings." Proc. London Math.
Soc. 123, 517-564, 2021. https://doi.org/10.1112/plms.12404.
Cite this as:Weisstein, Eric W., with contributions by Ed Pegg, Jr.. "Substitution Tiling." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SubstitutionTiling.html Subject classifications | |||||||||||||||||||||||||||||||||
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,
is an 
counts the children of type 
in the expanded tile of type 
.
,
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.
,
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satisfies
and 
times that of their parent. The number 
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is the 
when 
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(top) and for the 
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of 
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