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The pictures and movies below are all my work, and some come from my mathematical research. I also created most of the other graphics on this site.
Substitution Rules and Tilings
Canonical substitution tilings:
These tilings can be constructed either using a substitution rule, or, using the canonical projection method, as a slice of a lattice in more than 2 dimensions. My Thesis characterises all such tilings in any number of dimensions.Rule
PatchThe Ammann-Beenker Substitution Rule: the projection is from a lattice in four dimensions to the plane. Rule
PatchThe Penrose Substitution Rule: the projection is from a lattice in five dimensions to the plane. Rule The Socolar Substitution Rule: the projection is from a lattice in six dimensions to the plane. Rule
Rule
RuleSome other canonical substitution rules:
the projections are from a lattice in four dimensions to the plane.Rule A substitution rule: the projection is from a lattice in five dimensions to the plane, the window is a polyhedron, but is not the canonical window. Flight Growing Some views of the Socolar-Steinhardt icosahedral tiling. This is a projection of a lattice in six dimensions to 3-space. It has the symmetries of the icosahedron. These quicktime movies are very much first attempts so are still a little rough around the edges. Growing is 4.0MB and flight is currently unavailable. Other substitution tilings: 4-fold
7-fold
doily
9-fold
doilyRhomb substitutions with 4, 7 and 9 fold rotational symmetry. These are part of a general class of substitution rules for any finite rotation described in Rhomb substitutions with order n rotational symmetry. The general case was constructed from the seven fold example, found by Chaim Goodman-Strauss. Rule
TilingPinwheel substitution rule with cubic scaling Rule
TilingPinwheel substitution rule with quartic scaling Nautilus
ConchTwo dual Rauzy fractals for a non-pisot substitution tiling, found with Pierre Arnoux and Shunji Ito (site in Japanese), (paper in preparation). Rule
Tiling
PictureThe chair tiling is a classic example of a one-tile substitution rule, often called a rep-tile. L-systems
These pictures are constructed from L-systems (Lindemeyer-systems), using lsysexp. L-systems are simple recursive algorithms developed for computer modelling of plants, see (the unfortunately out of print) Algorithmic Beauty of Plants.Image Penrose Fire Image Flow snake space-filling curve Image Robinson Smoke Image Luggage. Possible prize for anyone who can say why! Regular Polytopes
Polytopes are the general term of the sequence:
{line segment, polygon, polyhedra, polychoron...}
The regular polytopes are the ones were every facet is equivalent. The central published reference is Coxeter's Regular Polytopes. There are also many excellent resources on the web, for example:
- Russell Towle's Mathematica notebook Regular Polytopes that was used along with POVRay to create the animations below.
- George Hart's Polyhedra
- Paul Bourke's pages on regular polytopes in 2D, 3D and 4D +
- Jim Mcneill's polyhedra pages.
- To make your own models: Zometool.
Regular Convex Polytopes Vertex (0.8) The 4-simplex or 8-cell, {3,3,3} Vertex (1.1)
Cell (0.9)The 4-cube, tesseract or 8-cell, {4,3,3} Vertex (0.8)
Cell (0.8)The 4-cross polytope, or 16-cell, {3,3,4} Vertex (1.1)
Cell (1.1)The 24-cell, {3,4,3} Vertex (1.8)
Cell (1.8)The 120-cell, {5,3,3} Vertex (1.8)
Cell (1.8)The 600-cell, {3,3,5} File Data and POV-Ray file to generate these images. Regular Star Polytopes (based on an idea of Russell Towle) Cell First(1.0) {3,5/2,5} Cell First(1.1) {5,5/2,3} Cell First(1.1) {5.5/2,5}
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