A perspective projection of a dodecahedral tessellation in H3. Note the recursive structure: each pentagon contains smaller pentagons, which contain smaller pentagons. This is an example of a subdivision rule arising from a finite universe (i.e. a closed 3-manifold).
In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals.[1] Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule.
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a finitesubdivisionrule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivisionrules in...
thirds' technique for creating the Cantor set is a subdivisionrule, as is barycentric subdivision. A function may be recursively defined in terms of...
done in several different ways, including matching rules, substitution tiling or finitesubdivisionrules, cut and project schemes, and coverings. Even constrained...
if none of the angles is 90° (so the tangent function's value is always finite). There are thousands of different constructions that find a special point...
better than you." Peer relationships, such as can be governed by the golden rule, are based on symmetry, whereas power relationships are based on asymmetry...
This process of removing middle thirds is a simple example of a finitesubdivisionrule. The complement of the Cantor ternary set is an example of a fractal...
analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA). The subdivision of a whole domain into simpler parts has several advantages:...
Kepler conjecture Quasicrystals Aperiodic tilings Periodic graph Finitesubdivisionrules Structural rigidity is a combinatorial theory for predicting the...
the Penrose tilings. Substitution tilings are special cases of finitesubdivisionrules, which do not require the tiles to be geometrically rigid. A tile...
hyperbolic groups with Gromov boundary homeomorphic to the 2-sphere. Finitesubdivisionrules, also in relation to Cannon's conjecture. Interactions with topological...
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often...
minus the area below the x-axis. A partition of an interval [a, b] is a finite sequence of numbers of the form a = x 0 < x 1 < x 2 < ⋯ < x i < ⋯ < x n...
is a unique identifier of the entity, to distinguish it from others in a finite set of geographic entities. In general the geocode is a human-readable and...
also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties...
preserve symmetries that already exist between the domain and range. With finite precision (or a discrete domain), this translates to removing bias. A rounding...
for which the objective function can be made to be better than any given finite value. Thus there is no optimal solution, because there is always a feasible...
respectively. V {\displaystyle V} and E {\displaystyle E} are usually taken to be finite, and many of the well-known results are not true (or are rather different)...
Farrell–Jones conjecture Finite lattice representation problem: is every finite lattice isomorphic to the congruence lattice of some finite algebra? Goncharov...
methods and non-gridded or mesh-free methods. In the common finite difference method and finite element method (FEM) the domain is completely gridded ("cut"...
Yenisei within the boundaries of the Taimyr Municipality District, a subdivision of Krasnoyarsk Krai, Russia Federation. Enets belongs to the Northern...
algorithms such as crack boundary tracing, digital straight segment subdivision, etc. One such rule maps the points, cracks, and faces to the top left coordinate...
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