The Five-Fold Maps study has been concerned with mapping aperiodic patterns onto three-dimensional forms, specifically forms with icosahedral symmetry. There are two-dimensional quasicrystalline tiling patterns which can be projected onto the surface of a sphere and polyhedral surfaces, without losing their essential mathematical properties. The icosahedron has five-fold axes which make it a natural carrier for such patterns.
Ammann bars and other five-fold decorative systems with related geometry are structural organizing principles that can be extended beyond surface decoration. It is assumed that these patterns also theoretically penetrate into the interior, dividing the volumes into fractal tetrahedral arrangements. The planar aperiodic systems translate to three-dimensional spatial organization as a result of the projection process.
As you map five-fold symmetry patterns onto curved surfaces and polyhedral surfaces there are varying degrees of distortion, depending upon the mapping method chosen. It is true that hierarchical subdivision follows aperiodic logic, when it is applied to the icosahedral form. The investigation of this process is ongoing.
Photography: Phillip C. Reiner
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