Wallpaper Patterns from Nonplanar Chain Mail Rings
Wallpaper Patterns from Nonplanar Chain Mail Rings
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Frank A. Farris, Santa Clara University
Frank A. Farris, Santa Clara University
This website gives an overview of my paper for Bridges 2020, with some additional materials.
Abstract: We explain techniques for creating mathematical chain mail, by which we mean patterns made from identical linked ring shapes. Most chain mail currently in existence uses circular, or at least planar, rings which are linked in a variety of ways. By contrast, we show how a small nonplanar “wiggle” in the shapes permits a marvelous variety of woven patterns. We give formulas general enough to capture every possible periodic chain mail pattern, but admit that the issue of weaving and self-avoidance of the rings is a decidedly empirical one. We show photographs of 3D-printed chain mail as well as virtual images, created and rendered in Rhino using the Grasshopper plugin. We mention color symmetry in chain mail, as well as possibilities for covering surfaces in mail.
Particularly twisty chain mail with p6 wallpaper symmetry (ignoring color). Respecting color, the symmetry group is p2.
The basic idea: A multi-parameter family of ring shapes, based on Fourier series, are placed in a lattice grid in the 3D design program Rhino. The parameters are wiggled until the rings link like chain mail.
The basic idea: A multi-parameter family of ring shapes, based on Fourier series, are placed in a lattice grid in the 3D design program Rhino. The parameters are wiggled until the rings link like chain mail.
The Fourier formulas guarantee the symmetry type of the configuration, but the self-avoidance and linking arise by experiment.
A successfully linking chain mail pattern.
A funny failure of the method. (If we printed this, it would fall apart.)
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