Icosahedral Symmetry on a Sphere

by pmoews, published

Icosahedral Symmetry on a Sphere by pmoews Dec 31, 2012

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Here is openSCAD code which covers a sphere with a pattern displaying icosahedral symmetry. The code uses the same technique as that described in thing:39424 Octahedral Symmetry on a Sphere.

The Platonic solids with five fold rotation axes are the icosahedron and the dodecahedron. They share symmetry elements; each has six 5 fold axes, ten 3 fold axes and fifteen 2 fold axes. We can use their vertices, mapped to a unit sphere, to locate planar patterns with three or five fold rotation axes and generate a sphere with icosahedral symmetry.

To illustrate the method rosettes, consisting of a central circle and having three or five equally spaced rays, were used. See the images at the left.

The rosettes with 3 rays are centered at the mapped vertices of a dodecahedron and the rosettes with 5 rays are centered at the mapped vertices of an icosahedron. In both cases the overall pattern has icosahedral symmetry. For these simple examples the code generates icosahedrons and dodecahedrons mapped to a sphere. See the image at the upgper left.

Stl files are:

dodecahedron_x3_small.stl and dodecahedron_x3_large.stl icosahedron_x5_small.stl and icosahedron_x5_large.stl

OpenSCAD code is included. Gen_rosettes.scad generates the dxf rosette files and gen_icosahedral_sphere.scad makes the stl files.

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The small stl files are spheres and need to be printed with external support. I used 3 shells and 5% infill on my Replicator and they printed in about 2 hours. The large stl files are truncated spheres. They can be printed directly on the build platform. They print in under 3 hours.

More interesting results come from more complex patterns. For example a rosette with 20 rays placed at the icosahedral vertices makes stars with five arms. Gen_rosettes.scad will generate a number of different rosettes for use in experiments.