Octahedral Symmetry on a Sphere

by pmoews, published

Octahedral Symmetry on a Sphere by pmoews Dec 28, 2012

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Here is openSCAD code which covers a sphere with a pattern that displays octahedral symmetry.

Any planar pattern with a three or four fold rotation axis at right angles to the origin can be used. Copies of the pattern are "bent" and subtracted from the surface of the sphere.

To illustrate the method groups of Sierpinski triangles arranged to have three and four fold rotation axes were used. The files are sierpinski_triangles_3.dxf and sierpinski_triangles_4.dxf. See the images at the left.

In the case of patterns with three fold axes the patterns are centered at the vertices of a cube mapped onto the sphere and in the case of the four fold axes the patterns are cented at the vertices of an octahedron mapped onto the sphere.

In either case the overall pattern will have octahedral symmetry; that is four 3 fold axis, three 4 fold axis, and six 2 fold axis. Three and four fold axes can be seen in the image at the upper left.

Two stl files were generated using the pattern with the 3 fold axis and two using the pattern with the 4 fold axis.

sierpinski_x3_small.stl and sierpinski_x3_large.stl sierpinski_x4_small.stl and sierpinski_x4_large.stl

The dxf files and the openSCAD code that generated them are included as is the openSCAD code that makes the spheres.

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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.

Note that in the sierpinski_x3 files there is little overlap of the patterns while in the sierpinski_x4 files the patterns overlap to give more complex results. Octahedral symmetry is still obeyed.

The openSCAD code is easily modified to accept other dxf files.