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Johnson Polyhedra Duals

by pmoews, published

Johnson Polyhedra Duals by pmoews Oct 9, 2012

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Description

Here are the 92 Duals of the Johnson Polyhedra, see thing:16508. They were constructed by the method described in thing:30958, Duals of Polyhedra.

You can read about duals on Wikipedia or on the dual page at George Hart's Virtual Polyhedra Site, see

georgehart.com/virtual-polyhedra/duality.html

Simply put if object A is transformed to object B by some process and if B is tranformed back to A by the same process, A and B are duals. In this case A and B are different but related polyhedra. The vertices of A become the faces of B and vice versa.

Ninety-two are too many polyhedra to show here though all are printable. It seemed better to present a few examples of the orginal Johnson polyhedra together with their duals to show how the process works. Compounds of the duals and their polyhedra are included as they are helpful in understanding the relationships. In the compounds the vertices of the original polyhedron protrude from the faces of the dual.

Stl files are:

Example 1: J9.stl, J9_dual.stl and compound_j9.stl

Example 2: J12.stl, J12_dual.stl and compound_j12.stl

Example 3: J16.stl, J16_dual.stl and compound_j16.stl


The fortran program that produced the openSCAD programs and the openSCAD programs that generated the duals are included in johnson_duals_scad.zip. The stl files of all the duals are included in johnson_duals_stl.zip

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Instructions

Four of the stl files need to be printed with external support. They are:
compound_j9.stl, J12.stl, compound_j12.stl, and compound_j16.stl.

Example 1. J9.stl is the Johnson polyhedron. It is an elongated pentagonal pyramid with 5 square faces, 5 triangular faces and a single pentagonal face. Its dual, J9_dual.stl,. is similar but only the pentagonal face is regular. Both polyhedra have 11 vertices. Vertices of J9 protrude from the dual in the compound.

Example 2. J12 is a triangular dipyramid composed of 6 equilateral triangles. It has only 5 vertices. Its dual is a triangular prism composed of two triangles and 3 rectangles with 6 vertices. The 5 vertices of J12 protrude from the dual in the compound.

Example 3. J16 is a elongated pentagonal dipyramid with 5 square and ten triangular faces. It has 12 vertices. J16_dual is an elongated pentagonal prism. It has two regular pentagonal faces joined by 10 identical trapezoids and has 15 vertices. The 12 vertices of J16 protrude from the dual in the compound.

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