## Symmetry Maps

In PPT mode, geodesic spheres are constructed using the notion of symmetry maps. Each map is formed from the net of the Base Polyhedron. There are two symmetry maps for each Principal Polyhedron – one for Class I (Alternate) spheres, and another for Class II (Triacon) spheres. Full Triacon facets are generated from the centroids of each PPT (Principle Polyhedron Triangle). Note that the Triacon map is NOT a net; it simply maps the Class II triangles onto the net of the base polyhedron. At first sight, this literal breakdown of the sphere might seem simplistic; but it provides a powerful way to access the symmetry of the system, especially with high subdivisions, when it is hard to see the original PPT.

 Icosahedron Alternate Symmetry Map Icosahedron Triacon Symmetry Map

In Class I maps, each PPT is numbered and subdivided into two right triangles which form enantiomorphous (mirror image) a-b pairs. In the Triacon maps, these pairs represent spherical Möbius triangles. There are six Möbius triangles for each PPT and four along each PPT edge. In the Icosahedron Triacon map, the Möbius triangle is equivalent to the Schwarz triangle. Schwarz triangles are the smallest repeating element of spherical tesselation. (For this reason, Buckminster Fuller called them ‘alpha particles’). Since there are six Schwarz triangles in every Icosahedron PPT, the Triacon symmetry map for the Icosadron has 120 Schwarz triangles consisting of 60 a-b pairs. Basic or Full Triacon subdivision is achieved by splitting/welding the pairs. This is done by clicking the particles on the Net window which is accessed from the Windows menu:

A Class I 6V icosahedron. PPT 6 was split into 6a and 6b by clicking the corresponding particles in the Net window.

A Class II 8V icosahedron. Particle 22 was split into 22a and 22b by clicking the corresponding particles in the Net window.

To weld particles back together, hold down the ‘w’ key and click again on either the left (a) or right (b) pair. Note that when clicking on a sphere in the OpenGL view with the group select tool, the corresponding particle is automatically entered in the Net window.

Cleary the Symmetry Map breaks up a geodesic sphere into its constituent ‘particles’ spheres may be truncated along certain great circle boundaries by turning particles on or off. Another way of accessing the symmetry of the system is via the Symmetry Table and this is explained on the next page.