The underlying symmetry of the sphere can by accessed via the Symmetry Map window. This feature, which was rescinded with PPT mode has now been re-implemented for Quad-Edge mode.

### Overview

The Symmetry Map is formed from the net of the Base Polyhedron. There are two 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 (Principal 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 distinguish the original PPTs.

Below: a Class I 6V icosahedron. PPT 6w was split into 6a and 6b by holding down the ‘s’ key and left-clicking the corresponding particles in the Symmetry Map window.

Below: a Class II 8V icosahedron. PPT 22w was split into 22a and 22b by holding down the ‘s’ key and left-clicking the corresponding particles in the Symmetry Map window.

### Summary of Operation

- To split a PPT, hold down the ‘s’ key and click on either the left (a) or right (b) particle.
- To weld a PPT back together, hold down the ‘w’ key and click again on either the left (a) or right (b) particle.
- Clicking on a PPT without a modifier key toggles its visibility.
- The button ‘SPLIT ALL’ splits all PPTs and colours the Left particle red and the Right particle blue.
- The button ‘WELD ALL’ welds all PPTs and colours the PPTs with their group colours.*
- During a Split, cells within the PPT are regrouped and renamed according to the particle names in the map. Eg the group 5w becomes two separate groups 5a and 5b.
- During a Weld, split groups are regrouped into their original welded group. Verify this behaviour using the Group Select tool.

*To set the group colours, choose the Group Select tool, then open the Attributes Window. Click the ‘SELECT ALL’ button for Cells. Click the random colour button. Cell groups will be assigned a random colour which is persistent across all frequencies.

### Explanation

In Class I (Alternate) maps, each PPT is numbered and subdivided into two right triangles which form enantiomorphous (mirror image) a-b pairs. In Class II (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.

Cleary the Symmetry Map breaks up a geodesic sphere into its constituent ‘particles’. Consequently, spheres may be cut along certain great circle boundaries by turning particles on or off. For example, the spherical Schwarz triangles of a Class II (Triacon) Icosahedron are also formed by the intersections of Great Circle Family 2:

### Usage Notes

The Symmetry Map window is only intended for convex cells that have been newly created by changing frequency. However it may be used after certain modelling operations such as Stellate. The images below show the results after splitting and welding. Note that only those eges formed by the split are removed during the weld; further, only vertices created by the split with no more than two edges are removed. This matches the functionality for the rescinded PPT mode, with all operations now performed on a closed quad-edge manifold.

Unless the envelope is a unit sphere, the above operations are best performed with 'Project to Envelope' OFF. You can re-project the vertices after splitting/welding the PPT's. Alternatively, work on the unit sphere and check 'Project bisection points' in the Symmetry Map Window.

#### Splitting and Welding PPTs on Truncated Spheres

Splitting and welding PTTs on a truncated sphere presents its own special problems. At the time of writing, this can be done on truncated spheres that do not have the ‘Make dual sit flat’ option. Otherwise truncation must be performed *after* splitting and welding the PPTs. One work around is:

- Truncate the sphere without the ‘Make dual sit flat’ option.
- Split the PPTs as required.
- Truncate the sphere
*again*using*exactly*the same parameters as in 1, except this time, check the ‘Make dual sit flat’ option. Note: you will receive a warning that the sphere has already been truncated, but it is safe to continue in this instance. - The dual can now be constructed.