### Method 1

Equal edge divisions *before* projection gives similarity of triangle shape.

The proceedure for obtaining Method 1 co-ordinates is as follows:

- Interpolate linearly along PPT edge according to the frequency. This subdvides the PPT edge into equal divisions.
- Connect subdivisions on left side of PPT with subdivisions on bottom to make left diagonals (red). These run parallel to the right side.
- Connect subdivisions on right side of PPT with subdivisions on bottom to make right diagonals(blue). These run parallel to the left side.
- Connect subdivisions on left side of PPT to right side of PPT to form horizontals (green). These run parallel to the bottom side.
- Project the grid intersections to the envelope.

### Method 2

Equal edge divisions *after* projection gives similarity of triangle size.

The proceedure for obtaining Method 2 is equivalent to spherical interpolation on the envelope and is as follows:

- Subdivide by spacing Theta coordinates equally down the spherical PPT edges.
- Project a ray from the polyhedron center to each spherical subdivision and find the intersection with the planar PPT edge. (Note that these subdivisions are not equal, as they are in Method 1).
- Connect subdivisions on left side of PPT with subdivisions on bottom to make left diagonals (red). These run parallel to the right side.
- Connect subdivisions on right side of PPT with subdivisions on bottom to make right diagonals(blue). These run parallel to the left side.
- Connect subdivisions on left side of PPT to right side of PPT to form horizontals (green). These run parallel to the bottom side.
- The intersections form small equilateral triangle windows within the grid. (Shown in yellow).
- Find the centers of these small equilateral windows and use as the grid vertices.
- Project the grid intersections to the envelope.

The proceedure for deriving Method 2 co-ordinates is based on that in ‘Geodesic Math’ (excerpt of an article by Joe Clinton) p.11, and ‘Structural Design Concepts For Future Space Missions’, pp 10-13, both by Jay Salsburg, Design Scientist. I am grateful to the author(s) for making these works available.

### Method 3

Method 3 allows greater economy of fabrication at a slight cost in overall symmetry. The number of unique struts in a Method 3 subdivision is equal to the frequency, i.e., 12 unique struts for a 12V sphere, 16 unique struts for a 16V sphere, 20 unique struts for a 20V sphere, etc.

**WARNING: Method 3 should only be used on perfect spheres. All economy of fabrication is lost when using ellipsoidal or egg type envelopes.**

The proceedure for obtaining Method 3 co-ordinates, as implemented in Geodesica, is different to that given in 'Geodesic Math and How to Use It' but the results are the same. My intention was to streamline the build process between the different methods and so keep the code base as small as possible. For example, the interior grid for Method 3 is built in exactly the same way as Method 1, using diagonal intersection points. This saves a plethora of rotations and preserves accuracy.

1. Subdivide Class II triangle median into *n* divisions where n = Frequency/2.

Note A. Whilst these divisions are equal on the spherical median, they are NOT equal on the planar median.

2. Construct horizontals at each median division to intersect with the Class II triangle edges. These horizontals are parallel with the base LR, ie kv is parallel to sL.

Note B. ‘Geodesic Math and How to Use It’ states that “the horizontal members cross the Class II triangle’s median at equally spaced intervals...”. [Page 67]. Note that the interpolation is done on the spherical median, so the divisions on the planar median are NOT equal.

3. Normalize the vectors s,k,v to give *sn, kn, vn*. Compute angle sigma between vectors *kn* and *vn*, and then rotate *sn* about the perpendicluar vector L.Cross(s), by sigma degrees to get vertex *bn*. Compute intersection of *bn* onto Class II triangle face to get the planar grid vertex b for the vertical line vb.

Note C. ‘Geodesic Math and How to Use It’ states that the “...broken lines are at 90° angles with bottom edge”. This is not the case on the planar Class II triangle. The vertical line vb is not perpendicular to kv.

4. Connect subdivisions on left side of Class II triangle with subdivisions on bottom to make left diagonals (blue). NOTE: These do not run parallel to the right side.

Connect subdivisions on right side of Class II triangle with subdivisions on bottom to make right diagonals (red). NOTE: These do not run parallel to the left side.

5. As with Method 1, the planar grid is formed from the diagonal intersections. Above shows both projected and planar grids with construction lines.

OBSERVATIONS. The Method 3 breakdown instructions on page 67 of ‘Geodesic Math and How to Use It’ are a little obscure. It was only after much head-scratching that I finally derived the code to make this work, as outlined above.

It is importnat to note that the underlying grid in the diagram on page 67 is not the planar Class II triangle grid, but the grid of the *projected* vertices. In other words, this is the grid of the subdivided spherical Class II triangle.

The animation below shows a Method 3 breakdown for a 10V Class II icosahedron. Note how the side edge divisions propagate down the the projected Class II triangle by ziz-zagging towards the base. Both diagonal and horizontal struts become shorter as they move downward and outward.