The 5 VEs in the 30-Verti

Marvin has shown me an interesting way to "fit" 5 VEs (a.k.a. Cuboctahedron) within the 30-Verti.

Figure 1 The 30-Verti

First consider a pentagon face of the 30-Verti. Draw a line across the face. You can continue the line all around the 30-Verti. These lines will pass through 6 of the pentagon faces to complete the cycle. Also, these lines, when projected onto a sphere, define a great circle.

Figure 2 Blue lines all around 30-Verti

Now divide the line into Golden Ratio sections. This is shown in the next Figure as a white ball. The green ball shows the half-way point along the blue line.

Figure 3 Golden Ratio division of blue line

Similarly, we divide all the other lines around the 30-Verti into Golden Ratio segments.

Figure 4 Golden Ratio Segments

It is then possible to place a VE in the 30-Verti such that 6 of the VE's vertice are exactly at these Golden Ratio segment points (white spheres).

Figure 5 Position and scale of 1 of the VEs
in the 30-Verti

Note that if we continue to add lines across the pentagons' faces we get pentagrams.

Figure 6 Pentagrams in the pentagons

The lines of the pentagrams define 10 Great Circles around the 30-Verti.

Note that the intersection point of any of the 2 lines in the pentagram divides the lines into Golden Ratio segments.

We can add 4 additional VEs to this matrix. All of the vertices (5*12=60) of these VEs will occur at the Golden Ratio segment points.

Figure 7 All 5 VEs in position

In the next Figure, I have made the VEs a little bigger so that their vertices stick up through the 30-Verti pentagon faces. These clearly shows the poisitioning of the VEs' vertices at the Golden Ratio segment points.

Figure 8

Note that the VE is a truncated Octahedron. So, we can consider the VE position, scale and orientation to also give the position, scale, orientation of the corresponding Octahedron.

Figure 9 Corresponding Octahedron to one of the VEs

We can add in the other 4 corresponding Octahedra.

Figure 10 5 Octahedra of the 5 VEs

Then we have a structure which defines another, larger, 30-Verti.

Figure 11 5 Octahedra define larger 30-Verti

Here is a Figure showing the size of the original 30-Verti and the new, larger 30-Verti.

Figure 11 Comparing 30-Verti scales

This work is copyrighted, 2003 by Robert W. Gray and Marvin Solit.
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