[r-t] Symmetries of a Dodecahedron
pabs at cantab.net
Sun Jun 16 20:05:48 BST 2019
I was intrigued by Brian Price's letter of 75 Years ago in this week's
issue of the RW, where he describes a graph of 60 in-course courses of
Major linked by Q-sets at W, M, H with different colours, drawn on the
surface of a dodecahedron. I have been trying to visualize this.
The possible rotations consist of
- lines through opposite vertices (10 pairs, order 3, 20 elements)
- lines through the centres of opposite faces (6 pairs, order 5, 24
- lines through the centres of opposite edges (15 pairs, order 2, 15
which along with the identity, give 60 elements in total, and a group
of order 60.
The group is generated by three rotations, W, M and H, where
WH and MH have order 2
WM has order 5
Looking at the effect of combining rotations about different axes, we
see that rotating about two axes with vertices separated by a single
edge is equivalent to rotating about a face, and rotating about two
axes with vertices separated by two edges is equivalent to rotating
about an edge. Thus we can choose our rotations to be about vertices of
the same face.
If we now choose an initial point anywhere on the dodecahedron except
on an axis of rotation, and label the plain course, we can apply
different combinations of the three generators that correspond to
other courses, labelling each point that finishes in the position of
our initial point with the course that would be reached with the
equivalent calling. We can then join these points with coloured lines
for different Q-sets.
To minimize the lengths of these lines, and the number of crossings,
the initial point should be close to the axes of the generators, i.e.
somewhere on the face with the rotation axes as vertices. A suitable
choice would be on the edge joining W and M, asymmetrically disposed
(there are then two points on each edge, giving a total of 60). Then
the W and M Q-sets are indeed triangles, linking points on edges with a
vertex in common. The lines joining the H Q-sets have to traverse at
least two faces, giving what could loosely be described as a circle.
The letter says that the M and H Q-sets are triangles, while W are
circles. Is this a mistake, or has anyone got an alternative
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