[r-t] Symmetries of a Dodecahedron
Peter Blight
liberiabells at gmail.com
Wed Jul 31 11:19:33 BST 2019
Brian Price's letter of 1944 was not originally published with a diagram
so it is difficult to be sure what his dodecahedron looked like.
However, arranging the courses in way proposed by Richard Pearce, (and
using coursing orders rather than course heads,) a possibility is shown
here:
<http://ringing.info/bdp/dodecahedron/sample.jpg> which matches Brian's
description: "three courses capable of being united by three bobs at M
lie at the corners of green triangles, those by three bobs at W on black
circles, and of bobs at H on red triangles" and "in fact there are no
actual course end numbers on my model".
Peter Blight
On 16/06/2019 20:05, Philip Saddleton wrote:
> 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
> elements
> - lines through the centres of opposite edges (15 pairs, order 2, 15
> elements
> 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
> interpretation?
>
> PABS
>
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