[r-t] Symmetries of a Dodecahedron
Philip Saddleton
pabs at cantab.net
Wed Jul 31 20:39:57 BST 2019
Yes, I think this is equivalent to what I came up with, though the
vertices of the triangles should each be the same distance from the
nearest vertex of the dodecahedron, and this distance should be the
radius of the circle. But why is the black shape a circle and not a
triangle?
Moving on, how does the rotation group correspond with permutations of
five objects? Looking at the photo we see that rotating about the axis
through the point in the centre keeps 4 and 6 fixed while permuting 2,
3 and 5. Similarly each vertex lies on an axis of rotation that keeps a
pair fixed. There are 20 vertices, so 10 possible axes, each
corresponding to a different pair - the four axes fixing each bell are
the diagonals of a cube, and there is a 1-1 correspondence between the
possible rotations of the dodecahedron and the permutaions of these
five cubes.
PABS
On Wed, 2019-07-31 at 11:19 +0100, Peter Blight wrote:
> 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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> >
>
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