**Analogical**

Train of Thought
** Part I: The 24-Cell**

From S. H. Cullinane,

Visualizing GL(2,p),

March 26, 1985–

From John Baez, “This Week’s Finds in Mathematical Physics (Week 198),” September 6, 2003:
Noam Elkies writes to John Baez:
Hello again,
You write:
[…]
“I’d like to wrap up with a few small comments about last Week. There I said a bit about a 24-element group called the ‘binary tetrahedral group’, a 24-element group called SL(2,Z/3), and the vertices of a regular polytope in 4 dimensions called the ’24-cell’. The most important fact is that these are all the same thing! And I’ve learned a bit more about this thing from here:”
[…]
Here’s yet another way to see this: the 24-cell is the subgroup of the unit quaternions (a.k.a. SU(2)) consisting of the elements of norm 1 in the Hurwitz quaternions – the ring of quaternions obtained from the Z-span of {1,i,j,k} by plugging up the holes at (1+i+j+k)/2 and its <1,i,j,k> translates. Call this ring A. Then this group maps injectively to A/3A, because for any g,g’ in the group |g-g’| is at most 2 so g-g’ is not in 3A unless g=g’. But for any odd prime p the (Z/pZ)-algebra A/pA is isomorphic with the algebra of 2*2 matrices with entries in Z/pZ, with the quaternion norm identified with the determinant. So our 24-element group injects into SL_{2}(Z/3Z) – which is barely large enough to accommodate it. So the injection must be an isomorphism.
Continuing a bit longer in this vein: this 24-element group then injects into SL_{2}(Z/pZ) for any odd prime p, but this injection is not an isomorphism once p>3. For instance, when p=5 the image has index 5 – which, however, does give us a map from SL_{2}(Z/5Z) to the symmetric group of order 5, using the action of SL_{2}(Z/5Z) by conjugation on the 5 conjugates of the 24-element group. This turns out to be one way to see the isomorphism of PSL_{2}(Z/5Z) with the alternating group A_{5}.
Likewise the octahedral and icosahedral groups S_{4} and A_{5} can be found in PSL_{2}(Z/7Z) and PSL_{2}(Z/11Z), which gives the permutation representations of those two groups on 7 and 11 letters respectively; and A_{5} is also an index-6 subgroup of PSL_{2}(F_{9}), which yields the identification of that group with A_{6}.
NDE
*The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics* – Gian-Carlo Rota |

*Like footprints erased in the sand….*

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Log24, May 27, 2004 —

“Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.”

“A very short space of time through very short times of space….

Am I walking into eternity along Sandymount strand?”

— James Joyce, Ulysses, Proteus chapter

*A very short space of time through very short times of space….*

*
** * “It is demonstrated that space-time should possess a discrete structure on Planck scales.”

— Peter Szekeres, abstract of Discrete Space-Time

“A theory…. predicts that space and time are indeed made of discrete pieces.”

— Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004

“… a fundamental discreteness of spacetime seems to be a prediction of the theory….”

— Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity

“Theories of discrete space-time structure are being studied from a variety of perspectives.”

— Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London

**Disclaimer:**
** ** The above speculations by physicists

are offered as curiosities.

I have no idea whether

any of them are correct.

Related material:

Stephen Wolfram offers a brief

History of Discrete Space.

For a discussion of space as discrete

by a non-physicist, see John Bigelow‘s

Space and Timaeus.

**Part III: Quaternions**

in a Discrete Space