Another possibility --

  A more modest object --
the 4x4 square.

Update of Aug. 20-21 --

Symmetries and Facets

Kostant's poetic comparison might be applied also to this object.

The natural rearrangements (symmetries) of the 4x4 array might also be described poetically as "thousands of facets, each facet offering a different view of... internal structure."

More precisely, there are 322,560 natural rearrangements-- which a poet might call facets*-- of the array, each offering a different view of the array's internal structure-- encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.

For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.

* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet--

The metaphor of rearrangements as facets breaks down, however, when we try to use it to compute, as above with the Platonic solids, the number of natural rearrangements, or symmetries, of the 4x4 array. Actually, the true analogy is between the 16 unit squares of the 4x4 array, regarded as the 16 points of a finite 4-space (which has finitely many symmetries), and the infinitely many points of Euclidean 4-space (which has infinitely many symmetries).

If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that

"The problem is-- the genius is-- given an infinite question, to think
of the right finite question to ask. Once you thought of the finite
answer, then you would know the right answer to the infinite question."

The Greeks, of course, answered the infinite questions first-- at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.

Seeing the Finite Structure

The following supplies some context for remarks of Halmos on combinatorics.

From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991--

Interviews with Halmos, "Paul Halmos by Parts," by Donald J. Albers--

"Part II: In Touch with God*"-- on pp. 27-28:

The Root of All Deep Mathematics

"Albers. In the conclusion of 'Fifty Years of Linear Algebra,' you wrote: 'I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight... I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need-- the pendulum needs-- to swing back, and will swing back in the combinatorial direction.' I always thought of you as an analyst.

Halmos: People call me an analyst, but I think I'm a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.

Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the 'somehow' would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions-- that's silly. The problem is-- the genius is-- given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.

Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it's sometimes deep, but it is nowhere near as fundamental as seeing the finite structure."

Walsh Series: An Introduction
to Dyadic Harmonic Analysis,
by F. Schipp et al.,
Taylor & Francis, 1990

Halmos's above remarks on combinatorics as a source of "deep mathematics" were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, "Operator Theory and Harmonic Analysis," pp. 31-56 in Twentieth Century Harmonic Analysis-- A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.

Walsh Series states that Walsh functions provide "the simplest non-trivial model for harmonic analysis."

The patterns on the faces of the cube on the cover of Walsh Series above illustrate both the Walsh functions of order 3 and the same structure in a different guise, subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see Symmetry of Walsh Functions.

Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find "the right finite question to ask," I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.

* See the Log24 entries following Halmos's death.

From the last link within the last link of yesterday's entry:

"Review the concepts of integritas, consonantia, and claritas in Aquinas...."

Review also the properties of the number six that appears in today's date.

For such properties, see the page of Log24 entries that end on September 6, 2006, with "Hamlet's Transformation."

Happy Feast of the Transfiguration.