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Theme and Variations

"...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case."

-- Paul Halmos in
I Want To Be a Mathematician

The great mathematician Hermann Weyl wrote a book, Symmetry, to make the point that visual symmetry is a special case of the concept known as invariance.

(An invariant is something that does not change under a set of transformations.  Invariance is the property of being an invariant.  Some felt that Einstein's relativity theory should be called Invariantentheorie, "theory of invariants.")

However you rearrange the quadrants of the diamond figure, the result has some ordinary or color-interchange symmetry:

So the diamond figure illustrates how symmetry itself can be invariant under a set of tranformations.

This would be of little interest unless this invariance of symmetry can be generalized.  Fortunately, it can.

-- Steven H. Cullinane

Related Material

Bach, BWV 1087

"What could be simpler? Four
scale-steps descend from Do.
Four such measures carry over
the course of four phrases, then home.

      ... the theme swells
to four seasons, four compass points, four winds,
forcing forth the four corners of the world
perfect for getting lost in....

What could be simpler? Not even music
yet, but only counting: Do, ti, la, sol....

Everything that ever summered forth starts
in identical springs, or four-note variations
on that repeated theme: four seasons,
four winds, four corners, four-chambered heart...

Look, speak, add to the variants (what could
be simpler?) now beyond control. How can we help
but hitch our all to these mere four notes?"

-- Richard Powers, "The Perpetual Calendar,"
from The Gold Bug Variations, 1991

From Diamond Theory, 1976 --


on the diamond theme:

Click on picture for details.


  1. Richard Powers, The Gold Bug Variations

  2. Charles Small's new transcription
    of The Goldberg Variations for string quartet

  3. Geometry of the 4x4 Square

  4. Poetry's Bones

"The more familiar we become with Four Quartets... the more we realize that the analogy with music goes much deeper than a comparison of the sections with the movements of a quartet, or than an identification of the four elements as 'thematic material.'  One is constantly reminded of music by the treatment of images...."

-- Helen Gardner, The Art of T. S. Eliot

Page created April 28, 2004

For a large downloadable folder
containing this and many related web pages,
see Notes on Finite Geometry.

However you
rearrange the
four quadrants
of the above
the result
always has
some ordinary or

24 variations on the diamond figure

This generalizes
as follows...

(Click on
figures for

2x2x2 Cube

4x4 Square

4x4x4 Cube

The groups
that leave
invariant in
these figures
are of order:

2x2 Square

2x2x2 Cube

4x4 Square

4x4x4 Cube
1.3 trillion