**Reciprocity**

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of *Himmelfarb*, New York Times Book Review, October 30, 1994

Last year's entry on this date:

The picture above is of the complete graph K_{6 }… Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M_{24}.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. "Reciprocity" in the sense of Lao Tzu. See

**Reciprocity and Reversal in Lao Tzu**.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of *Shu* (Analects 15:23 or 24) described in

*Shu*: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K_{6} graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate. The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K_{6}, applied to the sixteen 4×4 Dirac matrices, in

**A Graphical Representation**

of the Dirac Algebra.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See

**The Jewel of Arithmetic** and

**The Golden Theorem**.