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Knight Moves:
The Relativity Theory
of Kindergarten Blocks

The following note was suggested by W. D. Joyner's discussion of the (vertex) "cross group" in his Mathematics of the Rubik's Cube.  Taken together with The Eightfold Cube,  the note suggests a visual explanation of why two embodiments of Klein's order-168 simple group-- PSL(2, 7) and GL(3, 2)-- are isomorphic.  The "knight moves" of the title are not those of chess, but are instead derived from a labeling of the subcubes of the 2x2x2 (eightfold) cube based on the Singer 7-cycle defined in Generating the Octad Generator:

The "knight" 7-cycle

Actions of PSL(2, 7) on the Eightfold Cube

"... die Schönheit... [ist] die richtige Übereinstimmung der Teile miteinander und mit dem Ganzen."

"Beauty is the proper conformity of the parts to one another and to the whole."
  -- Werner Heisenberg, "Die Bedeutung des Schönen in der exakten Naturwissenschaft," address delivered to the Bavarian Academy of Fine Arts, Munich, 9 Oct. 1970, reprinted in Heisenberg's Across the Frontiers, translated by Peter Heath, Harper & Row, 1974

"After the analysis which discovers the second quality [symmetry, following wholeness] the mind makes the only possible synthesis and discovers the third quality [claritas or radiance]. This is the moment which I call epiphany. First we recognise that the object is one integral thing, then we recognise that it is an organised composite structure, a thing in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany."

-- James Joyce, Stephen Hero, as quoted by Jorn Barger

An important subset of the above
thirty-five 8-set partitions
may also be pictured
as follows:

A set of 7 partitions of the 2x2x2 cube that is invariant under PSL(2, 7) acting on the 'knight' coordinatization

These seven partitions may also
be viewed as the seven points of
the finite projective plane
over the 2-element field.
See The Eightfold Cube.

Related quotations:

  Notions of Relativity
and Kindergarten Blocks

Part I:

A page from a 1906 book on
kindergarten education whose
first edition was published in 1869:

Page from 'The Paradise of Childhood' showing 7 partitions of a 2x2x2 cube

Part II:

Joseph Payne, "Froebel and the Kindergarten System of Education" (lecture, Feb. 25, 1874) in Lectures on the Science and Art of Education, with Other Lectures, New York and Chicago, E. L. Kellogg & Co., New Edition, 1890, pp. 330-331--

"Then the examination of the cube, especially its surfaces, edges, and angles, which any child can observe for himself, suggest [sic] new sensations and their resulting perceptions. At the same time, notions of space, time, form, motion, relativity in general, take their place in the mind, as the unshaped blocks which, when fitly compacted together, will lay the firm foundation of the understanding. These elementary notions, as the very groundwork of mathematics, will be seen to have their use as time goes on.

The third Gift is a large cube, making a whole, which is divisible into eight small ones. The form is recognized as that of the cube before seen; the size is different. But the new experiences consist in notions of relativity-- of the whole in its relation to the parts, of the parts in their relation to the whole; and thus the child acquires the notion and the names, and both in immediate connection with the sensible objects, of halves, quarters, eighths, and of how many of the small divisions make one of the larger.  But in connection with the third Gift a new faculty is called forth-- Imagination-- and with it the instinct of construction is awakened."

Part III:

A somewhat different use
of the term "relativity"--

The Relativity Problem

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

-- Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
For some related remarks on coordinatizations that apply to the eightfold cube as well as to larger structures, see Finite Relativity.

From some 1949 remarks of Weyl:
"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."
-- Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society, Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541

Page created January 16, 2008.