Finite
Geometry Notes |

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:

"... 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

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.

"Beauty is the proper conformity of the parts to one another and to the whole."

-- Werner Heisenberg, "

"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

-- James Joyce,

An important subset of the above

thirty-five 8-set partitions

may also be pictured

as follows:

thirty-five 8-set partitions

may also be pictured

as follows:

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.

Notions of Relativity

and Kindergarten Blocks

A page from a 1906 book on

kindergarten education whose

first edition was published in 1869:

Joseph Payne, "Froebel and the Kindergarten System of Education" (lecture, Feb. 25, 1874) in

"Then the examination of the cube, especially its surfaces, edges, and angles, which any child can observe for himself, suggest [

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."

A somewhat different use

of the term "relativity"--

of the term "relativity"--

"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,

For some related remarks on coordinatizations that
apply to the eightfold cube as well as to larger structures, see Finite
Relativity.

*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

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,

Page created January 16, 2008.