Finite Relativity
From the journal of Steven H. Cullinane:
Today, Feb. 20, 2004, is the 18th birthday of my note "The
Relativity Problem in Finite Geometry" shown below.
That note begins with a quotation from Weyl:
"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
Here is another quotation from Weyl, on the profound branch of
mathematics known as Galois theory, which he says
"... is nothing else but the relativity theory for the set
Sigma, a set which, by its discrete and finite character, is
conceptually so much simpler than the infinite set of points in space
or space-time dealt with by ordinary relativity theory."
-- Weyl, Symmetry, Princeton University Press, 1952, p. 138
This second quotation applies equally well to the much less
profound, but more accessible, part of mathematics described in Diamond
Theory
and in my note below.
On the order of AGL(4,2):
See Groups
and
Symmetry, by Phill
Schultz.
See especially Part 19,
Linear Groups Over Other Fields.
As Schultz demonstrates, the order of AGL(4,2), the affine
group
in the four-dimensional space over the two-element field, is
(24)(24 - 1)(24 - 21)(24
- 22)(24 - 23) =
(16)(16 - 1)(16 - 2)(16 - 4)(16 - 8) =
(16)(15)(14)(12)(8)
= 322,560.
This group can be generated by arbitrarily mixing
permutations of rows and columns in the 4x4 array
with permutations of the array's four quadrants.
For a proof, see Binary
Coordinate
Systems.
Related material:
Invariants
"What modern painters are trying to do,
if they only knew it, is paint invariants."
-- James J. Gibson in Leonardo
(Vol. 11, pp. 227-235.
Pergamon Press Ltd., 1978)
An example of invariant structure:
The three line diagrams above result from the three partitions, into
pairs of 2-element sets, of the 4-element set from which the entries of
the bottom colored figure are drawn. Taken as a set, these three
line diagrams describe the structure of the bottom colored
figure. After coordinatizing the figure in a suitable manner, we
find that the line diagrams are invariant under the group of 16 binary
translations acting on the colored figure.
For another sort of invariance of the colored figure, try applying a
symmetry of the square to each of the set of four diagonally-divided
squares from which the figure's entries are drawn, and observe the induced effect on
the figure itself.
A more remarkable invariance -- that of symmetry itself --
is observed if we arbitrarily and repeatedly permute rows and/or
columns and/or 2x2 quadrants of the colored figure above. Each
resulting figure has some ordinary or
color-interchange symmetry. The cause of this symmetry-invariance
in the colored patterns is the symmetry-invariance of the line diagrams
under a group of 322,560 binary affine transformations.
For more details on the above invariant structures, see
Geometry of
the 4x4 Square, the Diamond
16 Puzzle, and Diamond
Theory.
Also perhaps relevant:
| Einstein wanted to know what was invariant
(the same) for all observers. Some felt that his relativity theory
should be called "theory of invariants." (See below.) |
For another perspective, see Block
Designs in Art and Mathematics.
Some further remarks
of Feb. 20, 2007,
the 21st birthday
of the above note:
Weyl's set Sigma is a finite set of complex numbers.
Some other sets with "discrete and finite character" are those of 4, 8,
16, or 64 points, arranged in squares and cubes. For
illustrations, see Finite
Geometry of the Square and Cube.
What Weyl calls "the relativity problem" for these sets involves fixing
"objectively" a class of equivalent coordinatizations. For what
Weyl's "objectively" means, see the article "Symmetry
and Symmetry Breaking," by Katherine Brading and Elena Castellani,
in
the
Stanford Encyclopedia of Philosophy:
"The
old and natural idea that what is objective should not depend upon the
particular perspective under which it is taken into consideration is
thus reformulated in the following group-theoretical terms: what is
objective is what is invariant with respect to the transformation
group of reference frames, or, quoting Hermann Weyl (1952, p. 132),
'objectivity means invariance with respect to the group of
automorphisms [of space-time].'[22]
22. The significance of the
notion of invariance and its group-theoretic treatment for the issue
of objectivity is explored in Born (1953), for example. For more
recent discussions see Kosso (2003) and Earman (2002, Sections 6 and
7).
References:
Born, M., 1953, "Physical Reality," Philosophical
Quarterly, 3, 139-149. Reprinted in E. Castellani (ed.), Interpreting
Bodies:
Classical
and Quantum Objects in Modern
Physics, Princeton, NJ: Princeton University Press, 1998,
pp. 155-167.
Earman, J., 2002, "Laws, Symmetry, and Symmetry Breaking;
Invariance, Conservation Principles, and Objectivity,' PSA 2002,
Proceedings of the Biennial Meeting of the Philosophy of Science
Association 2002, forthcoming [Abstract/Preprint
available
online]
Kosso, P., 2003, "Symmetry, objectivity, and design," in
K. Brading and E. Castellani (eds.), Symmetries in Physics:
Philosophical Reflections, Cambridge: Cambridge University Press,
pp. 410-421.
Weyl, H., 1952, Symmetry, Princeton, NJ: Princeton
University Press.
On relativity theory as "theory of invariants"--
See Archives Henri
Poincaré (research unit UMR 7117, at Université
Nancy
2, of the CNRS)--
"Minkowski, Mathematicians, and the Mathematical Theory of
Relativity," by Scott Walter, in The Expanding Worlds of
General Relativity
(Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T.
Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 45-86--
"Developing his ideas before Göttingen mathematicians in April
1909,
Klein pointed out that the new theory based on the Lorentz group (which
he preferred to call 'Invariantentheorie') could have come from pure
mathematics (1910: 19). He felt that the new theory was anticipated by
the ideas on geometry and groups that he had introduced in 1872,
otherwise known as the Erlangen program (see Gray 1989: 229)."
References:
Gray, Jeremy J. (1989). Ideas of Space. 2d ed. Oxford: Oxford
University Press.
Klein, Felix. (1910). "Über die geometrischen Grundlagen der
Lorentzgruppe." Jahresbericht der deutschen Mathematiker-Vereinigung
19: 281-300. [Reprinted: Physikalische Zeitschrift 12
(1911): 17-27].
Update of Feb. 20, 2008:
For some remarks on the relativity problem in the simpler case of
a
2x2x2 cube rather than a 4x4 square, see Knight
Moves: Geometry of the Eightfold Cube.
Update of May 16, 2008:
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
Update of February 20, 2010:
Weyl on what he calls the relativity problem–
"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,
1949, "Relativity Theory as a Stimulus in Mathematical Research"
"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, 1946, The
Classical
Groups, Princeton University Press, p. 16
Twenty-four years ago a note of Feb. 20, 1986,
supplied an example of such coordinatizations in finite geometry. In
that note, the group of mediating transformations acted directly
on
coordinates within a 4×4 array. When the 4×4 array is
embedded
in a 4×6
array, a larger and more interesting group, M24 (containing the original
group), acts on the larger
array.
There is no obvious solution to Weyl's relativity problem for M24.
That
is, there is no obvious way to apply exactly 24 distinct
transformable coordinates (or symbol-strings) to the 24 array
elements in such a way that the natural group of mediating
transformations of the 24 symbol-strings is M24.
There is, however, an assignment of symbol-strings that yields a family
of
sets with automorphism group M24.
R.D. Carmichael in 1931 on his construction of the Steiner system
S(5,8,24)–
"The linear fractional group modulo 23
of
order 24•23•11 is often represented as a doubly transitive group of
degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group
contains a subgroup of order 8 each element of which transforms into
itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the
whole group transforms this set into 3•23•11 sets of eight each. This
configuration of octuples has the remarkable property that any given
set of five of the 24 symbols occurs in one and just one of these
octuples. The largest permutation group Γ on the 24
symbols, each element of which leaves this configuration invariant, is
a five-fold transitive group of degree 24 and order 24•23•22•21•20•48.
This is the Mathieu group of degree 24."
– R. D. Carmichael, 1931, "Tactical
Configurations of Rank Two," in American Journal of Mathematics,
Vol.
53, No. 1 (Jan., 1931), pp. 217-240
Page created Feb. 21, 2004