June 4, 2005
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Drama of the Diagonal
The 4×4 Square:
French PerspectivesEarendil_Silmarils:
Les Anamorphoses:
“Pour construire un dessin en perspective,
le peintre trace sur sa toile des repères:
la ligne d’horizon (1),
le point de fuite principal (2)
où se rencontre les lignes de fuite (3)
et le point de fuite des diagonales (4).”
_______________________________
Serge Mehl,
Perspective &
Géométrie Projective:
“… la géométrie projective était souvent
synonyme de géométrie supérieure.
Elle s’opposait à la géométrie
euclidienne: élémentaire…
La géométrie projective, certes supérieure
car assez ardue, permet d’établir
de façon élégante des résultats de
la géométrie élémentaire.”
Similarly…
Finite projective geometry
(in particular, Galois geometry)
is certainly superior to
the elementary geometry of
quilt-pattern symmetry
and allows us to establish
de façon élégante
some results of that
elementary geometry.
Other Related Material…
from algebra rather than
geometry, and from a German
rather than from the French:
“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 U. Press, 1946
Evariste Galois
Weyl also says that the profound branch
of mathematics known as Galois theory“… 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 U. Press, 1952
Metaphor and Algebra…“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.”– attributed, in varying forms, to
Max Black, Models and Metaphors, 1962For metaphor and
algebra combined, see
“Symmetry invariance
in a diamond ring,”A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.
More on Max Black…“When approaching unfamiliar territory, we often, as
observed earlier, try to describe or frame the novel situation using
metaphors based on relations perceived in a familiar domain, and by
using our powers of association, and our ability to exploit the
structural similarity, we go on to conjecture new features for
consideration, often not noticed at the outset. The metaphor works,
according to Max Black, by transferring the associated ideas and
implications of the secondary to the primary system, and by selecting,
emphasising and suppressing features of the primary in such a way that
new slants on it are illuminated.”— Paul Thompson, University College, Oxford,
The Nature and Role of Intuition
in Mathematical EpistemologyA New Slant…That intuition, metaphor (i.e., analogy), and association may lead
us astray is well known. The examples of French perspective above
show what might happen if someone ignorant of finite geometry were to
associate the phrase “4×4 square” with the phrase “projective
geometry.” The results are ridiculously inappropriate, but at
least the second example does, literally, illuminate “new slants”–
i.e., diagonals– within the perspective drawing of the 4×4 square.Similarly, analogy led the ancient Greeks to believe that the diagonal
of a square is commensurate with the side… until someone gave them a new slant on the subject.
