SPS Lecture: Symmetry in Art and Biology

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SPS Lecture Symmetry in Art and Biology
ART
MIND
NATURE
BIOLOGY
sublime creations
perceptual dualities
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Rotational/Reflection Symmetry
Bowl Amratia, 4200-3600BC, Hahn, p. 3
Algae, starfish, sand stars Hanh, p. 39, 168,
168.
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Translational Symmetry
Wristband Mesin, Ukrain, ca. 11000 B. C., may
show the dentinal structure of mammoth tusks,
Hahn, p. 3
Honeycomb exhibiting hexagonal tiling, Hahn, p.
70
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Planar Tiles, Devlin, 1997, p. 164
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The Moors used all 17 Wallpaper Patterns, first
discovered by the ancient Egyptians, to decorate
the Alhambra in Granada, Spain
http//www.clarku.edu/djoyce/wallpaper
http//www.red2000.com/spain/alhamb.html
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Rigid Transformations

• RigidTransformations T Plane ? Plane such that
  distance(T(a),T(b)) distance(a,b), a,b in
  Plane
• Transform every figure X subset Plane into a
  congruent figure T(X) T(p) p in X
• Form a group with multiplication defined by
  composition ToS(a) T(S(a)), a group is closed
  under multiplication, contains the identity I,
  contains the inverse of each element, and
  multiplication is associative (ToS)oU To(SoU)
• Is a continuous group (or Lie group) can be
  parameterized (locally) by real numbers, this can
  be seen since every rigid transformation can be
  specified by a rotation about a specified point
  followed by a translation

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Symmetry Groups

• X is invariant under T if T(X) X
• The set of transformations under which a figure X
  is invariant forms the symmetry group of X
• Sym(general figure) I
• Sym(isoceles triangle a,b,c d(a,b) not d(b,c))
  I, A where A is the flip transformation
  A(a) a, A(b) c, A(c) b
• Sym(equilateral triangle) I, R, RoR, A, B, C
  where R is rotation by 120 degrees and A, B, C
  are flips such that A(a) a, B(b) b, C(c) c
• Sym(circle) R_angle addition modulo 360
  degrees U F_angle addition modulo 180 degrees
  

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Wallpaper Patterns

• F Plane ? red, yellow, green, blue
• Figures X_red p F(p) red, X_yellow, etc.
  
• Sym(F) T T(X_c X_c for every color c
• Lattice group generated by two linearly
  independent translation vectors
• Admissible colorings symmetry groups are
  discrete and contain a lattice subgroup
• Two colorings are equivalent if they have the
  same symmetry groups.
• Wallpaper pattern equivalence class of
  admissible planar colorings

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Plato, Devlin, 1997, p. 112-113
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Spatial Tiles, Devlin, 1997, p. 165
Pomegranate seeds grow to rhombic
dodecahedrons (green) from spheres in
a bicubic lattice
There are 230 three-dimensional wallpaper
patterns classified by crystallographers in the
19-th century
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Golden Ratio
It appears in natures angles, e.g. plantains,
pinecones, peacocks and snails, and Nautilus
Hahn, p. 175, 454, 455, 456.
Its remarkable number theoretic properties give
it optimal circle partitioning properties, Lawton
1.
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Growth Processes
of a population of immortal cells that divide at
the end of each calendar year is described by
iterating a linear transformation
cells at least one year old
cells
Fibbonaci sequence, whose ratio of successive
terms approaches the golden ratio
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Projection, Devlin, 1997, p. 130
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Durer, Devlin, 1997, p. 168
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Desargues Theorem, Devlin, 1997, p. 134-135
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Cross Ratio
a
b
c
d
Preserved under projective transformations
Implied by Desargues Theorem
Symmetry is group if linear fractional
transformations related to string theories
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Peron Frobenius Theorem
Nonnegative matrices (mild assumptions) have
unique positive eigenvectors
Implied by cross ratio invariance
Implies that general growth processes
yield symmetries (stationary equilibria)
Similar limits of spatial growth patterns
yield fractal tiles with amazing properties,
Lawton 2
More general transformation groups
(conformal) are related to biological growth,
Thompson
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PenroseAperiodic Tiling
Exhibits five fold local rotational symmetry,
ratio of fat to thin tiles approaches the Golden
Ratio, Devlin, p. 168
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References
Devlin, Keith, Mathematics The Science of
Patterns, Scientific American Library, Division
of HPHLP, New York, 1997
Grossman, Israel and Magnus, Wilhelm, Groups and
their Graphs, The Mathematical Association of
America, 1992
Hahn, Werner, Symmetry as a Developmental
Principle in Nature and Art, World Scientific,
Singapore, 1998
Lawton, Wayne, Kroneckers theorem and rational
approximation of algebraic, The Fibonacci
Quarterly, volume 21, number 2, pages 143-146,
May 1983
Lawton, Wayne and Resnikoff, Howard, Fractal
tiling for multiple mirror telescopes, U. S.
Patent 4,904,073, 27, February 1990
Thompson, DArcy Wentworth, Growth and Form, Vol.
I and II, Cambridge University Press, Cambridge,
1952