M'C' Escher and Infinity and Geometry and Tessellations and'

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M.C. Escher and Infinityand Geometryand
Tessellationsand.

• Valerie R. Hammans
• 6 December 2007

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Why this?
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Maurits Cornelis Escher (1898-1972)

• Born in Leeuwarden, Netherlands
• Father was a civil engineer
• Spent most of youth in Arnhem
• Failed high school exams.
• Enrolled in School for Architecture and
  Decorative Arts in Haarlem
• Within a week decided he wanted to study graphic
  art
• Moved to Italy
• met wife
• lived there for eleven years

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• One of his woodcuts he used for the trees in
  Puddle and also in Pintea of Cavi

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Other info.

• Made over 448 lithographs, woodcuts, wood
  engravings
• Over 2000 sketches and drawings.
• Left handed

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More of his Life

• In 1922, on his visit to Spain, he became
  fascinated with Division Plane
• Was visiting Alhambra, 14 century Moorish castle.
  
• In Switzerland, during WWII, he completed 62 of
  137 Regular Division Drawings

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?
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Math?

• In 1956 he was featured in Time
• Greatest admires were mathematicians.
• Visualization of math principles.
• Had no formal math beyond high school
• After this adoration, he read more about math,
  dealing with plane and projective geometry.
• Goes on to use non-Euclidean geometry
• Love of impossible figures
• Encompasses geometry of space and logic of space

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Division Plane

• Deals with tessellations

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What is a tessellation?

• are arrangements of closed shapes that
  completely cover the plane without overlapping
  and without leaving gaps.
• Usually polygons or similar shapes, like square
  tiles on a floor.

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His quote on tessellations

• In mathematical quarters, the regular division
  of the plane has been considered theoretically .
  . . Does this mean that it is an exclusively
  mathematical question? In my opinion, it does
  not. Mathematicians have opened the gate
  leading to an extensive domain, but they have not
  entered this domain themselves. By their very
  nature thay are more interested in the way in
  which the gate is opened than in the garden lying
  behind it.

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Tessellations Cont.

• Was thought to just be triangle, squares,
  rectangles, and hexagons
• Yet, he took his basic problems and applied
  reflections, glide reflections, translations, and
  rotations
• Also distorted the shapes of animals, birds, and
  others figures.
• Rule three, four, or six-fold symmetry

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Polyhedra

• AMAZING!
• Platonic solids tetrahedron (4), cube (6),
  octahedron (8), dodecahedron (12), and
  icosahedron (20)
• What could make these better?
• Intersect them
• Stellating (replace each face with a pyramid)

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Topology

• Just becoming something of an interest to the
  world in his lifetime.
• Topology is the mathematical study of the
  properties that are preserved through
  deformations, twisting, and stretchings of
  objects. Tearing, however, is not allowed.
• Leads to Knots and Knot theory

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Logic of Space?

• Spatial relation among physical objects that are
  necessary
• Optical illusion
• Use lights and shadows
• Concave and convex
• Visual clues
• Vanishing points
• Points of infinity by Alberi, and others led to
  projective geometry

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Self-Reference

• Escher has many pictures of himself in orbs.
• The illusion is that the picture seems to draw
  itself or that it will reflect a personal
  reflection.
• Seems to have intersecting worlds
• every part of the world seems to contain, and be
  contained in, every other part
• A reflection of the artist, the artist reflected
  in his work.

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References

• http//en.wikipedia.org/wiki/Regular_Division_of_t
  he_Plane
• http//www.mcescher.com/
• http//www.mathacademy.com/pr/minitext/escher/
• MathWorld.Wolfram.com
• M. C. Escher by Sandra Forty
• M.C. Escher Visions of Symmetry by Doris
  Schattschneider