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```					M.C. Escher and Infinity
and Geometry
and Tessellations
and….
Valerie R. Hammans
6 December 2007
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
One of his woodcuts he used for the trees in
Puddle and also in Pintea of Cavi
Other info.
• Made over 448 lithographs,
woodcuts, wood engravings
• Over 2000 sketches and
drawings.
• Left handed
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
Math?
• In 1956 he was featured in Time
• Greatest admires were mathematicians.
• Visualization of math principles.
• Had no formal math beyond high school
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
Division Plane

Deals with tessellations
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.
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.”
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
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)
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
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
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.”
References
•   http://en.wikipedia.org/wiki/Regular_Division_of_the_Plane
•   http://www.mcescher.com/