M. C. Escher
“For me it remains an open question whether [this work]
pertains to the realm of mathematics or to that of art.”
The Life of Escher
• Lived 1898 to 1972 in
Holland
• Was never a good
student, even in math
• Grew to enjoy graphic
art and travel
• Became fascinated
with geometry and
symmetry
The Life of Escher
• Learned relation to math from brother
– Crystallography
• Developed systematic approach for tiling
and use of space in planes
• Became mathematician through his
discoveries in art
• By exploiting many features of geometry,
he opened new domain of mathematical art
At first I had no idea at all of the possibility
of systematically building up my figures. I
did not know ... this was possible for
someone untrained in mathematics, and
especially as a result of my putting forward
my own layman's theory, which forced me
to think through the possibilities.
-Escher, 1958
Escher’s Work
• Began with landscapes
• Worked through drawings,
lithographs, and woodcuts
• After studying ideas of
planes and geometry,
developed works with:
–
–
–
–
–
Tessellations
Polyhedron
The Shape of Space
The Logic of Space
Self-Reference
Tessellations
• Defined as regular divisions of
a plane; closed shapes that do
not overlap nor leave gaps
• Previously only known for
triangles, squares, and
hexagons
• Escher discovered use of
irregular polygons:
– Reflections, translations, and
rotations
– Use of three, four, or six fold
symmetry
Polyhedron
• Platonic solids: polyhedron with same polygonal faces
• Intersecting or stellating for new forms
• To make your own polyhedron: Platonic Solid Model
The Shape of Space
Dimensions
Hyperbolic space
Topology
The Logic of Space
• The geometry of space determines its logic,
and likewise the logic of space often
determines its geometry
Light and shadow
Vanishing
points
Self-Reference
• Important concept because of artificial
intelligence’s inability to process
Although Escher was not trained
as a mathematician, his geometric
theoretical discoveries have made
a tremendous impact on both the
mathematic and artistic worlds.
Student Objectives
• Learn about:
– Two-dimensional shapes: sides and angles
– Geometric concepts: symmetry, congruency
– Patterns: translations, reflections, rotations
• Personal expression – visual art
• Recognize role in real world
Related Materials
• In order to convey the features of Escher’s
mathematical work, many resources may be
used:
–
–
–
–
–
Tangram-tiles
Protractors
Collect data of polygons and angles for tiling
Patterns such as in flooring, quilts, mosaics
Creating tessellations: manual, software
Concluding Thoughts
From M. C. Escher
Only those who attempt the absurd will achieve the impossible. I think
it's in my basement... let me go upstairs and check.
By keenly confronting the enigmas that surround us, and by
considering and analysing the observations that I have made, I ended
up in the domain of mathematics, Although I am absolutely without
training in the exact sciences, I often seem to have more in common
with mathematicians than with my fellow artists.
The laws of mathematics are not merely human inventions or creations.
They simply 'are'; they exist quite independently of the human
intellect. The most that any(one) ... can do is to find that they are there
and to take cognizance of them.