The Work of Maurits
Cornelis (M.C.) Escher
Presented by:
Tiana Taylor
Tonja Hudson
Cheryll Crowe
“For me it remains an open question whether
[this work] pertains to the realm of mathematics
or to that of art.” - M.C. Escher
Small Group
Discussion - Guiding
Questions
1) Give examples of how Escher’s work can be utilized at the
following levels: elementary, middle / high school, beyond.
2) Do you believe Escher’s work is more in the realm of
mathematics or art?
3) How can technology be used to create Escher-inspired
art? Other technology/software?
4) Knowing there are other artists/mathematicians with
Escher’s vision, how can you use this information in
constructing lessons in mathematics?
Topics touched by Escher
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Tessellations
Isometric drawings
Transformations
Impossible figures
Non-Euclidean
geometry
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Symmetry
Polygons
Limiting Idea
Infinity
Polyhedra
Spherical geometry
History of M.C. Escher
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Maurits Cornelis Escher was born in Holland in 1898.
Poor student who had to repeat two grades
Architecture to Graphic Arts to professional work
Inspired by Italian architecture
Two categories:
– Geometry of Space
• Tessellations
• Polyhedra
• Hyperbolic Space
– Logic of Space
• Visual Paradoxes
• Impossible Drawings
• MC Escher died in 1972 and Snakes was his last
contribution to the world of art and mathematics.
Snakes (1969)
Artist and/or Mathematician
“The ideas that are basic to [my works] often bear
witness to my amazement and wonder at the
laws of nature which operate in the world around
us. He who wonders discovers that this is in
itself a wonder. By keenly confronting the
enigmas that surround us, and by considering
and analyzing the observations that I had made,
I ended up in the domain of mathematics.
Although I am absolutely without training or
knowledge in the exact sciences, I often seem to
have more in common with mathematicians than
with my fellow artists.”
~ M. C. Escher
Escher at the Elementary
Level (K-5)
NCTM Standards
Pre-K – Grade 2 Connections
• Recognize 2D and 3D shapes
• Investigate and predict results of putting
together and taking apart shapes
• Recognize and apply slides, flips, and
turns
• Recognize and represent shapes from
different perspectives
NCTM Standards
Grade 3-5 Connections
• Investigate, describe, and reason about the
results of combining and transforming shapes
• Explore congruence and similarity
• Predict and describe results of sliding, flipping,
and turning 2D shapes
• Identify and describe line and rotational
symmetry
• Recognize geometric ideas and relationships
and apply them in everyday life
Escher-Inspired Lego Designs
M.C. Escher "Relativity"
http://imp-world.narod.ru/art/lego/
Escher-Inspired Lego Designs cont.
M.C. Escher "Waterfall"
http://imp-world.narod.ru/art/lego/
Escher for the Middle and
High School (6-12)
NCTM Standards
Grade 6-8 Connections
• Examine the congruence, similarity, and
line or rotational symmetry of objects using
transformations
• Recognize and apply geometric ideas and
relationships in areas outside the
mathematics classroom, such as art and
everyday life
NCTM Standards
Grade 9-12 Connections
• Analyze properties and determine attributes of
2D and 3D objects
• Explore relationships among classes of 2D and
3D geometric objects
• Understand and represent translations,
reflections, rotations, and dilations of objects in
the plane
• Draw and construct representation of 2D and 3D
geometric objects using a variety of tools
Creating Escher’s Impossible Figures
Using Computer Software
Impossible Puzzle 1.10 – creates impossible
figures from triangles
http://imp-world.narod.ru/programs/index.html
Creating Escher’s Impossible Figures
Using Computer Software
Impossible Constructor 1.25 – creates impossible
figures from cubes
http://imp-world.narod.ru/programs/index.html
Escher and Beyond
Escher and the Droste Effect
• Did Escher leave the
center blank on purpose?
• Is it possible to render
this image via computer
programming?
• Hendrik Lendstra & Bart
de Smit
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http://www.msri.org/people/members/sara/ar
ticles/siamescher.pdf
http://escherdroste.math.leidenuniv.nl/index.
php?menu=intro
Beyond Escher
• Victor
Acevedo
• 1977 Escher
inspired
pilgrimage to
Alhambra in
Granada
Synapse - Cuboctahedronic Periphery
Beyond Escher
• Dr. Helaman Ferguson
• Sculpture
• “celebrates the
remarkable achievements
of mathematics as an
abstract art form”
Thirteenth Eigenfunction on the
Helge Koch fractal snowflake curve
Beyond Escher
• Dr. Robert Fathauer
• In 1993 founded
Tessellations
• Researcher for
NASA’s Jet
Propulsion Laboratory
• Fractal Iteration
Fractal Knots No. 1
Beyond Escher
• Dr. S. Jan Abas
• School of Computer
Science at the
University of Wales
• Islamic Geometric
Patterns: his heritage
Islamic Pattern in
Hyperbolic Space
Small Group
Discussion - Guiding
Questions
1) Give examples of how Escher’s work can be utilized at the
following levels: elementary, middle / high school, beyond.
2) Do you believe Escher’s work is more in the realm of
mathematics or art?
3) How can technology be used to create Escher-inspired
art? Other technology/software?
4) Knowing there are other artists/mathematicians with
Escher’s vision, how can you use this information in
constructing lessons in mathematics?
References
Abas, S. J. (2001). Islamic geometrical patterns for the teaching of mathematics of
symmetry [Special issue of Symmetry: Culture and Science]. Symmetry in
Ethnomathematics, 12(1-2), 53-65. Budapest, Hungary: International Symmetry
Foundation.
Abas, S. J. (2000). Symmetry 2000. Retrieved November 10, 2007, from
http://www.bangor.ac.uk/~mas009/pcont.htm
Alexeev, V. (n.d.). Programs. Retrieved November 6, 2007, from
http://imp-world.narod.ru/programs/index.html
Alexeev, V. (n.d.). Impossible world: Legos. Retrieved November 6, 2007 from
http://imp-world.narod.ru/art/lego/
Art-Baarn, C. (n.d.). Escher and the droste effect. Retrieved November 10, 2007, from
http://escherdroste.math.leidenuniv.nl/index.php?menu=intro
De Smit, B. & Lenstra Jr., H. W. (2003). The mathematical structure of Escher’s
print gallery, Notices of the AMS, 50(4).
Fathauer, R. (n.d.) The iteration (fractal) art of Robert Fathauer. Retrieved November 10,
2007, from http://members.cox.net/tessellations/IterationArt.html
Ferguson, H. (2003). Helaman Ferguson Sculpture. Retrieved November 10, 2007, from
http://www.helasculpt.com/index.html
Gershon E. (2007). Escher for real. Retrieved November 6, 2007, from
http://www.cs.technion.ac.il/~gershon/EscherForReal/Penrose.gif
Impossible Staircase (n.d.) Retrieved November 6, 2007, from
http://www.timhunkin.com/ page_pictures/a119_f3.jpg
References
Mathematics Behind the Art of M.C. Escher (n.d.). Retrieved November 6, 2007, from
http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/
main3.html
M.C. Escher Company B.V. (2007). The official M.C. Escher website. Retrieved
November 6, 2007, from http://www.mcescher.com/
NNDB (2007). M.C. Escher. Retrieved November 6, 2007, from
http://www.nndb.com/people/308/000030218/
Platonic Realms (2007). The Mathematical Art of M.C. Escher. Retrieved November 10,
2007, from http://www.mathacademy.com/pr/minitext/escher/index.asp
Robinson, S. (2002). M.C. Escher: More mathematics than meets the eye.
SIAM News, 35(8).