The Work of Maurits Cornelis M'C' Escher

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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
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Small Group Discussion - Guiding Questions
1) Give examples of how Eschers work can be
utilized at the following levels elementary,
middle / high school, beyond. 2) Do you believe
Eschers 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 Eschers vision, how can you use this
information in constructing lessons in
mathematics?
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Topics touched by Escher

• Tessellations
• Isometric drawings
• Transformations
• Impossible figures
• Non-Euclidean geometry

• Symmetry
• Polygons
• Limiting Idea
• Infinity
• Polyhedra
• Spherical geometry

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History of M.C. Escher

• 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)
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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

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Escher at the Elementary Level (K-5)
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NCTM StandardsPre-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

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NCTM StandardsGrade 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

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Escher-Inspired Lego Designs
M.C. Escher "Relativity"
http//imp-world.narod.ru/art/lego/
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Escher-Inspired Lego Designs cont.
M.C. Escher "Waterfall"
http//imp-world.narod.ru/art/lego/
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Escher for the Middle and High School (6-12)
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NCTM StandardsGrade 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

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NCTM StandardsGrade 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

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Creating Eschers Impossible Figures Using
Computer Software
Impossible Puzzle 1.10 creates
impossible figures from triangles
http//imp-world.narod.ru/programs/index.html
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Creating Eschers Impossible FiguresUsing
Computer Software
Impossible Constructor 1.25 creates
impossible figures from cubes
http//imp-world.narod.ru/programs/index.html
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Escher and Beyond
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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
• http//www.msri.org/people/members/sara/articles/s
  iamescher.pdf
• http//escherdroste.math.leidenuniv.nl/index.php?m
  enuintro

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Beyond Escher

• Victor Acevedo
• 1977 Escher inspired pilgrimage to Alhambra in
  Granada

Synapse - Cuboctahedronic Periphery
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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
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Beyond Escher

• Dr. Robert Fathauer
• In 1993 founded Tessellations
• Researcher for NASAs Jet Propulsion Laboratory
• Fractal Iteration

Fractal Knots No. 1
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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
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Small Group Discussion - Guiding Questions
1) Give examples of how Eschers work can be
utilized at the following levels elementary,
middle / high school, beyond. 2) Do you believe
Eschers 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 Eschers vision, how can you use this
information in constructing lessons in
mathematics?
23
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 A
lexeev, 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
?menuintro De Smit, B. Lenstra Jr., H. W.
(2003). The mathematical structure of
Eschers 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/te
ssellations/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/EscherFor
Real/Penrose.gif Impossible Staircase (n.d.)
Retrieved November 6, 2007, from
http//www.timhunkin.com/ page_pictures/a119_f3
.jpg

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References
Mathematics Behind the Art of M.C. Escher (n.d.).
Retrieved November 6, 2007, from
http//www.math.nus.edu.sg/aslaksen/gem-project
s/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/ Platoni
c Realms (2007). The Mathematical Art of M.C.
Escher. Retrieved November 10, 2007, from
http//www.mathacademy.com/pr/minitext/escher/inde
x.asp Robinson, S. (2002). M.C. Escher More
mathematics than meets the eye. SIAM News,
35(8).