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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

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M c escher

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
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 escher1
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
Escher’s Work 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.

  • 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
Tessellations 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.

  • 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
Polyhedron 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.

  • 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
The Shape of Space 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.

Dimensions

Hyperbolic space

Topology


The logic of space
The Logic of Space 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.

  • The geometry of space determines its logic, and likewise the logic of space often determines its geometry

    Light and shadow

    Vanishing

    points


Self reference
Self-Reference 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.

  • 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
Student Objectives geometric theoretical discoveries have made a tremendous impact on both the mathematic and artistic worlds.

  • 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
Related Materials geometric theoretical discoveries have made a tremendous impact on both the mathematic and artistic worlds.

  • 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
Concluding Thoughts geometric theoretical discoveries have made a tremendous impact on both the mathematic and artistic worlds.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.