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 ThoughtsFrom 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.