Art as a Mathform The Intersection of Antipodal Worlds

Game Plan 1)Introduction 2)Artists doing Math 3)Mathematicians doing Art Lily Pads by Laurent Davidson StabiloMobile Aluminum and Steel 21.5” high 41” wide 22” deep

And so Begins our Quest…

Definitions Disclaimer: 1)I am NOT an artist

Definitions Disclaimer: 1)I am NOT an artist. 2)I don’t like art.

Definitions Disclaimer: 1)I am NOT an artist. 2)I don’t like art. 3)I am a Mathematician. 4)I love Math and try to find it in all things.

Math & Art Differences How would a mathematician describe art? Boring Too abstract Doesn’t make any sense All artists are weirdos The Moon-Woman Jackson Pollock

Math & Art Differences How would a mathematician describe art? Boring Too abstract Doesn’t make any sense All artists are weirdos How would an artist describe math? Boring Too abstract Doesn’t make any sense All mathematicians are weirdos

Math & Art Similarities How would a mathematician describe math? Abstract representation of our world Makes sense to “most” people Means different things to different people Experience joy of creation in making something that has never been made before The results are beautiful

Math & Art Similarities How would a mathematician describe math? Abstract representation of our world Makes sense to “most” people Means different things to different people Experience joy of creation in making something that has never been made before The results are beautiful How would an artist describe art? Abstract representation of our world Makes sense to “most” people Means different things to different people Experience joy of creation in making something that has never been made before The results are beautiful

Artists Doing Math The Golden Ratio Perspective (Projective Geometry) Impossible Art Space-Filling (Tilings)

The Golden Ratio Discovered by Pythagoreans in 5 th century B.C. The Golden Ratio by Mario Livio

The Golden Ratio Discovered by Pythagoreans in 5 th century B.C. The Golden Ratio by Mario Livio

The Golden Ratio Discovered by Pythagoreans in 5 th century B.C. The Golden Ratio by Mario Livio b a

The Golden Ratio Discovered by Pythagoreans in 5 th century B.C. The Golden Ratio by Mario Livio b c

The Golden Ratio Discovered by Pythagoreans in 5 th century B.C. The Golden Ratio by Mario Livio c d

The Golden Ratio Discovered by Pythagoreans in 5 th century B.C. The Golden Ratio by Mario Livio e d

The Golden Ratio Euclid’s Elements (300 B.C.) The Extreme and Mean Ratio: ABC

The Golden Ratio Euclid’s Elements (300 B.C.) The Extreme and Mean Ratio: ABC x1

The Golden Ratio Euclid’s Elements (300 B.C.) The Extreme and Mean Ratio: ABC x1

The Golden Ratio Simplify: Solve using Quadratic Formula: The Golden Ratio:

The Golden Ratio Simplify: Solve using Quadratic Formula: The Golden Ratio:

The Golden Ratio can be found in nature via Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … The ratios of successive Fibonaccis head towards  Formula for the n th Fibonacci number: Logarithmic Spirals Ram’s horns, elephant tusks, seashells, whirlpools, hurricanes, galaxies… Peregrine Falcon Golden Ratio in Nature

Golden Ratio in Art Great Pyramid at Giza

Golden Ratio in Art Great Pyramid at Giza Parthenon

Golden Ratio in Art Great Pyramid at Giza Parthenon

Golden Ratio in Art Great Pyramid at Giza Parthenon

Golden Ratio in Art Great Pyramid at Giza Parthenon Leonardo da Vinci’s Saint Jerome

Golden Ratio in Art Great Pyramid at Giza Parthenon Leonardo da Vinci’s Saint Jerome

Golden Ratio in Art Great Pyramid at Giza Parthenon Leonardo da Vinci’s Saint Jerome

Golden Ratio in Art Great Pyramid at Giza Parthenon Leonardo da Vinci’s Saint Jerome Michelangelo’s Holy Family

Golden Ratio in Art Great Pyramid at Giza Parthenon Leonardo da Vinci’s Saint Jerome Michelangelo’s Holy Family

Golden Ratio in Art Great Pyramid at Giza Parthenon Leonardo da Vinci’s Saint Jerome Michelangelo’s Holy Family Leonardo da Vinci’s Mona Lisa

Golden Ratio in Art Great Pyramid at Giza Parthenon Leonardo da Vinci’s Saint Jerome Michelangelo’s Holy Family Leonardo da Vinci’s Mona Lisa Salvador Dali’s Sacrament of the Last Supper

Renaissance Art  Three of the best known Renaissance artists also made contributions to mathematics:  Piero della Francesca (ca ):  On Perspective in Painting  Short Book on the Five Regular Solids  Treatise on the Abacus  Leonardo da Vinci ( )  Illustrator of The Divine Proportion (Luca Pacioli)  Quadrature of the Circle (Squaring the Circle)  Areas of regions bounded by curves  Albrecht Durer ( )  Treatise on Measurement with Compass and Ruler  One of first Math books published in German  Earliest Nets of Polyhedra  Tiling of the plane

Albrecht Durer Melencolia I

Putting it in Perspective

Putting it in Perspective Pre-Renaissance subjects were depicted according to status in Church or social hierarchy Represent a scene in true and objective way Projective Geometry: what properties of an object are preserved under a projection? –Parallel lines intersect at horizon (vanishing point) –Circles become ellipses –Squares become trapezoids Horizon Vanishing point Vanishing point

Putting it in Perspective Dimensions should decrease at same rate as we move towards the horizon Compare heights of objects Similar Triangles preserve ratios of corresponding sides

Man: dpdp HmHm d hmhm Column: dpdp HcHc d hchc  

Man: dpdp HmHm d hmhm Column: dpdp HcHc d hchc  

and So we must have Cross-multiplying gives us

Piero della Francesca The Flagellation

Piero della Francesca The Flagellation

Sandro Botticelli The Annunciation

Impossible Art Roger Penrose 1950s –Impossible Triangle

Impossible Art Roger Penrose 1950s –Impossible Triangle –Tribar

Impossible Art Roger Penrose 1950 –Impossible Triangle –Tribar –Tribox

Impossible Art Roger Penrose 1950s  Impossible Triangle  Tribar  Tribox  M.C. Escher ( )  Waterfall

Impossible Art Roger Penrose 1950s  Impossible Triangle  Tribar  Tribox  M.C. Escher ( )  Waterfall  Belvedere

Impossible Art Roger Penrose 1950s  Impossible Triangle  Tribar  Tribox  M.C. Escher ( )  Waterfall  Belvedere  Cube With Ribbons

Impossible Art  Escher For Real

Impossible Art  Escher For Real

Impossible Art  Escher For Real

Impossible Art  Escher For Real

Impossible Art  Escher For Real

Impossible Art  Escher For Real

Impossible Art  Escher For Real

Major Themes Impossible Art Tessellations  Space Filling  Tilings  Metamorphosis II

Major Themes Impossible Art Tessellations  Space Filling  Tilings  Metamorphosis II  Metamorphosis III

Major Themes Impossible Art Tessellations  Space Filling  Tilings  Metamorphosis II  Metamorphosis III  Penrose Tiling

Major Themes Impossible Art Tessellations  Space Filling  Tilings  Metamorphosis II  Metamorphosis III  Limits  Circle Limit III

Major Themes Impossible Art Tessellations  Space Filling  Tilings  Metamorphosis II  Metamorphosis III  Limits  Circle Limit III  Circle Limit IV

Mathematicians Doing Art Larry Frazier Triple Bocote Blush

Mathematicians Doing Art Larry Frazier

Mathematicians Doing Art Helaman Ferguson Umbilic Torus NC

Mathematicians Doing Art Ken Leap ConfluenceSalter’s Lune

Mathematicians Doing Art Harriet Brisson Magic Cube

Movie Math

“Let no one who is not a mathematician read my works.” -Leonardo da Vinci

Sources Hofstadter, Douglas R. Godel, Escher, Bach: An Eternal Golden Braid. Random House, New York Maor, Eli. To Infinity and Beyond: A Cultural History of the Infinite. Princeton University Press, New Jersey Livio, Mario. The Golden Ratio. Random House, New York Peterson, Ivars. Fragments of Infinity: A Kaleidoscope of Math and Art. John Wiley & Sons, Inc. New York

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