Art as a Mathform The Intersection of Antipodal Worlds .

Art as a MathformThe Intersection of Antipodal Worlds

http://www.mcescher.com

Game Plan1) Introduction

2) Artists doing Math

3) Mathematicians doing Art

http://www.highlands-gallery.com/Laurent_Davidson2.cfm

Lily Padsby Laurent DavidsonStabiloMobileAluminum and Steel 21.5” high 41” wide 22” deep

And so Begins our Quest…

Definitions

• Disclaimer: 1) I am NOT an artist

http://www.kenleap.com/

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 Pollock1942

http://www.ibiblio.org/wm/paint/auth/pollock/pollock.moon-woman.jpg

Math & Art Differences

How would a mathematician describe art?

• Boring

• Too abstract


• All artists are weirdos

How would an artist describe math?

• Boring

• Too abstract


• All mathematicians are weirdos

Math & Art SimilaritiesHow 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 SimilaritiesHow 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


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


Artists Doing Math

• The Golden Ratio

• Perspective (Projective Geometry)

• Impossible Art

• Space-Filling (Tilings)

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

• The Golden Ratio by Mario Livio



b

a

b

a



b

c

c

b

b

a



d

c

c

b

b

a

c

d



e

d

e

d

d

c

c

b

b

a

The Golden Ratio

segmentlarger

line whole

segmentshorter

segmentlarger

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

A BC

AC

AB

CB

AC

The Golden Ratio

segmentlarger

line whole

segmentshorter

segmentlarger


A BC

x 1

AC

AB

CB

AC

The Golden Ratio

segmentlarger

line whole

segmentshorter

segmentlarger

AC

AB

CB

AC


A BC

x 1

x

xx 1

1

The Golden Ratio

01

1

1

1

2

2

xx

xx

x

xx

2

51,

2

51 x

Simplify:

Solve using Quadratic Formula:

The Golden Ratio:

2

51

The Golden Ratio

01

1

1

1

2

2

xx

xx

x

xx

2

51,

2

51 x

Simplify:

Solve using Quadratic Formula:

The Golden Ratio:

61803.12

51

1

• 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 nth Fibonacci number:

• Logarithmic Spirals

• Ram’s horns, elephant tusks, seashells, whirlpools, hurricanes, galaxies…• Peregrine Falcon

nn

nF2

51

2

51

5

1

Golden Ratio in Nature

Golden Ratio in Art

• Great Pyramid at Giza

http://people.bath.ac.uk/jaj21/disprovingmyth.html

Golden Ratio in Art


• Parthenon

http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm

Golden Ratio in Art


• Parthenon

• Leonardo da Vinci’s Saint Jerome


Golden Ratio in Art


• Parthenon


• Michelangelo’s Holy Family


Golden Ratio in Art


• Parthenon



• Leonardo da Vinci’s Mona Lisa

http://library.thinkquest.org/27890/applications6.html

Golden Ratio in Art


• Parthenon



• Leonardo da Vinci’s Mona Lisa

• Salvador Dali’s Sacrament of the Last Supper

http://plus.maths.org/issue22/features/golden/

Renaissance Art Three of the best known Renaissance artists also made contributions to mathematics:

Piero della Francesca (ca. 1412-1492): On Perspective in Painting Short Book on the Five Regular Solids Treatise on the Abacus

Leonardo da Vinci (1452-1519) Illustrator of The Divine Proportion (Luca Pacioli) Quadrature of the Circle (Squaring the Circle) Areas of regions bounded by curves

Albrecht Durer (1471-1528) Treatise on Measurement with Compass and Ruler One of first Math books published in German Earliest Nets of Polyhedra Tiling of the plane

http://www.intriguing.com/mp/

Albrecht Durer Melencolia I

http://www.ibiblio.org/wm/paint/auth/durer/

Putting it in Perspective

http://www.intriguing.com/mp/


• 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

HorizonVanishing

pointVanishing

point


http://plus.maths.org/issue23/features/criminisi/

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

Man:dp

Hm

d

hm

Column:

dp

Hc

d

hc

Man:

d

d

h

H p

m

m

dp

Hm

d

hm

Column:

dp

Hc

d

hc

d

d

h

H p

c

c

d

d

h

H p

m

m d

d

h

H p

c

c

c

c

m

m

h

H

h

H

c

mcm H

Hhh

and

So we must have

Cross-multiplying gives us

Piero della Francesca The Flagellation

www.artchive.com

Sandro Botticelli The Annunciation

http://www.kap.pdx.edu/trow/winter01/perspective/persp-images.htm

Impossible Art

• Roger Penrose 1950s

– Impossible Triangle

http://mathworld.wolfram.com/PenroseTriangle.html

Impossible Art

• Roger Penrose 1950s

– Impossible Triangle

– Tribar

http://icl.pku.edu.cn/yujs/MathWorld/math/t/t317.htm

Impossible Art

• Roger Penrose 1950– Impossible Triangle – Tribar – Tribox

http://icl.pku.edu.cn/yujs/MathWorld/math/t/t318.htm

Impossible Art

• Roger Penrose 1950s Impossible Triangle Tribar Tribox

M.C. Escher (1898-1972) Waterfall

http://www.mathacademy.com/pr/minitext/escher/index.asp

Impossible Art


M.C. Escher (1898-1972) Waterfall Belvedere

http://www.mcescher.com/

Impossible Art


M.C. Escher (1898-1972) Waterfall Belvedere Cube With Ribbons

http://www.mathacademy.com/pr/minitext/escher/index.asp

Impossible Art

Escher For Real

http://www.cs.technion.ac.il/~gershon/EscherForReal/

Major Themes

• Impossible Art

• Tessellations Space Filling Tilings Metamorphosis II http://www.mcescher.com/

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

http://goldennumber.net/penrose.htm

Major Themes

• Impossible Art


Limits Circle Limit III


Major Themes

• Impossible Art


Limits Circle Limit III Circle Limit IV


Mathematicians Doing Art

• Larry Frazier Triple Bocote Blush

http://www.highlands-gallery.com/Larry_Frazier2.cfm


• Larry Frazier

http://www.highlands-gallery.com/Larry_Frazier2.cfm


• Helaman Ferguson Umbilic Torus NC

http://www.angelo.edu/dept/mathematics/gallery.htm


• Ken Leap Confluence Salter’s Lune



• Harriet Brisson Magic Cube

http://www.harrietbrisson.com

Movie Math

www.pixar.com

“Let no one who is not a mathematician read my works.”

-Leonardo da Vinci

http://www.georgehart.com/virtual-polyhedra/leonardo.html

Sources• Hofstadter, Douglas R. Godel, Escher, Bach: An Eternal Golden Braid.

Random House, New York 1979.

• Maor, Eli. To Infinity and Beyond: A Cultural History of the Infinite. Princeton University Press, New Jersey 1991.

• Livio, Mario. The Golden Ratio. Random House, New York 2002.

• Peterson, Ivars. Fragments of Infinity: A Kaleidoscope of Math and Art. John Wiley & Sons, Inc. New York 2001.


Your Moment of Zen

Art as a Mathform The Intersection of Antipodal Worlds .

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