art as a mathform the intersection of antipodal worlds
TRANSCRIPT
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.
http://www.kenleap.com/
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.
http://www.kenleap.com/
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
• 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 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
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 5th century B.C.
• The Golden Ratio by Mario Livio
The Golden Ratio• Discovered by Pythagoreans in 5th century B.C.
• The Golden Ratio by Mario Livio
The Golden Ratio• Discovered by Pythagoreans in 5th century B.C.
• The Golden Ratio by Mario Livio
b
a
b
a
The Golden Ratio• Discovered by Pythagoreans in 5th century B.C.
• The Golden Ratio by Mario Livio
b
c
c
b
b
a
The Golden Ratio• Discovered by Pythagoreans in 5th century B.C.
• The Golden Ratio by Mario Livio
d
c
c
b
b
a
c
d
The Golden Ratio• Discovered by Pythagoreans in 5th century B.C.
• The Golden Ratio by Mario Livio
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
•Euclid’s Elements (300 B.C.) •The Extreme and Mean Ratio:
A BC
x 1
AC
AB
CB
AC
The Golden Ratio
segmentlarger
line whole
segmentshorter
segmentlarger
AC
AB
CB
AC
•Euclid’s Elements (300 B.C.) •The Extreme and Mean Ratio:
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
• Great Pyramid at Giza
• Parthenon
http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm
Golden Ratio in Art
• Great Pyramid at Giza
• Parthenon
http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm
Golden Ratio in Art
• Great Pyramid at Giza
• Parthenon
http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm
Golden Ratio in Art
• Great Pyramid at Giza
• Parthenon
• Leonardo da Vinci’s Saint Jerome
http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm
Golden Ratio in Art
• Great Pyramid at Giza
• Parthenon
• Leonardo da Vinci’s Saint Jerome
http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm
Golden Ratio in Art
• Great Pyramid at Giza
• Parthenon
• Leonardo da Vinci’s Saint Jerome
http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm
Golden Ratio in Art
• Great Pyramid at Giza
• Parthenon
• Leonardo da Vinci’s Saint Jerome
• Michelangelo’s Holy Family
http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm
Golden Ratio in Art
• Great Pyramid at Giza
• Parthenon
• Leonardo da Vinci’s Saint Jerome
• Michelangelo’s Holy Family
http://ccins.camosun.bc.ca/~jbritton/goldslide/jbgoldslide.htm
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
http://library.thinkquest.org/27890/applications6.html
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
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/
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
HorizonVanishing
pointVanishing
point
Putting it in Perspective
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
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
• Roger Penrose 1950s Impossible Triangle Tribar Tribox
M.C. Escher (1898-1972) Waterfall Belvedere
http://www.mcescher.com/
Impossible Art
• Roger Penrose 1950s Impossible Triangle Tribar Tribox
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/
Impossible Art
Escher For Real
http://www.cs.technion.ac.il/~gershon/EscherForReal/
Impossible Art
Escher For Real
http://www.cs.technion.ac.il/~gershon/EscherForReal/
Impossible Art
Escher For Real
http://www.cs.technion.ac.il/~gershon/EscherForReal/
Impossible Art
Escher For Real
http://www.cs.technion.ac.il/~gershon/EscherForReal/
Impossible Art
Escher For Real
http://www.cs.technion.ac.il/~gershon/EscherForReal/
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
http://www.mcescher.com/
Major Themes
• Impossible Art
• Tessellations Space Filling Tilings Metamorphosis II Metamorphosis III Penrose Tiling
http://goldennumber.net/penrose.htm
Major Themes
• Impossible Art
• Tessellations Space Filling Tilings Metamorphosis II Metamorphosis III
Limits Circle Limit III
http://www.mcescher.com/
Major Themes
• Impossible Art
• Tessellations Space Filling Tilings Metamorphosis II Metamorphosis III
Limits Circle Limit III Circle Limit IV
http://www.mcescher.com/
Mathematicians Doing Art
• Larry Frazier Triple Bocote Blush
http://www.highlands-gallery.com/Larry_Frazier2.cfm
Mathematicians Doing Art
• Larry Frazier
http://www.highlands-gallery.com/Larry_Frazier2.cfm
Mathematicians Doing Art
• Helaman Ferguson Umbilic Torus NC
http://www.angelo.edu/dept/mathematics/gallery.htm
Mathematicians Doing Art
• Ken Leap Confluence Salter’s Lune
http://www.kenleap.com/
Mathematicians Doing Art
• 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.
http://www.mcescher.com/
Your Moment of Zen