mathematics and art: making beautiful music together d.n. seppala-holtzman st. joseph’s college
TRANSCRIPT
![Page 1: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/1.jpg)
Mathematics and Art:Making Beautiful Music Together
D.N. Seppala-HoltzmanSt. Joseph’s College
![Page 2: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/2.jpg)
Math & Art: the Connection Many people think that
mathematics and art are poles apart, the first cold and precise, the second emotional and imprecisely defined. In fact, the two come together more as a collaboration than as a collision.
![Page 3: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/3.jpg)
Math & Art: Common Themes Proportions Patterns Perspective Projections Impossible Objects Infinity and Limits
![Page 4: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/4.jpg)
The Divine Proportion The Divine Proportion, better known
as the Golden Ratio, is usually denoted by the Greek letter Phi: .
is defined to be the ratio obtained by dividing a line segment into two unequal pieces such that the entire segment is to the longer piece as the longer piece is to the shorter.
![Page 5: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/5.jpg)
A Line Segment in Golden Ratio
![Page 6: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/6.jpg)
: The Quadratic Equation The definition of leads to the
following equation, if the line is divided into segments of lengths a and b:a b a
a b
![Page 7: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/7.jpg)
The Golden Quadratic II Cross multiplication yields:
2 2a ab b
![Page 8: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/8.jpg)
The Golden Quadratic III Setting equal to the quotient a/b
and manipulating this equation shows that satisfies the quadratic equation:
2 1 0
![Page 9: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/9.jpg)
The Golden Quadratic IV Applying the quadratic formula to
this simple equation and taking to be the positive solution yields:
1 51.618
2
![Page 10: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/10.jpg)
Properties of is irrational Its reciprocal, 1/ , is one less than
Its square, 2, is one more than
![Page 11: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/11.jpg)
Is an Infinite Square Root
1 1 1 1 .....
![Page 12: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/12.jpg)
Φ is an Infinite Continued Fraction
11
11
11
11 ...
1
![Page 13: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/13.jpg)
Constructing Begin with a 2 by 2 square.
Connect the midpoint of one side of the square to a corner. Rotate this line segment until it provides an extension of the side of the square which was bisected. The result is called a Golden Rectangle. The ratio of its width to its height is .
![Page 14: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/14.jpg)
Constructing
A
B
C
AB=AC
![Page 15: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/15.jpg)
Properties of a Golden Rectangle If one chops off the largest possible
square from a Golden Rectangle, one gets a smaller Golden Rectangle.
If one constructs a square on the longer side of a Golden Rectangle, one gets a larger Golden Rectangle.
Both constructions can go on forever.
![Page 16: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/16.jpg)
The Golden Spiral In this infinite process of chopping
off squares to get smaller and smaller Golden Rectangles, if one were to connect alternate, non-adjacent vertices of the squares, one gets a Golden Spiral.
![Page 17: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/17.jpg)
The Golden Spiral
![Page 18: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/18.jpg)
The Golden Spiral II
![Page 19: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/19.jpg)
The Golden Triangle An isosceles triangle with two base
angles of 72 degrees and an apex angle of 36 degrees is called a Golden Triangle.
The ratio of the legs to the base is .
The regular pentagon with its diagonals is simply filled with golden ratios and triangles.
![Page 20: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/20.jpg)
The Golden Triangle
![Page 21: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/21.jpg)
A Close Relative:Ratio of Sides to Base is 1 to Φ
![Page 22: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/22.jpg)
Golden Spirals From Triangles As with the Golden Rectangle,
Golden Triangles can be cut to produce an infinite, nested set of Golden Triangles.
One does this by repeatedly bisecting one of the base angles.
Also, as in the case of the Golden Rectangle, a Golden Spiral results.
![Page 23: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/23.jpg)
Chopping Golden Triangles
![Page 24: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/24.jpg)
Spirals from Triangles
![Page 25: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/25.jpg)
In Nature There are physical reasons that
and all things golden frequently appear in nature.
Golden Spirals are common in many plants and a few animals, as well.
![Page 26: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/26.jpg)
Sunflowers
![Page 27: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/27.jpg)
Pinecones
![Page 28: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/28.jpg)
Pineapples
![Page 29: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/29.jpg)
The Chambered Nautilus
![Page 30: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/30.jpg)
Angel Fish
![Page 31: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/31.jpg)
Tiger
![Page 32: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/32.jpg)
Human Face I
![Page 33: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/33.jpg)
Human Face II
![Page 34: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/34.jpg)
Le Corbusier’s Man
![Page 35: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/35.jpg)
A Golden Solar System?
![Page 36: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/36.jpg)
In Art & Architecture For centuries, people seem to have
found to have a natural, nearly universal, aesthetic appeal.
Indeed, it has had near religious significance to some.
Occurrences of abound in art and architecture throughout the ages.
![Page 37: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/37.jpg)
The Pyramids of Giza
![Page 38: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/38.jpg)
The Pyramids and
![Page 39: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/39.jpg)
The Pyramids were laid out in a Golden Spiral
![Page 40: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/40.jpg)
The Parthenon
![Page 41: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/41.jpg)
The Parthenon II
![Page 42: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/42.jpg)
The Parthenon III
![Page 43: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/43.jpg)
Cathedral of Chartres
![Page 44: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/44.jpg)
Cathedral of Notre Dame
![Page 45: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/45.jpg)
Michelangelo’s David
![Page 46: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/46.jpg)
Michelangelo’s Holy Family
![Page 47: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/47.jpg)
Rafael’s The Crucifixion
![Page 48: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/48.jpg)
Da Vinci’s Mona Lisa
![Page 49: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/49.jpg)
Mona Lisa II
![Page 50: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/50.jpg)
Da Vinci’s Study of Facial Proportions
![Page 51: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/51.jpg)
Da Vinci’s St. Jerome
![Page 52: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/52.jpg)
Da Vinci’s The Annunciation
![Page 53: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/53.jpg)
Da Vinci’s Study of Human Proportions
![Page 54: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/54.jpg)
Rembrandt’s Self Portrait
![Page 55: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/55.jpg)
Seurat’s Parade
![Page 56: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/56.jpg)
Seurat’s Bathers
![Page 57: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/57.jpg)
Turner’s Norham Castle at Sunrise
![Page 58: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/58.jpg)
Mondriaan’s Broadway Boogie-Woogie
![Page 59: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/59.jpg)
Hopper’s Early Sunday Morning
![Page 60: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/60.jpg)
Dali’s The Sacrament of the Last Supper
![Page 61: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/61.jpg)
Literally an (Almost) Golden Rectangle
![Page 62: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/62.jpg)
Patterns Another subject common to art
and mathematics is patterns. These usually take the form of a
tiling or tessellation of the plane. Many artists have been fascinated
by tilings, perhaps none more than M.C. Escher.
![Page 63: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/63.jpg)
Patterns & Other Mathematical Objects In addition to tilings, other
mathematical connections with art include fractals, infinity and impossible objects.
Real fractals are infinitely self-similar objects with a fractional dimension.
Quasi-fractals approximate real ones.
![Page 64: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/64.jpg)
Fractals Some art is actually created by
mathematics. Fractals and related objects are
infinitely complex pictures created by mathematical formulae.
![Page 65: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/65.jpg)
The Koch Snowflake (real fractal)
![Page 66: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/66.jpg)
The Mandelbrot Set (Quasi)
![Page 67: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/67.jpg)
Blow-up 1
![Page 68: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/68.jpg)
Blow-up 2
![Page 69: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/69.jpg)
Blow-up 3
![Page 70: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/70.jpg)
Blow-up 4
![Page 71: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/71.jpg)
Blow-up 5
![Page 72: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/72.jpg)
Blow-up 6
![Page 73: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/73.jpg)
Blow-up 7
![Page 74: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/74.jpg)
Fractals Occur in Nature (the coastline)
![Page 75: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/75.jpg)
Another Quasi-Fractal
![Page 76: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/76.jpg)
Yet Another Quasi-Fractal
![Page 77: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/77.jpg)
And Another Quasi-Fractal
![Page 78: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/78.jpg)
Tessellations There are many ways to tile the
plane. One can use identical tiles, each
being a regular polygon: triangles, squares and hexagons.
Regular tilings beget new ones by making identical substitutions on corresponding edges.
![Page 79: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/79.jpg)
Regular Tilings
![Page 80: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/80.jpg)
New Tiling From Old
![Page 81: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/81.jpg)
Maurits Cornelis Escher (1898-1972) Escher is nearly every mathematician’s
favorite artist. Although, he himself, knew very little
formal mathematics, he seemed fascinated by many of the same things which traditionally interest mathematicians: tilings, geometry,impossible objects and infinity.
Indeed, several famous mathematicians have sought him out.
![Page 82: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/82.jpg)
M.C. Escher A visit to the Alhambra in Granada
(Spain) in 1922 made a major impression on the young Escher.
He found the tilings fascinating.
![Page 83: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/83.jpg)
The Alhambra
![Page 84: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/84.jpg)
An Escher Tiling
![Page 85: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/85.jpg)
Escher’s Butterflies
![Page 86: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/86.jpg)
Escher’s Lizards
![Page 87: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/87.jpg)
Escher’s Sky & Water
![Page 88: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/88.jpg)
M.C. Escher Escher produced many, many
different types of tilings. He was also fascinated by
impossible objects, self reference and infinity.
![Page 89: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/89.jpg)
Escher’s Hands
![Page 90: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/90.jpg)
Escher’s Circle Limit
![Page 91: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/91.jpg)
Escher’s Waterfall
![Page 92: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/92.jpg)
Escher’s Ascending & Descending
![Page 93: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/93.jpg)
Escher’s Belvedere
![Page 94: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/94.jpg)
Escher’s Impossible Box
![Page 95: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/95.jpg)
Penrose’s Impossible Triangle
![Page 96: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/96.jpg)
Roger Penrose Roger Penrose is a mathematical
physicist at Oxford University. His interests are many and they
include cosmology (he is an expert on black holes), mathematics and the nature of comprehension.
He is the author of The Emperor’s New Mind.
![Page 97: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/97.jpg)
Penrose Tiles In 1974, Penrose solved a difficult
outstanding problem in mathematics that had to do with producing tilings of the plane that had 5-fold symmetry and were non-periodic.
There are two roughly equivalent forms: the kite and dart model and the dual rhombus model.
![Page 98: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/98.jpg)
Dual Rhombus Model
![Page 99: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/99.jpg)
Kite and Dart Model
![Page 100: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/100.jpg)
Kites & Darts II
![Page 101: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/101.jpg)
Kites & Darts III
Kite Dart
72 72
72
144
36 36
72
216
![Page 102: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/102.jpg)
Kite & Dart Tilings
![Page 103: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/103.jpg)
Rhombus Tiling
![Page 104: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/104.jpg)
Rhombus Tiling II
![Page 105: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/105.jpg)
Rhombus Tiling III
![Page 106: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/106.jpg)
Penrose Tilings There are infinitely many ways to
tile the plane with kites and darts. None of these are periodic. Every finite region in any kite-dart
tiling sits somewhere inside every other infinite tiling.
In every kite-dart tiling of the plane, the ratio of kites to darts is .
![Page 107: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/107.jpg)
Luca Pacioli (1445-1514) Pacioli was a Franciscan monk and
a mathematician. He published De Divina
Proportione in which he called Φ the Divine Proportion.
Pacioli: “Without mathematics, there is no art.”
![Page 108: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/108.jpg)
Jacopo de Barbari’s Pacioli
![Page 109: Mathematics and Art: Making Beautiful Music Together D.N. Seppala-Holtzman St. Joseph’s College](https://reader034.vdocuments.us/reader034/viewer/2022051218/5697bfa81a28abf838c99549/html5/thumbnails/109.jpg)
In Conclusion Although one might argue that
Pacioli somewhat overstated his case when he said that “without mathematics, there is no art,” it should, nevertheless, be quite clear that art and mathematics are intimately intertwined.