1. Julie Chatfield Jen Larthey Kristen Olsen Lisa Trenson
Geometry and Art
Fractals
Origami
Julie Chatfield
Jen Larthey
Kristen Olsen
Lisa Trenson
Wycinanki
Tessellations

2. Geometry and Art....
Fractals
Origami
Wycinanki
Tessellations

3. Benoit Mandelbrot
Fractals
By: Jen Larthey

4. Background Information
Born in Poland
November 20,1924
Father
Baught and sold clothes
Mother
doctor
2 Uncles
Introduced him to mathematics
Moved to France – 1936
Taught by Szolem Mandelbrot
Married Aliette Kagan
Moved to United States in 1958
Worked for IBM

5. Fractal Geometry Mandelbrot came up with the name in 1970’s
Repetitive in shape but not size
Closer you look the more there are
He showed how fractals occur in math and nature
Fractals – self-similar objects
They have a fractional dimension

6. Spiral Fractals Two Dimensional Spirals
Spiral – a curve that turns around some central point or axis, getting closer or farther from it
Two Dimensional Spirals
r is a continuous monotonic function of θ.
Archimedean Spiral
Hyperbolic spiral
Logarithmic spiral
Fermat’s spiral
Lituus

7. Archimedean and Hyperbolic Spirals
r = a + bθ
a and b are real numbers
Changing a will turn the spiral and b controls distance between arms
Hyperbolic
Transcendental plane curve
Inverse of Archimedean

8. Logarithmic Spirals Fermat’s Spiral Lituus Spiral
Equiangular spiral
a rotates the spiral and b controls how tight or in what direction it is wrapped
Also known as a parabolic spiral
A type of Archimedean spiral
Lituus Spiral
Angle Is inversely proportional to the square of the radius

9. Mandelbrot Set
A fractal that is defined as the set of points c in the complex number plane for which the iteratively defined sequence zn+1 = zn^2 + c with z^0 = 0 does not tend to infinity
Created as an index to the Julia sets
Each point in the complex plane corresponds to a different Julia set
Mandelbrot Set
Julia Set

10. Generating Fractals: Herter- Heighway Dragon
3 iterations
1 iteration
2 iterations
4 iterations
6 iterations
5 iterations
7 iterations
8 iterations
18 iterations
10 iterations
9 iterations
11 iterations

11. Fractal Art
algorithmic approach for producing computer generated art using fractal mathematics
Movies use computer generated graphics
Computer generated imagery
Computer Film Company
Industrial Light and Magic
PIXAR
Machinima

12. Origami and Geometry
By: Julie Chatfield

13. What is Origami?
Origami is a form of visual/sculptural representation that is defined primarily by the folding of the medium (usually paper).
Literally, “oru” means fold and “kami” means paper.

14. What is Origami’s relationship to Geometry?
Kawasaki’s Thereom:
This thereom states if you add up the angle measurements of every other angle around a point, the sum will be 180 degrees.
A1 + A3 +A5… +A2n-1=180
For example, the Traditional Waterbomb base is a folding technique of Origami with a crease pattern that has eight congruent right triangles.

15. Humiaki Huzita
“In the geometry of paper-folding, a straight line becomes a crease of fold.”
An Italian-Japanese mathematician
Formulated the 6 axioms of paper-folding

16. There exists a single fold connecting two distinct points. (p1 and p2)
This is like geometry because two points make up one line.

17. 2. Given two points, P1 and P2, there exists a unique fold that maps P1 onto P2.
3. Given two creases, L1 and L2, there exists a unique fold that maps L1 onto L2.
This relates to a perpendicular bisector in geometry.
This relates to an angle bisector in geometry.

18. 4. Given a point P and a crease L, there exists a unique fold through P perpendicular to L.
This is similar to the patty paper constructions we used to create the midpoint of a segment.

19. For given points P1 and P2 and a crease L, there exists a fold that passes through P1 and maps P2 onto L.
This is similar to finding the center of an angle in geometry.

20. 6. Given two points, P1 and P2, and two creases, L1 and L2, there exists a unique fold that maps P1 into L1 and P2 into L2.

21. Wycinanki Polish Papercutting By: Lisa Trenson

22. Background
In Poland, Folk paper cutouts were used in the 1800’s by Polish peasants to decorate their houses
Sheepherders cut designs out of bark and leather in bad weather. Paper was used more once it became widely available.
Tapestries and painted decorations seen in homes of affluence allowed inspiration which translated into paper cuts used in peasant cottages

23. Background continued
Few farm houses had glass windows. Peasant farmers hung sheep skins over the window openings to keep out elements. Took sheep shears and snipped small openings in the skins to let some light and air in which were eventually recognized as decorative along with functional.

24. Background continued
Used by many members of a family and decorated the inside and outside of their houses
Hung on whitewashed walls and along wooden ceiling beams to make the house more cheery
Originated with Polish, Ukranian, and Byelerussian peasants
In Poland, Wycinanki can be identified just by looking at the design

25. Design
“Wycinanki” pronounced Vee-chee-non-kee is the polish word for ‘paper-cut design’
Intricate designs cut with scissors. Complexity of the designs created by repeating symmetrical patterns and folk motifs inspired by nature
birds, cocks, trees, flowers, small animals, etc.
Symmetrical cutouts with nature designs and geometric shapes (a lot of roosters)
Layered sometimes to make a more intricate design
different colored cutouts places one on top of another

26. Styles of Wycinanki Kurpie Cut:
symmetrical design cut from a single piece of colored paper folded one time. Spruce trees and birds are the most popular motifs. Arranged randomly on walls instead of wallpaper.
Lowitz:
Many layers of brightly colored paper cut and arranged. Express themes or tell stories of village activities. Colors blended visually to give richness and dimension. Displayed tandem style over windows, doorways, and on main walls of one story rural houses.

27. Styles of Wycinanki continued
Gwiazdy:
Circular medallion which includes doily type designs as well as the bird and flower paper cuts that have a symmetrical center axis.
Riband:
Center medallion with serrated edges sometimes from which two streamers dangle at a slight angle. Color overlays for wall decoration. One of earliest forms.

28. Relation to Holidays
Originally Easter-oriented, but later became big part of Christmas primarily in Poland.
Used on furniture cupboards, cradles, shelves, and coverlets
Developed in area north of Warsaw
Sometimes used as ornaments for Christmas
Replace old designs with new ones during Easter and Christmas.
Sometimes makes symmetrical Christmas tree shape

29. M.C. Escher and Tessellations
by Kristen Olsen

30. Self Portrait

31. Background History
Maurits Cornelius Escher was born on June 17th, 1898 in Leeuwarden, Netherlands
He was the youngest of four, and lived with his mother and father.
After he got through school, he went to the School for Architecture and Decorative Arts
After Graduation, he traveled through Italy, where he met his wife, Jetta Umiker
They lived together in Rome until 1935
Escher took a yearly visit to Italy to get inspirations for his work

32. “At high school in Arnhem, I was extremely poor at arithmetic and algebra because I had, and still have, great difficulty with the abstractions of numbers and letters. When, later, in stereometry [solid geometry], an appeal was made to my imagination, it went a bit better, but in school I never excelled in that subject. But our path through life can take strange turns.”
M.C. Escher
Fish Design
(left)
Circle Limit IV
(right)

33. Escher’s Work
One of the world’s most famous graphic artists
Most famous for his “impossibe structures”
Also created realistic pieces
He played with architecture, perspectives and impossible spaces
Illustrated books, designed tapestries, stamps and murals
448 Lithograpgs, Woodcuts and Wood engravings
2000 Drawings and Sketches

34. Escher’s first work featuring division of the plane, Eight Heads
His final work, a woodcut titled Snakes, took him 6 months to create, and it was unveiled in 1969.

35. Ascending and Descending
Impossible Structures
Ascending and Descending
Relativeity
Metamorphisis I
Metamorphisis II
Metamorphisis III
Sky and Water I
Reptiles

36. Ascending and Descending

37. Realistic Work
CASTROVALVA
ATRANI

38. Escher modified this to create many of his art pieces.
Tessellations
Tessellations are created by translating, reflecting and rotating polygons in a plane
Escher modified this to create many of his art pieces.
Day and Night

39. Bibliography http://classes.yale.edu/fractals/Welcome.html