Geometry and Art

1. Geometry and Art Origami Fractals Wycinanki Tessellations

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

3. Benoit Mandelbrot Fractals

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

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 – 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 Spiral Fractals

7. Archimedean and Hyperbolic Spirals • Archimedean • 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. Fermat’s Spiral Logarithmic Spirals • Also known as a parabolic spiral • A type of Archimedean spiral • Equiangular spiral • a rotates the spiral and b controls how tight or in what direction it is wrapped 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 4 iterations 2 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

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. 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. What is Origami’s relationship to Geometry?

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. WycinankiPolish Papercutting

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. 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. Styles of Wycinanki

27. 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. Styles of Wycinanki continued

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

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. 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. 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. http://classes.yale.edu/fractals/Welcome.html Bibliography http://www-gap.dcs.stand.ac.uk/~history/Mathematicians/Mandelbrot.html http://www.origami.com/ http://www.mcescher.com/ http://www.exploratorium.edu/complexity/CompLexicon/mandelbrot.html http://www.dsha.k12.wi.us/clarice/wyci.htm