4Background Information Born in PolandNovember 20,1924FatherBaught and sold clothesMotherdoctor2 UnclesIntroduced him to mathematicsMoved to France – 1936Taught by Szolem MandelbrotMarried Aliette KaganMoved to United States in 1958Worked for IBM
5Fractal Geometry Mandelbrot came up with the name in 1970’s Repetitive in shape but not sizeCloser you look the more there areHe showed how fractals occur in math and natureFractals – self-similar objectsThey have a fractional dimension
6Spiral Fractals Two Dimensional Spirals Spiral – a curve that turns around some central point or axis, getting closer or farther from itTwo Dimensional Spiralsr is a continuous monotonic function of θ.Archimedean SpiralHyperbolic spiralLogarithmic spiralFermat’s spiralLituus
7Archimedean and Hyperbolic Spirals r = a + bθa and b are real numbersChanging a will turn the spiral and b controls distance between armsHyperbolicTranscendental plane curveInverse of Archimedean
8Logarithmic Spirals Fermat’s Spiral Lituus Spiral Equiangular spirala rotates the spiral and b controls how tight or in what direction it is wrappedAlso known as a parabolic spiralA type of Archimedean spiralLituus SpiralAngle Is inversely proportional to the square of the radius
9Mandelbrot SetA 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 infinityCreated as an index to the Julia setsEach point in the complex plane corresponds to a different Julia setMandelbrot SetJulia Set
11Fractal Artalgorithmic approach for producing computer generated art using fractal mathematicsMovies use computer generated graphicsComputer generated imageryComputer Film CompanyIndustrial Light and MagicPIXARMachinima
13What 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.
14What 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=180For example, the Traditional Waterbomb base is a folding technique of Origami with a crease pattern that has eight congruent right triangles.
15Humiaki Huzita“In the geometry of paper-folding, a straight line becomes a crease of fold.”An Italian-Japanese mathematicianFormulated the 6 axioms of paper-folding
16There exists a single fold connecting two distinct points. (p1 and p2) This is like geometry because two points make up one line.
172. 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.
184. 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.
19For 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.
206. 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.
22BackgroundIn Poland, Folk paper cutouts were used in the 1800’s by Polish peasants to decorate their housesSheepherders 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
23Background continuedFew 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.
24Background continuedUsed by many members of a family and decorated the inside and outside of their housesHung on whitewashed walls and along wooden ceiling beams to make the house more cheeryOriginated with Polish, Ukranian, and Byelerussian peasantsIn Poland, Wycinanki can be identified just by looking at the design
25Design“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 naturebirds, 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 designdifferent colored cutouts places one on top of another
26Styles 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.
27Styles 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.
28Relation to HolidaysOriginally Easter-oriented, but later became big part of Christmas primarily in Poland.Used on furniture cupboards, cradles, shelves, and coverletsDeveloped in area north of WarsawSometimes used as ornaments for ChristmasReplace old designs with new ones during Easter and Christmas.Sometimes makes symmetrical Christmas tree shape
31Background HistoryMaurits Cornelius Escher was born on June 17th, 1898 in Leeuwarden, NetherlandsHe 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 ArtsAfter Graduation, he traveled through Italy, where he met his wife, Jetta UmikerThey lived together in Rome until 1935Escher 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. EscherFish Design(left)Circle Limit IV(right)
33Escher’s WorkOne of the world’s most famous graphic artistsMost famous for his “impossibe structures”Also created realistic piecesHe played with architecture, perspectives and impossible spacesIllustrated books, designed tapestries, stamps and murals448 Lithograpgs, Woodcuts and Wood engravings2000 Drawings and Sketches
34Escher’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.
35Ascending and Descending Impossible StructuresAscending and DescendingRelativeityMetamorphisis IMetamorphisis IIMetamorphisis IIISky and Water IReptiles
38Escher modified this to create many of his art pieces. TessellationsTessellations are created by translating, reflecting and rotating polygons in a planeEscher modified this to create many of his art pieces.Day and Night