Geometry and Art

Julie Chatfield

Jen Larthey

Kristen Olsen

Lisa Trenson

Fractals Origami

Wycinanki Tessellations

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 1958Worked for IBM

Fractal Geometry• He showed how fractals

occur in math and nature

• Fractals – self-similar objects

• They have a fractional dimension

• Mandelbrot came up with the name in 1970’s

• Repetitive in shape but not size

• Closer you look the more there are

Spiral Fractals• 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

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

Logarithmic Spirals– Equiangular spiral

– a rotates the spiral and b controls how tight or in what direction it is wrapped

Fermat’s Spiral• Also known as a parabolic

spiral

• A type of Archimedean spiral

Lituus Spiral• Angle Is inversely

proportional to the square of the radius

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

Generating Fractals: Herter- Heighway Dragon

1 iteration2 iterations

3 iterations4 iterations

5 iterations 6 iterations7 iterations

8 iterations

9 iterations

10 iterations

11 iterations

18 iterations

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

Origami and Geometry

By: Julie Chatfield

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.

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.

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

1. There exists a single fold connecting two distinct points. (p1 and p2)

• This is like geometry because two points make up one line.

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.

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.

5. 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.

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.

WycinankiWycinankiPolish PapercuttingPolish Papercutting

By: Lisa TrensonBy: Lisa Trenson

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

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.

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

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

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.

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.

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

M.C. Escher and Tessellations

by Kristen Olsen

Self Portrait

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

“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)

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

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.

Impossible Structures

Ascending and DescendingRelativeity

Metamorphisis IMetamorphisis IIMetamorphisis IIISky and Water I

Reptiles

Ascending and Descending

Realistic Work

CASTROVALVA ATRANI

Tessellations

Day and Night

Tessellations are created by translating, reflecting and rotating polygons in a plane

Escher modified this to create many of his art pieces.

Bibliography

http://www.mcescher.com/

http://www-gap.dcs.stand.ac.uk/~history/Mathematicians/Mandelbrot.html

http://www.exploratorium.edu/complexity/CompLexicon/mandelbrot.html

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

http://www.origami.com/

http://www.dsha.k12.wi.us/clarice/wyci.htm