1. Math and Art

2. outline

3. Math in art Math can commonly be seen in art through the use of: Patterns Fractals Tesselations John Biggers, Four Seasons Jock Cooper, Fractal Art

4. Artist: Leonardo da vinci Born April 15, 1452 Died May 2, 1519 Educated by Verrocchio The original Renaissance man Made significant contributions to: Art Math Science Engineering

5. Mathematics in Da vinci’s art? "In the Renaissance, you don't find others doing paintings with geometric underlayings to them, whether it's [da Vinci’s] unerring eye... or whether it was purposeful, we'll never know.“ – Bülent Atalay

6. The Golden Rectangle & triangle The golden ratio: 1: 1.618 (approximate) A golden rectangle is one whose side lengths are in the golden ratio. A golden triangle is an isosceles triangle in which the two longer sides have equal lengths and in which the ratio of this length to the third, smaller side is the golden ratio.

7. Artist: Helaman fergus0n Born: August 11, 1940 Most of his works are sculpture. Was raised by a distant cousin and her husband in Palmyra, New York. His 9 th grade math teacher recognized Ferguson’s talent for both math and art. Assigned him projects such as creating a wire- frame model of a hyperbolic paraboloid. Was a math professor at BYU from 1972-1988

8. Ferguson’s math-sculpture Ferguson is very purposeful about creating mathematical art. Most of his sculptures are related to topology. Torus with Cross-Cap and Vector Field Whaledream II “I find that sculpture is a very powerful way to convey mathematics, and mathematics is a very powerful design language for sculpture.” -Helaman Ferguson

9. Topology

10. Double Torus Stonehenge

11. Four Canoes Each ring represents a Klein surface.Klein surface Only has one side Can be represented as a “bottle” or “figure-eight” The tiled floor that Four Canoes rests on is a 2-D way to visualize a Klein surface. Whether the Klein tessellation can be extended to infinity has not yet been proved.

12. Artist: Peter Wang

13. How he creates double walled bowls

14. Phyllotaxis Spiral In botany, phyllotaxis is the arrangement of leaves on a stem or axis. Spiral is just one type of phyllotaxis. It can be represented in polar coordinates by the following: r = c √n, Θ = n ∙ 137.5º Θ is the angle, r is the radius n is the index number of the floret c is a constant scaling factor It is a form of Fermat’s Spiral. The number of left and right spirals are successive Fibonacci numbers.

15. The Golden Angle The golden angle is the smaller of the two angles that are created by dividing the circumference of a circle into two arcs where the lengths are in proportion equal to the golden ratio. It is equal to approximately 137.5º or 2.399 radians

16. Discussion