870 likes | 885 Views
Delve into the creation process of Keizo Ushio's Oushi Zokei, a stunning art piece combining mathematical concepts with artistic design. Follow the intricate journey from inception to installation, exploring the fusion of geometry and fine details. Discover the innovative techniques used to craft geometric masterpieces and explore the theoretical implications of twisted toroids and Möbius bands in art. Engage in hands-on activities like splitting Möbius strips and double Möbius strips, offering a unique perspective on the intersection of mathematics and art.
E N D
DeYoung Museum, June12, 2013 MATHEMATICAL TREASURE HUNTS Tracking Twisted Toroids Carlo H. Séquin University of California, Berkeley
What came first: Art or Mathematics ? • Question posed Nov. 16, 2006 by Dr. Ivan Sutherland“father” of computer graphics (SKETCHPAD, 1963).
Early “Free-Form” Art Cave paintings, Lascaux Venus von Willendorf
Regular, Geometric Art • Early art: Patterns on bones, pots, weavings... • Mathematics (geometry) to help make things fit:
Geometry ! • Descriptive Geometry – love since high school
40 Years of Geometry and Design CCD TV Camera Soda Hall (for CS) RISC 1 Computer Chip Octa-Gear (Cyberbuild)
ISAMA, San Sebastian 1999 Keizo Ushio and his “OUSHI ZOKEI”
The Making of “Oushi Zokei” (1) Fukusima, March’04 Transport, April’04
The Making of “Oushi Zokei” (2) Keizo’s studio, 04-16-04 Work starts, 04-30-04
The Making of “Oushi Zokei” (3) Drilling starts, 05-06-04 A cylinder, 05-07-04
The Making of “Oushi Zokei” (4) Shaping the torus with a water jet, May 2004
The Making of “Oushi Zokei” (5) A smooth torus, June 2004
The Making of “Oushi Zokei” (6) Drilling holes on spiral path, August 2004
The Making of “Oushi Zokei” (7) Drilling completed, August 30, 2004
The Making of “Oushi Zokei” (8) Rearranging the two parts, September 17, 2004
The Making of “Oushi Zokei” (9) Installation on foundation rock, October 2004
The Making of “Oushi Zokei” (10) Transportation, November 8, 2004
The Making of “Oushi Zokei” (11) Installation in Ono City, November 8, 2004
The Making of “Oushi Zokei” (12) Intriguing geometry – fine details !
Schematic Model of 2-Link Torus • Knife blades rotate through 360 degreesas it sweep once around the torus ring. 360°
From George Hart’s web page:http://www.georgehart.com/bagel/bagel.html . . . and Adding Cream Cheese
Schematic Model of 2-Link Torus • 2 knife blades rotate through 360 degreesas they sweep once around the torus ring. 360°
Generalize this to 3-Link Torus • Use a 3-blade“knife” 360°
Generalization to 4-Link Torus • Use a 4-blade knife, square cross section
Generalization to 6-Link Torus 6 triangles forming a hexagonal cross section
Keizo Ushio’s Multi-Loop Cuts • There is a second parameter: • If we change twist angle of the cutting knife, torus may not get split into separate rings! 180° 360°540°
Cutting with a Multi-Blade Knife • Use a knife with b blades, • Twist knife through t * 360° / b. b = 2, t = 1; b = 3, t = 1; b = 3, t = 2.
Cutting with a Multi-Blade Knife ... • results in a(t, b)-torus link; • each component is a (t/g, b/g)-torus knot, • where g = GCD (t, b). b = 4, t = 2 two double loops.
ART: Focus on thecutting space !Use “thick knife”. “Moebius Space” (Séquin, 2000)
It is a Möbius Band ! • A closed ribbon with a 180° flip; • A single-sided surface with a single edge:
Changing Shapes of a Möbius Band • Using a “magic” surface material that can pass through itself. +180°(ccw), 0°, –180°, –540°(cw) Apparent twist (compared to a rotation-minimizing frame) Regular Homotopies
Twisted Möbius Bands in Art Web Max Bill M.C. Escher M.C. Escher
Splitting Other Stuff What if we started with something more intricate than a torus ?. . . and then split that shape . . .
Splitting Möbius Bands (not just tori) Keizo Ushio 1990
Splitting Möbius Bands M.C.Escher FDM-model, thin FDM-model, thick
Splitting a Band with a Twist of 540°by Keizo Ushio (1994) Bondi, 2001
Another Way to Split the Möbius Band Metal band available from Valett Design: conrad@valett.de
SOME HANDS-ON ACTIVITIES • Splitting Möbius Strips • Double-layer Möbius Strips • Escher’s Split Möbius Band
Activity #1: Möbius Strips For people who have not previously played with physical Möbius strips. • Take an 11” long white paper strip; bend it into a loop; • But before joining the end, flip one end an odd number of times: +/– 180°or 540°; • Compare results among students:How many different bands do you find? • Take a marker pen and draw a line ¼” offfrom one of the edges . . .Continue the line until it closes (What happens?) • Cut the strip lengthwise down the middle . . .(What happens? -- Discuss with neighbors!)
Activity #2: Double Möbius Strips • Take TWO 11” long, 2-color paper strips;put them on top of each other so touching colors match;bend sandwich into a loop; join after 1 or 3 flips( tape the two layers individually! ). • Convince yourself that strips are separate by passing a pencil or small paper piece around the whole loop. • Separate (open-up) the two loops. • Put the configuration back together.