Courtesy of Erik D. Demaine, Martin L. Demaine, and Anna Lubiw. Used with permission. Demaine, Demaine, Lubiw

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1 Courtesy of Erik D. Demaine, Martin L. Demaine, and Anna Lubiw. Used with permission. Demaine, Demaine, Lubiw

2 Hyperbolic Paraboloid Courtesy of Jenna Fizel. Used with permission. [Albers at Bauhaus, ] 2

3 Circular Variation from Bauhaus [Albers at Bauhaus, ] 3

4 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. Virtual Origami Demaine, Demaine, Fizel, Ochsendorf

5 Virtual Origami Demaine, Demaine, Fizel, Ochsendorf 2006 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. 5

6 Black Hexagon Demaine, Demaine, Fizel 2006 Courtesy of Erik Demaine, Martin Demaine, and Jenna Fizel. Used with permission. 6

7 Hyparhedra: Platonic Solids [Demaine, Demaine, Lubiw 1999] 7

8 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. Virtual Origami Demaine, Demaine, Fizel, Ochsendorf

9 Computational Origami Erik & Martin Demaine MoMA, 2008 Elephant hide paper ~9 x15 x7 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also 9

10 Peel Gallery, Houston Nov Demaine & Demaine 2009 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also 10

11 Natural Cycles Erik & Martin Demaine JMM Exhibition of Mathematical Art, San Francisco, 2010 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also 11

12 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also Demaine & Demaine

13 Hyperbolic Paraboloid Courtesy of Jenna Fizel. Used with permission. [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 13

14 θ = 30 n = 16 Courtesy of Erik D. Demaine, Martin L. Demaine, Vi Hart, Gregory N. Price, and Tomohiro Tachi. Used with permission. [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 14

15 [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 15

16 Courtesy of Erik D. Demaine, Martin L. Demaine, Vi Hart, Gregory N. Price, and Tomohiro Tachi. Used with permission. [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 16

17 θ = 8 n = 16 Courtesy of Erik D. Demaine, Martin L. Demaine, Vi Hart, Gregory N. Price, and Tomohiro Tachi. Used with permission. [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 17

18 Courtesy of Erik D. Demaine, Martin L. Demaine, Vi Hart, Gregory N. Price, and Tomohiro Tachi. Used with permission. θ = n θ = n θ = n [Demaine, Demaine, Hart, Price, Tachi 2009/2010] Courtesy of Erik D. Demaine, Martin L. Demaine, Vi Hart, Gregory N. Price, and Tomohiro Tachi. Used with permission. 18

19 actual parabolic fits [Demaine, Demaine, Hart, Price, Tachi 2009/2010] Courtesy of Erik D. Demaine, Martin L. Demaine, Vi Hart, Gregory N. Price, and Tomohiro Tachi. Used with permission. 19

20 Image removed due to copyright restrictions. Refer to: 20

21 Image removed due to copyright restrictions. Refer to: and 21

22 [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 22

23 [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 23

24 [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 24

25 Photo courtesy of gin_able on Flickr. Used with permission. Under CC-BY-NC. 25

26 Photo courtesy of themacinator on Flickr. Used with permission. Under CC-BY. 26

27 Circular Variation from Bauhaus [Albers at Bauhaus, ] 27

28 Huffman Family (May 2010) Linda Marilyn Elise 28

29 Tessellations circles Courtesy of Erik D. Demaine, Martin L. Demaine, and Duks Koschitz. Used with permission. design by David Huffman; recreated folding by Demaine, Demaine, Koschitz

30 Tessellations Arches parabolas & lines Courtesy of Erik D. Demaine, Martin L. Demaine, and Duks Koschitz. Used with permission. design by David Huffman; recreated folding by Demaine, Demaine, Koschitz

31 Cone reflected 7 times circles & nonconics Courtesy of Erik D. Demaine, Martin L. Demaine, and Duks Koschitz. Used with permission. design by David Huffman; virtual model by Demaine, Demaine, Koschitz

32 Hexagonal column with cusps (two variations) circles & lines design by David Huffman; recreated crease pattern Courtesy of Erik D. Demaine, Martin L. Demaine, and Duks Koschitz. Used with permission. design by David Huffman; recreated folding by Demaine, Demaine, Koschitz

33 Hexagonal column with cusps circles & lines Courtesy of Erik D. Demaine, Martin L. Demaine, and Duks Koschitz. Used with permission. design by David Huffman; recreated folding by Demaine, Demaine, Koschitz

34 4-lobed cloverleaf ellipses & lines Courtesy of Erik D. Demaine, Martin L. Demaine, and Duks Koschitz. Used with permission. design by David Huffman; design recreated by David folding Huffman; by Demaine, recreated Demaine, crease Koschitz pattern

35 4-lobed cloverleaf ellipses & lines Courtesy of Erik D. Demaine, Martin L. Demaine, and Duks Koschitz. Used with permission. design by David Huffman; recreated folding by Demaine, Demaine, Koschitz

36 One column parabolas & lines Courtesy of Erik D. Demaine, Martin L. Demaine, and Duks Koschitz. Used with permission. design by David Huffman; recreated folding by Demaine, Demaine, Koschitz

37 One column parabolas & lines design Courtesy by David of Erik D. Huffman; Demaine, Martin L. Demaine, and Duks Koschitz. Used with permission. design by David recreated Huffman; folding recreated folding by Demaine, Demaine, Koschitz

38 Huffman Family (May 2010) Linda Marilyn Elise 38

39 MIT OpenCourseWare Geometric Folding Algorithms: Linkages, Origami, Polyhedra Fall 2012 For information about citing these materials or our Terms of Use, visit: