By John Montroll
N this interesting consultant for paperfolders, origami specialist John Montroll offers basic instructions and obviously certain diagrams for developing awesome polyhedra. step by step directions convey the way to create 34 diversified types. Grouped in response to point of hassle, the versions diversity from the straightforward Triangular Diamond and the Pyramid, to the extra advanced Icosahedron and the hugely tough Dimpled Snub dice and the impressive Stella Octangula.
A problem to devotees of the traditional jap artwork of paperfolding, those multifaceted marvels also will attract scholars and a person drawn to geometrical configurations.
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Extra resources for A Constellation of Origami Polyhedra
1. The 1st equation in (11) is nothing but the condition that the Gaussian curvature of H13 is identically equal to −1. The 2nd and 3rd equations in (11) are respectively equivalent to the Codazzi equations hxx;y = hxy;x and hxy;y = hyy;x . We now assume that M is an oriented spacelike minimal surface in H13 . M is minimal if and only if hxx + hyy = 0, (12) September 4, 2013 36 17:10 WSPC - Proceedings Trim Size: 9in x 6in main T. Ichiyama & S. Udagawa which gives hxx hyy − (hxy )2 = − (hxx )2 + (hxy )2 .
For γ = 0 and γ = 1, Meeks’ M¨obius strip  and L´opez’ Klein bottle  satisfy deg(g) = γ + 3, respectively. But, for γ ≥ 2, no examples with deg(g) = γ + 3 are known. So, it is interesting to give a minimal surface satisfying deg(g) = γ + 3 with an antiholomorphic involution without fixed points. References 1. C. C. Chen and F. Gackstatter, Elliptische und hyperelliptische Funktionen und vollstandige Minimalflachen vom Enneperschen Typ, Math. Ann. 259 (1982), 359-369. 2. A. Costa, Examples of a Complete Minimal Immersion in of Genus One and Three Embedded Ends, Bil.
179 (1982), 337–344. 38. S. Maeda and H. Tanabe, Totally geodesic immersions of K¨ ahler manifolds and K¨ ahler Frenet curves, Math. Z. 252 (2006), 787–795. 39. S. Maeda and K. Tsukada, Isotropic immersions into a real space form, Canad. Math. Bull. 37 (1994), 245–253. 40. S. Maeda and S. Udagawa, Characterization of parallel isometric immersions of space forms into space forms in the class of isotropic immersions, Canadian J. Math. 61 (2009), 641-655. Received August 7, 2012. jp Dedicated to Professor Sadahiro Maeda on his 60th birthday We have shown  that if the projective developing map of a regular curve in the sphere is injective then the curve has no self-intersection.