By John Montroll
Step by step directions and over 1,000 transparent diagrams exhibit starting and skilled paperfolders find out how to create 27 awesome polyhedra from one sheet of paper. Graded in line with hassle, the initiatives variety from an easy dice, tetrahedron and octahedron to a tough rhombic dodecahedron, sunken icosahedron, and an antidiamond with pentagonal base.
It's easily outstanding to work out a fancy form as a dodecahedron(12 sided polyhedron) or an icosahedron (20 sided polyhedron) shape on your fingers after a large amount of twiddling and scratching your head thinking about how you can get to the following step...
Scanning caliber is sweet. All textual content (and the photographs of course!) is clear
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Additional info for A Plethora of Polyhedra in Origami
3 Analytic geometry of the ellipsoidal system 13 √ √ Since the argument of 3 A belongs to the interval (0, π/3) it follows that 3y < 3x, which implies that κ2 < κ3 < κ1 . Consequently, ρ 2 = κ1 , μ2 = κ3 , ν 2 = κ2 . 6. 47) while all three eccentricities tend to zero as ρ → ∞. The limiting value of e12 corresponds to the eccentricity of the focal ellipse, the limiting values of e23 and e31 state that the corresponding ellipses degenerate to the two axes of the focal ellipse, and the vanishing of all three eccentricities as ρ → ∞ implies that the ellipsoid deforms to a sphere at infinity.
51) ρ 2 (μ2 − ν 2 ) + μ2 (ν 2 − ρ 2 ) + ν 2 (ρ 2 − μ2 ) = 0. 52) by Au, where A and B are arbitrary constants, and add the resulting equations to obtain the identity (μ2 − ν 2 )(Aρ 2 + B)u + (ν 2 − ρ 2 )(Aμ2 + B)u + (ρ 2 − μ2 )(Aν 2 + B)u = 0. 54) where each bracket depends on a single ellipsoidal variable and involves the same two arbitrary constants A and B. 58) ∂ 2 N (ν) + (Aν 2 + B)N (ν) = 0. 61) (ν 2 − h 23 )(ν 2 − h 22 )N (ν) + ν(2ν 2 − h 23 − h 22 )N (ν) + (Aν 2 + B)N (ν) = 0. 62) are identical .
131) Therefore, the values of μ2 and ν 2 are the roots of the quadratic equation x 2 − (h 21 sin2 ϑe cos2 ϕe + h 22 cos2 ϑe + h 23 )x + h 23 h 22 cos ϑe = 0. 138) where B= h 21 x22 ρ 2 − h 23 + h 22 x12 + h 23 . 138) provide the ellipsoidal variables μ and ν in terms of the Cartesian variables x1 , x2 and the ellipsoidal variable ρ. Hence, it is a ρ-dependent transformation. 135), the transformation is dependent only on the eccentric angles. 43). The ellipto-spherical coordinate system is useful in reducing expressions from the ellipsoidal to the spherical case, as the ellipsoidal geometry deforms continuously to the spherical one.
A Plethora of Polyhedra in Origami by John Montroll