Download A Constellation of Origami Polyhedra by John Montroll PDF

By John Montroll

ISBN-10: 0486439585

ISBN-13: 9780486439587

N this attention-grabbing advisor for paperfolders, origami professional John Montroll offers basic instructions and obviously precise diagrams for growing outstanding polyhedra. step by step directions exhibit the right way to create 34 diversified types. Grouped in accordance with point of trouble, the versions variety from the straightforward Triangular Diamond and the Pyramid, to the extra complicated Icosahedron and the hugely difficult Dimpled Snub dice and the marvelous Stella Octangula.

A problem to devotees of the traditional eastern artwork of paperfolding, those multifaceted marvels also will entice scholars and someone drawn to geometrical configurations.

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I , dxi(dk) = Ajl ... i , ' The evaluation of a tensor field on one-forms and vector fields can be described in terms of coordinates by writing everything out in components. For example, suppose A is a (1, 2 ) tensor. Write 0 = Ok dxk for an arbitrary one-form and X = X i d i , Y = Y j d j for arbitrary vector fields. jOkXiyj. i, j , k i, j , k For a fixed coordinate system, the components of a sum of tensors are just the sum of the components. ) The components of a tensor product are given by ( A 8 B ) ~.

Ar,X I , . . , x s ) = A ( @ , . . ,@, x , ,. . ,X,)(p), where 8', . . , O r are any one-forms on M such that Oilp= ui (1 I i Ir) and X , , . . ,X , are any vector fields such that X i, 1 = xi (1 5 j I s). It is easy to check that the function A , is R-multilinear; then by Definition 1 it is an (r, s) tensor over T,(M). We can thus consider A E 2L(M) as afield smoothly assigning to each p E M the tensor A , . Just as a vector field is a smooth section of the tangent bundle T M , such a field p + A , is a smooth section of the (r, s ) tensor bundle-the latter obtained, roughly speaking, by replacing each T,(M) in T M by the space T,(M): of (r, s) tensors over T,(M).

Corollary. If a, 8:I -+ M are integral curves of V such that a(a) = P(a) for some a E I , then a = 8. 50. Proof By continuity the agreement set A = { t E I : a(t) = P(t)} is closed. If A is also open, then since A is nonempty, A = I . Fix t E A. Then s -+ a(t + s) and s + p(t + s) are integral curves of V that agree at s = 0. W Hence by Proposition 49 they agree for s sufficiently near 0. Consider the collection of all integral curves a : I , -+ M of V that start at is, for which a(0) = p . For any two such, the corollary shows that a = on I , n I , .

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A Constellation of Origami Polyhedra by John Montroll


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