A Constellation of Origami Polyhedra by John Montroll PDF

By John Montroll

ISBN-10: 0486439585

ISBN-13: 9780486439587

N this attention-grabbing consultant for paperfolders, origami specialist John Montroll presents basic instructions and obviously particular diagrams for growing outstanding polyhedra. step by step directions express find out how to create 34 assorted types. Grouped in accordance with point of trouble, the versions diversity from the easy Triangular Diamond and the Pyramid, to the extra complicated Icosahedron and the hugely demanding Dimpled Snub dice and the remarkable Stella Octangula.

A problem to devotees of the traditional eastern paintings of paperfolding, those multifaceted marvels also will attract scholars and an individual attracted to geometrical configurations.

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Additional resources for A Constellation of Origami Polyhedra

Example text

1 C 2 2 1 3 2 2 2 3 3 2 /. 7 since at most one factor in the last expression can be non-positive for 1 ; 2 ; 3 > 0. 12. u` // for fj; k; `g D f1; 2; 3g, and let h1 , h2 , h3 denote the corresponding horocycles which we assume have pairwise distinct centers. Then there is a point equidistant to h1 , h2 , h3 if and only if 1 , 2 , 3 satisfy the strict triangle inequalities, and in this case, the equidistant point is unique. Furthermore in this case, let ˛j denote the h-length of the sector corresponding to hj and let j be the geodesic connecting the centers of hk , h` , for fj; k; `g D f1; 2; 3g.

4 3 taken with a positive sign if At the risk of proving beyond any doubt that we have too much spare time, let us remark that there is a musical instrument, the “hormonica”, based on this observation as follows. Begin with the Farey tesselation as the untuned instrument regarded as drawn before you on the computer screen. Perform a sequence of flips by serially selecting edges to produce another tesselation of the Poincaré disk likewise displayed on the computer screen. Choose some basic frequency, say middle C, to represent unity, so that any natural number may be interpreted as a multiple of this frequency.

Given two further positive real e numbers c, d , we claim that there is a unique point D xu C yv C zw 2 LC so that h ; u0 i D d 2 , h ; v 0 i D c 2 , and ; w lie on opposite sides of plane through the origin containing u0 and v 0 . x C z/; a d 2 D h ; u0 i D which give y D we find 0D bd 2 ae abc 2 d 2 C z2 abe 2 z, x D  z ac 2 be z. , 0 D xy Cxz Cyz, à  bd 2 ac 2 ac 2 C C be ae be à z zC  bd 2 ae à z zD c2d 2 e2 z2: Thus, z D ˙ cd , and we must take the minus sign to have on the correct side of the e plane through the origin containing u0 and v 0 .

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A Constellation of Origami Polyhedra by John Montroll

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