By John Montroll
N this interesting consultant for paperfolders, origami specialist John Montroll offers basic instructions and obviously specific diagrams for growing awesome polyhedra. step by step directions convey the best way to create 34 assorted types. Grouped in line with point of hassle, the types diversity from the straightforward Triangular Diamond and the Pyramid, to the extra complicated Icosahedron and the hugely tough Dimpled Snub dice and the terrific Stella Octangula.
A problem to devotees of the traditional eastern artwork of paperfolding, those multifaceted marvels also will entice scholars and a person drawn to geometrical configurations.
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Extra resources for A Constellation of Origami Polyhedra
Let µ be the plane determined by R and p and m its corresponding line through S. Consider the plane ν joining m with R (if m passes through R, take any plane through m). Let P be the point of intersection of ν with the line p of the surface. Point P cannot coincide with R by assumption, but possibly is P = S. Now take l = RP. It is claimed that l is contained in its corresponding plane. Namely, since P is on p it is a point of the surface, hence it must be on l and its corresponding plane. The line l is contained in µ, hence its corresponding plane must contain m.
The following definitions imply that the beginning-elements of all arcs involved contain each other; the same is true of the end-elements. Definition A cone-segment consists of Definition A plane curve-segment cona plane-arc and a line-arc, both passing sists of a point-arc and a line-arc, both through one point, such that every plane lying in one plane, such that every point of the plane-arc passes through exactly of the point-arc lies in exactly one line one line of the line-arc, its line of conof the line-arc, its tangent line, and tact, and every line of the line-arc lies every line of the line-arc passes through in exactly one plane of the plane-arc, its exactly one point of the point-arc, its tangent plane.
Assume, plane ε does not pass through R, then every plane through R has in addition to R at least a range of points in common with the surface S not passing through R. 3A, then every plane has also a range of points through R in common with the surface S. 1A(2), the only case where more than two ranges of points through R belong to the surface is the case where these ranges lie in their corresponding planes through S and hence through RS, forming a pencil of lines in a plane through R. Hence the surface S contains a plane through R.