| Near-miss Johnson Solid |
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| CATEGORIES ABOUT NEAR-MISS JOHNSON SOLID | |
| polyhedra | |
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The set of ''near-misses'' is not exactly defined, but can be loosely defined as convex polyhedra that can be approximately constructed from rigid regular polygon faces as a physical model. Because of the "fuzziness" of this definition, the exact number of ''near-misses'' is not known. EXAMPLES These four convex polyhedra can be modelled physically with regular polygon faces with various degrees of adjustment. If they could be built exactly, they would be ''Johnson solids''. POSSIBLE VERTEX FIGURES The ''near-misses'', like all convex polyhedra made of regular polygons, have a Countably Infinite set of Vertex Figure s that they can use, defined by a positive Angle Defect . A secondary constraint for the ''triples'' requires the angle sum of the two smaller polygons to exceed the angle of the larger one. The set of polygons that can create convex vertex figures include:
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Permutation s of these polygon lists further extend possible vertex figures. Each vertex figure has an angle defect, and a convex polyhedron will have a combined angle defect of 720 degrees. These vertex figures and angle defect sums contrain the possible existence of convex polyhedra of regular or near regular polygon faces. See Vertex Configuration for the convex vertex figures used in the regular and semiregular solids. SEE ALSO
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