Solid Geometry Object Instruction Manual

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Solid Geometry Object Instruction Manual Swivel-Snaps® Solid Geometry Object Instruction Manual Fundamental Shapes Type Definition/Figure 2D Layout Edges all same length; same number edges at every vertex; all faces same Platonic Solids shape and size. Cube 6 12 Tetrahedron 4 6 Octahedron 8 12 Cannot be made with one 74 piece Dodecahedron Swivel-Snaps kit. Icosahedron 20 30 2 | P a g e Copyright Creative Toys LLC, all rights reserved Swivel-Snaps® Solid Geometry Object Instruction Manual Pyramids Triangular sides. Polygon base. Triangle Base For equilateral triangles, See Platonic solids Pyramid this is a tetrahedron. above Equilateral Square Base 3 1 8 Pyramid Pentagonal or 5 5 Base Pyramid (base not included) Archimedean Same as Platonic solids except two different face types. Solids Cuboctahedron 8 6 24 Rhombicub- 8 18 48 octahedron Two kits needed Snub 32 6 60 hexahedron Two kits needed Other Archimedean solids with pentagon faces cannot be made with Others the 74 piece Swivel-Snaps® kit. 3 | P a g e Copyright Creative Toys LLC, all rights reserved Swivel-Snaps® Solid Geometry Object Instruction Manual Convex All faces equilateral triangles; no adjacent faces in same plane. Deltahedra Regular tetrahedron, octahedron, These are also Platonic solids. See above. Deltahedra icosahedron Johnson Five shown below These are not Platonic solids. Deltahedra Triangular 6 9 Bipyramid Pentagonal 10 15 Bipyramid Snub 12 18 Disphenoid Triaugmented Triangular 14 21 Prism Gyroelongated Square 16 24 Bipyramid 4 | P a g e Copyright Creative Toys LLC, all rights reserved Swivel-Snaps® Solid Geometry Object Instruction Manual Composite Shapes Type Definition/Figure 3D Layout All faces equilateral triangles; two or more adjacent faces in the same plane. A Coplanar Convex few of an infinite number shown below. Deltahedra Combine fundamental shape convex deltahedra (see above) 3 tetrahedrons + 4 Triangular 16 18 Frustrum Add a tetrahedron Tetrahedron to the top of a 19 24 Augmentation triangular frustum 2 tetrahedrons + 3 square base pyramids without Triangular squares + 6 Bipyramid 26 36 Augmentation Two kits needed 6 square based pyramids + 8 Tetrakis Cuboctahedron 32 6 48 Augmentation Two kits needed 5 | P a g e Copyright Creative Toys LLC, all rights reserved Swivel-Snaps® Solid Geometry Object Instruction Manual Non-convex All faces equilateral triangles. One example of an infinite number shown below. Deltahedra 8 tetrahedrons (without bottoms) Stella Octangula 24 48 Two kits needed Other Solid Geometry Objects Try making some of the other infinite number of coplanar convex, and non-convex, deltahedra mentioned above. Try searching online for other solid geometry objects you can make with Swivel-Snaps®. 6 | P a g e Copyright Creative Toys LLC, all rights reserved .

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Uniform Panoploid Tetracombs
Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs.
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