About: Prism-Expanded Dissected Cuboctahedron

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The Prism-Expanded Dissected Cuboctahedron is a that has a convex hull, but not all faces of that hull are regular. This particular toroid was discovered by Alex Doskey, which he named the Prism-Expanded Dissected Cuboctahedron. He explains the process used to create this polyhedron (and other "Doskey Toroids") on his Prism Expansions page. The name is derived by starting with a "dissected cuboctahedron", which is the cuboctahedron dissected into 6 square-based pyramids and 8 triangle-based pyramids (tetrahedra). Alex Doskey notates this dissection as B4 ≈ 6Y4 ⊕ 8Y3. The pyramids from the dissection are then separated from each other by one unit length, then a triangular prism is inserted between each pair corresponding faces (for a total of 24 prisms inserted).

Attributes	Values
rdfs:label	Prism-Expanded Dissected Cuboctahedron
rdfs:comment	The Prism-Expanded Dissected Cuboctahedron is a that has a convex hull, but not all faces of that hull are regular. This particular toroid was discovered by Alex Doskey, which he named the Prism-Expanded Dissected Cuboctahedron. He explains the process used to create this polyhedron (and other "Doskey Toroids") on his Prism Expansions page. The name is derived by starting with a "dissected cuboctahedron", which is the cuboctahedron dissected into 6 square-based pyramids and 8 triangle-based pyramids (tetrahedra). Alex Doskey notates this dissection as B4 ≈ 6Y4 ⊕ 8Y3. The pyramids from the dissection are then separated from each other by one unit length, then a triangular prism is inserted between each pair corresponding faces (for a total of 24 prisms inserted).
Rods	168(xsd:integer)
Squares	78(xsd:integer)
dcterms:subject	Stewart Toroid Polyhedron Stereo View
PageTitle	Prism-Expanded Dissected Cuboctahedron
filename	Drilled Prism-Expanded Cuboctahedron 1 .jpg
Type	Stewart Toroid
Caption	Prism-Expanded Dissected Cuboctahedron
dbkwik:geomag/prop...iPageUsesTemplate	dbkwik:resource/GVJoJNZVdzfd8RbMw3vdAw==
Author	--02-06
Title	Prism-Expanded Dissected Cuboctahedron
Spheres	62(xsd:integer)
Triangles	8(xsd:integer)
abstract	The Prism-Expanded Dissected Cuboctahedron is a that has a convex hull, but not all faces of that hull are regular. This particular toroid was discovered by Alex Doskey, which he named the Prism-Expanded Dissected Cuboctahedron. He explains the process used to create this polyhedron (and other "Doskey Toroids") on his Prism Expansions page. The name is derived by starting with a "dissected cuboctahedron", which is the cuboctahedron dissected into 6 square-based pyramids and 8 triangle-based pyramids (tetrahedra). Alex Doskey notates this dissection as B4 ≈ 6Y4 ⊕ 8Y3. The pyramids from the dissection are then separated from each other by one unit length, then a triangular prism is inserted between each pair corresponding faces (for a total of 24 prisms inserted). For another description of this dissection, see Jim McNeill's Dissection of the Expanded Rhombic Dodecahedron page. So, to summarize, this object may be composed entirely from: * 6 square-based pyramids -- one for every square in a cuboctahedron * 8 triange-based pyramids (tetrahedra) -- one for every triangle in a cuboctahedron * 24 triangular prisms -- one for every edge in a cuboctahedron (or equivalently, for every edge in its dual, the rhombic dodecahedron, whichever way you want to think about it) One can view a mathematical construction of this object by starting with the skeleton of a rhombic dodecahedron (just the edges, not the entire solid). For every edge in that skeleton, attach the square:square edge of a triangular prism, with the prism aligned away from the skeleton. Then fill in the spaces between those prisms wtih square-based pyramids or triangle-based pyramids (tetrahedra) as appropriate. There it is! Using this composition, the numbers of rods, spheres, and panels may be quickly totalled: * 6 square-based pyramids: 4 red rods, 4 silver rods, 1 clear square, 5 spheres * 8 triangle-based pyramids: 3 blue rods, 3 silver rods, 1 clear triangle, 4 spheres * 24 triangular prisms: 3 rods, 3 squares (2 pink, 1 clear), (end triangles with spheres already counted) 6( 8 rods + 1 square + 5 spheres ) + 8( 6 rods + 1 triangle + 4 spheres ) + 24( 3 rods + 3 squares ) -------------------------------------------------------------- = 48+48+72 rods + 6+72 squares + 8 triangles + 30+32 spheres = 168 rods + 78 squares + 8 triangles + 62 spheres Equivalently, the object may be considered as: * An Expanded Rhombic Dodecahedron * Minus 12 rhombic prism holes, in which the ratio of the diagonals of each rhombus is . * Minus 1 central rhombic dodecahedron hole NOTE: Interestingly, this model is one way to use Geomags to exactly construct a model that features the usually unattainable rhombi. Of course those rhombi are present only as holes, but they're exact nonetheless.

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