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Topic: Truncated octahedron

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Truncated cube

Truncated cuboctahedron

Truncated tetrahedron

Cuboctahedron

Octahedron

Andreini tessellation

Tetrahedron

Archimedean solid

Platonic solid

Hexagon

Crystal

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	[No title] (Site not responding. Last check: 2007-10-14)
		octahedron +------------------------------------------------------------ The octahedron is a polyhedron with 6 vertices and 8 faces.
		truncated tetrahedron +------------------------------------------------------------ The truncated tetrahedron is a polyhedron with 12 vertices and 8 faces.
		truncated octahedron +------------------------------------------------------------ The truncated octahedron is a polyhedron with 24 vertices and 14 faces.
	www.math.harvard.edu /~knill/sofia/data/polyhedra.txt (2272 words)

	Index: Platonic and Archimedean Solids (69-79)
		The five regular convex polyhedra, or Platonic solids, are the tetrahedron, cube, octahedron, dodecahedron, icosahedron (75 - 79), with 4, 6, 8, 12, and 20 faces, respectively.
		A polyhedron and its dual have the same number of edges (12 for a cube and an octahedron, but the numbers of vertices and faces are interchanged).
		Thus we obtain the truncated cube (69), the truncated tetrahedron (70), the truncated octahedron (74).
	math.arizona.edu /~models/Platonic_and_Archimedean_Solids (376 words)

	Truncated Octahedron (Site not responding. Last check: 2007-10-14)
		A truncated octahedron contains six squares and eight hexagons.
		In this drawing, each nucleotide is shown with a colored dot corresponding to the backbone, and a white dot corresponding to the base.
		In addition to the 36 edges of the truncated octahedron, each vertex contains a hairpin of DNA extending from it.
	seemanlab4.chem.nyu.edu /nano-oct.html (183 words)

	Truncated Trickery: Truncatering
		The truncated square is formed in much the same way: Remove the corners to make equilateral triangles, and the original square faces of the cube become octagons.
		This happens because the dual of the cube is the octahedron, and truncating along planes that meet in the centers of the faces is the same as constructing the dual of the original shape.
		Surprisingly, truncating along the edges and corners of the tetrahedron yields four new triangular faces on the vertexes and six square faces along the edges.
	www.theory.org /geotopo/tt/html/truncatering.html (794 words)

	Octahedron - Wikipedia, the free encyclopedia
		An octahedron (plural: octahedra) is a polyhedron with eight faces.
		Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles).
		If each edge of an octahedron is replaced by a one ohm resistor, the resistance between opposite vertices is 1/2 ohms, and that between adjacent vertices 5/12 ohms.
	en.wikipedia.org /wiki/Octahedron (595 words)

	Minimum inventory maximum diversity building system - Patent 3974600 (Site not responding. Last check: 2007-10-14)
		The portions to be truncated are indicated in dashed outline and the solid lines remaining define a truncated tetrahedron 58, and the truncated octahedron 60.
		The geometry of this dwelling structure is composed of truncated octahedron space units.
		As has been noted, the truncated octahedron 60 is the simplest and most economical in terms of surface area to volume ratios of the space filling systems that have been discussed.
	www.freepatentsonline.com /3974600.html (14075 words)

	Platonic and Archimedean
		The others are the cube, the octahedron (with 8 equilateral triangle faces, made by gluing together the bases of two square pyramids with equal edge lengths), the icosahedron (with 20 equilateral triangular faces), and the dodecahedron (with 12 pentagonal faces).
		A polyhedron and its dual have the same number of edges (12 for a cube and an octahedron, for example) but the numbers of vertices and faces are interchanged.
		Thus we obtain the truncated cube, the truncated tetrahedron, the truncated octahedron.
	mcraefamily.com /MathHelp/GeometrySolidPlatonic.htm (607 words)

	Truncated octahedron - Wikipedia, the free encyclopedia
		All permutations of (0, ±1, ±2) are Cartesian coordinates of the vertices of a truncated octahedron centered at the origin.
		The truncated octahedron can also be represented by even more symmetric coordinates in four dimensions: all permutations of (1,2,3,4) form the vertices of a truncated octahedron in the three-dimensional subspace x+y+z+w=10.
		Truncated octahedra are able to tessellate 3-dimensional space, forming an convex uniform honeycomb.
	en.wikipedia.org /wiki/Truncated_octahedron (274 words)

	Fifth Foresight Conference: DNA Nanotechnology
		A similar approach was taken with the proof of the truncated octahedron synthesis: The presence of the six square strands was demonstrated first.
		The cube is a 3-connected object, as is the truncated octahedron.
		The cube was constructed from 3-arm branched junctions, but the truncated octahedron was constructed from 4-arm branched junctions, because we had originally planned to link the truncated octahedra together.
	www.foresight.org /Conferences/MNT05/Papers/Seeman (9193 words)

	Octahedron -- from Wolfram MathWorld
		An octahedron is a polyhedron having eight faces.
		The dual polyhedron of an octahedron with unit edge lengths is a cube with edge lengths
		The centers of the faces of an octahedron form a cube, and the centers of the faces of a cube form an octahedron (Steinhaus 1999, pp.
	mathworld.wolfram.com /Octahedron.html (633 words)

	11011110: Zonohedra and cubic partial cubes
		The truncated octahedron is also known as the permutohedron, as its vertices can be given coordinates in 4-dimensional space that are the permutations of the vector (1,2,3,4).
		The Truncated Rhombic Dodecahedron is the first example of a cubic zonohedron not resulting from the inflation operation (in inflated graphs, every vertex belongs to a quadrilateral face, while this graph has vertices all of whose adjacent faces are hexagons).
		The Truncated Small Rhombicosidodecahedron is the result of applying the inflation operation to the Rhombic Triacontahedron or to the Icosidodecahedron.
	11011110.livejournal.com /12756.html (614 words)

	Trunc.Octahedron (Site not responding. Last check: 2007-10-14)
		When the octahedron is truncated it creates a square face replacing the vertex, and changes the triangular face to a hexagonal face.
		The octahedron is the dual of the hexahedron (cube).
		It should be noted here that the truncations are not regular polyhedra but rather square based prisims.
	www.ul.ie /~cahird/polyhedronmode/truncate1.htm (84 words)

	[No title]
		Not surprising there are five truncated solids: truncated tetrahedron, truncated cube, truncated octahedron, truncated dodecahedron and truncated icosahedron.
		The polyhedron symbol for the octahedron is 3,3,3,3 and the symbol for the truncated octahedron in 6,6,4.
		Design a net for the truncated octahedron that is based on the octahedron net assigned to you.
	people.wcsu.edu /burnsd/Mat242/Math_242_week10_lab.doc (753 words)

	Truncated Octahedron -- from Wolfram MathWorld
		The dual polyhedron of the truncated octahedron is the tetrakis hexahedron.
		The volume of the truncated octahedron is then given by the volume of the
		The truncated octahedron is a space-filling polyhedron (Steinhaus 1999, pp.
	mathworld.wolfram.com /TruncatedOctahedron.html (189 words)

	Chapter 20 updates - Dissections: Plane & Fancy (Site not responding. Last check: 2007-10-14)
		The truncated octahedron is cut into four identical pieces as on the left in that figure.
		The interior of truncated octahedron is a rectangular block.
		It is closely related to his dissection of a truncated octahedron to a cube.
	www.cs.purdue.edu /homes/gnf/book/Booknews/ch20.html (1287 words)

	Life on Mars
		This hexa-octahedron has the form of a regular octahedron which is expanded in the direction of one of its 3-fold axes, producing six new parallelogram faces.
		This truncation can be done so the six new faces lie in the planes of a cube.
		The six truncation faces (in the planes of a cube) are constructed as g1-g1-g1-b1-b1 (adjoin a g1 square and half a b1 square).
	www.georgehart.com /zomebook/life-on-mars.html (736 words)

	Trunc.Hexahedron (Site not responding. Last check: 2007-10-14)
		The notation describing this solid states that a triangle (3), a square (4), a triangle (3) and a square (4) meet at each of the vertices.
		This solid however can also be formed through the assemblage of the hexahedron and its dual, the octahedron using a common centre.
		This truncated hexahedron or more commonly known as the cuboctahedron which is also one of the quasi-regular polyhedra is derived from the compound of the hexahedron and the octahedron as is shown above.
	www.ul.ie /~cahird/polyhedronmode/truncate2.htm (201 words)

	Octahedron and truncated octahedron (Site not responding. Last check: 2007-10-14)
		To cut off the corners of the octahedron, we move in the same distance from each corner along the edges.
		This process shortens the side lengths of the edges in the original octahedron by the same amount.
		Notice that it also doubles the number of edges -- changing the green triangular faces of the octahedron (left) into green hexagonal faces in the truncated octahedron (right).
	www.geom.uiuc.edu /~sudzi/polyhedra/archimedean/octa_trunc.html (124 words)

	Polymorf - Knowhere - Archimedean symmetry (Site not responding. Last check: 2007-10-14)
		and octahedron can be progressively truncated to yield the cuboctahedron.
		truncated cube and the truncated octahedron are generated in the process.
		Exercise: Truncating the vertices of which polyhedron produces the triangle
	www.polymorf.net /lesson11.htm (99 words)

	Archimedean Polyhedra (Site not responding. Last check: 2007-10-14)
		Except for the truncated tetrahedron, lower right, all the Archimedean polyhedra are modifications of the cube-octahedron pair or the dodecahedron-icosahedron pair.
		The tetrahedron and truncated tetrahedron are shown in the last pair.
		Further truncation of the tetrahedron would eventually lead to eight triangular faces, an octahedron.
	www.uwgb.edu /dutchs/symmetry/archpol.htm (336 words)

	Icosidodecahedron, the rhombic triacontahedron, and the dual (Site not responding. Last check: 2007-10-14)
		Truncating the 5 Platonic solids produces the 13 Archimedean solids.
		Truncations of the five Platonic solids produce 5 of the Archimedean solids.
		Further truncations produce 4 more from the cube and 4 more from the dodecahedron.
	www.thearchimedeandual.com /platonic/Greek/icosidodecahedron.htm (315 words)

	Examples of Integrated, Multi-set Concept Schemes: Polygons and polyhedra
		(Pugh, p.2-3) The cube and octahedron are duals, as are the icosahedron and the dodecahedron.
		- octahedron, tetrahedron - triangular prism, cubic prism/cube
		- octahedron, truncated cube - triangular prism, dodecahedral prism
	www.laetusinpraesens.org /docs80s/84nsetsx/x18.php (1214 words)

	polyhedra.mathmos.net - Truncated octahedron (Site not responding. Last check: 2007-10-14)
		The truncated icosahedron is one of the thirteen archimedian solids.
		It is also the unit cell for face centred cubic (FCC) packing.
		It can be created by slicing suitable sections off the vertices of either a cube or an octahedron and thus may be inscribed in either solid.
	polyhedra.mathmos.net /entry/truncatedoctahedron.html (61 words)

	New Page 1
		A Cayley diagram of S4 as the truncated octahedron.
		A Cayley diagram of A4 as a truncated tetrahedron.
		Another way of seeing A4 as a subgroup of S4: A Cayley diagram of S4 as a truncated tetrahedron inside another truncated tetrahedron, namely the coset of A4 and A4 itself.
	www.cs.indiana.edu /~anmaguit/s4gallery2 (1172 words)

	Zonohedra and Zonotopes
		This is the dual to the cuboctahedron, an Archimedean solid formed by combining the six squares of a cube with the eight triangles of an octahedron.
		This is the Minkowski sum of a cube, a truncated octahedron, and a rhombic dodecahedron.
		We demonstrate for k=4; although it is hard to tell from the coordinates, the result is isometric to the truncated octahedron.
	www.ics.uci.edu /~eppstein/junkyard/ukraine/ukraine.html (2108 words)

	Truncated Trickery
		Cube, Double Truncated Cube, Double Truncated Octahedron, & Octagonal Prism
		Truncated Tetrahedron, Truncated Cube, and Double Truncated Octahedron
		Truncated Tetrahedron, Truncated Cube, & Double Truncated Octahedron
	www.theory.org /geotopo/tt/vrml/vrml.html (176 words)

	Solutions to Truncated Octahedron's Hamiltonian Path
		Listing of all 18 Hamiltonian Paths of truncated octahedron, with the criteria that path must start 1234, 2134, 2314, 3214, 3124, 1324, 1342
		() thus these are also the eighteen (lexicographically) first Hamiltonian paths of the total 344 possible such paths through the truncated* octahedron.
		Note that the antipodal vertex of 1234 in the truncated octahedron is 4321, and for any vertex x, the antipodal vertex is x(14)(23) i.e., its four-element permutation list reversed.
	ndirty.cute.fi /~karttu/matikka/permgraf/troctahe.htm (349 words)

	Parallel Expansions
		Keep in mind that if two base polyhedra can both be expanded to form the same new polyhedron, then we have a new way of transforming between the two base polyhedra.
		For example, the cube and octahedron can both be expanded to form the rhombicuboctahedron.
		So if we want to transform from cube to octahedron, we can do so by expanding the cube to form the rhombicuboctahedron, and then contracting the rhombicuboctahedron to form the octahedron:
	galileo.spaceports.com /~adoskey/polyhedra/ParallelExpansions.html (409 words)

	Ned Seeman's Home Page (Site not responding. Last check: 2007-10-14)
		A major effort in our laboratory is devoted to DNA Nanotechnology.The attachment of specific sticky ends to a DNA branched junction enables the construction of stick figures, whose edges are double-stranded DNA.
		This approach has already been used to assemble a cube, a truncated octahedron, nanomechanical devices and 2-D crystals from DNA.
		Ultimate goals for this approach include the assembly of a biochip computer, nanorobotics and the rational synthesis of periodic matter.
	seemanlab4.chem.nyu.edu (379 words)

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