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Topic: Truncated tetrahedron

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Triakis tetrahedron

Truncated octahedron

Augmented truncated tetrahedron

Tetrahedron

Cuboctahedron

Octahedron

Andreini tessellation

List of polygons, polyhedra and polytopes

Archimedean solid

Hexagon

Group action

Magic square

Melancholic

David Hilbert

Phlegmatic

	Piero della Francesca's Polyhedra
		He rediscovered the five Archimedean solids which are truncated Platonic solids: the truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron.
		These six solids are employed in new and substantial geometric problems --- calculating the edge length of the solid given the radius of their circumsphere.
		Above are Piero's drawings of the truncated tetrahedron.
	www.georgehart.com /virtual-polyhedra/piero.html (579 words)

	Krystyna Burczyk's Origami Gallery - truncated tetrahedron (Site not responding. Last check: )
		Truncated tetrahedron - 12 modules of Lewis Simon.
		There are tetrahedrons with edge length 1 truncated from a tetrahedron with edge length 4.
		There are tetrahedrons with edge length 1 truncated from a tetrahedron with edge length 5.
	www1.zetosa.com.pl /~burczyk/origami/g1-06-en.htm (114 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex.
		A regular tetrahedron is one in which the four triangles are regular, or "equilateral," and is one of the Platonic solids.
		A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=tetrahedron (1401 words)

	NationMaster - Encyclopedia: Octahedron (Site not responding. Last check: )
		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).
		The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids.
		A disdyakis dodecahedron, or hexakis octahedron, is the Catalan solid whose Archimedean dual is the truncated cuboctahedron.
	www.nationmaster.com /encyclopedia/Octahedron (2079 words)

	Tetrahedron - LoveToKnow 1911
		This is one of the Platonic solids, and is treated in the article Polyhedron, as is also the derived Archimedean solid named the "truncated tetrahedron"; in addition, the regular tetrahedron has important crystallographic relations, being the hemihedral form of the regular octahedron and consequently a form of the cubic system.
		The bisphenoids (the hemihedral forms of the tetragonal and rhombic bipyramids)., and the trigonal pyramid of the hexagonal system, are examples of non-regular tetrahedra (see Crystallography).
		This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces.
	www.1911encyclopedia.org /Tetrahedron (305 words)

	Keith Critchlow Archimedian Solids (Site not responding. Last check: )
		The top of the cuboctahedron is illustrated by the solid lines forming its four equilateral triange and three square faces and the bottom by the broken lines forming the remaining four triangle faces and three square faces.
		The 12 white vectors of Buckminster Fuller's Dymaxion (Vector Equilibrium) pass from the center of the truncted tetrahedron each one through one of its 12 vertices and on through each of the Archimedians residing at the 12 vertices of the cuboctahedron.
		The truncated tetrahedron is the only semi-regular solid figure with 12 independent axis (The Twelve Degrees of Freedom) passing through its vertices from its center.
	www.porcelainia.com /aexp.html (248 words)

	Truncated tetrahedron (Site not responding. Last check: )
		Canonical coordinate s for the vertices of a truncated tetrahedron centered at the origin are (±3, ±1, ±1), (±1, ±3, ±1), (±1, ±1, ±3), where the ± has the same parity for each coordinate, that is, all coordinates have an even number of minuses (or all have an odd number).
		A famous depiction of a truncated tetrahedron is in Albrecht Durer 's engraving, "Melencolia I" See illustration at entry'' Melancholy.
		(geometry) a polyhedron with four faces; the regular tetrahedron, the faces of which are equal equilateral triangles, is one of the Platonic solids.
	www.serebella.com /encyclopedia/article-Truncated_tetrahedron.html (181 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.
		Thus we obtain the truncated cube (69), the truncated tetrahedron (70), the truncated octahedron (74).
		Truncating a cuboctahedron and adjusting the resulting faces to make them squares gives the truncated cuboctahedron (71).
	math.arizona.edu /~models/Platonic_and_Archimedean_Solids (376 words)

	Truncated Trickery: Truncatering
		For the tetrahedron, this is 4+4, yielding a new 8 sided object, with regular triangular faces.
		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)

	4th Dimension Hcube-slices: Projections of Hypercube Slices
		Each face of the tetrahedron is a slice of one of the corners of one of the four cubes that meet at this vertex.
		At this point, the corners of the tetrahedron begin to be cut off as the slice starts to pass through the other four cubes in the hypercube.
		Note that the final tetrahedron is inverted with respect to the one at the beginning of the slicing sequence.
	www.math.union.edu /~dpvc/math/4D/hcube-slices/welcome.html (695 words)

	Octonion Mirrorhouse
		Let the floor be the distinguished third generator E and the 3 walls be I,J,K (with a specific assignment) Then, by reflection through the E floor, the reflected I J K become i j k and we now have all 7 imaginary octonions, and 1 of the 480 different multiplications.
		The icosahedron is only qualitatively an icosahedron, and is not exact, since the internal angle of the pentagonal vertex figure of the reflected icosahedron is not 108 degrees, but is 109.47 degrees (the octahedral dihedral angle), and the vertex angle is not 72 degrees, but is 70.53 degrees (the tetrahedral dihedral angle).
		The truncated tetrahedron with an icosahedron at each vertex combines two types of symmetries: tetrahedral, related to the square and the ratio sqrt(2) and the continued fraction 1 + sqrt(2) = 2 + 1/ 2 + 1/ 2 + 1/ 2 + 1/...
	www.valdostamuseum.org /hamsmith/miroct.html (1912 words)

	Archimedean Polyhedra (Site not responding. Last check: )
		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.
		For example, in the top row we see a truncated tetrahedron with vertices 3.6.6 (n=3), a truncated cube with 3.8.8 (n=4), a truncated dodecahedron with 3.10.10 (n=5).
	www.uwsp.edu /geo/projects/geoweb/participants/Dutch/symmetry/archpol.htm (348 words)

	What the Origami Means
		The truncated polyhedra are formed by slicing the corners off of the tetrahedron, the octahedron, and the icosahedron.
		The other four vertices of the cube form the vertices of another tetrahedron; this can be seen by looking at the model of two intersecting tetrahedra among the dual polyhedra models above, in which the edges of the two tetrahedra are the diagonals of the square faces of the circumscribed cube.
		Four of the eight vertices of this central cube are the centers of the four octahedra and form the vertices of a tetrahedron parallel to the tetrahedron circumscribing the model.
	www.amherst.edu /~sgoldstine/origami/displaytext.html (2729 words)

	Platonic and Archimedean (Site not responding. Last check: )
		Thus we obtain the truncated cube, the truncated tetrahedron, the truncated octahedron.
		Truncating a cuboctahedron and adjusting the resulting faces to make them squares gives the truncated cuboctahedron.
		The seven other Archimedean solids are the truncated dodecahedron, truncated icosahedron, cuboctahedron, rhombicosidodecahedron, truncated icosidodecahedron, snub cube, and snub dodecahedron.
	mcraefamily.com /MathHelp/GeometrySolidPlatonic.htm (607 words)

	Math Forum - Ask Dr. Math Archives: College Polyhedra
		I am weatherproofing my home, and have to mitre boards in a pyramid with a rectangular - not square - base, and an apex that is directly over the centre of one edge of the base.
		Recall that the opposite edges of an orthocentric tetrahedron are perpendicular.
		The volume of a tetrahedron is one-third the distance from a vertex to the opposite face, times the area of that face.
	www.mathforum.org /library/drmath/sets/college_polyhedra.html (603 words)

	Truncated tetrahedron : Information and resources about Truncated tetrahedron : School Work Guru
		Truncated tetrahedron : Information and resources about Truncated tetrahedron : School Work Guru
		Canonical coordinates for the vertices of a truncated tetrahedron centered at the origin are (±3, ±1, ±1), (±1, ±3, ±1), (±1, ±1, ±3), where the ± has the same parity for each coordinate, that is, all coordinates have an even number of minuses (or all have an odd number).
		It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.
	www.schoolworkguru.org /encyclopedia/t/tr/truncated_tetrahedron.html (151 words)

	[No title]
		It is generated by truncating the vertices of a tetrahedron at 1/3 edge-length.
		Interlock the tetrahedrons together by making each vertex of one tetrahedron emerge from the face of the other.
		The tetrahedron, cube, octahedron, icosahedron and dodecahedron are known as the platonic solids.
	www.lycos.com /info/tetrahedron.html (394 words)

	The International Bone Rollers' Guild
		While there are only five "perfect" regular convex polyhedra, there are a number of solids that can be formed using only regular polygons that still meet the criteria of being convex and having all vertices be identical, as long as we allow more than one type of polygon to be used in the same polyhedron.
		In the Truncated Tetrahedron, for example, you will find a triangle-hexagon-hexagon combination at each vertex.
		The Truncated Cube would be represented as (3,8,8), meaning that a triangle and two octagons appear at each vertex.
	members.aol.com /dicetalk/polyh2.htm (454 words)

	Truncated Tetrahedron
		The above animation shows how the regular tetrahedron is truncated to create the first of the thirteen archimedean solids.
		It is also regular tetrahedrons that make up the sections which are truncated.
		It should be noted that for the first five truncations, the sides of the pieces of the solids that are being truncated are always one-third of the original side of the platonic solid.
	www.ul.ie /~cahird/polyhedronmode/truncate.htm (97 words)

	TETRAHEDRON (Gr. riepa... - Online Information article about TETRAHEDRON (Gr. riepa...
		POLYHEDRON, as is also the derived Archimedean solid named the " truncated tetrahedron "; in addition, the regular tetrahedron has important crystallographic relations, being the hemihedral See also:
		Tetrahedral co-ordinates " are a system of quadriplanar co-ordinates, the fundamental planes being the faces of a tetrahedron, and the co-ordinates the perpendicular distances of the point from the faces, a See also:
		If the perpendiculars from the vertices to the opposite faces of a tetrahedron be concurrent, then a sphere passes through the four feet of the perpendiculars, and consequently through the centre of gravity of each of the four faces, and through the See also:
	encyclopedia.jrank.org /TAV_THE/TETRAHEDRON_Gr_riepa_four_Ebpa_.html (503 words)

	Truncated Tetrahedron
		-The truncated tetrahedron is an Archimedean (semi-regular) solid.
		-Below, we have a tetrahedron on the left, and its truncated form on the right.
		The tetrahedron is made from solid walnut, while the truncated model is mahogany.
	home.cc.umanitoba.ca /~gunderso/pages/archimedean_solids/truncated_tetrahedron.htm (66 words)

	Archimedean Semi-Regular Polyhedra
		All but one of these polyhedra were gradually rediscovered during the Renaissance by various artists, and Kepler finally reconstructed the entire set in 1619.
		Truncation adds a new face for each previously existing vertex, and replaces n-sided polygons with 2n-sided ones, e.g., octagons instead of squares.
		There is also another class of polyhedra in which the same regular polygons appear at each vertex: the prisms and antiprisms, which have vertex types (4, 4, n) and (3, 3, 3, n) respectively.
	www.georgehart.com /virtual-polyhedra/archimedean-info.html (628 words)

	Uniform polyhedra --- List
		The notation in parentheses is a Wythoff symbol which characterizes the derivation of each.
		compound of small stellated truncated dodecahedron and dual
		compound of great stellated truncated dodecahedron and dual
	www.georgehart.com /virtual-polyhedra/uniform-index.html (58 words)

	Cool math .com - Polyhedra - Truncated Tetrahedron
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		Properties of the truncated tetrahedron: Number of faces, edges and dihedral angle measure
		The truncated tetrahedron is created by truncating (cutting off) the tips of the tetrahedron one third of the way into each edge
	www.coolmath.com /reference/polyhedra-truncated-tetrahedron.html (76 words)

	Excite - Search: Tetrahedron
		tetrahedron (plural: tetrahedra) is a polyhedron composed of four...
		tetrahedron is the diagonal of the cube, as can be seen in Figure's 3 and 4.
		tetrahedron can be constructed with two different kinds of generating triangles;...
	msxml.excite.com /info.xcite/search/web/Tetrahedron/1/20/1/-/-/0/1/1/1/1/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/-/1/-/-/-/-/-/-/-/-/-/-/0/302349/right (233 words)

	Tetrahedron Information
		Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper.
		The proper rotations and reflections in the symmetry group of the regular tetrahedron
		An irregular tetrahedron (3-sided Pyramid (geometry)) with equilateral base and the top vertex above the center has 6 isometries, like an equilateral triangle.
	www.bookrags.com /Tetrahedron (1029 words)

	Reference.com/Encyclopedia/Cube
		The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron.
		The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself; the other symmetries of the cube map the two to each other.
		One such regular tetrahedron has a volume of ⅓ of that of the cube.
	www.reference.com /browse/wiki/Cube_%28geometry%29 (635 words)

	Tetrahedron Net - FullSearch.co.uk (Site not responding. Last check: )
		Tetrahedron Cut out along the solid lines, and then score the dashed lines and fold back.
		Finite generalized tetrahedron groups with a high-power relator An ``ordinary'' tetrahedron group is a group with a presentation
		A mathematical model is presented of the flow, mass fraction and temperature profiles in the zirconia tube of the SOFC, when the tube is initially heated from one end.
	www.fullsearch.co.uk /search-Tetrahedron+Net (223 words)

	Truncated tetrahedron
		Already truncated tetrahedron locks wore go spell mary cunnilingus photo position technique.
		Has from cardboard truncated tetrahedron if handlebar james are fl cream pie movie.
		Seemed truncated tetrahedron severity rocks at pierre leave title time there is only one god.
	truncated-tetrahedron.search-buy.biz (918 words)

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