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# Topic: Archimedean solid

###### In the News (Sat 20 Jul 19)

 Welcome To Archimedean 1.0 The dual of the dual of a Platonic solid is the original Platonic solid. The dual of an Archimedean solid is neither Archimedean or Platonic and does not have a dual. As the degree of truncation increases, Platonic solids pass through three different Archimedean solid stages before ariving at the final stage which is simply the inscribed dual. users.erols.com /quantime/Archimedean.html   (1172 words)

 Tetrahedron - LoveToKnow 1911   (Site not responding. Last check: ) If the faces be all equal equilateral triangles the solid is termed the "regular" tetrahedron. 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). www.1911encyclopedia.org /Tetrahedron   (305 words)

 Archimedean Semi-Regular Polyhedra These solids were described by Archimedes, although his original writings on the topic are lost and only known of second-hand. A key characteristic of the Archimedean solids is that each face is a regular polygon, and around every vertex, the same polygons appear in the same sequence, e.g., hexagon-hexagon-triangle in the truncated tetrahedron, shown above. The Archimedean solid (shown at right) in which three squares and a triangle meet at each vertex is the rhombicuboctahedron. www.georgehart.com /virtual-polyhedra/archimedean-info.html   (628 words)

 Archimedean solid A convex semi-regular polyhedron –; a solid made from regular polygonal sides of two or more types that meet in a uniform pattern around each corner. The duals of the Archimedean solids (made by replacing each face with a vertex, and each vertex with a face) are commonly known as Catalan solids. Apart from the Platonic and Archimedean solids, the only other convex uniform polyhedra with regular faces are prisms and antiprisms. www.daviddarling.info /encyclopedia/A/Archimedean_solid.html   (158 words)

 Archimedean solid - Education - Information - Educational Resources - Encyclopedia - Music Compare to Platonic solids, which are face-uniform (in addition to being vertex-uniform), and to Johnson solids, which need not be vertex-uniform. The Archimedean solids are known to have been discussed by Archimedes, although the complete record is lost. All edges of an Archimedean solid have the same length, since the faces are regular polygons, and the edges of a regular polygon have the same length. www.music.us /education/A/Archimedean-solid.htm   (530 words)

 Ritz Pumpen RITZ is one of the foremost large-scale manufacturers of a comprehensive range of pumps for both clean water and effluent services. Single-Stage submersible pumps in close-coupled design for pumping sewage and wastewater as well as sludge with maximum 15% solids. Archimedean screw pumps for pumping clean or heavily contaminated liquids. www.tyhsin.com /ritz.html   (137 words)

 Archimedean solid - Definition, explanation In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices, and from the Johnson solids, whose regular polygonal faces do not meet in identical vertices. The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. www.calsky.com /lexikon/en/txt/a/ar/archimedean_solid.php   (332 words)

 Some of the Archimedean Solids After these, the most basic solid shapes, there is a family of shapes whose faces are regular polygons which is one step less uniform than them, known as the Archimedean solids. The three solids depicted above are the dodecahedron, a Platonic solid, and the two Archimedean solids known as the truncated dodecahedron and the icosidodecahedron. Yet another solid belongs to this branch of the family of Archimedean solids, the snub dodecahedron. members.shaw.ca /quadibloc/math/acsint.htm   (615 words)

 What the Origami Means I   (Site not responding. Last check: ) The Greek names of the solids refer to the number of faces they have: "tetra-" means four, "octa-" means eight, "dodeca-" means twelve, and "icosa-" means twenty. The same solid results if the faces of an octahedron are pulled apart; to understand why, see the description of dual polyhedra below. Now, once again imagine pulling apart the faces of a cube, and imagine that the original square faces of the cube and the triangles that were once the vertices of the cube are rigid like pieces of metal and are joined at the corners. www.amherst.edu /~sgoldstine/origami/plato.html   (498 words)

 Constructing Platonic Solids in the Classroom The five Platonic solids are the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. These solids are perfectly symmetrical in that each face of a solid is identical to every other face of the solid, each vertex is identical to every other vertex, and each edge is identical to every other edge. Paper patterns of the five platonic solids are available as a postscript file or as two GIF files: plato1 and plato2. www.davidparker.com /janine/mathpage/platonic.html   (1586 words)

 Archimedean Solid -- from Wolfram MathWorld The 13 Archimedean solids are the convex polyhedra that have a similar arrangement of nonintersecting Two additional solids (the small rhombicosidodecahedron and small rhombicuboctahedron) can be obtained by expansion of a Platonic solid, and two further solids (the great rhombicosidodecahedron and great rhombicuboctahedron) can be obtained by expansion of one of the previous 9 Archimedean solids (Stott 1910; Ball and Coxeter 1987, pp. Since the Archimedean solids are convex, the convex hull of each Archimedean solid is the solid itself. mathworld.wolfram.com /ArchimedeanSolid.html   (1156 words)

 Polyhedron Summary The Platonic solids are within the larger grouping known as regular polyhedrons, in which the polygons of each are regular and congruent (that is, all polygons are identical in size and shape and all edges are identical in length), and are characterized by the same number of polygons meeting at each vertex. a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Archimedean solids give rise to regular graphs: 7 Archimedean solids are degree 3, 4 solids are degree 4, and the remaining 2 are chiral pairs of degree 5. www.bookrags.com /Polyhedron   (2968 words)

 Constructing Platonic Solids in the Classroom   (Site not responding. Last check: ) The five Platonic solids are the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. These solids are perfectly symmetrical in that each face of a solid is identical to every other face of the solid, each vertex is identical to every other vertex, and each edge is identical to every other edge. Paper patterns of the five platonic solids are available as a postscript file or as two GIF files: plato1 and plato2. www.dpgraph.com /janine/mathpage/platonic.html   (1586 words)

 Pedagoguery Software: Poly’s Polyhedra The Archimedean solids were defined historically by Archimedes, although we have lost his writings. A common heuristic for the Archimedean solids is that the arrangement of faces surrounding each vertex must be the same for all vertices. The Catalan solids are duals of Archimedean solids. www.peda.com /poly/poly.html   (549 words)

 Polyhedron, Polyhedra, Polytopes - Numericana Catalan solids are the duals of Archimedean polyhedra. All the vertices in a platonic solid are equivalent. We may focus on the n-dimensional equivalent of the Platonic solids, namely the regular convex polytopes, whose hyperfaces are regular convex polytopes of a lower dimension, given the fact that the concept reduces to that of a regular polygon [equiangular and equilateral] in dimension 2. home.att.net /~numericana/answer/polyhedra.htm   (5442 words)

 Jerry L. Atwood In these reports we began by presenting the idea of self-assembly in the context of spherical hosts and then, after summarizing the Platonic and Archimedean solids, we provided examples of cubic symmetry-based hosts, from both the laboratory and nature, with structures that conform to these polyhedra. The Platonic solids comprise a family of five convex uniform polyhedra (Table 1) which possess cubic symmetry and are made of the same regular polygons (equilateral triangle, square, pentagon) arranged in space such that the vertices, edges, and three coordinate directions of each solid are equivalent In addition to the Platonic solids, there exists a family of 13 convex uniform polyhedra known as the Archimedean solids. www.chem.missouri.edu /faculty/Atwood/research.html   (798 words)

 Platonic and Archimedean There are five Platonic Solids and thirteen Archimedean Solids, which are convex polyhedra whose faces are all regular polygons. The "base case" is a solid with two faces glued back-to-back which is "flattened" to a single polygon. 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)

 Archimedean solid (cube/octahedron) Note on the surfaces: On this solid there are (in fact) two types of squares. One type, square (c), is because the base solid is a cube. This solid is not symmetrical under reflection, because the surface order is not symmetrical: When mirroring (4-3e-3v-3e-3e) one finds (4-3e-3e-3v-3e). www.phys.uu.nl /~beugelng/polyhedra/archimedean1.html   (370 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. Archimedean polyhedra and Archimedean tilings are closely related as shown below. A solid with 20 equilateral triangle faces is an icosahedron. www.uwgb.edu /dutchs/symmetry/archpol.htm   (336 words)

 Platonic and Archimedean Polyhedra The Platonic Solids, discovered by the Pythagoreans but described by Plato (in the Timaeus) and used by him for his theory of the 4 elements, consist of surfaces of a single kind of regular polygon, with identical vertices. Blue Archimedean Solids are produced from green ones by continuing the trucation until edges disappear and half the vertices merge. Purple Archimedean Solids result when, in the five triangles per vertex of the Platonic Icosahedron, one triangle is replaced by either a square (the Snub Cube) or a pentagon (the Snub Dodecahedron). www.friesian.com /polyhedr.htm   (633 words)

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 Rhombic polyhedra   (Site not responding. Last check: ) By rhombic solid we mean a polyhedron that consists of prolate and oblate golden rhombohedra. Some combinations of Archimedean solids are equidecomposable to some combinations of their approximates, and can be dissected to a cube. In fact there are basically two solutions depending on the length of edges of the Archimedean solid, which may be shorter or larger diagonal of the rhombus. torina.fe.uni-lj.si /~izidor/articles/visual40A/Visual40A.html   (412 words)

 Cayley Graphs   (Site not responding. Last check: ) The Platonic and Archimedean solids are uniform, that is to say, they have so much rotational symmetry that one vertex can be mapped onto each of the others. The (convex) uniform polyhedra consist of the Platonic and Archimedean solids, the prisms, and the anti-prisms. This is because you can enlarge any one face of the solid until it encircles all the rest, and the surface of the solid then becomes (projected on) the flat plane. www.geocities.com /jaapsch/puzzles/cayley.htm   (4449 words)

 The HyperSphere   (Site not responding. Last check: ) Similarly to the case in three dimensions, there is a family of Platonic and Archimedean solids that can be viewed on the four dimensional sphere. The solid "faces" of this hyper-dodecahedron have been pulled away from each other slightly, which makes it easier to see that they are all reasonably similar dodecahedron. Considering that the largest Archimedean solid, the hyper truncated icosahedron, has over 14,000 faces, this object alone could contain within it an entire menagerie of never-seen before mathematical beasties. www.swiss.ai.mit.edu /~rfrankel/fourd/FourDArt.html   (3236 words)

 Abstract Only after he knew what the diagrams would have to be as graphs did he seek the specific groups associated with them. We speculate that he must have been delighted to find that permutation groups determined by the Platonic solids define all the groups so associated. Figure 11, we see that Maschke associated the dihedral groups with the rotations of a dihedron (a solid determined by a regular n-gon on the equator of a sphere, with an additional vertex at each pole). www.lcsc.edu /csteenbe/abstract.htm   (3076 words)

 Polyhedral Cartography On those polyhedra where you can draw an equator and prime meridian that symmetrically divide the solid (such as a cube with the poles at the center of a face), you can mark off longitude and latitude lines using equal distances. on a tetrahedron, though the equator may not divide the solid symmetrically, it is easy to find lines at which to mark off latitude and longitude lines). This solid has 30 square faces, 20 triangular faces, and 12 pentagonal faces. www.lausd.k12.ca.us /NHHS_Highly_Gifted_Mag/mathweb/polycart.html   (393 words)

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