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Topic: Catalan solids

	Archimedean Duals
		When one takes the duals of the Archimedean solids, one gets an interesting set of new polyhedra that are called the Archimedean duals or the Catalan polyhedra, after the French mathematician Eugene Catalan, who first described them in 1865.
		The relationships between the Archimedean solids and their respective duals is nicely brought out by studying a compound of a solid and its dual.
		The edges cross at right angles at a point which is the midpoint of the Archimedean solid's edge, but not always the midpoint of the dual solid's edge.
	www.georgehart.com /virtual-polyhedra/archimedean-duals-info.html (436 words)

	Archimedean solid
		An Archimedean solid is a semi-regular (ie vertex-uniform, but not face-uniform) convex polyhedron with regular polygons for faces.
		Compare to Platonic solids, which are face-uniform, and Johnson solids, which need not be vertex-uniform.
		These solids were known to be discussed by Archimedes, although the complete record is lost.
	www.ebroadcast.com.au /lookup/encyclopedia/ar/Archimedean_solid.html (498 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)

	Catalan solid at AllExperts
		The Catalan solids are named for the Belgian mathematician, EugÃ¨ne Catalan who first described them in 1865.
		Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons.
		Additionally, two of the Catalan solids are edge-uniform: the rhombic dodecahedron and the rhombic triacontahedron.
	en.allexperts.com /e/c/ca/catalan_solid.htm (244 words)

	3D Puzzles - Regularity
		The number of edges of an Archimedic or Catalan polyhedron is however variable, therefore are not al regularity indexes the same (although they are all 'half-regular polyhedra').
		The Catalan polyhedra are the dual counterparts of the Archimedic solids.
		All three polyhedra aplies to at least three of the five rules of regularity (solids are convex, vertexes are congruent and dihedral angles are congruent).
	www.3dpuzzles.nl /polyhedra/regelmaat/engexamples.htm (456 words)

	Results for Catalan (Site not responding. Last check: )
		This paper describes the history of the Catalan campaign to win the.cat domain against political opposition from the former conservative Spanish government...
		THE intensity of the debate in Spain about the Catalan statute of autonomy had been a cause of concern over the past few weeks.
		Catalan is Associate Professor in the Department of Chemical...
	www.gogoglo.com /buscar/searchold/Catalan/130 (179 words)

	DICE : Encyclopedia Entry
		The platonic solids are commonly used to make dice of 4, 6, 8, 12, and 20 faces.
		The six-die versions add the pentagonal trapezohedron, in which the faces (identical to one another as to angles and edge lengths) each have two different lengths of side, and three different sizes of angle; the corners at which multiple faces meet are also of two different kinds.
		Catalan solids: 13 Archimedean duals: (12, 24, 30, 48, 60, 120 sides)
	bibleocean.com /OmniDefinition/Dice (4140 words)

	uniform polyhedra (Site not responding. Last check: )
		The five regular and the thirteen archimedean solids are convex; among the non convex ones appear the Kepler and Poinsot polyhedra.
		The 75-(5+13)-4=53 non convex and non regular uniform polyhedra are analogous to the archimedean solids but with faces which intersect each other (9 among them have faces which contain the centre of the polyhedron); they can be truncated and "snubbed".
		Their 53-9=44 finite duals are equifacial and their vertices are regular (as are the Catalan's solids).
	www.ac-noumea.nc /maths/amc/polyhedr/p_unif_.htm (177 words)

	Variations of Fair Dice (Site not responding. Last check: )
		Like platonic solids, catalan solids are all convex.
		As a result, catalan solids are not vertex-uniform.
		The platonic solids are the most common as well as some trapezhedrons.
	www.bath.ac.uk /~ma3mrp/variations.html (466 words)

	LLLI \| FAQ: When Should My Baby Start Solids?
		Most solid foods are lower in calories than human milk, of lower nutritional value, and can be difficult for young babies to digest.
		Some parents have found introducing solids before baby is ready to be a waste of time, energy and money.
		Gradual introduction of iron-enriched solid foods in the second half of the first year should complement the breast milk diet.
	www.llli.org /FAQ/solids.html (0 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.
		The convex ones consist of the prisms and antiprisms and the Archimedean solids.
		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)

	Variations of Fair Dice (Site not responding. Last check: )
		Like platonic solids, catalan solids are all convex.
		As a result, catalan solids are not vertex-uniform.
		The platonic solids are the most common as well as some trapezhedrons.
	people.bath.ac.uk /ma3mrp/variations.html (466 words)

	Contents
		This absolutely proves that we like the macrocosm (the planets, and stars) or the microcosm (of atomic and subatomic particles) all were created using the PHI design...
		Plato was mightily impressed by these five definite shapes that constitute the only perfectly symmetrical arrangements of a set of (non-planar) points in space, and late in life he expounded a complete "theory of everything" (in the treatise called Timaeus) based explicitly on these five solids.
		Plato conceived the four classical elements as atoms with the geometrical shapes of four of the five platonic solids.
	milan.milanovic.org /math/english/contents.html (0 words)

	LLLI \| FAQ: When Should My Baby Start Solids?
		Most solid foods are lower in calories than human milk, of lower nutritional value, and can be difficult for young babies to digest.
		Some parents have found introducing solids before baby is ready to be a waste of time, energy and money.
		Gradual introduction of iron-enriched solid foods in the second half of the first year should complement the breast milk diet.
	www.lalecheleague.org /FAQ/solids.html (615 words)

	Discovery Channel's Cosmeo
		Catalan numbers, Catalan solids, Catalan minimal surface, and Catalan's conjecture are all named for their discoverer, Belgian mathematician Eugene Catalan.
		Catalan numbers are a sequence of natural numbers that represent the number of ways a polygon with n sides can be divided into triangles without intersecting diagonals.
		Catalan proposed his famous conjecture in 1844, but it took mathematicians until 2002 to finally prove it.
	www.cosmeo.com /viewTodayInHistoryEvents.cfm?guidAssetId=960eb1dc-36df-4928-a537-0cdad9f17075&eventId=1175 (161 words)

	VICHY CATALAN
		A slice of antiquity, the Vichy spring in the Catalan area near Girona has hosted Europe's most famous rulers, conquerers, and health-seekers further back than recorded documents go.
		With an astonishing 3,052 milligrams per liter of Total Dissolved Solids (the highest of any water sold by Aqua Maestro), there's nothing by way minerals and salts missing in Vichy - except Nitrates, which are splendidly near ZERO.
		In Spain, it says on the bottle, "Vichy Catalan te ayuda a controller el cholesterol." Of course, that's not FDA, but they believe it there.
	www.aquamaestro.com /innerview.asp?catid=49 (211 words)

	Polyhedron at AllExperts
		In 1966, he published a list of 92 convex solids, now known as the Johnson solids, and gave them their names and numbers.
		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.
	en.allexperts.com /e/p/po/polyhedron.htm (1964 words)

	Untitled Document
		(C) The Thirteen Catalan Solids (dual of Archimedean Solids) Yes-No-No-No In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid.
		The Catalan solids are named for the Belgian mathematician, Eugène Catalan who first described them in 1865.The Catalan solids are all convex.
		The Johnson solids are convex polyhedra having regular faces and uniform edges which are not Platonic solids, Archimedean solids, prisms, or antiprisms.
	www.artangent.com /sygma.htm (488 words)

	Numerical Constants - Mathematics & Physics - Numericana
		Catalan's Constant G : The alternating sum of the reciprocal odd squares.
		Catalan's name has also been given to the Catalan solids (the duals of the Archimedean solids) and the famous integer sequence of Catalan numbers:
		Catalan number is C(2n,n) / (n+1) [see choice numbers] This quantity often occurs in combinatorics: For example, it's the number of different binary trees with n internal nodes.
	home.att.net /~numericana/answer/constants.htm (5451 words)

	Applelinks.com Macintosh News
		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.
		Although all of the Archimedean solids have this property, so does the elongated square gyrobicupola (Johnson solid which is not an Archimedean solid.
	www.applelinks.com /articles/1999/12/19991213132426.shtml (477 words)

	Computer Graphics
		Recall that the Archimedean Solids are convex polyhedra composed entirely of regular polygons, but unlike the Platonic solids they are not each composed entirely of one type of regular polygon.
		Exercise: Check that you can see that any archimedean solid has a unique midradius which each edge is tangent to, even in the case of a snub cube and a snub dodecahedron.
		We have briefly explored geometry calculations for Platonic, Archimedean and Catalan Solids, demonstrating a need to find views which help us analyse symmetry and showing that trigonometry can often be avoided.
	soi.city.ac.uk /~dcd/ig2/week06/l09.htm (1732 words)

	The International Bone Rollers' Guild
		Our POLYHEDRAL adventurese have thus far introduced us to the five regular Platonian Solids, the thirteen Catalan Solids, and two infinite series of solids known as the Dipyramids and the Trapezohedra.
		By twisting and stretching the outlines of the Platonian solids, we will discover eleven new types of solids that qualify as isohedra.
		Catalan's Rhombic Dodecahedron can also be bent into a new variation, known as the Trapezoidal Dodecahedron (right).
	members.aol.com /dicetalk/polymore.htm (420 words)

	Polyhedron information - Search.com
		There are 58 stellations of the icosahedron, including the great icosahedron (one of the Kepler-Poinsot solids), and the 2nd and final stellations of the icosahedron.
		Here we add the rule that all of the original faces must "contribute" to the stellation, so the cube is not considered a stellation of the cuboctahedron.
		Seventeen of the nonconvex uniform polyhedra are stellations of Archimedean solids.
	domainhelp.search.com /reference/Polyhedron (2197 words)

	Egyéb érdekes testek (Site not responding. Last check: )
		All of the Archimedean solids are uniform polyhedra with regular faces.
		For example the dual of the Archimedean solid which was mentioned in the previous paragraph is the rhombic dodecahedron.
		Moving the mouse to the cube it will be transformed to the mentioned solid.
	mbuttons.bolyai.hu /abrak/VRML/palyazat/egyeb_ang.htm (310 words)

	Rhombic polyhedra (Site not responding. Last check: )
		We think that solving mazes on convex solids is a nice space visualization experience.
		We have a fl dot on one face a solid and a gray dot on another face.
		The aim is to move from the fl dot to the gray one.
	torina.fe.uni-lj.si /~izidor/visual13/PolyhedraMazes.html (80 words)

	Download Poly (16-bit) v1.05 software - Softlookup Downloads
		Poly enbles you to manipulate a polyhedral solid and print out a flattened version, which can be cut out, folded, and taped to produce a three-dimensional model.
		You can design Platonic solids (tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons), Archimedean solids (truncated tetrahedrons, cuboctahedrons, truncated cubes, and more), and prisms and anti-prisms (triangular prisms, pentagonal prisms, hexagonal prisms, and more).
		In addition, Poly can generate Johnson solids (square pyramid, pentagonal pyramid, triangular cupola, and more), Catalan solids (Triakis tetrahedron, rhombic dodecahedron, Triakis octahedron, and more), and dipyramids and deltohedrons (triangular dipyramid, pentagonal dipyramid, hexagonal dipyramid, and more).
	www.softlookup.com /download.asp?id=16053 (314 words)

	Archimedean solids and Catalan solids, the convex semi-regular polyhedra
		Archimedean solids and Catalan solids, the convex semi-regular polyhedra
		The Archimedean solids are the 13 convex semi-regular polyhedra, excluding the infinite set of prisms and antiprisms.
		The duals of these models are called the Catalan solids.
	web.aanet.com.au /robertw/Archimedean.html (0 words)

	Reference.com/Encyclopedia/Catalan solid
		In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid.
		This is because the dual Archimedean solids are vertex-uniform and not face-uniform.
		Eugène Catalan Mémoire sur la Théorie des Polyèdres.
	www.reference.com /browse/wiki/Catalan_solid (207 words)

	The International Bone Rollers' Guild (Site not responding. Last check: )
		For the most part, however, the Archemedean Solids, Prisms, and Anti-Prisms are not suitable for use as dice...
		The Archimedian Duals are also known as a the "Catalan Solids".
		Their facets are not "perfect" regular polygons, but all the facets of each solid are identical.
	members.aol.com /dicetalk/polyh3.htm (590 words)

	Catalan Rustic Wood Square Coffee Cocktail Table w/ Storage
		The beautiful solid wood construction of this coffee table set carries an elegance and weight that really sets these pieces apart.
		Features include all hardwood solids in a rich lumber finish and deep chiseled legs in pilaster shapes.
		The "Catalan" collection includes matching Sofa/Console Table, Cocktail Table, Rustic Server/Hutch that are available in our website.
	www.efurniturehouse.com /browseproducts/Catalan-Rustic-Wood-Square-Coffee-Cocktail-Table-w--Storage.html (272 words)

	Polyhedra
		They can be generated from the platonics by slicing off the corners (truncating), expanding them and filling in the gaps with regular polyhedra and a transformation called the snub.
		The Catalan Solidsare a transformation of the Archimedians.
		An antiprism, a pyramid, a prism and the associated Catalan solids are shown.
	members.ozemail.com.au /~llan/polhed.html (0 words)

	Computer Graphics
		Recall that the Archimedean Solids are convex polyhedra composed entirely of regular polygons, but unlike the Platonic solids they are not each composed entirely of one type of regular polygon.
		Exercise: Check that you can see that any archimedean solid has a unique midradius which each edge is tangent to, even in the case of a snub cube and a snub dodecahedron.
		We have briefly explored geometry calculations for Platonic, Archimedean and Catalan Solids, demonstrating a need to find views which help us analyse symmetry and showing that trigonometry can often be avoided.
	www.soi.city.ac.uk /~dcd/ig/s5pol2/laCalcP/l09.htm (1721 words)

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