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Encyclopedia > Archimedean solid

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. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ... In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... Wiktionary has a definition of: Polygon For other use please see Polygon (disambiguation) A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. ... The Elongated square gyrobicupola, a Johnson solid A Johnson solid is a convex polyhedron each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism or antiprism. ...


Origin of name

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. During the Renaissance, artists and mathematicians valued pure forms and rediscovered all of these forms. This search was completed around 1619 by Johannes Kepler, who defined prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot solids. Archimedes of Syracuse (circa 287 BC - 212 BC), was a Greek mathematician, astronomer, philosopher, physicist and engineer. ... By Region: Italian Renaissance Northern Renaissance -French Renaissance -German Renaissance -English Renaissance The Renaissance was an influential cultural movement which brought about a period of scientific revolution and artistic transformation, at the dawn of modern European history. ... An artist is someone who employs creative talent to produce works of art. ... A mathematician is a person whose area of study and research is mathematics. ... Events May 13 - Dutch statesman Johan van Oldenbarnevelt is executed in The Hague after having been accused of treason. ... Johannes Kepler Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German astronomer, mathematician and astrologer. ... A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ...


Classification

There are 13 Archimedean solids (15 if the mirror images of two enantiomorphs, see below, are counted separately). Here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a vertex configuration of (4,6,8) means that a square, hexagon, and octagon meet at a vertex (with the order taken to be clockwise around the vertex). In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly cant be mapped to its mirror images by rotations and translations alone. ...

Name and picture Faces Edges Vertices Vertex configuration Symmetry group
cuboctahedron

( Video)
 14  8 triangles
6 squares
24 12 3,4,3,4 Oh
icosidodecahedron

( Video)
32 20 triangles
12 pentagons
60 30 3,5,3,5 Ih
truncated tetrahedron

( Video)
8 4 triangles
4 hexagons
18 12 3,6,6 Td
truncated cube
or truncated hexahedron

( Video)
14 8 triangles
6 octagons
36 24 3,8,8 Oh
truncated octahedron

( Video)
14 6 squares
8 hexagons
36 24 4,6,6 Oh
truncated dodecahedron

( Video)
32 20 triangles
12 decagons
90 60 3,10,10 Ih
truncated icosahedron
or commonly football (soccer ball)

( Video)
32 12 pentagons
20 hexagons
90 60 5,6,6 Ih
rhombicuboctahedron
or small rhombicuboctahedron

( Video)
26 8 triangles
18 squares
48 24 3,4,4,4 Oh
truncated cuboctahedron
or great rhombicuboctahedron

( Video)
26 12 squares
8 hexagons
6 octagons
72 48 4,6,8 Oh
rhombicosidodecahedron
or small rhombicosidodecahedron

( Video)
62 20 triangles
30 squares
12 pentagons
120 60 3,4,5,4 Ih
truncated icosidodecahedron
or great rhombicosidodecahedron

( Video)
62 30 squares
20 hexagons
12 decagons
180 120 4,6,10 Ih
snub cube
or snub cuboctahedron (2 chiral forms)

( Video)

( Video)
38 32 triangles
6 squares
60 24 3,3,3,3,4 O
snub dodecahedron
or snub icosidodecahedron (2 chiral forms)

( Video)

( Video)
92 80 triangles
12 pentagons
150 60 3,3,3,3,5 I

The first two solids (cuboctahedron and icosidodecahedron) are edge-uniform and are called quasi-regular. The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Download high resolution version (818x804, 71 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning cuboctahedron, made by me using POV-Ray, see image:poly. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In plane geometry, a square is a polygon with four equal sides and equal angles. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... Download high resolution version (841x861, 76 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning icosidodecahedron, made by me using POV-Ray, see image:poly. ... In geometry, a pentagon is any five-sided polygon. ... The truncated tetrahedron is an Archimedean solid. ... Download high resolution version (867x773, 50 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made using POV-Ray. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Download high resolution version (819x855, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see :image:poly. ... An octagon is a polygon that has eight sides. ... The truncated octahedron is an Archimedean solid. ... Download high resolution version (857x789, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see :image:poly. ... The truncated dodecahedron is an Archimedean solid. ... Download high resolution version (876x802, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated dodecahedron, made by me using POV-Ray, see :image:poly. ... An image of a Regular Decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and all angles equal to 144°. Its Schläfli symbol is {10}. The area of a regular decagon... The truncated icosahedron is an Archimedean solid. ... Download high resolution version (878x874, 80 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated icosahedron, made by me using POV-Ray, see :image:poly. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Download high resolution version (823x836, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see :image:poly. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Download high resolution version (855x868, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see image:poly. ... The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ... Download high resolution version (858x871, 113 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see image:poly. ... The truncated icosidodecahedron, or great rhombicosidodecahedron, is an Archimedean solid. ... Download high resolution version (868x869, 105 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see image:poly. ... The snub cube, or snub cuboctahedron, is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a cube or an octahedron. ... Download high resolution version (837x817, 96 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (843x833, 97 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see image:poly. ... The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a dodecahedron or an icosahedron. ... Download high resolution version (853x866, 121 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning snub dodecahedron (ccw), made by me using POV-Ray, see image:poly. ... Download high resolution version (872x877, 116 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning snub dodecahedron (cw), made by me using POV-Ray, see image:poly. ...


The last two (snub cube and snub dodecahedron) are known as chiral, as they come in a left-handed (Latin: levomorph or laevomorph) form and right-handed (Latin: dextromorph) form. When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of chemical compounds). In geometry, the mirror image of an object or two-dimensional figure is the virtual image formed by a plane mirror; it is of the same size as the original object, yet different, unless the object or figure has mirror-image symmetry (also known in the terminology of modern physics... A chemical compound is a chemical substance formed from two or more elements, with a fixed ratio determining the composition. ...


The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... A rhombic dodecahedron In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. ... A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... The trapezohedra are the Dual polyhedrons of the antiprisms. ...


External links

  • Paper models of Archimedean solids (http://www.software3d.com/Archimedean.html)
  • The Uniform Polyhedra (http://www.mathconsult.ch/showroom/unipoly/)
  • Virtual Reality Polyhedra (http://www.georgehart.com/virtual-polyhedra/vp.html) The Encyclopedia of Polyhedra
  • Penultimate Modular Origami (http://www.cs.utk.edu/~plank/plank/origami/penultimate/intro.html)
  • Interactive 3D polyhedra (http://ibiblio.org/e-notes/3Dapp/Convex.htm) in Java

  Results from FactBites:
 
Welcome To Archimedean 1.0 (1172 words)
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.
  More results at FactBites »

 
 

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