"Identical vertices" means that each two vertices are symmetric to each other: A global isometry of the entire solid takes one vertex to the other while laying the solid directly on its initial position. Branko Grünbaum (2009) observed that a 14th polyhedron, the elongated square gyrobicupola (or pseudo-rhombicuboctahedron), meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a local isometry is required. Grünbaum pointed out a frequent error in which authors define Archimedean solids using this local definition but omit the 14th polyhedron. If only 13 polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods.
Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, even though their faces are regular polygons and their symmetry groups act transitively on their vertices. Excluding these two infinite families, there are 13 Archimedean solids. All the Archimedean solids (but not the elongated square gyrobicupola) can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.
The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra.[2] During the Renaissance, artists and mathematicians valued pure forms with high symmetry, and by around 1620 Johannes Kepler had completed the rediscovery of the 13 polyhedra,[3] as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot polyhedra. (See Schreiber, Fischer & Sternath 2008 for more information about the rediscovery of the Archimedean solids during the renaissance.)
Kepler may have also found the elongated square gyrobicupola (pseudorhombicuboctahedron): at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, and the first clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by Duncan Sommerville.[2]
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).
The snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form (Latin: levomorph or laevomorph) and right-handed form (Latin: dextromorph). 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 certain chemical compounds.)
The different Archimedean and Platonic solids can be related to each other using a handful of general constructions. Starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated (see table below), different Platonic and Archimedean (and other) solids can be created. If the truncation is exactly deep enough such that each pair of faces from adjacent vertices shares exactly one point, it is known as a rectification. An expansion, or cantellation, involves moving each face away from the center (by the same distance so as to preserve the symmetry of the Platonic solid) and taking the convex hull. Expansion with twisting also involves rotating the faces, thus splitting each rectangle corresponding to an edge into two triangles by one of the diagonals of the rectangle. The last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as the rectification of the rectification. Likewise, the cantitruncation can be viewed as the truncation of the rectification.
Note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron. Also, partially because the tetrahedron is self-dual, only one Archimedean solid that has at most tetrahedral symmetry. (All Platonic solids have at least tetrahedral symmetry, as tetrahedral symmetry is a symmetry operation of (i.e. is included in) octahedral and isohedral symmetries, which is demonstrated by the fact that an octahedron can be viewed as a rectified tetrahedron, and an icosahedron can be used as a snub tetrahedron.)
^Field J., Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler, Archive for History of Exact Sciences, 50, 1997, 227
Grünbaum, Branko (2009), "An enduring error", Elemente der Mathematik, 64 (3): 89–101, doi:10.4171/EM/120, MR 2520469. Reprinted in Pitici, Mircea, ed. (2011), The Best Writing on Mathematics 2010, Princeton University Press, pp. 18–31.
Malkevitch, Joseph (1988), "Milestones in the history of polyhedra", in Senechal, M.; Fleck, G. (eds.), Shaping Space: A Polyhedral Approach, Boston: Birkhäuser, pp. 80–92.
General references
Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette. 89 (514): 76–81. doi:10.1017/S0025557200176818. S2CID 125675814..
Pugh, Anthony (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN0-520-03056-7. Chapter 2
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN0-486-23729-X. (Section 3–9)
Schreiber, Peter; Fischer, Gisela; Sternath, Maria Luise (2008). "New light on the rediscovery of the Archimedean solids during the renaissance". Archive for History of Exact Sciences. 62 (4): 457–467. Bibcode:2008AHES...62..457S. doi:10.1007/s00407-008-0024-z. ISSN 0003-9519. S2CID 122216140..
Paper models of Archimedean Solids and Catalan Solids
Free paper models(nets) of Archimedean solids
The Uniform Polyhedra by Dr. R. Mäder
Archimedean Solids at Visual Polyhedra by David I. McCooey
Virtual Reality Polyhedra, The Encyclopedia of Polyhedra by George W. Hart
Penultimate Modular Origami by James S. Plank
in Java
Solid Body Viewer is an interactive 3D polyhedron viewer which allows you to save the model in svg, stl or obj format.
Stella: Polyhedron Navigator: Software used to create many of the images on this page.
Paper Models of Archimedean (and other) Polyhedra
March 10, 2023
archimedean, solid, geometry, solids, first, enumerated, archimedes, they, convex, uniform, polyhedra, composed, regular, polygons, meeting, identical, vertices, excluding, five, platonic, solids, which, composed, only, type, polygon, excluding, prisms, antipr. In geometry an Archimedean solid is one of the 13 solids first enumerated by Archimedes They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices excluding the five Platonic solids which are composed of only one type of polygon excluding the prisms and antiprisms and excluding the pseudorhombicuboctahedron 1 They are a subset of the Johnson solids whose regular polygonal faces do not need to meet in identical vertices Truncated tetrahedron cuboctahedron and truncated icosidodecahedron The first and the last one can be described as the smallest and the largest Archimedean solid respectively Rhombicuboctahedron and pseudo rhombicuboctahedron Identical vertices means that each two vertices are symmetric to each other A global isometry of the entire solid takes one vertex to the other while laying the solid directly on its initial position Branko Grunbaum 2009 observed that a 14th polyhedron the elongated square gyrobicupola or pseudo rhombicuboctahedron meets a weaker definition of an Archimedean solid in which identical vertices means merely that the faces surrounding each vertex are of the same types i e each vertex looks the same from close up so only a local isometry is required Grunbaum pointed out a frequent error in which authors define Archimedean solids using this local definition but omit the 14th polyhedron If only 13 polyhedra are to be listed the definition must use global symmetries of the polyhedron rather than local neighborhoods Prisms and antiprisms whose symmetry groups are the dihedral groups are generally not considered to be Archimedean solids even though their faces are regular polygons and their symmetry groups act transitively on their vertices Excluding these two infinite families there are 13 Archimedean solids All the Archimedean solids but not the elongated square gyrobicupola can be made via Wythoff constructions from the Platonic solids with tetrahedral octahedral and icosahedral symmetry Contents 1 Origin of name 2 Classification 3 Properties 3 1 Chirality 4 Construction of Archimedean solids 5 Stereographic projection 6 See also 7 Citations 7 1 Works cited 8 General references 9 External linksOrigin of name EditThe Archimedean solids take their name from Archimedes who discussed them in a now lost work Pappus refers to it stating that Archimedes listed 13 polyhedra 2 During the Renaissance artists and mathematicians valued pure forms with high symmetry and by around 1620 Johannes Kepler had completed the rediscovery of the 13 polyhedra 3 as well as defining the prisms antiprisms and the non convex solids known as Kepler Poinsot polyhedra See Schreiber Fischer amp Sternath 2008 for more information about the rediscovery of the Archimedean solids during the renaissance Kepler may have also found the elongated square gyrobicupola pseudorhombicuboctahedron at least he once stated that there were 14 Archimedean solids However his published enumeration only includes the 13 uniform polyhedra and the first clear statement of the pseudorhombicuboctahedron s existence was made in 1905 by Duncan Sommerville 2 Classification EditThere are 13 Archimedean solids not counting the elongated square gyrobicupola 15 if the mirror images of two enantiomorphs the snub cube and snub dodecahedron 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 Name alternative name SchlafliCoxeter Transparent Solid Net Vertexconf fig Faces Edges Vert Volume unit edges Pointgroup SphericityTruncated tetrahedron t 3 3 3 6 6 8 4 triangles4 hexagons 18 12 2 710576 Td 0 7754132Cuboctahedron rhombitetratetrahedron triangular gyrobicupola r 4 3 or rr 3 3 or 3 4 3 4 14 8 triangles6 squares 24 12 2 357023 Oh 0 9049972Truncated cube t 4 3 3 8 8 14 8 triangles6 octagons 36 24 13 599663 Oh 0 8494937Truncated octahedron truncated tetratetrahedron t 3 4 or tr 3 3 or 4 6 6 14 6 squares8 hexagons 36 24 11 313709 Oh 0 9099178Rhombicuboctahedron small rhombicuboctahedron elongated square orthobicupola rr 4 3 3 4 4 4 26 8 triangles18 squares 48 24 8 714045 Oh 0 9540796Truncated cuboctahedron great rhombicuboctahedron tr 4 3 4 6 8 26 12 squares8 hexagons6 octagons 72 48 41 798990 Oh 0 9431657Snub cube snub cuboctahedron sr 4 3 3 3 3 3 4 38 32 triangles6 squares 60 24 7 889295 O 0 9651814Icosidodecahedron pentagonal gyrobirotunda r 5 3 3 5 3 5 32 20 triangles12 pentagons 60 30 13 835526 Ih 0 9510243Truncated dodecahedron t 5 3 3 10 10 32 20 triangles12 decagons 90 60 85 039665 Ih 0 9260125Truncated icosahedron t 3 5 5 6 6 32 12 pentagons20 hexagons 90 60 55 287731 Ih 0 9666219Rhombicosidodecahedron small rhombicosidodecahedron rr 5 3 3 4 5 4 62 20 triangles30 squares12 pentagons 120 60 41 615324 Ih 0 9792370Truncated icosidodecahedron great rhombicosidodecahedron tr 5 3 4 6 10 62 30 squares20 hexagons12 decagons 180 120 206 803399 Ih 0 9703127Snub dodecahedron snub icosidodecahedron sr 5 3 3 3 3 3 5 92 80 triangles12 pentagons 150 60 37 616650 I 0 9820114Some definitions of Semiregular polyhedron include one more figure the Elongated square gyrobicupola or pseudo rhombicuboctahedron 4 Properties EditThe number of vertices is 720 divided by the vertex angle defect The cuboctahedron and icosidodecahedron are edge uniform and are called quasi regular 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 Chirality Edit The snub cube and snub dodecahedron are known as chiral as they come in a left handed form Latin levomorph or laevomorph and right handed form Latin dextromorph 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 certain chemical compounds Construction of Archimedean solids EditFurther information Uniform polyhedron and Conway polyhedron notation The Archimedean solids can be constructed as generator positions in a kaleidoscope The different Archimedean and Platonic solids can be related to each other using a handful of general constructions Starting with a Platonic solid truncation involves cutting away of corners To preserve symmetry the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners Depending on how much is truncated see table below different Platonic and Archimedean and other solids can be created If the truncation is exactly deep enough such that each pair of faces from adjacent vertices shares exactly one point it is known as a rectification An expansion or cantellation involves moving each face away from the center by the same distance so as to preserve the symmetry of the Platonic solid and taking the convex hull Expansion with twisting also involves rotating the faces thus splitting each rectangle corresponding to an edge into two triangles by one of the diagonals of the rectangle The last construction we use here is truncation of both corners and edges Ignoring scaling expansion can also be viewed as the rectification of the rectification Likewise the cantitruncation can be viewed as the truncation of the rectification Construction of Archimedean Solids Symmetry Tetrahedral Octahedral Icosahedral Starting solidOperation Symbol p q Tetrahedron 3 3 Cube 4 3 Octahedron 3 4 Dodecahedron 5 3 Icosahedron 3 5 Truncation t t p q truncated tetrahedron truncated cube truncated octahedron truncated dodecahedron truncated icosahedron Rectification r Ambo a r p q tetratetrahedron octahedron cuboctahedron icosidodecahedron Bitruncation 2t Dual kis dk 2t p q truncated tetrahedron truncated octahedron truncated cube truncated icosahedron truncated dodecahedron Birectification 2r Dual d 2r p q tetrahedron octahedron cube icosahedron dodecahedron cantellation rr Expansion e rr p q rhombitetratetrahedron cuboctahedron rhombicuboctahedron rhombicosidodecahedron Snub rectified sr Snub s sr p q snub tetratetrahedron icosahedron snub cuboctahedron snub icosidodecahedron Cantitruncation tr Bevel b tr p q truncated tetratetrahedron truncated octahedron truncated cuboctahedron truncated icosidodecahedron Note the duality between the cube and the octahedron and between the dodecahedron and the icosahedron Also partially because the tetrahedron is self dual only one Archimedean solid that has at most tetrahedral symmetry All Platonic solids have at least tetrahedral symmetry as tetrahedral symmetry is a symmetry operation of i e is included in octahedral and isohedral symmetries which is demonstrated by the fact that an octahedron can be viewed as a rectified tetrahedron and an icosahedron can be used as a snub tetrahedron Stereographic projection Edittruncated tetrahedron truncated cube truncated octahedron truncated dodecahedron truncated icosahedron triangle centered hexagon centered octagon centered triangle centered square centered hexagon centered Decagon centered Triangle centered pentagon centered hexagon centeredcuboctahedron icosidodecahedron rhombicuboctahedron rhombicosidodecahedron square centered triangle centered vertex centered pentagon centered triangle centered square centered square centered triangle centered Pentagon centered Triangle centered Square centeredtruncated cuboctahedron truncated icosidodecahedron snub cube square centered hexagon centered octagon centered decagon centered hexagon centered square centered square centeredSee also EditAperiodic tiling Archimedean graph Icosahedral twins List of uniform polyhedra Prince Rupert s cube Generalizations Quasicrystal Regular polyhedron Semiregular polyhedron Toroidal polyhedron Uniform polyhedronCitations Edit Steckles Katie The Unwanted Shape YouTube Retrieved 20 January 2022 a b Grunbaum 2009 Field J Rediscovering the Archimedean Polyhedra Piero della Francesca Luca Pacioli Leonardo da Vinci Albrecht Durer Daniele Barbaro and Johannes Kepler Archive for History of Exact Sciences 50 1997 227 Malkevitch 1988 p 85 Works cited Edit Grunbaum Branko 2009 An enduring error Elemente der Mathematik 64 3 89 101 doi 10 4171 EM 120 MR 2520469 Reprinted in Pitici Mircea ed 2011 The Best Writing on Mathematics 2010 Princeton University Press pp 18 31 Malkevitch Joseph 1988 Milestones in the history of polyhedra in Senechal M Fleck G eds Shaping Space A Polyhedral Approach Boston Birkhauser pp 80 92 General references EditJayatilake Udaya March 2005 Calculations on face and vertex regular polyhedra Mathematical Gazette 89 514 76 81 doi 10 1017 S0025557200176818 S2CID 125675814 Pugh Anthony 1976 Polyhedra A visual approach California University of California Press Berkeley ISBN 0 520 03056 7 Chapter 2 Williams Robert 1979 The Geometrical Foundation of Natural Structure A Source Book of Design Dover Publications Inc ISBN 0 486 23729 X Section 3 9 Schreiber Peter Fischer Gisela Sternath Maria Luise 2008 New light on the rediscovery of the Archimedean solids during the renaissance Archive for History of Exact Sciences 62 4 457 467 Bibcode 2008AHES 62 457S doi 10 1007 s00407 008 0024 z ISSN 0003 9519 S2CID 122216140 External links EditWeisstein Eric W Archimedean solid MathWorld Archimedean Solids by Eric W Weisstein Wolfram Demonstrations Project Paper models of Archimedean Solids and Catalan Solids Free paper models nets of Archimedean solids The Uniform Polyhedra by Dr R Mader Archimedean Solids at Visual Polyhedra by David I McCooey Virtual Reality Polyhedra The Encyclopedia of Polyhedra by George W Hart Penultimate Modular Origami by James S Plank Interactive 3D polyhedra in Java Solid Body Viewer is an interactive 3D polyhedron viewer which allows you to save the model in svg stl or obj format Stella Polyhedron Navigator Software used to create many of the images on this page Paper Models of Archimedean and other Polyhedra Retrieved from https en wikipedia org w index php title Archimedean solid amp oldid 1121154209, wikipedia, wiki, book, books, library,
