The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric vertices. Some Archimedean solids were portrayed in the works of artists and mathematicians during the Renaissance.
The elongated square gyrobicupola or pseudorhombicuboctahedron is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive.
The solids
The Archimedean solids have a single vertex configuration and highly symmetric properties. A vertex configuration indicates which regular polygons meet at each vertex. For instance, the configuration 
| Name | Solids | Vertex configurations | Faces | Edges | Vertices | Point group |
|---|---|---|---|---|---|---|
| Truncated tetrahedron |  | 3.6.6 | 4 triangles 4 hexagons | 18 | 12 | Td |
| Cuboctahedron |  | 3.4.3.4 | 8 triangles 6 squares | 24 | 12 | Oh |
| Truncated cube |  | 3.8.8 | 8 triangles 6 octagons | 36 | 24 | Oh |
| Truncated octahedron |  | 4.6.6 | 6 squares 8 hexagons | 36 | 24 | Oh |
| Rhombicuboctahedron |  | 3.4.4.4 | 8 triangles 18 squares | 48 | 24 | Oh |
| Truncated cuboctahedron |  | 4.6.8 | 12 squares 8 hexagons 6 octagons | 72 | 48 | Oh |
| Snub cube |  | 3.3.3.3.4 | 32 triangles 6 squares | 60 | 24 | O |
| Icosidodecahedron |  | 3.5.3.5 | 20 triangles 12 pentagons | 60 | 30 | Ih |
| Truncated dodecahedron |  | 3.10.10 | 20 triangles 12 decagons | 90 | 60 | Ih |
| Truncated icosahedron |  | 5.6.6 | 12 pentagons 20 hexagons | 90 | 60 | Ih |
| Rhombicosidodecahedron |  | 3.4.5.4 | 20 triangles 30 squares 12 pentagons | 120 | 60 | Ih |
| Truncated icosidodecahedron |  | 4.6.10 | 30 squares 20 hexagons 12 decagons | 180 | 120 | Ih |
| Snub dodecahedron |  | 3.3.3.3.5 | 80 triangles 12 pentagons | 150 | 60 | I |
The construction of some Archimedean solids begins from the Platonic solids. The truncation involves cutting away 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, and an example can be found in truncated icosahedron constructed by cutting off all the icosahedron's vertices, having the same symmetry as the icosahedron, the icosahedral symmetry. 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. Expansion involves moving each face away from the center (by the same distance to preserve the symmetry of the Platonic solid) and taking the convex hull. An example is the rhombicuboctahedron, constructed by separating the cube or octahedron's faces from the centroid and filling them with squares.Snub is a construction process of polyhedra by separating the polyhedron faces, twisting their faces in certain angles, and filling them up with equilateral triangles. Examples can be found in snub cube and snub dodecahedron. The resulting construction of these solids gives the property of chiral, meaning they are not identical when reflected in a mirror. However, not all of them can be constructed in such a way, or they could be constructed alternatively. For example, the icosidodecahedron can be constructed by attaching two pentagonal rotunda base-to-base, or rhombicuboctahedron that can be constructed alternatively by attaching two square cupolas on the bases of octagonal prism.
At least ten of the Archimedean solids have the Rupert property: each can pass through a copy of itself, of the same size. They are the cuboctahedron, truncated octahedron, truncated cube, rhombicuboctahedron, icosidodecahedron, truncated cuboctahedron, truncated icosahedron, truncated dodecahedron, and the truncated tetrahedron.
The dual polyhedron of an Archimedean solid is a Catalan solid.
Background of discovery
The names of Archimedean solids were taken from Ancient Greek mathematician Archimedes, who discussed them in a now-lost work. Although they were not credited to Archimedes originally, Pappus of Alexandria in the fifth section of his titled compendium Synagoge referring that Archimedes listed thirteen polyhedra and briefly described them in terms of how many faces of each kind these polyhedra have.
During the Renaissance, artists and mathematicians valued pure forms with high symmetry. Some Archimedean solids appeared in Piero della Francesca's De quinque corporibus regularibus, in attempting to study and copy the works of Archimedes, as well as include citations to Archimedes. Yet, he did not credit those shapes to Archimedes and know of Archimedes' work but rather appeared to be an independent rediscovery. Other appearance of the solids appeared in the works of Wenzel Jamnitzer's Perspectiva Corporum Regularium, and both Summa de arithmetica and Divina proportione by Luca Pacioli, drawn by Leonardo da Vinci. The net of Archimedean solids appeared in Albrecht Dürer's Underweysung der Messung, copied from the Pacioli's work. By around 1620, Johannes Kepler in his Harmonices Mundi had completed the rediscovery of the thirteen polyhedra, as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler–Poinsot polyhedra.
Kepler may have also found another solid known as elongated square gyrobicupola or pseudorhombicuboctahedron. Kepler once stated that there were fourteen Archimedean solids, yet his published enumeration only includes the thirteen uniform polyhedra. The first clear statement of such solid existence was made by Duncan Sommerville in 1905. The solid appeared when some mathematicians mistakenly constructed the rhombicuboctahedron: two square cupolas attached to the octagonal prism, with one of them rotated in forty-five degrees. The thirteen solids have the property of vertex-transitive, meaning any two vertices of those can be translated onto the other one, but the elongated square gyrobicupola does not. Grünbaum (2009) observed that it meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the parts of the polyhedron near any two vertices look the same (they have the same shapes of faces meeting around each vertex in the same order and forming the same angles). Grünbaum pointed out a frequent error in which authors define Archimedean solids using some form of this local definition but omit the fourteenth polyhedron. If only thirteen polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods. In the aftermath, the elongated square gyrobicupola was withdrawn from the Archimedean solids and included into the Johnson solid instead, a convex polyhedron in which all of the faces are regular polygons.
See also
- Archimedean graph, planar graphs resembling the thirteen Archimedean solids.
- Conway polyhedron notation
References
Footnotes
- Diudea (2018), p. 39.
- Kinsey, Moore & Prassidis (2011), p. 380.
-
- Rovenski (2010), p. 116
- Malkevitch (1988), p. 85
- Williams (1979).
- Berman (1971).
- Koca & Koca (2013), p. 47–50.
-
- Chancey & O'Brien (1997), p. 13
- Koca & Koca (2013), p. 48
- Viana et al. (2019), p. 1123, See Fig. 6.
- Koca & Koca (2013), p. 49.
-
- Cromwell (1997), p. 156
- Grünbaum (2009)
- Field (1997), p. 248
- Banker (2005).
- Field (1997), p. 248.
-
- Cromwell (1997), p. 156
- Field (1997), p. 253–254
- Schreiber, Fischer & Sternath (2008).
- Grünbaum (2009).
-
- Cromwell (1997), p. 91
- Berman (1971)
Works cited
- Banker, James R. (March 2005), "A manuscript of the works of Archimedes in the hand of Piero della Francesca", The Burlington Magazine, 147 (1224): 165–169, JSTOR 20073883, S2CID 190211171.
- Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245.
- Chai, Ying; Yuan, Liping; Zamfirescu, Tudor (2018), "Rupert Property of Archimedean Solids", The American Mathematical Monthly, 125 (6): 497–504, doi:10.1080/00029890.2018.1449505, S2CID 125508192.
- Chancey, C. C.; O'Brien, M. C. M. (1997), The Jahn-Teller Effect in C60 and Other Icosahedral Complexes, Princeton University Press, ISBN 978-0-691-22534-0.
- Cromwell, Peter R. (1997), Polyhedra, Cambridge University Press, ISBN 978-0-521-55432-9.
- Diudea, M. V. (2018), Multi-shell Polyhedral Clusters, Carbon Materials: Chemistry and Physics, vol. 10, Springer, doi:10.1007/978-3-319-64123-2, ISBN 978-3-319-64123-2.
- Field, J. V. (1997), "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 (3–4): 241–289, doi:10.1007/BF00374595, JSTOR 41134110, MR 1457069, S2CID 118516740.
- Grünbaum, Branko (2009), "An enduring error" (PDF), 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.
- Hoffmann, Balazs (2019), "Rupert properties of polyhedra and the generalized Nieuwland constant", Journal for Geometry and Graphics, 23 (1): 29–35
- Kinsey, L. Christine; Moore, Teresa E.; Prassidis, Efstratios (2011), Geometry and Symmetry, John Wiley & Sons, ISBN 978-0-470-49949-8.
- Koca, M.; Koca, N. O. (2013), "Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes", Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010, World Scientific.
- Lavau, Gérard (2019), "The Truncated Tetrahedron is Rupert", The American Mathematical Monthly, 126 (10): 929–932, doi:10.1080/00029890.2019.1656958, S2CID 213502432.
- 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.
- Rovenski, Vladimir (2010), Modeling of Curves and Surfaces with MATLAB®, Springer Undergraduate Texts in Mathematics and Technology, Springer, doi:10.1007/978-0-387-71278-9, ISBN 978-0-387-71278-9.
- 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, JSTOR 41134285, S2CID 122216140.
- Viana, Vera; Xavier, João Pedro; Aires, Ana Paula; Campos, Helena (2019), "Interactive Expansion of Achiral Polyhedra", in Cocchiarella, Luigi (ed.), ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics 40th Anniversary - Milan, Italy, August 3-7, 2018, Springer, doi:10.1007/978-3-319-95588-9, ISBN 978-3-319-95587-2.
- Williams, Robert (1979), The Geometrical Foundation of Natural Structure: A Source Book of Design, Dover Publications, Inc., ISBN 978-0-486-23729-9.
Further reading
- Viana, Vera (2024), "Archimedean solids in the fifteenth and sixteenth centuries", Archive for History of Exact Sciences, 78 (6): 631–715, doi:10.1007/s00407-024-00331-7.
- Williams, Kim; Monteleone, Cosimo (2021), Daniele Barbaro's Perspective of 1568, p. 19–20, doi:10.1007/978-3-030-76687-0, ISBN 978-3-030-76687-0.
External links
- Weisstein, 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. 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
- 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
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons but not all alike and whose vertices are all symmetric to each other The solids were named after Archimedes although he did not claim credit for them They belong to the class of uniform polyhedra the polyhedra with regular faces and symmetric vertices Some Archimedean solids were portrayed in the works of artists and mathematicians during the Renaissance The Archimedean solids Two of them are chiral with both forms shown making 15 models in all The elongated square gyrobicupola or pseudo rhombi cub octa hedron is an extra polyhedron with regular faces and congruent vertices but it is not generally counted as an Archimedean solid because it is not vertex transitive The solidsThe Archimedean solids have a single vertex configuration and highly symmetric properties A vertex configuration indicates which regular polygons meet at each vertex For instance the configuration 3 5 3 5 displaystyle 3 cdot 5 cdot 3 cdot 5 indicates a polyhedron in which each vertex is met by alternating two triangles and two pentagons Highly symmetric properties in this case mean the symmetry group of each solid were derived from the Platonic solids resulting from their construction Some sources say the Archimedean solids are synonymous with the semiregular polyhedron Yet the definition of a semiregular polyhedron may also include the infinite prisms and antiprisms including the elongated square gyrobicupola The thirteen Archimedean solids Name Solids Vertex configurations Faces Edges Vertices Point groupTruncated tetrahedron 3 6 6 4 triangles 4 hexagons 18 12 TdCuboctahedron 3 4 3 4 8 triangles 6 squares 24 12 OhTruncated cube 3 8 8 8 triangles 6 octagons 36 24 OhTruncated octahedron 4 6 6 6 squares 8 hexagons 36 24 OhRhombicuboctahedron 3 4 4 4 8 triangles 18 squares 48 24 OhTruncated cuboctahedron 4 6 8 12 squares 8 hexagons 6 octagons 72 48 OhSnub cube 3 3 3 3 4 32 triangles 6 squares 60 24 OIcosidodecahedron 3 5 3 5 20 triangles 12 pentagons 60 30 IhTruncated dodecahedron 3 10 10 20 triangles 12 decagons 90 60 IhTruncated icosahedron 5 6 6 12 pentagons 20 hexagons 90 60 IhRhombicosidodecahedron 3 4 5 4 20 triangles 30 squares 12 pentagons 120 60 IhTruncated icosidodecahedron 4 6 10 30 squares 20 hexagons 12 decagons 180 120 IhSnub dodecahedron 3 3 3 3 5 80 triangles 12 pentagons 150 60 I The construction of some Archimedean solids begins from the Platonic solids The truncation involves cutting away 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 and an example can be found in truncated icosahedron constructed by cutting off all the icosahedron s vertices having the same symmetry as the icosahedron the icosahedral symmetry 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 Expansion involves moving each face away from the center by the same distance to preserve the symmetry of the Platonic solid and taking the convex hull An example is the rhombicuboctahedron constructed by separating the cube or octahedron s faces from the centroid and filling them with squares Snub is a construction process of polyhedra by separating the polyhedron faces twisting their faces in certain angles and filling them up with equilateral triangles Examples can be found in snub cube and snub dodecahedron The resulting construction of these solids gives the property of chiral meaning they are not identical when reflected in a mirror However not all of them can be constructed in such a way or they could be constructed alternatively For example the icosidodecahedron can be constructed by attaching two pentagonal rotunda base to base or rhombicuboctahedron that can be constructed alternatively by attaching two square cupolas on the bases of octagonal prism At least ten of the Archimedean solids have the Rupert property each can pass through a copy of itself of the same size They are the cuboctahedron truncated octahedron truncated cube rhombicuboctahedron icosidodecahedron truncated cuboctahedron truncated icosahedron truncated dodecahedron and the truncated tetrahedron The dual polyhedron of an Archimedean solid is a Catalan solid Background of discoveryThe names of Archimedean solids were taken from Ancient Greek mathematician Archimedes who discussed them in a now lost work Although they were not credited to Archimedes originally Pappus of Alexandria in the fifth section of his titled compendium Synagoge referring that Archimedes listed thirteen polyhedra and briefly described them in terms of how many faces of each kind these polyhedra have Truncated icosahedron in De quinque corporibus regularibusRhombicuboctahedron drawn by Leonardo da Vinci colorized Cuboctahedron in Perspectiva Corporum Regularium During the Renaissance artists and mathematicians valued pure forms with high symmetry Some Archimedean solids appeared in Piero della Francesca s De quinque corporibus regularibus in attempting to study and copy the works of Archimedes as well as include citations to Archimedes Yet he did not credit those shapes to Archimedes and know of Archimedes work but rather appeared to be an independent rediscovery Other appearance of the solids appeared in the works of Wenzel Jamnitzer s Perspectiva Corporum Regularium and both Summa de arithmetica and Divina proportione by Luca Pacioli drawn by Leonardo da Vinci The net of Archimedean solids appeared in Albrecht Durer s Underweysung der Messung copied from the Pacioli s work By around 1620 Johannes Kepler in his Harmonices Mundi had completed the rediscovery of the thirteen polyhedra as well as defining the prisms antiprisms and the non convex solids known as Kepler Poinsot polyhedra The elongated square gyrobicupola a polyhedron where mathematicians mistakenly constructed the rhombicuboctahedron Kepler may have also found another solid known as elongated square gyrobicupola or pseudorhombicuboctahedron Kepler once stated that there were fourteen Archimedean solids yet his published enumeration only includes the thirteen uniform polyhedra The first clear statement of such solid existence was made by Duncan Sommerville in 1905 The solid appeared when some mathematicians mistakenly constructed the rhombicuboctahedron two square cupolas attached to the octagonal prism with one of them rotated in forty five degrees The thirteen solids have the property of vertex transitive meaning any two vertices of those can be translated onto the other one but the elongated square gyrobicupola does not Grunbaum 2009 observed that it meets a weaker definition of an Archimedean solid in which identical vertices means merely that the parts of the polyhedron near any two vertices look the same they have the same shapes of faces meeting around each vertex in the same order and forming the same angles Grunbaum pointed out a frequent error in which authors define Archimedean solids using some form of this local definition but omit the fourteenth polyhedron If only thirteen polyhedra are to be listed the definition must use global symmetries of the polyhedron rather than local neighborhoods In the aftermath the elongated square gyrobicupola was withdrawn from the Archimedean solids and included into the Johnson solid instead a convex polyhedron in which all of the faces are regular polygons See alsoArchimedean graph planar graphs resembling the thirteen Archimedean solids Conway polyhedron notationReferencesFootnotes Diudea 2018 p 39 Kinsey Moore amp Prassidis 2011 p 380 Rovenski 2010 p 116Malkevitch 1988 p 85 Williams 1979 Berman 1971 Koca amp Koca 2013 p 47 50 Chancey amp O Brien 1997 p 13Koca amp Koca 2013 p 48 Viana et al 2019 p 1123 See Fig 6 Koca amp Koca 2013 p 49 Chai Yuan amp Zamfirescu 2018 Hoffmann 2019 Lavau 2019 Cromwell 1997 p 156Grunbaum 2009 Field 1997 p 248 Banker 2005 Field 1997 p 248 Cromwell 1997 p 156Field 1997 p 253 254 Schreiber Fischer amp Sternath 2008 Grunbaum 2009 Cromwell 1997 p 91Berman 1971 Works cited Banker James R March 2005 A manuscript of the works of Archimedes in the hand of Piero della Francesca The Burlington Magazine 147 1224 165 169 JSTOR 20073883 S2CID 190211171 Berman Martin 1971 Regular faced convex polyhedra Journal of the Franklin Institute 291 5 329 352 doi 10 1016 0016 0032 71 90071 8 MR 0290245 Chai Ying Yuan Liping Zamfirescu Tudor 2018 Rupert Property of Archimedean Solids The American Mathematical Monthly 125 6 497 504 doi 10 1080 00029890 2018 1449505 S2CID 125508192 Chancey C C O Brien M C M 1997 The Jahn Teller Effect in C60 and Other Icosahedral Complexes Princeton University Press ISBN 978 0 691 22534 0 Cromwell Peter R 1997 Polyhedra Cambridge University Press ISBN 978 0 521 55432 9 Diudea M V 2018 Multi shell Polyhedral Clusters Carbon Materials Chemistry and Physics vol 10 Springer doi 10 1007 978 3 319 64123 2 ISBN 978 3 319 64123 2 Field J V 1997 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 3 4 241 289 doi 10 1007 BF00374595 JSTOR 41134110 MR 1457069 S2CID 118516740 Grunbaum Branko 2009 An enduring error PDF 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 Hoffmann Balazs 2019 Rupert properties of polyhedra and the generalized Nieuwland constant Journal for Geometry and Graphics 23 1 29 35 Kinsey L Christine Moore Teresa E Prassidis Efstratios 2011 Geometry and Symmetry John Wiley amp Sons ISBN 978 0 470 49949 8 Koca M Koca N O 2013 Coxeter groups quaternions symmetries of polyhedra and 4D polytopes Mathematical Physics Proceedings of the 13th Regional Conference Antalya Turkey 27 31 October 2010 World Scientific Lavau Gerard 2019 The Truncated Tetrahedron is Rupert The American Mathematical Monthly 126 10 929 932 doi 10 1080 00029890 2019 1656958 S2CID 213502432 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 Rovenski Vladimir 2010 Modeling of Curves and Surfaces with MATLAB Springer Undergraduate Texts in Mathematics and Technology Springer doi 10 1007 978 0 387 71278 9 ISBN 978 0 387 71278 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 JSTOR 41134285 S2CID 122216140 Viana Vera Xavier Joao Pedro Aires Ana Paula Campos Helena 2019 Interactive Expansion of Achiral Polyhedra in Cocchiarella Luigi ed ICGG 2018 Proceedings of the 18th International Conference on Geometry and Graphics 40th Anniversary Milan Italy August 3 7 2018 Springer doi 10 1007 978 3 319 95588 9 ISBN 978 3 319 95587 2 Williams Robert 1979 The Geometrical Foundation of Natural Structure A Source Book of Design Dover Publications Inc ISBN 978 0 486 23729 9 Further reading Viana Vera 2024 Archimedean solids in the fifteenth and sixteenth centuries Archive for History of Exact Sciences 78 6 631 715 doi 10 1007 s00407 024 00331 7 Williams Kim Monteleone Cosimo 2021 Daniele Barbaro s Perspective of 1568 p 19 20 doi 10 1007 978 3 030 76687 0 ISBN 978 3 030 76687 0 External linksWeisstein 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
