Archimedean solid

The Archimedean solids. Two of them are chiral, with both forms shown, making 15 models in all.

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 the vertex configuration and highly symmetric properties. Vertex configuration means a polyhedron whose two or more polygonal faces meet at the vertex. For instance, the $3\cdot 5\cdot 3\cdot 5$ means 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 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.

There are at least for known ten solids that have the Rupert property, a polyhedron that can pass through a copy of itself with the same or similar size. They are the cuboctahedron, truncated octahedron, truncated cube, rhombicuboctahedron, icosidodecahedron, truncated cuboctahedron, truncated icosahedron, truncated dodecahedron, and the truncated tetrahedron. The Catalan solids are the dual polyhedron of Archimedean solids.

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.

Truncated icosahedron in De quinque corporibus regularibus

Rhombicuboctahedron 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 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.

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. 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

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, 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, 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, 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