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<title>Archimedean solid</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Archimedean solid</span></span>

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<p>The <b>Archimedean solids</b> are a set of thirteen <a href="Convex_polyhedra" class="mw-redirect" title="Convex polyhedra">convex polyhedra</a> whose faces are regular polygons and are <a href="Vertex-transitive" class="mw-redirect" title="Vertex-transitive">vertex-transitive</a>, although they are not face-transitive. The solids were named after <a href="Archimedes" title="Archimedes">Archimedes</a>, although he did not claim credit for them. They belong to the class of <a href="Uniform_polyhedra" class="mw-redirect" title="Uniform polyhedra">uniform polyhedra</a>, the polyhedra with regular faces and symmetric vertices. Some Archimedean solids were portrayed in the works of artists and mathematicians during the <a href="Renaissance" title="Renaissance">Renaissance</a>.

</p><p>The <a href="Elongated_square_gyrobicupola" title="Elongated square gyrobicupola">elongated square gyrobicupola</a> or <i>pseudorhombicuboctahedron</i> is an extra polyhedron with regular faces and congruent vertices. Still, it is not generally counted as an Archimedean solid because it is not <a href="Vertex-transitive" class="mw-redirect" title="Vertex-transitive">vertex-transitive</a>.

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<div class="mw-heading mw-heading2"><h2 id="The_solids">The solids</h2></div>

<p>The Archimedean solids have a single <a href="Vertex_configuration" title="Vertex configuration">vertex configuration</a> and highly symmetric properties. A vertex configuration indicates which regular polygons meet at each vertex. For instance, the configuration <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot 5\cdot 3\cdot 5}">

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<annotation encoding="application/x-tex">{\displaystyle 3\cdot 5\cdot 3\cdot 5}</annotation>

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</math></span><img src="./e071baa3c879e017e15fe5266f2fcec314174ed9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.687ex; height:2.176ex;" alt="{\displaystyle 3\cdot 5\cdot 3\cdot 5}" loading="lazy"></span> indicates a polyhedron in which each vertex is met by alternating two triangles and two pentagons. Highly symmetric properties in this case mean the <a href="Symmetry_group" title="Symmetry group">symmetry group</a> of each solid was derived from the <a href="Platonic_solids" class="mw-redirect" title="Platonic solids">Platonic solids</a>, resulting from their construction.<sup id="cite_ref-FOOTNOTEDiudea2018[httpsbooksgooglecombooksidp_06DwAAQBAJpgPA39_39]_1-0" class="reference"><a href="#cite_note-FOOTNOTEDiudea2018[httpsbooksgooglecombooksidp_06DwAAQBAJpgPA39_39]-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Some sources say the Archimedean solids are synonymous with the <a href="Semiregular_polyhedron" title="Semiregular polyhedron">semiregular polyhedron</a>.<sup id="cite_ref-FOOTNOTEKinseyMoorePrassidis2011[httpsbooksgooglecombooksidfFpuDwAAQBAJpgPA380_380]_2-0" class="reference"><a href="#cite_note-FOOTNOTEKinseyMoorePrassidis2011[httpsbooksgooglecombooksidfFpuDwAAQBAJpgPA380_380]-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Yet, the definition of a semiregular polyhedron may also include the infinite <a href="Prism_(geometry)" title="Prism (geometry)">prisms</a> and <a href="Antiprism" title="Antiprism">antiprisms</a>, including the <a href="Elongated_square_gyrobicupola" title="Elongated square gyrobicupola">elongated square gyrobicupola</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>The <a href="Skeleton_(topology)" class="mw-redirect" title="Skeleton (topology)">skeleton</a> of Archimedean solids can be drawn in a <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>, named <a href="Archimedean_graph" title="Archimedean graph">Archimedean graph</a>. Such graphs are <a href="Regular_graph" title="Regular graph">regular</a>, <a href="Polyhedral_graph" title="Polyhedral graph">polyhedral</a> (and therefore by necessity also <a href="K-vertex-connected_graph" title="K-vertex-connected graph">3-vertex-connected</a> <a href="Planar_graph" title="Planar graph">planar graphs</a>), and also <a href="Hamiltonian_graph" class="mw-redirect" title="Hamiltonian graph">Hamiltonian graphs</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<caption>The thirteen Archimedean solids

</caption>

<th>Name

</th>

<th class="unsortable">Solids

</th>

<th><a href="Vertex_configuration" title="Vertex configuration">Vertex configurations</a><sup id="cite_ref-FOOTNOTEWilliams1979_5-0" class="reference"><a href="#cite_note-FOOTNOTEWilliams1979-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</th>

<th>Faces<sup id="cite_ref-FOOTNOTEBerman1971_6-0" class="reference"><a href="#cite_note-FOOTNOTEBerman1971-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</th>

<th>Edges<sup id="cite_ref-FOOTNOTEBerman1971_6-1" class="reference"><a href="#cite_note-FOOTNOTEBerman1971-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</th>

<th>Vertices<sup id="cite_ref-FOOTNOTEBerman1971_6-2" class="reference"><a href="#cite_note-FOOTNOTEBerman1971-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</th>

<th><a href="List_of_spherical_symmetry_groups#Polyhedral_symmetry" title="List of spherical symmetry groups">Point<br>group</a><sup id="cite_ref-FOOTNOTEKocaKoca2013[httpsbooksgooglecombooksidILnBkuSxXGECpgPA48_47–50]_7-0" class="reference"><a href="#cite_note-FOOTNOTEKocaKoca2013[httpsbooksgooglecombooksidILnBkuSxXGECpgPA48_47–50]-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</th></tr>

<tr>

<td><a href="Truncated_tetrahedron" title="Truncated tetrahedron">Truncated tetrahedron</a>

</td>

</td>

</td>

<td>4 triangles<br>4 <a href="Hexagon" title="Hexagon">hexagons</a>

</td>

<td>18

</td>

<td>12

</td>

</td></tr>

<tr>

<td><a href="Cuboctahedron" title="Cuboctahedron">Cuboctahedron</a>

</td>

</td>

</td>

<td>8 <a href="Triangle" title="Triangle">triangles</a><br>6 <a href="Square" title="Square">squares</a>

</td>

<td>24

</td>

<td>12

</td>

</td></tr>

<tr>

<td><a href="Truncated_cube" title="Truncated cube">Truncated cube</a>

</td>

</td>

</td>

<td>8 triangles<br>6 <a href="Octagon" title="Octagon">octagons</a>

</td>

<td>36

</td>

<td>24

</td>

</td></tr>

<tr>

<td><a href="Truncated_octahedron" title="Truncated octahedron">Truncated octahedron</a>

</td>

</td>

</td>

<td>6 squares<br>8 hexagons

</td>

<td>36

</td>

<td>24

</td>

</td></tr>

<tr>

<td><a href="Rhombicuboctahedron" title="Rhombicuboctahedron">Rhombicuboctahedron</a>

</td>

</td>

</td>

<td>8 triangles<br>18 squares

</td>

<td>48

</td>

<td>24

</td>

</td></tr>

<tr>

<td><a href="Truncated_cuboctahedron" title="Truncated cuboctahedron">Truncated cuboctahedron</a>

</td>

</td>

</td>

<td>12 squares<br>8 hexagons<br>6 octagons

</td>

<td>72

</td>

<td>48

</td>

</td></tr>

<tr>

</td>

</td>

</td>

<td>32 triangles<br>6 squares

</td>

<td>60

</td>

<td>24

</td>

<td>O

</td></tr>

<tr>

<td><a href="Icosidodecahedron" title="Icosidodecahedron">Icosidodecahedron</a>

</td>

</td>

</td>

<td>20 triangles<br>12 <a href="Pentagon" title="Pentagon">pentagons</a>

</td>

<td>60

</td>

<td>30

</td>

</td></tr>

<tr>

<td><a href="Truncated_dodecahedron" title="Truncated dodecahedron">Truncated dodecahedron</a>

</td>

</td>

</td>

<td>20 triangles<br>12 <a href="Decagon" title="Decagon">decagons</a>

</td>

<td>90

</td>

<td>60

</td>

</td></tr>

<tr>

<td><a href="Truncated_icosahedron" title="Truncated icosahedron">Truncated icosahedron</a>

</td>

</td>

</td>

<td>12 pentagons<br>20 hexagons

</td>

<td>90

</td>

<td>60

</td>

</td></tr>

<tr>

<td><a href="Rhombicosidodecahedron" title="Rhombicosidodecahedron">Rhombicosidodecahedron</a>

</td>

</td>

</td>

<td>20 triangles<br>30 squares<br>12 pentagons

</td>

<td>120

</td>

<td>60

</td>

</td></tr>

<tr>

<td><a href="Truncated_icosidodecahedron" title="Truncated icosidodecahedron">Truncated icosidodecahedron</a>

</td>

</td>

</td>

<td>30 squares<br>20 hexagons<br>12 decagons

</td>

<td>180

</td>

<td>120

</td>

</td></tr>

<tr>

<td><a href="Snub_dodecahedron" title="Snub dodecahedron">Snub dodecahedron</a>

</td>

</td>

</td>

<td>80 triangles<br>12 pentagons

</td>

<td>150

</td>

<td>60

</td>

<td>I

</td></tr></tbody></table>

<p>The construction of some Archimedean solids begins from the Platonic solids. The <a href="Truncation_(geometry)" title="Truncation (geometry)">truncation</a> 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 <a href="Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a> constructed by cutting off all the <a href="Regular_icosahedron" title="Regular icosahedron">icosahedron</a>'s vertices, having the same symmetry as the icosahedron, the <a href="Icosahedral_symmetry" title="Icosahedral symmetry">icosahedral symmetry</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> 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 <a href="Rectification_(geometry)" title="Rectification (geometry)">rectification</a>. <a href="Expansion_(geometry)" title="Expansion (geometry)">Expansion</a> 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, which is constructed by separating the cube or octahedron's faces from their centroids and filling them with squares.<sup id="cite_ref-FOOTNOTEVianaXavierAiresCampos20191123See_Fig._6_9-0" class="reference"><a href="#cite_note-FOOTNOTEVianaXavierAiresCampos20191123See_Fig._6-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="Snub_(geometry)" title="Snub (geometry)">Snub</a> is a construction process of polyhedra by separating the polyhedron faces, twisting their faces in certain angles, and filling them up with <a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a>. Examples can be found in <a href="Snub_cube" title="Snub cube">snub cube</a> and <a href="Snub_dodecahedron" title="Snub dodecahedron">snub dodecahedron</a>. The resulting construction of these solids gives the property of <a href="Chirality_(mathematics)" title="Chirality (mathematics)">chirality</a>, meaning they are not identical when reflected in a mirror.<sup id="cite_ref-FOOTNOTEKocaKoca2013[httpsbooksgooglecombooksidILnBkuSxXGECpgPA49_49]_10-0" class="reference"><a href="#cite_note-FOOTNOTEKocaKoca2013[httpsbooksgooglecombooksidILnBkuSxXGECpgPA49_49]-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> However, not all of them can be constructed in such a way, or they could be constructed alternatively. For example, the <a href="Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a> can be constructed by attaching two <a href="Pentagonal_rotunda" title="Pentagonal rotunda">pentagonal rotunda</a> bases-to-base, or a rhombicuboctahedron that can be constructed alternatively by attaching two <a href="Square_cupola" title="Square cupola">square cupolas</a> on the bases of an octagonal prism.<sup id="cite_ref-FOOTNOTEBerman1971_6-3" class="reference"><a href="#cite_note-FOOTNOTEBerman1971-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>At least ten of the Archimedean solids have the <a href="Rupert_property" class="mw-redirect" title="Rupert property">Rupert property</a>: 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.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p><p>The <a href="Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> of an Archimedean solid is a <a href="Catalan_solid" title="Catalan solid">Catalan solid</a>.<sup id="cite_ref-FOOTNOTEDiudea2018[httpsbooksgooglecombooksidp_06DwAAQBAJpgPA39_39]_1-1" class="reference"><a href="#cite_note-FOOTNOTEDiudea2018[httpsbooksgooglecombooksidp_06DwAAQBAJpgPA39_39]-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Background_of_discovery">Background of discovery</h2></div>

<p>The names of Archimedean solids were taken from the Ancient Greek mathematician <a href="Archimedes" title="Archimedes">Archimedes</a>, who discussed them in a now-lost work. Although they were not credited to Archimedes originally, <a href="Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus of Alexandria</a> in the fifth section of his titled compendium <i>Synagoge</i>, referring to Archimedes, listed thirteen polyhedra and briefly described them in terms of how many faces of each kind these polyhedra have.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:442px;max-width:442px"><div class="trow"><div class="tsingle" style="width:153px;max-width:153px"><div class="thumbimage" style="height:145px;overflow:hidden"><span typeof="mw:File"></span></div><div class="thumbcaption">Truncated icosahedron in <i><a href="De_quinque_corporibus_regularibus" title="De quinque corporibus regularibus">De quinque corporibus regularibus</a></i></div></div><div class="tsingle" style="width:136px;max-width:136px"><div class="thumbimage" style="height:145px;overflow:hidden"><span typeof="mw:File"></span></div><div class="thumbcaption">Rhombicuboctahedron drawn by <a href="Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a></div></div><div class="tsingle" style="width:147px;max-width:147px"><div class="thumbimage" style="height:145px;overflow:hidden"><span typeof="mw:File"></span></div><div class="thumbcaption">Cuboctahedron in <i><a href="Perspectiva_Corporum_Regularium" class="mw-redirect" title="Perspectiva Corporum Regularium">Perspectiva Corporum Regularium</a></i></div></div></div></div></div>

<p>During the <a href="Renaissance" title="Renaissance">Renaissance</a>, artists and mathematicians valued pure forms with high symmetry. Some Archimedean solids appeared in <a href="Piero_della_Francesca" title="Piero della Francesca">Piero della Francesca</a>'s <i><a href="De_quinque_corporibus_regularibus" title="De quinque corporibus regularibus">De quinque corporibus regularibus</a></i>, in attempting to study and copy the works of Archimedes, as well as include citations to Archimedes.<sup id="cite_ref-FOOTNOTEBanker2005_13-0" class="reference"><a href="#cite_note-FOOTNOTEBanker2005-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Yet, he did not credit those shapes to Archimedes and knew of Archimedes' work, but rather appeared to be an independent rediscovery.<sup id="cite_ref-FOOTNOTEField1997248_14-0" class="reference"><a href="#cite_note-FOOTNOTEField1997248-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Other appearance of the solids appeared in the works of <a href="Wenzel_Jamnitzer" title="Wenzel Jamnitzer">Wenzel Jamnitzer</a>'s <i><a href="Perspectiva_Corporum_Regularium" class="mw-redirect" title="Perspectiva Corporum Regularium">Perspectiva Corporum Regularium</a></i>, and both <i><a href="Summa_de_arithmetica" title="Summa de arithmetica">Summa de arithmetica</a></i> and <i><a href="Divina_proportione" title="Divina proportione">Divina proportione</a></i> by <a href="Luca_Pacioli" title="Luca Pacioli">Luca Pacioli</a>, drawn by <a href="Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The <a href="Net_(polyhedron)" title="Net (polyhedron)">net</a> of Archimedean solids appeared in <a href="Albrecht_D%C3%BCrer" title="Albrecht Dürer">Albrecht Dürer</a>'s <i>Underweysung der Messung</i>, copied from the Pacioli's work. By around 1620, <a href="Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> in his <i><a href="Harmonices_Mundi" class="mw-redirect" title="Harmonices Mundi">Harmonices Mundi</a></i> had completed the rediscovery of the thirteen polyhedra, as well as defining the <a href="Prism_(geometry)" title="Prism (geometry)">prisms</a>, <a href="Antiprisms" class="mw-redirect" title="Antiprisms">antiprisms</a>, and the non-convex solids known as <a href="Kepler%E2%80%93Poinsot_polyhedra" class="mw-redirect" title="Kepler–Poinsot polyhedra">Kepler–Poinsot polyhedra</a>.<sup id="cite_ref-FOOTNOTESchreiberFischerSternath2008_16-0" class="reference"><a href="#cite_note-FOOTNOTESchreiberFischerSternath2008-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>

</p>

<div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:144px;max-width:144px"><div class="thumbimage" style="height:140px;overflow:hidden"><span typeof="mw:File"></span></div></div><div class="tsingle" style="width:144px;max-width:144px"><div class="thumbimage" style="height:140px;overflow:hidden"><span typeof="mw:File"></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption"><a href="Rhombicuboctahedron" title="Rhombicuboctahedron">Rhombicuboctahedron</a> and <a href="Elongated_square_gyrobicupola" title="Elongated square gyrobicupola">elongated square gyrobicupola</a>. The latter is not vertex-transitive, and thus not Archimedean.</div></div></div></div>

<p>Kepler may have also found another solid known as <a href="Elongated_square_gyrobicupola" title="Elongated square gyrobicupola">elongated square gyrobicupola</a> or <i>pseudorhombicuboctahedron</i>. 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 <a href="Duncan_Sommerville" title="Duncan Sommerville">Duncan Sommerville</a> in 1905.<sup id="cite_ref-FOOTNOTEGrünbaum2009_17-0" class="reference"><a href="#cite_note-FOOTNOTEGrünbaum2009-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The solid appeared when some mathematicians mistakenly constructed the <a href="Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a>: two <a href="Square_cupola" title="Square cupola">square cupolas</a> attached to the <a href="Octagonal_prism" title="Octagonal prism">octagonal prism</a>, with one of them rotated forty-five degrees.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The thirteen solids have the property of <a href="Vertex-transitive" class="mw-redirect" title="Vertex-transitive">vertex-transitive</a>, meaning any two vertices of those can be translated onto the other one, but the elongated square gyrobicupola does not. <a href="#CITEREFGrünbaum2009">Grünbaum (2009)</a> 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 in the <a href="Johnson_solid" title="Johnson solid">Johnson solids</a> instead, a convex polyhedron in which all of the faces are regular polygons.<sup id="cite_ref-FOOTNOTEGrünbaum2009_17-1" class="reference"><a href="#cite_note-FOOTNOTEGrünbaum2009-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Conway_polyhedron_notation" title="Conway polyhedron notation">Conway polyhedron notation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Footnotes">Footnotes</h3></div>

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</style><div class="reflist reflist-columns references-column-width" style="column-width: 20em;">

<li id="cite_note-FOOTNOTEDiudea2018[httpsbooksgooglecombooksidp_06DwAAQBAJpgPA39_39]-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEDiudea2018[httpsbooksgooglecombooksidp_06DwAAQBAJpgPA39_39]_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDiudea2018[httpsbooksgooglecombooksidp_06DwAAQBAJpgPA39_39]_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDiudea2018">Diudea (2018)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39">39</a>.</span>

</li>

<li id="cite_note-FOOTNOTEKinseyMoorePrassidis2011[httpsbooksgooglecombooksidfFpuDwAAQBAJpgPA380_380]-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKinseyMoorePrassidis2011[httpsbooksgooglecombooksidfFpuDwAAQBAJpgPA380_380]_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKinseyMoorePrassidis2011">Kinsey, Moore & Prassidis (2011)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fFpuDwAAQBAJ&pg=PA380">380</a>.</span>

</li>

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</style><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFRovenski2010">Rovenski (2010)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BhVCYqqP69kC&pg=PA116">116</a></li><li><a href="#CITEREFMalkevitch1988">Malkevitch (1988)</a>, p. 85</li></ul></div></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">An Atlas of Graphs, p. 267-270</span>

</li>

<li id="cite_note-FOOTNOTEWilliams1979-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWilliams1979_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWilliams1979">Williams (1979)</a>.</span>

</li>

<li id="cite_note-FOOTNOTEBerman1971-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBerman1971_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBerman1971_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBerman1971_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBerman1971_6-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBerman1971">Berman (1971)</a>.</span>

</li>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFChanceyO'Brien1997">Chancey & O'Brien (1997)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wcQIEAAAQBAJ&pg=PA13">13</a></li><li><a href="#CITEREFKocaKoca2013">Koca & Koca (2013)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48">48</a></li></ul></div></span>

</li>

<li id="cite_note-FOOTNOTEVianaXavierAiresCampos20191123See_Fig._6-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVianaXavierAiresCampos20191123See_Fig._6_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVianaXavierAiresCampos2019">Viana et al. (2019)</a>, p. 1123, See Fig. 6.</span>

</li>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFChaiYuanZamfirescu2018">Chai, Yuan & Zamfirescu (2018)</a></li><li><a href="#CITEREFHoffmann2019">Hoffmann (2019)</a></li><li><a href="#CITEREFLavau2019">Lavau (2019)</a></li></ul></div></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFCromwell1997">Cromwell (1997)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OJowej1QWpoC&pg=PA156">156</a></li><li><a href="#CITEREFGrünbaum2009">Grünbaum (2009)</a></li><li><a href="#CITEREFField1997">Field (1997)</a>, p. 248</li></ul></div></span>

</li>

<li id="cite_note-FOOTNOTEBanker2005-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBanker2005_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBanker2005">Banker (2005)</a>.</span>

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<li id="cite_note-FOOTNOTEField1997248-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEField1997248_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFField1997">Field (1997)</a>, p. 248.</span>

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<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFCromwell1997">Cromwell (1997)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OJowej1QWpoC&pg=PA156">156</a></li><li><a href="#CITEREFField1997">Field (1997)</a>, p. 253–254</li></ul></div></span>

</li>

<li id="cite_note-FOOTNOTESchreiberFischerSternath2008-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESchreiberFischerSternath2008_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSchreiberFischerSternath2008">Schreiber, Fischer & Sternath (2008)</a>.</span>

</li>

<li id="cite_note-FOOTNOTEGrünbaum2009-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEGrünbaum2009_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGrünbaum2009_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGrünbaum2009">Grünbaum (2009)</a>.</span>

</li>

<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFCromwell1997">Cromwell (1997)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OJowej1QWpoC&pg=PA91">91</a></li><li><a href="#CITEREFBerman1971">Berman (1971)</a></li></ul></div></span>

</li>

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<div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3></div>

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</style><cite id="CITEREFBanker2005" class="citation cs2">Banker, James R. (March 2005), "A manuscript of the works of Archimedes in the hand of Piero della Francesca", <i><a href="The_Burlington_Magazine" title="The Burlington Magazine">The Burlington Magazine</a></i>, <b>147</b> (1224): <span class="nowrap">165–</span>169, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/20073883">20073883</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:190211171">190211171</a></cite>.</li>

<li><cite id="CITEREFBerman1971" class="citation cs2">Berman, Martin (1971), "Regular-faced convex polyhedra", <i>Journal of the Franklin Institute</i>, <b>291</b> (5): <span class="nowrap">329–</span>352, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0016-0032%2871%2990071-8">10.1016/0016-0032(71)90071-8</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0290245">0290245</a></cite>.</li>

<li><cite id="CITEREFChaiYuanZamfirescu2018" class="citation cs2">Chai, Ying; Yuan, Liping; Zamfirescu, Tudor (2018), "Rupert Property of Archimedean Solids", <i><a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>, <b>125</b> (6): <span class="nowrap">497–</span>504, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2018.1449505">10.1080/00029890.2018.1449505</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125508192">125508192</a></cite>.</li>

<li><cite id="CITEREFChanceyO'Brien1997" class="citation cs2">Chancey, C. C.; O'Brien, M. C. M. (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wcQIEAAAQBAJ"><i>The Jahn-Teller Effect in C<sub>60</sub> and Other Icosahedral Complexes</i></a>, <a href="Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-22534-0</bdi></cite>.</li>

<li><cite id="CITEREFCromwell1997" class="citation cs2">Cromwell, Peter R. (1997), <a rel="nofollow" class="external text" href="https://archive.org/details/polyhedra0000crom"><i>Polyhedra</i></a>, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-55432-9</bdi></cite>.</li>

<li><cite id="CITEREFDiudea2018" class="citation cs2">Diudea, M. V. (2018), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=p_06DwAAQBAJ"><i>Multi-shell Polyhedral Clusters</i></a>, Carbon Materials: Chemistry and Physics, vol. 10, <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-64123-2">10.1007/978-3-319-64123-2</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-64123-2</bdi></cite>.</li>

<li><cite id="CITEREFField1997" class="citation cs2"><a href="Judith_V._Field" title="Judith V. Field">Field, J. V.</a> (1997), "Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler", <i><a href="Archive_for_History_of_Exact_Sciences" title="Archive for History of Exact Sciences">Archive for History of Exact Sciences</a></i>, <b>50</b> (<span class="nowrap">3–</span>4): <span class="nowrap">241–</span>289, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00374595">10.1007/BF00374595</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41134110">41134110</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1457069">1457069</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118516740">118516740</a></cite>.</li>

<li><cite id="CITEREFGrünbaum2009" class="citation cs2"><a href="Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, Branko</a> (2009), <a rel="nofollow" class="external text" href="https://digital.lib.washington.edu/dspace/bitstream/handle/1773/4592/An_enduring_error.pdf">"An enduring error"</a> <span class="cs1-format">(PDF)</span>, <i><a href="Elemente_der_Mathematik" title="Elemente der Mathematik">Elemente der Mathematik</a></i>, <b>64</b> (3): <span class="nowrap">89–</span>101, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4171%2FEM%2F120">10.4171/EM/120</a></span>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2520469">2520469</a></cite>. Reprinted in <cite id="CITEREFPitici2011" class="citation cs2">Pitici, Mircea, ed. (2011), <i>The Best Writing on Mathematics 2010</i>, Princeton University Press, pp. <span class="nowrap">18–</span>31</cite>.</li>

<li><cite id="CITEREFHoffmann2019" class="citation cs2">Hoffmann, Balazs (2019), <a rel="nofollow" class="external text" href="http://www.heldermann.de/JGG/JGG23/JGG231/jgg23003.htm">"Rupert properties of polyhedra and the generalized Nieuwland constant"</a>, <i>Journal for Geometry and Graphics</i>, <b>23</b> (1): <span class="nowrap">29–</span>35</cite></li>

<li><cite id="CITEREFKinseyMoorePrassidis2011" class="citation cs2"><a href="L._Christine_Kinsey" title="L. Christine Kinsey">Kinsey, L. Christine</a>; Moore, Teresa E.; Prassidis, Efstratios (2011), <i>Geometry and Symmetry</i>, <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-49949-8</bdi></cite>.</li>

<li><cite id="CITEREFKocaKoca2013" class="citation cs2">Koca, M.; Koca, N. O. (2013), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48">"Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes"</a>, <i>Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010</i>, World Scientific</cite>.</li>

<li><cite id="CITEREFLavau2019" class="citation cs2">Lavau, Gérard (2019), "The Truncated Tetrahedron is Rupert", <i><a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>, <b>126</b> (10): <span class="nowrap">929–</span>932, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.2019.1656958">10.1080/00029890.2019.1656958</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:213502432">213502432</a></cite>.</li>

<li><cite id="CITEREFMalkevitch1988" class="citation cs2">Malkevitch, Joseph (1988), "Milestones in the history of polyhedra", in <a href="Marjorie_Senechal" title="Marjorie Senechal">Senechal, M.</a>; Fleck, G. (eds.), <i>Shaping Space: A Polyhedral Approach</i>, Boston: Birkhäuser, pp. <span class="nowrap">80–</span>92</cite>.</li>

<li><cite id="CITEREFRovenski2010" class="citation cs2">Rovenski, Vladimir (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BhVCYqqP69kC"><i>Modeling of Curves and Surfaces with MATLAB®</i></a>, Springer Undergraduate Texts in Mathematics and Technology, Springer, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-71278-9">10.1007/978-0-387-71278-9</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-71278-9</bdi></cite>.</li>

<li><cite id="CITEREFSchreiberFischerSternath2008" class="citation cs2">Schreiber, Peter; Fischer, Gisela; Sternath, Maria Luise (2008), "New light on the rediscovery of the Archimedean solids during the Renaissance", <i>Archive for History of Exact Sciences</i>, <b>62</b> (4): <span class="nowrap">457–</span>467, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008AHES...62..457S">2008AHES...62..457S</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00407-008-0024-z">10.1007/s00407-008-0024-z</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-9519">0003-9519</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41134285">41134285</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122216140">122216140</a></cite>.</li>

<li><cite id="CITEREFVianaXavierAiresCampos2019" class="citation cs2">Viana, Vera; Xavier, João Pedro; Aires, Ana Paula; Campos, Helena (2019), "Interactive Expansion of Achiral Polyhedra", in Cocchiarella, Luigi (ed.), <i>ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics 40th Anniversary - Milan, Italy, August 3-7, 2018</i>, Springer, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-95588-9">10.1007/978-3-319-95588-9</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-95587-2</bdi></cite>.</li>

<li><cite id="CITEREFWilliams1979" class="citation cs2"><a href="Robert_Williams_(geometer)" title="Robert Williams (geometer)">Williams, Robert</a> (1979), <a rel="nofollow" class="external text" href="https://archive.org/details/geometricalfound00will/page/90"><i>The Geometrical Foundation of Natural Structure: A Source Book of Design</i></a>, Dover Publications, Inc., <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-23729-9</bdi></cite>.</li></ul>

</div>

<div class="mw-heading mw-heading3"><h3 id="Further_reading">Further reading</h3></div>

<ul><li><cite id="CITEREFViana2024" class="citation cs2">Viana, Vera (2024), "Archimedean solids in the fifteenth and sixteenth centuries", <i>Archive for History of Exact Sciences</i>, <b>78</b> (6): <span class="nowrap">631–</span>715, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00407-024-00331-7">10.1007/s00407-024-00331-7</a></span></cite>.</li>

<li><cite id="CITEREFWilliamsMonteleone2021" class="citation cs2">Williams, Kim; Monteleone, Cosimo (2021), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=w5RBEAAAQBAJ&pg=PA19"><i>Daniele Barbaro's Perspective of 1568</i></a>, p. 19–20, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-76687-0">10.1007/978-3-030-76687-0</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-030-76687-0</bdi></cite>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="citation mathworld" id="Reference-Mathworld-Archimedean_solid"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ArchimedeanSolid.html">"Archimedean solid"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li>

<li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/ArchimedeanSolids/">Archimedean Solids</a> by <a href="Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <a href="Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>.</li>

<li><a rel="nofollow" class="external text" href="http://www.software3d.com/Archimedean.php">Paper models of Archimedean Solids and Catalan Solids</a></li>

<li><a rel="nofollow" class="external text" href="http://www.korthalsaltes.com/cuadros.php?type=a">Free paper models(nets) of Archimedean solids</a></li>

<li><a rel="nofollow" class="external text" href="http://www.mathconsult.ch/showroom/unipoly/">The Uniform Polyhedra</a> by Dr. R. Mäder</li>

<li><a rel="nofollow" class="external text" href="http://dmccooey.com/polyhedra/Archimedean.html">Archimedean Solids</a> at Visual Polyhedra by David I. McCooey</li>

<li><a rel="nofollow" class="external text" href="http://www.georgehart.com/virtual-polyhedra/vp.html">Virtual Reality Polyhedra</a>, <i>The Encyclopedia of Polyhedra</i> by George W. Hart</li>

<li><a rel="nofollow" class="external text" href="http://www.cs.utk.edu/~plank/plank/origami/penultimate/intro.html">Penultimate Modular Origami</a> by James S. Plank</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20050403235101/http://ibiblio.org/e-notes/3Dapp/Convex.htm">Interactive 3D polyhedra</a> in Java</li>

<li><a rel="nofollow" class="external text" href="https://archive.today/20130411004747/http://kovacsv.github.com/JSModeler/documentation/examples/solids.html">Solid Body Viewer</a> is an interactive 3D polyhedron viewer that allows you to save the model in SVG, STL, or OBJ format.</li>

<li><a rel="nofollow" class="external text" href="http://www.software3d.com/Stella.php">Stella: Polyhedron Navigator</a>: Software used to create many of the images on this page.</li>

<li><a rel="nofollow" class="external text" href="http://www.polyedergarten.de/">Paper Models of Archimedean (and other) Polyhedra</a></li></ul>

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</style><div id="Archimedes193" style="font-size:114%;margin:0 4em"><a href="Archimedes" title="Archimedes">Archimedes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Written works</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><i><a href="Measurement_of_a_Circle" title="Measurement of a Circle">Measurement of a Circle</a></i></li>

<li><i><a href="The_Sand_Reckoner" title="The Sand Reckoner">The Sand Reckoner</a></i></li>

<li><i><a href="On_the_Equilibrium_of_Planes" title="On the Equilibrium of Planes">On the Equilibrium of Planes</a></i></li>

<li><i><a href="Quadrature_of_the_Parabola" title="Quadrature of the Parabola">Quadrature of the Parabola</a></i></li>

<li><i><a href="On_the_Sphere_and_Cylinder" title="On the Sphere and Cylinder">On the Sphere and Cylinder</a></i></li>

<li><i><a href="On_Spirals" title="On Spirals">On Spirals</a></i></li>

<li><i><a href="On_Conoids_and_Spheroids" title="On Conoids and Spheroids">On Conoids and Spheroids</a></i></li>

<li><i><a href="On_Floating_Bodies" title="On Floating Bodies">On Floating Bodies</a></i></li>

<li><i><a href="Ostomachion" title="Ostomachion">Ostomachion</a></i></li>

<li><i><a href="The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li>

<li><i><a href="Book_of_Lemmas" title="Book of Lemmas">Book of Lemmas</a></i> (apocryphal)</li></ul>

</div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discoveries and inventions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Archimedes's_cattle_problem" title="Archimedes's cattle problem">Archimedes's cattle problem</a></li>

<li><a href="Archimedes'_principle" title="Archimedes' principle">Archimedes' principle</a></li>

<li><a href="Archimedes's_screw" class="mw-redirect" title="Archimedes's screw">Archimedes's screw</a></li>

<li><a href="Claw_of_Archimedes" title="Claw of Archimedes">Claw of Archimedes</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Archimedes'_heat_ray" title="Archimedes' heat ray">Archimedes' heat ray</a></li>

<li><a href="Archimedes_Palimpsest" title="Archimedes Palimpsest">Archimedes Palimpsest</a></li>

<li><a href="List_of_things_named_after_Archimedes" title="List of things named after Archimedes">List of things named after Archimedes</a></li>

<li><a href="Pseudo-Archimedes" title="Pseudo-Archimedes">Pseudo-Archimedes</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related people</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euclid" title="Euclid">Euclid</a></li>

<li><a href="Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a></li>

<li><a href="Apollonius_of_Perga" title="Apollonius of Perga">Apollonius of Perga</a></li>

<li><a href="Hero_of_Alexandria" title="Hero of Alexandria">Hero of Alexandria</a></li>

<li><a href="Eutocius_of_Ascalon" title="Eutocius of Ascalon">Eutocius of Ascalon</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

</div></td></tr></tbody></table></div>

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.mw-parser-output .nobold{font-weight:normal}

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</style></div><div role="navigation" class="navbox" aria-labelledby="Convex_polyhedra686" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex_polyhedra686" style="font-size:114%;margin:0 4em">Convex <a href="Polyhedron" title="Polyhedron">polyhedra</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Platonic_solid" title="Platonic solid">Platonic solids</a> <span class="nobold">(<a href="Regular_polyhedron" title="Regular polyhedron">regular</a>)</span></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tetrahedron#Regular_tetrahedron" title="Tetrahedron">tetrahedron</a></li>

<li><a href="Octahedron" title="Octahedron">octahedron</a></li>

<li><a href="Regular_dodecahedron" title="Regular dodecahedron">dodecahedron</a></li>

<li><a href="Regular_icosahedron" title="Regular icosahedron">icosahedron</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><br><span class="nobold">(<a href="Semiregular_polyhedron" title="Semiregular polyhedron">semiregular</a> or <a href="Uniform_polyhedron" title="Uniform polyhedron">uniform</a>)</span></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Truncated_tetrahedron" title="Truncated tetrahedron">truncated tetrahedron</a></li>

<li><a href="Cuboctahedron" title="Cuboctahedron">cuboctahedron</a></li>

<li><a href="Truncated_cube" title="Truncated cube">truncated cube</a></li>

<li><a href="Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a></li>

<li><a href="Rhombicuboctahedron" title="Rhombicuboctahedron">rhombicuboctahedron</a></li>

<li><a href="Truncated_cuboctahedron" title="Truncated cuboctahedron">truncated cuboctahedron</a></li>

<li><a href="Icosidodecahedron" title="Icosidodecahedron">icosidodecahedron</a></li>

<li><a href="Truncated_dodecahedron" title="Truncated dodecahedron">truncated dodecahedron</a></li>

<li><a href="Truncated_icosahedron" title="Truncated icosahedron">truncated icosahedron</a></li>

<li><a href="Rhombicosidodecahedron" title="Rhombicosidodecahedron">rhombicosidodecahedron</a></li>

<li><a href="Truncated_icosidodecahedron" title="Truncated icosidodecahedron">truncated icosidodecahedron</a></li>

<li><a href="Snub_dodecahedron" title="Snub dodecahedron">snub dodecahedron</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Catalan_solid" title="Catalan solid">Catalan solids</a><br><span class="nobold">(duals of Archimedean)</span></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Triakis_tetrahedron" title="Triakis tetrahedron">triakis tetrahedron</a></li>

<li><a href="Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a></li>

<li><a href="Triakis_octahedron" title="Triakis octahedron">triakis octahedron</a></li>

<li><a href="Tetrakis_hexahedron" title="Tetrakis hexahedron">tetrakis hexahedron</a></li>

<li><a href="Deltoidal_icositetrahedron" title="Deltoidal icositetrahedron">deltoidal icositetrahedron</a></li>

<li><a href="Disdyakis_dodecahedron" title="Disdyakis dodecahedron">disdyakis dodecahedron</a></li>

<li><a href="Pentagonal_icositetrahedron" title="Pentagonal icositetrahedron">pentagonal icositetrahedron</a></li>

<li><a href="Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a></li>

<li><a href="Triakis_icosahedron" title="Triakis icosahedron">triakis icosahedron</a></li>

<li><a href="Pentakis_dodecahedron" title="Pentakis dodecahedron">pentakis dodecahedron</a></li>

<li><a href="Deltoidal_hexecontahedron" title="Deltoidal hexecontahedron">deltoidal hexecontahedron</a></li>

<li><a href="Disdyakis_triacontahedron" title="Disdyakis triacontahedron">disdyakis triacontahedron</a></li>

<li><a href="Pentagonal_hexecontahedron" title="Pentagonal hexecontahedron">pentagonal hexecontahedron</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral regular</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><i><a href="Dihedron" title="Dihedron">dihedron</a></i></li>

<li><i><a href="Hosohedron" title="Hosohedron">hosohedron</a></i></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral uniform</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Prism_(geometry)" title="Prism (geometry)">prisms</a></li>

<li><a href="Antiprism" title="Antiprism">antiprisms</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">duals:</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bipyramid" title="Bipyramid">bipyramids</a></li>

<li><a href="Trapezohedron" title="Trapezohedron">trapezohedra</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dihedral others</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a></li>

<li><a href="Truncated_trapezohedron" title="Truncated trapezohedron">truncated trapezohedra</a></li>

<li><a href="Gyroelongated_bipyramid" title="Gyroelongated bipyramid">gyroelongated bipyramid</a></li>

<li><a href="Cupola_(geometry)" title="Cupola (geometry)">cupola</a></li>

<li><a href="Bicupola_(geometry)" class="mw-redirect" title="Bicupola (geometry)">bicupola</a></li>

<li><a href="Frustum" title="Frustum">frustum</a></li>

<li><a href="Bifrustum" title="Bifrustum">bifrustum</a></li>

<li><a href="Rotunda_(geometry)" title="Rotunda (geometry)">rotunda</a></li>

<li><a href="Birotunda" title="Birotunda">birotunda</a></li>

<li><a href="Prismatoid" title="Prismatoid">prismatoid</a></li>

<li><a href="Scutoid" title="Scutoid">scutoid</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Degenerate polyhedra are in <i>italics</i>.</div></td></tr></tbody></table></div>

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