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Triaugmented triangular prism

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3">Triaugmented triangular prism</th></tr><tr><td colspan="2" class="infobox-image"></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="Deltahedron" title="Deltahedron">Deltahedron</a>, <a href="Johnson_solid" title="Johnson solid">Johnson</a> <a href="Biaugmented_triangular_prism" title="Biaugmented triangular prism">J50</a> – J51 – <a href="Augmented_pentagonal_prism" title="Augmented pentagonal prism">J52</a></td></tr><tr><th scope="row" class="infobox-label"><a href="Face_(geometry)" title="Face (geometry)">Faces</a></th><td class="infobox-data">14 <a href="Triangle" title="Triangle">triangles</a></td></tr><tr><th scope="row" class="infobox-label"><a href="Edge_(geometry)" title="Edge (geometry)">Edges</a></th><td class="infobox-data">21</td></tr><tr><th scope="row" class="infobox-label"><a href="Vertex_(geometry)" title="Vertex (geometry)">Vertices</a></th><td class="infobox-data">9</td></tr><tr><th scope="row" class="infobox-label"><a href="Vertex_configuration" title="Vertex configuration">Vertex configuration</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3^{4}+6\times 3^{5}}">

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<annotation encoding="application/x-tex">{\displaystyle 3\times 3^{4}+6\times 3^{5}}</annotation>

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</math><img src="./5b6e2074adb6ee067e194f3282bb50fb6d4ff883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.279ex; height:2.843ex;" alt="{\displaystyle 3\times 3^{4}+6\times 3^{5}}" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label"><a href="List_of_spherical_symmetry_groups" title="List of spherical symmetry groups">Symmetry group</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{3\mathrm {h} }}">

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<annotation encoding="application/x-tex">{\displaystyle D_{3\mathrm {h} }}</annotation>

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</math><img src="./c8f35493f21e2d3868ab7b52b08499661eee4276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.892ex; height:2.509ex;" alt="{\displaystyle D_{3\mathrm {h} }}" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label"><a href="Dihedral_angle" title="Dihedral angle">Dihedral angle</a> (<a href="Degree_(angle)" title="Degree (angle)">degrees</a>)</th><td class="infobox-data">109.5° 144.7° 169.5°</td></tr><tr><th scope="row" class="infobox-label"><a href="Dual_polyhedron" title="Dual polyhedron">Dual polyhedron</a></th><td class="infobox-data"><a href="Associahedron" title="Associahedron">Associahedron K5</a></td></tr><tr><th scope="row" class="infobox-label">Properties</th><td class="infobox-data"><a href="Convex_polytope" title="Convex polytope">convex</a>, <a href="Composite_polyhedron" title="Composite polyhedron">composite</a></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e7dcc3"><a href="Net_(polyhedron)" title="Net (polyhedron)">Net</a></th></tr><tr><td colspan="2" class="infobox-full-data"></td></tr></tbody></table>

The triaugmented triangular prism, in geometry, is a <a href="Convex_polyhedron" class="mw-redirect" title="Convex polyhedron">convex polyhedron</a> with 14 <a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a> as its faces. It can be constructed from a <a href="Triangular_prism" title="Triangular prism">triangular prism</a> by attaching <a href="Equilateral_square_pyramid" class="mw-redirect" title="Equilateral square pyramid">equilateral square pyramids</a> to each of its three square faces. The same shape is also called the tetrakis triangular prism,<a href="#cite_note-shdc-1">[1]</a> tricapped trigonal prism,<a href="#cite_note-kepert-2">[2]</a> tetracaidecadeltahedron,<a href="#cite_note-burgiel-3">[3]</a><a href="#cite_note-pugh-4">[4]</a> or tetrakaidecadeltahedron;<a href="#cite_note-shdc-1">[1]</a> these last names mean a polyhedron with 14 triangular faces. It is an example of a <a href="Deltahedron" title="Deltahedron">deltahedron</a>, <a href="Composite_polyhedron" title="Composite polyhedron">composite polyhedron</a>, and <a href="Johnson_solid" title="Johnson solid">Johnson solid</a>.

The edges and vertices of the triaugmented triangular prism form a <a href="Maximal_planar_graph" class="mw-redirect" title="Maximal planar graph">maximal planar graph</a> with 9 vertices and 21 edges, called the Fritsch graph. It was used by Rudolf and Gerda Fritsch to show that <a href="Alfred_Kempe" title="Alfred Kempe">Alfred Kempe</a>'s attempted proof of the <a href="Four_color_theorem" title="Four color theorem">four color theorem</a> was incorrect. The Fritsch graph is one of only six graphs in which every <a href="Neighbourhood_(graph_theory)" title="Neighbourhood (graph theory)">neighborhood</a> is a 4- or 5-vertex cycle.

The <a href="Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> of the triaugmented triangular prism is an <a href="Associahedron" title="Associahedron">associahedron</a>, a polyhedron with four quadrilateral faces and six pentagons whose vertices represent the 14 triangulations of a <a href="Regular_hexagon" class="mw-redirect" title="Regular hexagon">regular hexagon</a>. In the same way, the nine vertices of the triaugmented triangular prism represent the nine diagonals of a hexagon, with two vertices connected by an edge when the corresponding two diagonals do not cross. Other applications of the triaugmented triangular prism appear in chemistry as the basis for the <a href="Tricapped_trigonal_prismatic_molecular_geometry" title="Tricapped trigonal prismatic molecular geometry">tricapped trigonal prismatic molecular geometry</a>, and in mathematical optimization as a solution to the <a href="Thomson_problem" title="Thomson problem">Thomson problem</a> and <a href="Tammes_problem" title="Tammes problem">Tammes problem</a>.

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

The triaugmented triangular prism is a <a href="Composite_polyhedron" title="Composite polyhedron">composite polyhedron</a>, meaning it can be constructed by attaching <a href="Equilateral_square_pyramid" class="mw-redirect" title="Equilateral square pyramid">equilateral square pyramids</a> to each of the three square faces of a <a href="Triangular_prism" title="Triangular prism">triangular prism</a>, a process called <a href="Augmentation_(geometry)" class="mw-redirect" title="Augmentation (geometry)">augmentation</a>.<a href="#cite_note-timofeenko-2009-5">[5]</a><a href="#cite_note-trigg-6">[6]</a> These pyramids cover each square, replacing it with four <a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a>, so that the resulting polyhedron has 14 equilateral triangles as its faces. A polyhedron with only equilateral triangles as faces is called a <a href="Deltahedron" title="Deltahedron">deltahedron</a>. There are only eight different <a href="Convex_set" title="Convex set">convex</a> deltahedra, one of which is the triaugmented triangular prism.<a href="#cite_note-fw47-7">[7]</a><a href="#cite_note-cundy-8">[8]</a> More generally, the convex polyhedra in which all faces are <a href="Regular_polygon" title="Regular polygon">regular polygons</a> are called the <a href="Johnson_solid" title="Johnson solid">Johnson solids</a>, and every convex deltahedron is a Johnson solid. The triaugmented triangular prism is numbered among the Johnson solids as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{51}}">

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<annotation encoding="application/x-tex">{\displaystyle J_{51}}</annotation>

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</math><img src="./08681f4eb2c120b643cea750a6b9f0acb0250855.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle J_{51}}" loading="lazy">.<a href="#cite_note-francis-9">[9]</a>

One possible system of <a href="Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> for the vertices of a triaugmented triangular prism, giving it edge length 2, is:<a href="#cite_note-shdc-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(0,{\frac {2}{\sqrt {3}}},\pm 1\right),\qquad &\left(\pm 1,-{\frac {1}{\sqrt {3}}},\pm 1\right),\\\left(0,-{\frac {1+{\sqrt {6}}}{\sqrt {3}}},0\right),\qquad &\left(\pm {\frac {1+{\sqrt {6}}}{2}},{\frac {1+{\sqrt {6}}}{2{\sqrt {3}}}},0\right).\\\end{aligned}}}</annotation>

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

A triaugmented triangular prism with edge length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> has surface area<a href="#cite_note-berman-10">[10]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {7{\sqrt {3}}}{2}}a^{2}\approx 6.062a^{2},}</annotation>

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the area of 14 equilateral triangles. Its volume,<a href="#cite_note-berman-10">[10]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2{\sqrt {2}}+{\sqrt {3}}}{4}}a^{3}\approx 1.140a^{3},}</annotation>

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can be derived by slicing it into a central prism and three square pyramids, and adding their volumes.<a href="#cite_note-berman-10">[10]</a>

The triaugmented triangular prism has two types of <a href="Closed_geodesic" title="Closed geodesic">closed geodesics</a>. These are paths on its surface that are locally straight: they avoid vertices of the polyhedron, follow line segments across the faces that they cross, and form <a href="Complementary_angles" class="mw-redirect" title="Complementary angles">complementary angles</a> on the two incident faces of each edge that they cross. One of the two types of closed geodesic runs parallel to the square base of a pyramid, through the eight faces surrounding the pyramid. For a polyhedron with unit-length sides, this geodesic has length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}">

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<annotation encoding="application/x-tex">{\displaystyle 4}</annotation>

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</math><img src="./295b4bf1de7cd3500e740e0f4f0635db22d87b42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}" loading="lazy">. The other type of closed geodesic crosses ten faces, and has length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {19}}\approx 4.36}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {19}}\approx 4.36}</annotation>

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</math><img src="./1deff19d20229059949260392d06a83fbdfc9dea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.493ex; height:2.843ex;" alt="{\displaystyle {\sqrt {19}}\approx 4.36}" loading="lazy">. For each type there is a continuous family of parallel geodesics, all of the same length.<a href="#cite_note-lptw-11">[11]</a>

The triaugmented triangular prism has the same <a href="Point_groups_in_three_dimensions" title="Point groups in three dimensions">three-dimensional symmetry group</a> as the triangular prism, the <a href="Dihedral_group" title="Dihedral group">dihedral group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{3\mathrm {h} }}">

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<annotation encoding="application/x-tex">{\displaystyle D_{3\mathrm {h} }}</annotation>

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</math><img src="./c8f35493f21e2d3868ab7b52b08499661eee4276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.892ex; height:2.509ex;" alt="{\displaystyle D_{3\mathrm {h} }}" loading="lazy"> of order twelve. Its <a href="Dihedral_angle" title="Dihedral angle">dihedral angles</a> can be calculated by adding the angles of the component pyramids and prism. The prism itself has square-triangle dihedral angles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}">

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<annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation>

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</math><img src="./2b44e3d874a0b229fded7ffce67a0677dd5b8b67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}" loading="lazy"> and square-square angles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /3}">

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

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</math><img src="./56c1a0cd8279cea58b0ccb583e75a0ee93975883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /3}" loading="lazy">. The triangle-triangle angles on the pyramid are the same as in the <a href="Regular_octahedron" title="Regular octahedron">regular octahedron</a>, and the square-triangle angles are half that. Therefore, for the triaugmented triangular prism, the dihedral angles incident to the degree-four vertices, on the edges of the prism triangles, and on the square-to-square prism edges are, respectively,<a href="#cite_note-johnson-12">[12]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arccos \left(-{\frac {1}{3}}\right)&\approx 109.5^{\circ },\\{\frac {\pi }{2}}+{\frac {1}{2}}\arccos \left(-{\frac {1}{3}}\right)&\approx 144.7^{\circ },\\{\frac {\pi }{3}}+\arccos \left(-{\frac {1}{3}}\right)&\approx 169.5^{\circ }.\\\end{aligned}}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Fritsch_graph">Fritsch graph</h2></div>

The graph of the triaugmented triangular prism has 9 vertices and 21 edges. It was used by <a href="#CITEREFFritschFritsch1998">Fritsch & Fritsch (1998)</a> as a small counterexample to <a href="Alfred_Kempe" title="Alfred Kempe">Alfred Kempe</a>'s false proof of the <a href="Four_color_theorem" title="Four color theorem">four color theorem</a> using <a href="Kempe_chain" title="Kempe chain">Kempe chains</a>, and its dual map was used as their book's cover illustration.<a href="#cite_note-ff98-13">[13]</a> Therefore, this graph has subsequently been named the Fritsch graph.<a href="#cite_note-involve-14">[14]</a> An even smaller counterexample, called the Soifer graph, is obtained by removing one edge from the Fritsch graph (the bottom edge in the illustration here).<a href="#cite_note-involve-14">[14]</a><a href="#cite_note-soifer-15">[15]</a>

The Fritsch graph is one of only six connected graphs in which the <a href="Neighbourhood_(graph_theory)" title="Neighbourhood (graph theory)">neighborhood</a> of every vertex is a cycle of length four or five. More generally, when every vertex in a graph has a cycle of length at least four as its neighborhood, the triangles of the graph automatically link up to form a <a href="Manifold" title="Manifold">topological surface</a> called a <a href="Triangulation_(topology)" title="Triangulation (topology)">Whitney triangulation</a>. These six graphs come from the six Whitney triangulations that, when their triangles are equilateral, have positive <a href="Angular_defect" title="Angular defect">angular defect</a> at every vertex. This makes them a combinatorial analogue of the positively curved smooth surfaces. They come from six of the eight deltahedra—excluding the two that have a vertex with a triangular neighborhood. As well as the Fritsch graph, the other five are the graphs of the <a href="Regular_octahedron" title="Regular octahedron">regular octahedron</a>, <a href="Regular_icosahedron" title="Regular icosahedron">regular icosahedron</a>, <a href="Pentagonal_bipyramid" title="Pentagonal bipyramid">pentagonal bipyramid</a>, <a href="Snub_disphenoid" title="Snub disphenoid">snub disphenoid</a>, and <a href="Gyroelongated_square_bipyramid" title="Gyroelongated square bipyramid">gyroelongated square bipyramid</a>.<a href="#cite_note-knill-16">[16]</a>

<div class="mw-heading mw-heading2"><h2 id="Dual_associahedron">Dual associahedron</h2></div>

The <a href="Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> of the triaugmented triangular prism has a face for each vertex of the triaugmented triangular prism, and a vertex for each face. It is an <a href="Enneahedron" title="Enneahedron">enneahedron</a> (that is, a nine-sided polyhedron)<a href="#cite_note-fr07-17">[17]</a> that can be realized with three non-adjacent <a href="Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a> faces, and six more faces that are congruent irregular <a href="Pentagon" title="Pentagon">pentagons</a>.<a href="#cite_note-as18-18">[18]</a> It is also known as an order-5 <a href="Associahedron" title="Associahedron">associahedron</a>, a polyhedron whose vertices represent the 14 triangulations of a <a href="Regular_hexagon" class="mw-redirect" title="Regular hexagon">regular hexagon</a>.<a href="#cite_note-fr07-17">[17]</a> A less-symmetric form of this dual polyhedron, obtained by slicing a <a href="Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a> into four congruent quarters by two planes that perpendicularly bisect two parallel families of its edges, is a <a href="Space-filling_polyhedron" title="Space-filling polyhedron">space-filling polyhedron</a>.<a href="#cite_note-goldberg-19">[19]</a>

More generally, when a polytope is the dual of an associahedron, its boundary (a <a href="Simplicial_complex" title="Simplicial complex">simplicial complex</a> of triangles, tetrahedra, or higher-dimensional simplices) is called a "cluster complex". In the case of the triaugmented triangular prism, it is a cluster complex of type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{3}}">

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</math><img src="./e3785baf5f5c5e129c758b7ffafc94d559d799df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{3}}" loading="lazy">, associated with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{3}}">

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

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</math><img src="./e3785baf5f5c5e129c758b7ffafc94d559d799df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{3}}" loading="lazy"> <a href="Cluster_algebra" title="Cluster algebra">cluster algebra</a>.<a href="#cite_note-bsw13-20">[20]</a> The connection with the associahedron provides a correspondence between the nine vertices of the triaugmented triangular prism and the nine diagonals of a hexagon. The edges of the triaugmented triangular prism correspond to pairs of diagonals that do not cross, and the triangular faces of the triaugmented triangular prism correspond to the triangulations of the hexagon (consisting of three non-crossing diagonals). The triangulations of other regular polygons correspond to polytopes in the same way, with dimension equal to the number of sides of the polygon minus three.<a href="#cite_note-fr07-17">[17]</a>

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

In the geometry of <a href="Chemical_compound" title="Chemical compound">chemical compounds</a>, it is common to visualize an <a href="Atom_cluster" class="mw-redirect" title="Atom cluster">atom cluster</a> surrounding a central atom as a polyhedron—the <a href="Convex_hull" title="Convex hull">convex hull</a> of the surrounding atoms' locations. The <a href="Tricapped_trigonal_prismatic_molecular_geometry" title="Tricapped trigonal prismatic molecular geometry">tricapped trigonal prismatic molecular geometry</a> describes clusters for which this polyhedron is a triaugmented triangular prism, although not necessarily one with equilateral triangle faces.<a href="#cite_note-kepert-2">[2]</a> For example, the <a href="Lanthanide" title="Lanthanide">lanthanides</a> from <a href="Lanthanum" title="Lanthanum">lanthanum</a> to <a href="Dysprosium" title="Dysprosium">dysprosium</a> dissolve in water to form <a href="Cation" class="mw-redirect" title="Cation">cations</a> surrounded by nine water molecules arranged as a triaugmented triangular prism.<a href="#cite_note-Persson2022-21">[21]</a>

In the <a href="Thomson_problem" title="Thomson problem">Thomson problem</a>, concerning the minimum-energy configuration of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> charged particles on a sphere, and for the <a href="Tammes_problem" title="Tammes problem">Tammes problem</a> of constructing a <a href="Spherical_code" title="Spherical code">spherical code</a> maximizing the smallest distance among the points, the minimum solution known for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=9}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=9}</annotation>

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</math><img src="./b1029ce8384fde9f4da54009c5a79f17a9758085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=9}" loading="lazy"> places the points at the vertices of a triaugmented triangular prism with non-equilateral faces, <a href="Circumscribed_sphere" title="Circumscribed sphere">inscribed in a sphere</a>. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is not known.<a href="#cite_note-whyte-22">[22]</a>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Triaugmented_triangular_prism" class="extiw external" title="commons:Category:Triaugmented triangular prism">Triaugmented triangular prism</a>.</div></div>

</div>

<ul><li><a href="Cs%C3%A1sz%C3%A1r_polyhedron" title="Császár polyhedron">Császár polyhedron</a> – Toroidal polyhedron with 14 triangle faces</li>

<li><a href="Steffen's_polyhedron" title="Steffen's polyhedron">Steffen's polyhedron</a> – Flexible polyhedron with 14 triangle faces</li></ul>

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

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</style><cite id="CITEREFSloaneHardinDuffConway1995" class="citation cs2"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a>; Hardin, R. H.; Duff, T. D. S.; <a href="John_Horton_Conway" title="John Horton Conway">Conway, J. H.</a> (1995), "Minimal-energy clusters of hard spheres", <a href="Discrete_%26_Computational_Geometry" title="Discrete & Computational Geometry">Discrete & Computational Geometry</a>, 14 (3): 237–259, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02570704">10.1007/BF02570704</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=1344734">1344734</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:26955765">26955765</a></cite>

</li>

<li id="cite_note-kepert-2">^ <a href="#cite_ref-kepert_2-0">a</a> <a href="#cite_ref-kepert_2-1">b</a> <cite id="CITEREFKepert1982" class="citation cs2">Kepert, David L. (1982), "Polyhedra", Inorganic Chemistry Concepts, vol. 6, Springer, pp. 7–21, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-68046-5_2">10.1007/978-3-642-68046-5_2</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-68048-9</bdi></cite>

</li>

<li id="cite_note-burgiel-3"><a href="#cite_ref-burgiel_3-0">^</a> <cite id="CITEREFBurgiel2015" class="citation cs2">Burgiel, Heidi (2015), <a rel="nofollow" class="external text" href="https://archive.bridgesmathart.org/2015/bridges2015-585.html">"Unit origami: star-building on deltahedra"</a>, in Delp, Kelly; Kaplan, Craig S.; McKenna, Douglas; Sarhangi, Reza (eds.), Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, Phoenix, Arizona: Tessellations Publishing, pp. 585–588, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-938664-15-1</bdi></cite>

</li>

<li id="cite_note-pugh-4"><a href="#cite_ref-pugh_4-0">^</a> <cite id="CITEREFPugh1976" class="citation cs2">Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, p. 31, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780520030565</bdi></cite>; see table, line 35

</li>

<li id="cite_note-timofeenko-2009-5"><a href="#cite_ref-timofeenko-2009_5-0">^</a> <cite id="CITEREFTimofeenko2009" class="citation cs2">Timofeenko, A. V. (2009), <a rel="nofollow" class="external text" href="https://www.interocitors.com/tmp/papers/timo-parquet.pdf">"Convex Polyhedra with Parquet Faces"</a> (PDF), Docklady Mathematics, 80 (2): 720–723, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2FS1064562409050238">10.1134/S1064562409050238</a></cite>

</li>

<li id="cite_note-trigg-6"><a href="#cite_ref-trigg_6-0">^</a> <cite id="CITEREFTrigg1978" class="citation cs2"><a href="Charles_W._Trigg" title="Charles W. Trigg">Trigg, Charles W.</a> (1978), "An infinite class of deltahedra", Mathematics Magazine, 51 (1): 55–57, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0025570X.1978.11976675">10.1080/0025570X.1978.11976675</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/2689647">2689647</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=1572246">1572246</a></cite>

</li>

<li id="cite_note-fw47-7"><a href="#cite_ref-fw47_7-0">^</a> <cite id="CITEREFFreudenthalvan_der_Waerden1947" class="citation cs2"><a href="Hans_Freudenthal" title="Hans Freudenthal">Freudenthal, H.</a>; <a href="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">van der Waerden, B. L.</a> (1947), "On an assertion of Euclid", <a href="Simon_Stevin_(journal)" title="Simon Stevin (journal)">Simon Stevin</a>, 25: 115–121, <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=0021687">0021687</a></cite>

</li>

<li id="cite_note-cundy-8"><a href="#cite_ref-cundy_8-0">^</a> <cite id="CITEREFCundy1952" class="citation cs2"><a href="Martyn_Cundy" title="Martyn Cundy">Cundy, H. Martyn</a> (December 1952), "Deltahedra", <a href="The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a>, 36 (318): 263–266, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3608204">10.2307/3608204</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/3608204">3608204</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=0051525">0051525</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:250435684">250435684</a></cite>

</li>

<li id="cite_note-francis-9"><a href="#cite_ref-francis_9-0">^</a> <cite id="CITEREFFrancis2013" class="citation cs2">Francis, Darryl (August 2013), <a rel="nofollow" class="external text" href="https://go.gale.com/ps/i.do?id=GALE%7CA340298118">"Johnson solids & their acronyms"</a>, Word Ways, 46 (3): 177</cite>

</li>

<li id="cite_note-berman-10">^ <a href="#cite_ref-berman_10-0">a</a> <a href="#cite_ref-berman_10-1">b</a> <a href="#cite_ref-berman_10-2">c</a> <cite id="CITEREFBerman1971" class="citation cs2">Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, <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>; see Table IV, line 71, p. 338

</li>

<li id="cite_note-lptw-11"><a href="#cite_ref-lptw_11-0">^</a> <cite id="CITEREFLawsonParishTraubWeyhaupt2013" class="citation cs2">Lawson, Kyle A.; Parish, James L.; Traub, Cynthia M.; Weyhaupt, Adam G. (2013), "Coloring graphs to classify simple closed geodesics on convex deltahedra", International Journal of Pure and Applied Mathematics, 89 (2): 123–139, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.12732%2Fijpam.v89i2.1">10.12732/ijpam.v89i2.1</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1286.05048">1286.05048</a></cite>

</li>

<li id="cite_note-johnson-12"><a href="#cite_ref-johnson_12-0">^</a> <cite id="CITEREFJohnson1966" class="citation cs2"><a href="Norman_Johnson_(mathematician)" title="Norman Johnson (mathematician)">Johnson, Norman W.</a> (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-1966-021-8">10.4153/CJM-1966-021-8</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=0185507">0185507</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:122006114">122006114</a></cite>; see Table III, line 51

</li>

<li id="cite_note-ff98-13"><a href="#cite_ref-ff98_13-0">^</a> <cite id="CITEREFFritschFritsch1998" class="citation cs2">Fritsch, Rudolf; Fritsch, Gerda (1998), The Four-Color Theorem: History, Topological Foundations, and Idea of Proof, New York: Springer-Verlag, pp. 175–176, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-1720-6">10.1007/978-1-4612-1720-6</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98497-6</bdi>, <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=1633950">1633950</a></cite>

</li>

<li id="cite_note-involve-14">^ <a href="#cite_ref-involve_14-0">a</a> <a href="#cite_ref-involve_14-1">b</a> <cite id="CITEREFGethnerKallichandaMentisBraudrick2009" class="citation cs2"><a href="Ellen_Gethner" title="Ellen Gethner">Gethner, Ellen</a>; Kallichanda, Bopanna; Mentis, Alexander; Braudrick, Sarah; Chawla, Sumeet; Clune, Andrew; Drummond, Rachel; Evans, Panagiota; Roche, William; Takano, Nao (October 2009), "How false is Kempe's proof of the Four Color Theorem? Part II", Involve: A Journal of Mathematics, 2 (3), Mathematical Sciences Publishers: 249–265, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Finvolve.2009.2.249">10.2140/involve.2009.2.249</a></cite>

</li>

<li id="cite_note-soifer-15"><a href="#cite_ref-soifer_15-0">^</a> <cite id="CITEREFSoifer2008" class="citation cs2"><a href="Alexander_Soifer" title="Alexander Soifer">Soifer, Alexander</a> (2008), <a href="The_Mathematical_Coloring_Book" title="The Mathematical Coloring Book">The Mathematical Coloring Book</a>, Springer-Verlag, pp. 181–182, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-74640-1</bdi></cite>

</li>

<li id="cite_note-knill-16"><a href="#cite_ref-knill_16-0">^</a> <cite id="CITEREFKnill2019" class="citation cs2">Knill, Oliver (2019), A simple sphere theorem for graphs, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1910.02708">1910.02708</a></cite>

</li>

<li id="cite_note-fr07-17">^ <a href="#cite_ref-fr07_17-0">a</a> <a href="#cite_ref-fr07_17-1">b</a> <a href="#cite_ref-fr07_17-2">c</a> <cite id="CITEREFFominReading2007" class="citation cs2">Fomin, Sergey; Reading, Nathan (2007), "Root systems and generalized associahedra", in Miller, Ezra; Reiner, Victor; Sturmfels, Bernd (eds.), Geometric combinatorics, IAS/Park City Mathematics Series, vol. 13, Providence, Rhode Island: American Mathematical Society, pp. 63–131, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0505518">math/0505518</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fpcms%2F013%2F03">10.1090/pcms/013/03</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-3736-8</bdi>, <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=2383126">2383126</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:11435731">11435731</a></cite>; see Definition 3.3, Figure 3.6, and related discussion

</li>

<li id="cite_note-as18-18"><a href="#cite_ref-as18_18-0">^</a> <cite id="CITEREFAmirSéquin2018" class="citation cs2">Amir, Yifat; <a href="Carlo_H._S%C3%A9quin" title="Carlo H. Séquin">Séquin, Carlo H.</a> (2018), <a rel="nofollow" class="external text" href="https://archive.bridgesmathart.org/2018/bridges2018-131.html">"Modular toroids constructed from nonahedra"</a>, in <a href="Eve_Torrence" title="Eve Torrence">Torrence, Eve</a>; Torrence, Bruce; <a href="Carlo_H._S%C3%A9quin" title="Carlo H. Séquin">Séquin, Carlo</a>; Fenyvesi, Kristóf (eds.), Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture, Phoenix, Arizona: Tessellations Publishing, pp. 131–138, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-938664-27-4</bdi></cite>

</li>

<li id="cite_note-goldberg-19"><a href="#cite_ref-goldberg_19-0">^</a> <cite id="CITEREFGoldberg1982" class="citation cs2">Goldberg, Michael (1982), "On the space-filling enneahedra", Geometriae Dedicata, 12 (3): 297–306, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00147314">10.1007/BF00147314</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=0661535">0661535</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:120914105">120914105</a></cite>; see polyhedron 9-IV, p. 301

</li>

<li id="cite_note-bsw13-20"><a href="#cite_ref-bsw13_20-0">^</a> <cite id="CITEREFBarceloSeversWhite2013" class="citation cs2"><a href="H%C3%A9l%C3%A8ne_Barcelo" title="Hélène Barcelo">Barcelo, Hélène</a>; Severs, Christopher; White, Jacob A. (2013), "The discrete fundamental group of the associahedron, and the exchange module", International Journal of Algebra and Computation, 23 (4): 745–762, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1012.2810">1012.2810</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0218196713400079">10.1142/S0218196713400079</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=3078054">3078054</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:14722555">14722555</a></cite>

</li>

<li id="cite_note-Persson2022-21"><a href="#cite_ref-Persson2022_21-0">^</a> <cite id="CITEREFPersson2022" class="citation cs2">Persson, Ingmar (2022), "Structures of Hydrated Metal Ions in Solid State and Aqueous Solution", Liquids, 2 (3): 210–242, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fliquids2030014">10.3390/liquids2030014</a></cite>

</li>

<li id="cite_note-whyte-22"><a href="#cite_ref-whyte_22-0">^</a> <cite id="CITEREFWhyte1952" class="citation cs2">Whyte, L. L. (1952), "Unique arrangements of points on a sphere", <a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>, 59 (9): 606–611, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.1952.11988207">10.1080/00029890.1952.11988207</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/2306764">2306764</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=0050303">0050303</a></cite>

</li>

</ol></div></div>

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</style><div id="Johnson_solids430" style="font-size:114%;margin:0 4em"><a href="Johnson_solid" title="Johnson solid">Johnson solids</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Pyramid_(geometry)" title="Pyramid (geometry)">Pyramids</a>, <a href="Cupola_(geometry)" title="Cupola (geometry)">cupolae</a> and <a href="Rotunda_(geometry)" title="Rotunda (geometry)">rotundae</a></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="Square_pyramid" title="Square pyramid">square pyramid</a></li>

<li><a href="Pentagonal_pyramid" title="Pentagonal pyramid">pentagonal pyramid</a></li>

<li><a href="Triangular_cupola" title="Triangular cupola">triangular cupola</a></li>

<li><a href="Square_cupola" title="Square cupola">square cupola</a></li>

<li><a href="Pentagonal_cupola" title="Pentagonal cupola">pentagonal cupola</a></li>

<li><a href="Pentagonal_rotunda" title="Pentagonal rotunda">pentagonal rotunda</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modified <a href="Pyramid_(geometry)" title="Pyramid (geometry)">pyramids</a></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="Elongated_triangular_pyramid" title="Elongated triangular pyramid">elongated triangular pyramid</a></li>

<li><a href="Elongated_square_pyramid" title="Elongated square pyramid">elongated square pyramid</a></li>

<li><a href="Elongated_pentagonal_pyramid" title="Elongated pentagonal pyramid">elongated pentagonal pyramid</a></li>

<li><a href="Gyroelongated_square_pyramid" title="Gyroelongated square pyramid">gyroelongated square pyramid</a></li>

<li><a href="Gyroelongated_pentagonal_pyramid" title="Gyroelongated pentagonal pyramid">gyroelongated pentagonal pyramid</a></li>

<li><a href="Triangular_bipyramid" title="Triangular bipyramid">triangular bipyramid</a></li>

<li><a href="Pentagonal_bipyramid" title="Pentagonal bipyramid">pentagonal bipyramid</a></li>

<li><a href="Elongated_triangular_bipyramid" title="Elongated triangular bipyramid">elongated triangular bipyramid</a></li>

<li><a href="Elongated_square_bipyramid" title="Elongated square bipyramid">elongated square bipyramid</a></li>

<li><a href="Elongated_pentagonal_bipyramid" title="Elongated pentagonal bipyramid">elongated pentagonal bipyramid</a></li>

<li><a href="Gyroelongated_square_bipyramid" title="Gyroelongated square bipyramid">gyroelongated square bipyramid</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modified <a href="Cupola_(geometry)" title="Cupola (geometry)">cupolae</a> and <a href="Rotunda_(geometry)" title="Rotunda (geometry)">rotundae</a></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="Elongated_triangular_cupola" title="Elongated triangular cupola">elongated triangular cupola</a></li>

<li><a href="Elongated_square_cupola" title="Elongated square cupola">elongated square cupola</a></li>

<li><a href="Elongated_pentagonal_cupola" title="Elongated pentagonal cupola">elongated pentagonal cupola</a></li>

<li><a href="Elongated_pentagonal_rotunda" title="Elongated pentagonal rotunda">elongated pentagonal rotunda</a></li>

<li><a href="Gyroelongated_triangular_cupola" title="Gyroelongated triangular cupola">gyroelongated triangular cupola</a></li>

<li><a href="Gyroelongated_square_cupola" title="Gyroelongated square cupola">gyroelongated square cupola</a></li>

<li><a href="Gyroelongated_pentagonal_cupola" title="Gyroelongated pentagonal cupola">gyroelongated pentagonal cupola</a></li>

<li><a href="Gyroelongated_pentagonal_rotunda" title="Gyroelongated pentagonal rotunda">gyroelongated pentagonal rotunda</a></li>

<li><a href="Gyrobifastigium" title="Gyrobifastigium">gyrobifastigium</a></li>

<li><a href="Triangular_orthobicupola" title="Triangular orthobicupola">triangular orthobicupola</a></li>

<li><a href="Square_orthobicupola" title="Square orthobicupola">square orthobicupola</a></li>

<li><a href="Square_gyrobicupola" title="Square gyrobicupola">square gyrobicupola</a></li>

<li><a href="Pentagonal_orthobicupola" title="Pentagonal orthobicupola">pentagonal orthobicupola</a></li>

<li><a href="Pentagonal_gyrobicupola" title="Pentagonal gyrobicupola">pentagonal gyrobicupola</a></li>

<li><a href="Pentagonal_orthocupolarotunda" title="Pentagonal orthocupolarotunda">pentagonal orthocupolarotunda</a></li>

<li><a href="Pentagonal_gyrocupolarotunda" title="Pentagonal gyrocupolarotunda">pentagonal gyrocupolarotunda</a></li>

<li><a href="Pentagonal_orthobirotunda" title="Pentagonal orthobirotunda">pentagonal orthobirotunda</a></li>

<li><a href="Elongated_triangular_orthobicupola" title="Elongated triangular orthobicupola">elongated triangular orthobicupola</a></li>

<li><a href="Elongated_triangular_gyrobicupola" title="Elongated triangular gyrobicupola">elongated triangular gyrobicupola</a></li>

<li><a href="Elongated_square_gyrobicupola" title="Elongated square gyrobicupola">elongated square gyrobicupola</a></li>

<li><a href="Elongated_pentagonal_orthobicupola" title="Elongated pentagonal orthobicupola">elongated pentagonal orthobicupola</a></li>

<li><a href="Elongated_pentagonal_gyrobicupola" title="Elongated pentagonal gyrobicupola">elongated pentagonal gyrobicupola</a></li>

<li><a href="Elongated_pentagonal_orthocupolarotunda" title="Elongated pentagonal orthocupolarotunda">elongated pentagonal orthocupolarotunda</a></li>

<li><a href="Elongated_pentagonal_gyrocupolarotunda" title="Elongated pentagonal gyrocupolarotunda">elongated pentagonal gyrocupolarotunda</a></li>

<li><a href="Elongated_pentagonal_orthobirotunda" title="Elongated pentagonal orthobirotunda">elongated pentagonal orthobirotunda</a></li>

<li><a href="Elongated_pentagonal_gyrobirotunda" title="Elongated pentagonal gyrobirotunda">elongated pentagonal gyrobirotunda</a></li>

<li><a href="Gyroelongated_triangular_bicupola" title="Gyroelongated triangular bicupola">gyroelongated triangular bicupola</a></li>

<li><a href="Gyroelongated_square_bicupola" title="Gyroelongated square bicupola">gyroelongated square bicupola</a></li>

<li><a href="Gyroelongated_pentagonal_bicupola" title="Gyroelongated pentagonal bicupola">gyroelongated pentagonal bicupola</a></li>

<li><a href="Gyroelongated_pentagonal_cupolarotunda" title="Gyroelongated pentagonal cupolarotunda">gyroelongated pentagonal cupolarotunda</a></li>

<li><a href="Gyroelongated_pentagonal_birotunda" title="Gyroelongated pentagonal birotunda">gyroelongated pentagonal birotunda</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Augmented <a href="Prism_(geometry)" title="Prism (geometry)">prisms</a></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="Augmented_triangular_prism" title="Augmented triangular prism">augmented triangular prism</a></li>

<li><a href="Biaugmented_triangular_prism" title="Biaugmented triangular prism">biaugmented triangular prism</a></li>

<li><a href="Augmented_pentagonal_prism" title="Augmented pentagonal prism">augmented pentagonal prism</a></li>

<li><a href="Biaugmented_pentagonal_prism" title="Biaugmented pentagonal prism">biaugmented pentagonal prism</a></li>

<li><a href="Augmented_hexagonal_prism" title="Augmented hexagonal prism">augmented hexagonal prism</a></li>

<li><a href="Parabiaugmented_hexagonal_prism" title="Parabiaugmented hexagonal prism">parabiaugmented hexagonal prism</a></li>

<li><a href="Metabiaugmented_hexagonal_prism" title="Metabiaugmented hexagonal prism">metabiaugmented hexagonal prism</a></li>

<li><a href="Triaugmented_hexagonal_prism" title="Triaugmented hexagonal prism">triaugmented hexagonal prism</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modified <a href="Platonic_solid" title="Platonic solid">Platonic solids</a></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="Augmented_dodecahedron" title="Augmented dodecahedron">augmented dodecahedron</a></li>

<li><a href="Parabiaugmented_dodecahedron" title="Parabiaugmented dodecahedron">parabiaugmented dodecahedron</a></li>

<li><a href="Metabiaugmented_dodecahedron" title="Metabiaugmented dodecahedron">metabiaugmented dodecahedron</a></li>

<li><a href="Triaugmented_dodecahedron" title="Triaugmented dodecahedron">triaugmented dodecahedron</a></li>

<li><a href="Metabidiminished_icosahedron" title="Metabidiminished icosahedron">metabidiminished icosahedron</a></li>

<li><a href="Tridiminished_icosahedron" title="Tridiminished icosahedron">tridiminished icosahedron</a></li>

<li><a href="Augmented_tridiminished_icosahedron" title="Augmented tridiminished icosahedron">augmented tridiminished icosahedron</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Modified <a href="Archimedean_solid" title="Archimedean solid">Archimedean solids</a></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="Augmented_truncated_tetrahedron" title="Augmented truncated tetrahedron">augmented truncated tetrahedron</a></li>

<li><a href="Augmented_truncated_cube" title="Augmented truncated cube">augmented truncated cube</a></li>

<li><a href="Biaugmented_truncated_cube" title="Biaugmented truncated cube">biaugmented truncated cube</a></li>

<li><a href="Augmented_truncated_dodecahedron" title="Augmented truncated dodecahedron">augmented truncated dodecahedron</a></li>

<li><a href="Parabiaugmented_truncated_dodecahedron" title="Parabiaugmented truncated dodecahedron">parabiaugmented truncated dodecahedron</a></li>

<li><a href="Metabiaugmented_truncated_dodecahedron" title="Metabiaugmented truncated dodecahedron">metabiaugmented truncated dodecahedron</a></li>

<li><a href="Triaugmented_truncated_dodecahedron" title="Triaugmented truncated dodecahedron">triaugmented truncated dodecahedron</a></li>

<li><a href="Gyrate_rhombicosidodecahedron" title="Gyrate rhombicosidodecahedron">gyrate rhombicosidodecahedron</a></li>

<li><a href="Parabigyrate_rhombicosidodecahedron" title="Parabigyrate rhombicosidodecahedron">parabigyrate rhombicosidodecahedron</a></li>

<li><a href="Metabigyrate_rhombicosidodecahedron" title="Metabigyrate rhombicosidodecahedron">metabigyrate rhombicosidodecahedron</a></li>

<li><a href="Trigyrate_rhombicosidodecahedron" title="Trigyrate rhombicosidodecahedron">trigyrate rhombicosidodecahedron</a></li>

<li><a href="Diminished_rhombicosidodecahedron" title="Diminished rhombicosidodecahedron">diminished rhombicosidodecahedron</a></li>

<li><a href="Paragyrate_diminished_rhombicosidodecahedron" title="Paragyrate diminished rhombicosidodecahedron">paragyrate diminished rhombicosidodecahedron</a></li>

<li><a href="Metagyrate_diminished_rhombicosidodecahedron" title="Metagyrate diminished rhombicosidodecahedron">metagyrate diminished rhombicosidodecahedron</a></li>

<li><a href="Bigyrate_diminished_rhombicosidodecahedron" title="Bigyrate diminished rhombicosidodecahedron">bigyrate diminished rhombicosidodecahedron</a></li>

<li><a href="Parabidiminished_rhombicosidodecahedron" title="Parabidiminished rhombicosidodecahedron">parabidiminished rhombicosidodecahedron</a></li>

<li><a href="Metabidiminished_rhombicosidodecahedron" title="Metabidiminished rhombicosidodecahedron">metabidiminished rhombicosidodecahedron</a></li>

<li><a href="Gyrate_bidiminished_rhombicosidodecahedron" title="Gyrate bidiminished rhombicosidodecahedron">gyrate bidiminished rhombicosidodecahedron</a></li>

<li><a href="Tridiminished_rhombicosidodecahedron" title="Tridiminished rhombicosidodecahedron">tridiminished rhombicosidodecahedron</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="Elementary_polyhedron" class="mw-redirect" title="Elementary polyhedron">elementary solids</a></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="Snub_disphenoid" title="Snub disphenoid">snub disphenoid</a></li>

<li><a href="Snub_square_antiprism" title="Snub square antiprism">snub square antiprism</a></li>

<li><a href="Sphenocorona" title="Sphenocorona">sphenocorona</a></li>

<li><a href="Augmented_sphenocorona" title="Augmented sphenocorona">augmented sphenocorona</a></li>

<li><a href="Sphenomegacorona" title="Sphenomegacorona">sphenomegacorona</a></li>

<li><a href="Hebesphenomegacorona" title="Hebesphenomegacorona">hebesphenomegacorona</a></li>

<li><a href="Disphenocingulum" title="Disphenocingulum">disphenocingulum</a></li>

<li><a href="Bilunabirotunda" title="Bilunabirotunda">bilunabirotunda</a></li>

<li><a href="Triangular_hebesphenorotunda" title="Triangular hebesphenorotunda">triangular hebesphenorotunda</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>(See also <a href="List_of_Johnson_solids" title="List of Johnson solids">List of Johnson solids</a>, a sortable table)</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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