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<title>Triangular prism</title>

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Triangular prism

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</style><div role="note" class="hatnote navigation-not-searchable">For the optical prism, see <a href="Triangular_prism_(optics)" class="mw-redirect" title="Triangular prism (optics)">Triangular prism (optics)</a>.</div>

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e7dcc3">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="Prism_(geometry)" title="Prism (geometry)">Prism</a> <a href="Semiregular_polyhedron" title="Semiregular polyhedron">Semiregular polyhedron</a> <a href="Uniform_polyhedron" title="Uniform polyhedron">Uniform polyhedron</a> <a href="Cupola_(geometry)" title="Cupola (geometry)">Cupola</a></td></tr><tr><th scope="row" class="infobox-label"><a href="Face_(geometry)" title="Face (geometry)">Faces</a></th><td class="infobox-data">2 <a href="Triangle" title="Triangle">triangles</a> 3 <a href="Square" title="Square">squares</a></td></tr><tr><th scope="row" class="infobox-label"><a href="Edge_(geometry)" title="Edge (geometry)">Edges</a></th><td class="infobox-data">9</td></tr><tr><th scope="row" class="infobox-label"><a href="Vertex_(geometry)" title="Vertex (geometry)">Vertices</a></th><td class="infobox-data">6</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">D3h</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">As a semi-regular: <div><ul><li>square-to-square: 60°</li><li>square-to-triangle: 90°</li></ul></div></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="Triangular_bipyramid" title="Triangular bipyramid">Triangular bipyramid</a></td></tr></tbody></table>

In <a href="Geometry" title="Geometry">geometry</a>, a triangular prism or trigonal prism<a href="#cite_note-FOOTNOTEKing1994[httpsbooksgooglecombooksidc3fsCAAAQBAJpgPA113_113]-1">[1]</a> is a <a href="Prism_(geometry)" title="Prism (geometry)">prism</a> with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both <a href="Semiregular_polyhedron" title="Semiregular polyhedron">semiregular</a> and <a href="Uniform_polyhedron" title="Uniform polyhedron">uniform</a>.

The triangular prism can be used in constructing another polyhedron. Examples are some of the <a href="Johnson_solid" title="Johnson solid">Johnson solids</a>, the truncated right triangular prism, and <a href="Sch%C3%B6nhardt_polyhedron" title="Schönhardt polyhedron">Schönhardt polyhedron</a>.

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

A triangular prism has 6 vertices, 9 edges, and 5 faces. Every prism has 2 congruent faces known as its bases, and the bases of a triangular prism are <a href="Triangle" title="Triangle">triangles</a>. The triangle has 3 vertices, each of which pairs with another triangle's vertex, making up another 3 edges. These edges form 3 <a href="Parallelogram" title="Parallelogram">parallelograms</a> as other faces.<a href="#cite_note-2">[2]</a> If the prism's edges are perpendicular to the base, the lateral faces are <a href="Rectangle" title="Rectangle">rectangles</a>. The prism is called a right triangular prism.<a href="#cite_note-FOOTNOTEKernBland193825-3">[3]</a> This prism may also be considered a special case of a <a href="Wedge_(geometry)" title="Wedge (geometry)">wedge</a>.<a href="#cite_note-FOOTNOTEHaul1893[httpsarchiveorgdetailsmensuration00hallgoogpagen57_45]-4">[4]</a> Topologically a <a href="Triangular_frustum" class="mw-redirect" title="Triangular frustum">triangular frustum</a> is the same polyhedron. Still, the two triangles are different sizes, and the sides are slanted trapezoids.

If the base is <a href="Equilateral_triangle" title="Equilateral triangle">equilateral</a> and the lateral faces are <a href="Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a>, then the right triangular prism is <a href="Semiregular_polyhedron" title="Semiregular polyhedron">semiregular</a>. A semiregular prism means that the number of its polygonal base's edges equals the number of its square faces.<a href="#cite_note-FOOTNOTEO'KeeffeHyde2020[httpsbooksgooglecombooksid_MjPDwAAQBAJpgPA139_139]-5">[5]</a> More generally, the triangular prism is <a href="Uniform_polyhedron" title="Uniform polyhedron">uniform</a>. This means that a triangular prism has <a href="Regular_polygon" title="Regular polygon">regular faces</a> and has an <a href="Isogonal_figure" title="Isogonal figure">isogonal</a> symmetry on vertices.<a href="#cite_note-6">[6]</a> The <a href="Point_groups_in_three_dimensions" title="Point groups in three dimensions">three-dimensional symmetry group</a> of a right triangular prism is <a href="Dihedral_group" title="Dihedral group">dihedral group</a> D3h of order 12: the appearance is unchanged if the triangular prism is rotated one- and two- thirds of a full angle around its <a href="Axis_of_symmetry" class="mw-redirect" title="Axis of symmetry">axis of symmetry</a> passing through the center's base, and reflecting across a horizontal plane. The <a href="Dual_polyhedron" title="Dual polyhedron">dual polyhedron</a> of a triangular prism is a <a href="Triangular_bipyramid" title="Triangular bipyramid">triangular bipyramid</a>. The triangular bipyramid has the same symmetry as the triangular prism.<a href="#cite_note-FOOTNOTEKing1994[httpsbooksgooglecombooksidc3fsCAAAQBAJpgPA113_113]-1">[1]</a> The dihedral angle between two adjacent square faces is the <a href="Internal_angle" class="mw-redirect" title="Internal angle">internal angle</a> of an equilateral triangle π/3 = 60°, and that between a square and a triangle is π/2 = 90°.<a href="#cite_note-FOOTNOTEJohnson1966-7">[7]</a>

The volume of any prism is the product of the area of the base and the distance between the two bases.<a href="#cite_note-FOOTNOTEKernBland193826-8">[8]</a> In the case of a triangular prism, its base is a triangle, so its volume can be calculated by multiplying the area of a triangle and the length of the prism:

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

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where b is the length of one side of the triangle, h is the length of an <a href="Altitude_(triangle)" title="Altitude (triangle)">altitude</a> drawn to that side, and l is the distance between the triangular faces.<a href="#cite_note-9">[9]</a> In the case of a right triangular prism, where all its edges are equal in length l, its volume can be calculated as the product of the equilateral triangle's area and length l:<a href="#cite_note-FOOTNOTEBerman1971-10">[10]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {3}}{2}}l^{2}\cdot l\approx 0.433l^{3}}</annotation>

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The triangular prism can be represented as the <a href="Prism_graph" title="Prism graph">prism graph</a> Π3. More generally, the prism graph Πn represents the n-sided prism.<a href="#cite_note-FOOTNOTEPisanskiServatius2013[httpsbooksgooglecombooksid3vnEcMCx0HkCpgPA21_21]-11">[11]</a> It is an example of <a href="Halin_graph" title="Halin graph">Halin graph</a>.<sup id="cite_ref-FOOTNOTESysłoProskurowski1983254Prop._4.3._Here_the_triangular_prism_is_identified_as_the_unique_graph_with_exactly_three_cycles_that_can_be_the_outer_cycle_of_a_realization_as_a_Halin_graph_12-0" class="reference"><a href="#cite_note-FOOTNOTESysłoProskurowski1983254Prop._4.3._Here_the_triangular_prism_is_identified_as_the_unique_graph_with_exactly_three_cycles_that_can_be_the_outer_cycle_of_a_realization_as_a_Halin_graph-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="Related_polyhedron">Related polyhedron</h2></div>

<div class="mw-heading mw-heading3"><h3 id="In_construction_of_polyhedron">In construction of polyhedron</h3></div>

Beyond the triangular bipyramid as its dual polyhedron, many other polyhedrons are related to the triangular prism. A <a href="Johnson_solid" title="Johnson solid">Johnson solid</a> is a convex polyhedron with regular faces, and this definition is sometimes omitted uniform polyhedrons such as <a href="Archimedean_solid" title="Archimedean solid">Archimedean solids</a>, <a href="Catalan_solid" title="Catalan solid">Catalan solids</a>, prisms and <a href="Antiprism" title="Antiprism">antiprisms</a>.<a href="#cite_note-13">[13]</a> There are 6 Johnson solids with their construction involving the triangular prism: <a href="Elongated_triangular_pyramid" title="Elongated triangular pyramid">elongated triangular pyramid</a>, <a href="Elongated_triangular_bipyramid" title="Elongated triangular bipyramid">elongated triangular bipyramid</a>, <a href="Gyrobifastigium" title="Gyrobifastigium">gyrobifastigium</a>, <a href="Augmented_triangular_prism" title="Augmented triangular prism">augmented triangular prism</a>, <a href="Biaugmented_triangular_prism" title="Biaugmented triangular prism">biaugmented triangular prism</a>, and <a href="Triaugmented_triangular_prism" title="Triaugmented triangular prism">triaugmented triangular prism</a>. The elongated triangular pyramid and the gyroelongated triangular pyramid are constructed by attaching <a href="Tetrahedron" title="Tetrahedron">tetrahedron</a> onto the base of a triangular prism. The augmented triangular prism, biaugmented triangular prism, and triaugmented triangular prism are constructed by attaching <a href="Equilateral_square_pyramid" class="mw-redirect" title="Equilateral square pyramid">equilateral square pyramids</a> onto the square face of the prism. The gyrobifastigium is constructed by attaching two triangular prisms along one of its square faces.<a href="#cite_note-14">[14]</a>

A truncated triangular prism is a triangular prism constructed by <a href="Truncation_(geometry)" title="Truncation (geometry)">truncating</a> its part at an oblique angle. As a result, the two bases are not parallel and every height has a different edge length. If the edges connecting bases are perpendicular to one of its bases, the prism is called a truncated right triangular prism. Given that A is the area of the triangular prism's base, and the three heights h1, h2, and h3, its volume can be determined in the following formula:<a href="#cite_note-FOOTNOTEKernBland193881-15">[15]</a>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {A(h_{1}+h_{2}+h_{3})}{3}}.}</annotation>

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<a href="Sch%C3%B6nhardt_polyhedron" title="Schönhardt polyhedron">Schönhardt polyhedron</a> is another polyhedron constructed from a triangular prism with equilateral triangle bases. This way, one of its bases rotates around the prism's centerline and breaks the square faces into <a href="Skew_polygon" title="Skew polygon">skew polygons</a>. Each square face can be re-triangulated with two triangles to form a non-convex dihedral angle.<a href="#cite_note-16">[16]</a> As a result, the Schönhardt polyhedron cannot be <a href="Triangulation_(geometry)" title="Triangulation (geometry)">triangulated</a> by a partition into tetrahedra. It is also that the Schönhardt polyhedron has no internal diagonals.<a href="#cite_note-FOOTNOTEBagemihl1948-17">[17]</a> It is named after German mathematician <a href="Erich_Sch%C3%B6nhardt" title="Erich Schönhardt">Erich Schönhardt</a>, who described it in 1928, although the related structure was exhibited by artist <a href="Karlis_Johansons" title="Karlis Johansons">Karlis Johansons</a> in 1921.<a href="#cite_note-18">[18]</a>

There are 4 uniform compounds of triangular prisms. They are <a href="Compound_of_four_triangular_prisms" title="Compound of four triangular prisms">compound of four triangular prisms</a>, <a href="Compound_of_eight_triangular_prisms" title="Compound of eight triangular prisms">compound of eight triangular prisms</a>, <a href="Compound_of_ten_triangular_prisms" title="Compound of ten triangular prisms">compound of ten triangular prisms</a>, <a href="Compound_of_twenty_triangular_prisms" title="Compound of twenty triangular prisms">compound of twenty triangular prisms</a>.<a href="#cite_note-FOOTNOTESkilling1976-19">[19]</a>

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

There are 9 uniform honeycombs that include triangular prism cells:

<dl><dd><a href="Gyroelongated_alternated_cubic_honeycomb" class="mw-redirect" title="Gyroelongated alternated cubic honeycomb">Gyroelongated alternated cubic honeycomb</a>, <a href="Elongated_alternated_cubic_honeycomb" class="mw-redirect" title="Elongated alternated cubic honeycomb">elongated alternated cubic honeycomb</a>, <a href="Gyrated_triangular_prismatic_honeycomb" class="mw-redirect" title="Gyrated triangular prismatic honeycomb">gyrated triangular prismatic honeycomb</a>, <a href="Snub_square_prismatic_honeycomb" class="mw-redirect" title="Snub square prismatic honeycomb">snub square prismatic honeycomb</a>, <a href="Triangular_prismatic_honeycomb" title="Triangular prismatic honeycomb">triangular prismatic honeycomb</a>, <a href="Triangular-hexagonal_prismatic_honeycomb" class="mw-redirect" title="Triangular-hexagonal prismatic honeycomb">triangular-hexagonal prismatic honeycomb</a>, <a href="Truncated_hexagonal_prismatic_honeycomb" class="mw-redirect" title="Truncated hexagonal prismatic honeycomb">truncated hexagonal prismatic honeycomb</a>, <a href="Rhombitriangular-hexagonal_prismatic_honeycomb" class="mw-redirect" title="Rhombitriangular-hexagonal prismatic honeycomb">rhombitriangular-hexagonal prismatic honeycomb</a>, <a href="Snub_triangular-hexagonal_prismatic_honeycomb" class="mw-redirect" title="Snub triangular-hexagonal prismatic honeycomb">snub triangular-hexagonal prismatic honeycomb</a>, <a href="Elongated_triangular_prismatic_honeycomb" class="mw-redirect" title="Elongated triangular prismatic honeycomb">elongated triangular prismatic honeycomb</a></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Related_polytopes">Related polytopes</h3></div>

The triangular prism is first in a dimensional series of <a href="Uniform_k21_polytope" class="mw-redirect" title="Uniform k21 polytope">semiregular polytopes</a>. Each progressive <a href="Uniform_polytope" title="Uniform polytope">uniform polytope</a> is constructed <a href="Vertex_figure" title="Vertex figure">vertex figure</a> of the previous polytope. <a href="Thorold_Gosset" title="Thorold Gosset">Thorold Gosset</a> identified this series in 1900 as containing all <a href="Regular_polytope" title="Regular polytope">regular polytope</a> facets, containing all <a href="Simplex" title="Simplex">simplexes</a> and <a href="Orthoplex" class="mw-redirect" title="Orthoplex">orthoplexes</a> (<a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a> and <a href="Square" title="Square">squares</a> in the case of the triangular prism). In <a href="Coxeter" class="mw-redirect" title="Coxeter">Coxeter</a>'s notation the triangular prism is given the symbol −121.

<th colspan="12"><a href="Uniform_k_21_polytope" title="Uniform k 21 polytope">k21 figures</a> in n dimensions

</th></tr>

<tr>

<th>Space

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<th colspan="6">Finite

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<th>Euclidean

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<th>Hyperbolic

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</th>

</th>

</th>

</th>

</th>

</th>

</th>

</th>

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<th><a href="Coxeter_group" title="Coxeter group">Coxeter group</a>

</th>

<td>E3=A2A1

</td>

<td>E4=A4

</td>

<td>E5=D5

</td>

</td>

</td>

</td>

<msub>

<mover>

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</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {E}}_{8}}</annotation>

</semantics>

</math><img src="./8f7960ec54a7dac08a847e38ee3137a3e95a9044.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:3.009ex;" alt="{\displaystyle {\tilde {E}}_{8}}" loading="lazy"> = E8+

</td>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {T}}_{8}}</annotation>

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</math><img src="./897b8ae2da2454683a692ed6b588068316ee4816.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.828ex; height:2.843ex;" alt="{\displaystyle {\bar {T}}_{8}}" loading="lazy"> = E8++

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<th><a href="Coxeter%E2%80%93Dynkin_diagram" title="Coxeter–Dynkin diagram">Coxeter diagram</a>

</th>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<th><a href="Coxeter_notation" title="Coxeter notation">Symmetry</a>

</th>

<td>[3−1,2,1]

</td>

<td>[30,2,1]

</td>

<td>[31,2,1]

</td>

<td>[32,2,1]

</td>

<td>[33,2,1]

</td>

<td>[34,2,1]

</td>

<td>[35,2,1]

</td>

<td>[36,2,1]

</td></tr>

<th><a href="Group_order" class="mw-redirect" title="Group order">Order</a>

</th>

<td>12

</td>

<td>120

</td>

<td>1,920

</td>

<td>51,840

</td>

<td>2,903,040

</td>

<td>696,729,600

</td>

<td colspan="2">∞

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<th>Graph

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<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>-

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<td>-

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<th>Name

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<td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

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

<div class="mw-heading mw-heading3"><h3 id="Four_dimensional_space">Four dimensional space</h3></div>

The triangular prism exists as cells of a number of four-dimensional <a href="Uniform_4-polytope" title="Uniform 4-polytope">uniform 4-polytopes</a>, including:

<th colspan="12">Four dimensional polytopes with triangular prisms

</th></tr>

<td><a href="Tetrahedral_prism" title="Tetrahedral prism">Tetrahedral prism</a>

</td>

<td><a href="Octahedral_prism" title="Octahedral prism">Octahedral prism</a>

</td>

<td><a href="Cuboctahedral_prism" class="mw-redirect" title="Cuboctahedral prism">Cuboctahedral prism</a>

</td>

<td><a href="Icosahedral_prism" class="mw-redirect" title="Icosahedral prism">Icosahedral prism</a>

</td>

<td><a href="Icosidodecahedral_prism" class="mw-redirect" title="Icosidodecahedral prism">Icosidodecahedral prism</a>

</td>

<td><a href="Truncated_dodecahedral_prism" class="mw-redirect" title="Truncated dodecahedral prism">Truncated dodecahedral prism</a>

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<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<td><a href="Rhombicosidodecahedral_prism" class="mw-redirect" title="Rhombicosidodecahedral prism">Rhomb-icosidodecahedral prism</a>

</td>

<td><a href="Rhombicuboctahedral_prism" title="Rhombicuboctahedral prism">Rhombi-cuboctahedral prism</a>

</td>

<td><a href="Truncated_cubic_prism" class="mw-redirect" title="Truncated cubic prism">Truncated cubic prism</a>

</td>

<td><a href="Snub_dodecahedral_prism" class="mw-redirect" title="Snub dodecahedral prism">Snub dodecahedral prism</a>

</td>

<td><a href="Uniform_antiprismatic_prism" title="Uniform antiprismatic prism">n-gonal antiprismatic prism</a>

</td></tr>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<td><a href="Cantellated_5-cell" title="Cantellated 5-cell">Cantellated 5-cell</a>

</td>

<td><a href="Cantitruncated_5-cell" class="mw-redirect" title="Cantitruncated 5-cell">Cantitruncated 5-cell</a>

</td>

<td><a href="Runcinated_5-cell" title="Runcinated 5-cell">Runcinated 5-cell</a>

</td>

<td><a href="Runcitruncated_5-cell" class="mw-redirect" title="Runcitruncated 5-cell">Runcitruncated 5-cell</a>

</td>

<td><a href="Cantellated_tesseract" title="Cantellated tesseract">Cantellated tesseract</a>

</td>

<td><a href="Cantitruncated_tesseract" class="mw-redirect" title="Cantitruncated tesseract">Cantitruncated tesseract</a>

</td>

<td><a href="Runcinated_tesseract" class="mw-redirect" title="Runcinated tesseract">Runcinated tesseract</a>

</td>

<td><a href="Runcitruncated_tesseract" class="mw-redirect" title="Runcitruncated tesseract">Runcitruncated tesseract</a>

</td></tr>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<td><a href="Cantellated_24-cell" class="mw-redirect" title="Cantellated 24-cell">Cantellated 24-cell</a>

</td>

<td><a href="Cantitruncated_24-cell" class="mw-redirect" title="Cantitruncated 24-cell">Cantitruncated 24-cell</a>

</td>

<td><a href="Runcinated_24-cell" class="mw-redirect" title="Runcinated 24-cell">Runcinated 24-cell</a>

</td>

<td><a href="Runcitruncated_24-cell" class="mw-redirect" title="Runcitruncated 24-cell">Runcitruncated 24-cell</a>

</td>

<td><a href="Cantellated_120-cell" title="Cantellated 120-cell">Cantellated 120-cell</a>

</td>

<td><a href="Cantitruncated_120-cell" class="mw-redirect" title="Cantitruncated 120-cell">Cantitruncated 120-cell</a>

</td>

<td><a href="Runcinated_120-cell" class="mw-redirect" title="Runcinated 120-cell">Runcinated 120-cell</a>

</td>

<td><a href="Runcitruncated_120-cell" class="mw-redirect" title="Runcitruncated 120-cell">Runcitruncated 120-cell</a>

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

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

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<li id="cite_note-FOOTNOTEKing1994[httpsbooksgooglecombooksidc3fsCAAAQBAJpgPA113_113]-1">^ <a href="#cite_ref-FOOTNOTEKing1994[httpsbooksgooglecombooksidc3fsCAAAQBAJpgPA113_113]_1-0">a</a> <a href="#cite_ref-FOOTNOTEKing1994[httpsbooksgooglecombooksidc3fsCAAAQBAJpgPA113_113]_1-1">b</a> <a href="#CITEREFKing1994">King (1994)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=c3fsCAAAQBAJ&pg=PA113">113</a>.

</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="#CITEREFKing1994">King (1994)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=c3fsCAAAQBAJ&pg=PA113">113</a></li><li><a href="#CITEREFBerman1971">Berman (1971)</a></li></ul></div>

</li>

<li id="cite_note-FOOTNOTEKernBland193825-3"><a href="#cite_ref-FOOTNOTEKernBland193825_3-0">^</a> <a href="#CITEREFKernBland1938">Kern & Bland (1938)</a>, p. 25.

</li>

<li id="cite_note-FOOTNOTEHaul1893[httpsarchiveorgdetailsmensuration00hallgoogpagen57_45]-4"><a href="#cite_ref-FOOTNOTEHaul1893[httpsarchiveorgdetailsmensuration00hallgoogpagen57_45]_4-0">^</a> <a href="#CITEREFHaul1893">Haul (1893)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/mensuration00hallgoog/page/n57">45</a>.

</li>

<li id="cite_note-FOOTNOTEO'KeeffeHyde2020[httpsbooksgooglecombooksid_MjPDwAAQBAJpgPA139_139]-5"><a href="#cite_ref-FOOTNOTEO'KeeffeHyde2020[httpsbooksgooglecombooksid_MjPDwAAQBAJpgPA139_139]_5-0">^</a> <a href="#CITEREFO'KeeffeHyde2020">O'Keeffe & Hyde (2020)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA139">139</a>.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <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="#CITEREFBermanWilliams2009">Berman & Williams (2009)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=05DEJ8Kh67AC&pg=PA100">100</a></li><li><a href="#CITEREFMesser2002">Messer (2002)</a></li></ul></div>

</li>

<li id="cite_note-FOOTNOTEJohnson1966-7"><a href="#cite_ref-FOOTNOTEJohnson1966_7-0">^</a> <a href="#CITEREFJohnson1966">Johnson (1966)</a>.

</li>

<li id="cite_note-FOOTNOTEKernBland193826-8"><a href="#cite_ref-FOOTNOTEKernBland193826_8-0">^</a> <a href="#CITEREFKernBland1938">Kern & Bland (1938)</a>, p. 26.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <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="#CITEREFKinseyMoorePrassidis2011">Kinsey, Moore & Prassidis (2011)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fFpuDwAAQBAJ&pg=PA389">389</a></li><li><a href="#CITEREFHaul1893">Haul (1893)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/mensuration00hallgoog/page/n57">45</a></li></ul></div>

</li>

<li id="cite_note-FOOTNOTEBerman1971-10"><a href="#cite_ref-FOOTNOTEBerman1971_10-0">^</a> <a href="#CITEREFBerman1971">Berman (1971)</a>.

</li>

<li id="cite_note-FOOTNOTEPisanskiServatius2013[httpsbooksgooglecombooksid3vnEcMCx0HkCpgPA21_21]-11"><a href="#cite_ref-FOOTNOTEPisanskiServatius2013[httpsbooksgooglecombooksid3vnEcMCx0HkCpgPA21_21]_11-0">^</a> <a href="#CITEREFPisanskiServatius2013">Pisanski & Servatius (2013)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3vnEcMCx0HkC&pg=PA21">21</a>.

</li>

<li id="cite_note-FOOTNOTESysłoProskurowski1983254Prop._4.3._Here_the_triangular_prism_is_identified_as_the_unique_graph_with_exactly_three_cycles_that_can_be_the_outer_cycle_of_a_realization_as_a_Halin_graph-12"><a href="#cite_ref-FOOTNOTESysłoProskurowski1983254Prop._4.3._Here_the_triangular_prism_is_identified_as_the_unique_graph_with_exactly_three_cycles_that_can_be_the_outer_cycle_of_a_realization_as_a_Halin_graph_12-0">^</a> <a href="#CITEREFSysłoProskurowski1983">Sysło & Proskurowski 1983</a>, p. 254, Prop. 4.3. Here the triangular prism is identified as the unique graph with exactly three cycles that can be the outer cycle of a realization as a Halin graph.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <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="#CITEREFTodesco2020">Todesco (2020)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wtIBEAAAQBAJ&pg=PA282">282</a></li><li><a href="#CITEREFWilliamsMonteleone2021">Williams & Monteleone (2021)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=w5RBEAAAQBAJ&pg=PA23">23</a></li></ul></div>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <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="#CITEREFRajwade2001">Rajwade (2001)</a></li><li><a href="#CITEREFBerman1971">Berman (1971)</a></li></ul></div>

</li>

<li id="cite_note-FOOTNOTEKernBland193881-15"><a href="#cite_ref-FOOTNOTEKernBland193881_15-0">^</a> <a href="#CITEREFKernBland1938">Kern & Bland (1938)</a>, p. 81.

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <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="#CITEREFSchönhardt1928">Schönhardt (1928)</a></li><li><a href="#CITEREFBezdekCarrigan2016">Bezdek & Carrigan (2016)</a></li></ul></div>

</li>

<li id="cite_note-FOOTNOTEBagemihl1948-17"><a href="#cite_ref-FOOTNOTEBagemihl1948_17-0">^</a> <a href="#CITEREFBagemihl1948">Bagemihl (1948)</a>.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <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="#CITEREFSchönhardt1928">Schönhardt (1928)</a></li><li><a href="#CITEREFBansodNandanwarBurša2014">Bansod, Nandanwar & Burša (2014)</a></li></ul></div>

</li>

<li id="cite_note-FOOTNOTESkilling1976-19"><a href="#cite_ref-FOOTNOTESkilling1976_19-0">^</a> <a href="#CITEREFSkilling1976">Skilling (1976)</a>.

</li>

</ol></div>

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

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</style><cite id="CITEREFBagemihl1948" class="citation journal cs1"><a href="Frederick_Bagemihl" title="Frederick Bagemihl">Bagemihl, F.</a> (1948). "On indecomposable polyhedra". <a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 55 (7): 411–413. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2306130">10.2307/2306130</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/2306130">2306130</a>.</cite></li>

<li><cite id="CITEREFBansodNandanwarBurša2014" class="citation journal cs1">Bansod, Yogesh Deepak; Nandanwar, Deepesh; Burša, Jiří (2014). <a rel="nofollow" class="external text" href="http://www.engineeringmechanics.cz/pdf/21_5_355.pdf">"Overview of tensegrity – I: Basic structures"</a> (PDF). Engineering Mechanics. 21 (5): 355–367.</cite></li>

<li><cite id="CITEREFBermanWilliams2009" class="citation book cs1">Berman, Leah Wrenn; Williams, Gordon (2009). "Exploring Polyhedra and Discovering Euler's Formula". In Hopkin, Brian (ed.). Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles. <a href="Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>.</cite></li>

<li><cite id="CITEREFBerman1971" class="citation journal cs1">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></li>

<li><cite id="CITEREFBezdekCarrigan2016" class="citation journal cs1">Bezdek, Andras; Carrigan, Braxton (2016). "On nontriangulable polyhedra". Beiträge zur Algebra und Geometrie. 57 (1): 51–66. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs13366-015-0248-4">10.1007/s13366-015-0248-4</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=3457762">3457762</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:118484882">118484882</a>.</cite></li>

<li><cite id="CITEREFHaul1893" class="citation book cs1">Haul, Wm. S. (1893). <a rel="nofollow" class="external text" href="https://archive.org/details/mensuration00hallgoog">Mensuration</a>. Ginn & Company.</cite></li>

<li><cite id="CITEREFKernBland1938" class="citation book cs1">Kern, William F.; Bland, James R. (1938). Solid Mensuration with proofs. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1035479">1035479</a>.</cite></li>

<li><cite id="CITEREFKing1994" class="citation book cs1">King, Robert B. (1994). "Polyhedral Dynamics". In Bonchev, Danail D.; Mekenyan, O.G. (eds.). Graph Theoretical Approaches to Chemical Reactivity. 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%2F978-94-011-1202-4">10.1007/978-94-011-1202-4</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-94-011-1202-4</bdi>.</cite></li>

<li><cite id="CITEREFKinseyMoorePrassidis2011" class="citation book cs1"><a href="L._Christine_Kinsey" title="L. Christine Kinsey">Kinsey, L. Christine</a>; Moore, Teresa E.; Prassidis, Efstratios (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fFpuDwAAQBAJ">Geometry and Symmetry</a>. <a href="John_Wiley_%26_Son" class="mw-redirect" title="John Wiley & Son">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="CITEREFJohnson1966" class="citation journal cs1"><a href="Norman_W._Johnson" class="mw-redirect" title="Norman W. Johnson">Johnson, Norman W.</a> (1966). <a rel="nofollow" class="external text" href="https://doi.org/10.4153%2Fcjm-1966-021-8">"Convex polyhedra with regular faces"</a>. <a href="Canadian_Journal_of_Mathematics" title="Canadian Journal of Mathematics">Canadian Journal of Mathematics</a>. 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>. <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:0132.14603">0132.14603</a>.</cite></li>

<li><cite id="CITEREFMesser2002" class="citation journal cs1">Messer, Peter W. (2002). "Closed-Form Expressions for Uniform Polyhedra and Their Duals". <a href="Discrete_%26_Computational_Geometry" title="Discrete & Computational Geometry">Discrete & Computational Geometry</a>. 27 (3): 353–375. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00454-001-0078-2">10.1007/s00454-001-0078-2</a>.</cite></li>

<li><cite id="CITEREFO'KeeffeHyde2020" class="citation book cs1">O'Keeffe, Michael; Hyde, Bruce G. (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_MjPDwAAQBAJ">Crystal Structures: Patterns and Symmetry</a>. <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-83654-6</bdi>.</cite></li>

<li><cite id="CITEREFPisanskiServatius2013" class="citation book cs1">Pisanski, Tomaž; Servatius, Brigitte (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3vnEcMCx0HkC">Configuration from a Graphical Viewpoint</a>. 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%2F978-0-8176-8364-1">10.1007/978-0-8176-8364-1</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-8363-4</bdi>.</cite></li>

<li><cite id="CITEREFRajwade2001" class="citation book cs1">Rajwade, A. R. (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=afJdDwAAQBAJ">Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem</a>. Texts and Readings in Mathematics. Hindustan Book Agency. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-93-86279-06-4">10.1007/978-93-86279-06-4</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-93-86279-06-4</bdi>.</cite></li>

<li><cite id="CITEREFSchönhardt1928" class="citation journal cs1"><a href="Erich_Sch%C3%B6nhardt" title="Erich Schönhardt">Schönhardt, E.</a> (1928). <a rel="nofollow" class="external text" href="https://eudml.org/doc/159218">"Über die Zerlegung von Dreieckspolyedern in Tetraeder"</a>. <a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a>. 98: 309–312. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01451597">10.1007/BF01451597</a>.</cite></li>

<li><cite id="CITEREFSkilling1976" class="citation cs2">Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", <a href="Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society" title="Mathematical Proceedings of the Cambridge Philosophical Society">Mathematical Proceedings of the Cambridge Philosophical Society</a>, 79 (3): 447–457, <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/1976MPCPS..79..447S">1976MPCPS..79..447S</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004100052440">10.1017/S0305004100052440</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=0397554">0397554</a></cite></li>

<li><cite id="CITEREFSysłoProskurowski1983" class="citation book cs1">Sysło, Maciej M.; Proskurowski, Andrzej (1983). "On Halin graphs". Graph Theory: Proceedings of a Conference held in Lagów, Poland, February 10–13, 1981. Lecture Notes in Mathematics. Vol. 1018. Springer-Verlag. pp. 248–256. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0071635">10.1007/BFb0071635</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-12687-4</bdi>.</cite>.</li>

<li><cite id="CITEREFTodesco2020" class="citation book cs1">Todesco, Gian Marco (2020). "Hyperbolic Honeycomb". In Emmer, Michele; Abate, Marco (eds.). Imagine Math 7: Between Culture and Mathematics. 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%2F978-3-030-42653-8">10.1007/978-3-030-42653-8</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-030-42653-8</bdi>.</cite></li>

<li><cite id="CITEREFWilliamsMonteleone2021" class="citation book cs1">Williams, Kim; Monteleone, Cosino (2021). Daniele Barbaro's Perspective of 1568. 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%2F978-3-030-76687-0">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>

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