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Stereohedron
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<h2 id="plesiohedra">Plesiohedra</h2> <p>A subset of stereohedra are called <a href="/facts/Plesiohedron/Lytb02ys">plesiohedrons</a>, defined as the <a href="/facts/Voronoi_diagram/AhQY5LD8">Voronoi cells</a> of a symmetric <a href="/facts/Delone_set/VKGgETLd">Delone set</a>. </p><p><a href="/facts/Parallelohedron/4B0xFmsg">Parallelohedrons</a> are plesiohedra which are space-filling by translation only. Edges here are colored as parallel vectors. </p> Parallelohedra<table><tbody><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td><a href="/facts/Cube/65lUaAqN">cube</a></td><td><a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a></td><td><a href="/facts/Rhombic_dodecahedron/X3npAlmr">rhombic dodecahedron</a></td><td><a href="/facts/Elongated_dodecahedron/IARRN28r">elongated dodecahedron</a></td><td><a href="/facts/Truncated_octahedron/GA98b4TH">truncated octahedron</a></td></tr></tbody></table> <h2 id="other-periodic-stereohedra">Other periodic stereohedra</h2> <p>The <a href="/facts/Catoptric_tessellation/rTvaehtP">catoptric tessellation</a> contain stereohedra cells. <a href="/facts/Dihedral_angles/C9WEJsuS">Dihedral angles</a> are integer divisors of 180°, and are colored by their order. The first three are the fundamental domains of C ~ 3 {\displaystyle {\tilde {C}}_{3}} , B ~ 3 {\displaystyle {\tilde {B}}_{3}} , and A ~ 3 {\displaystyle {\tilde {A}}_{3}} symmetry, represented by <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagrams</a>: , and . B ~ 3 {\displaystyle {\tilde {B}}_{3}} is a half symmetry of C ~ 3 {\displaystyle {\tilde {C}}_{3}} , and A ~ 3 {\displaystyle {\tilde {A}}_{3}} is a quarter symmetry. </p><p>Any space-filling stereohedra with symmetry elements can be <a href="/facts/Dissection_(geometry)/lslQIYpS">dissected</a> into smaller identical cells which are also stereohedra. The name modifiers below, half, quarter, and eighth represent such dissections. </p> Catoptric cells<table><tbody><tr><th>Faces</th><th colspan="4">4</th><th colspan="3">5</th><th colspan="4">6</th><th>8</th><th>12</th></tr><tr><th>Type</th><td colspan="4"><a href="/facts/Tetrahedra/7mkSrl6j">Tetrahedra</a></td><td colspan="3"><a href="/facts/Square_pyramid/wDetDohM">Square pyramid</a></td><td colspan="2"><a href="/facts/Triangular_bipyramid/fU4kIBTV">Triangular bipyramid</a></td><td colspan="2"><a href="/facts/Cube/65lUaAqN">Cube</a></td><td><a href="/facts/Octahedron/G38q92xW">Octahedron</a></td><td><a href="/facts/Rhombic_dodecahedron/X3npAlmr">Rhombic dodecahedron</a></td></tr><tr><th>Images</th><td>1/48 (1)</td><td>1/24 (2)</td><td>1/12 (4)</td><td>1/12 (4)</td><td>1/24 (2)</td><td>1/6 (8)</td><td>1/6 (8)</td><td>1/12 (4)</td><td>1/4 (12)</td><td>1 (48)</td><td>1/2 (24)</td><td>1/3 (16)</td><td>2 (96)</td></tr><tr><th>Symmetry(order)</th><td>C11</td><td>C1v2</td><td>D2d4</td><td>C1v2</td><td>C1v2</td><td>C4v8</td><td>C2v4</td><td>C2v4</td><td>C3v6</td><td>Oh48</td><td>D3d12</td><td>D4h16</td><td>Oh48</td></tr><tr><th>Honeycomb</th><td>Eighth pyramidille</td><td>Triangular pyramidille</td><td>Oblate tetrahedrille</td><td>Half pyramidille</td><td>Square quarter pyramidille</td><td>Pyramidille</td><td>Half oblate octahedrille</td><td>Quarter oblate octahedrille</td><td>Quarter cubille</td><td>Cubille</td><td>Oblate cubille</td><td>Oblate octahedrille</td><td>Dodecahedrille</td></tr></tbody></table> <p>Other convex polyhedra that are stereohedra but not parallelohedra nor plesiohedra include the <a href="/facts/Gyrobifastigium/nGDBBx9K">gyrobifastigium</a>. </p> Others<table><tbody><tr><th>Faces</th><th colspan="2">8</th><th>10</th><th>12</th></tr><tr><th>Symmetry(order)</th><td colspan="3">D2d (8)</td><td>D4h (16)</td></tr><tr><th>Images</th><td></td><td></td><td></td><td></td></tr><tr><th>Cell</th><td><a href="/facts/Gyrobifastigium/nGDBBx9K">Gyrobifastigium</a></td><td><a href="/facts/Elongated_gyrobifastigium/Sl7t5vdC">Elongatedgyrobifastigium</a></td><td><a href="/facts/Ten_of_diamonds_decahedron/yuwfQEqu">Ten of diamonds</a></td><td><a href="/facts/Elongated_square_bipyramid/CytNtzub">Elongatedsquare bipyramid</a></td></tr></tbody></table> <ul><li>Ivanov, A. B. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Stereohedron">"Stereohedron"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li> <li><a href="/facts/Boris_Delaunay/70vafQlf">B. N. Delone</a>, N. N. Sandakova, <i>Theory of stereohedra</i> Trudy Mat. Inst. Steklov., 64 (1961) pp. 28–51 (Russian)</li> <li>Goldberg, Michael <i>Three Infinite Families of Tetrahedral Space-Fillers</i> Journal of Combinatorial Theory A, 16, pp. 348–354, 1974.</li> <li>Goldberg, Michael <i>The space-filling pentahedra</i>, Journal of Combinatorial Theory, Series A Volume 13, Issue 3, November 1972, Pages 437-443 <a href="http://www.sciencedirect.com/science/article/pii/0097316572900775">[1]</a> <a href="http://documentslide.com/documents/the-space-filling-pentahedra.html">PDF</a></li> <li>Goldberg, Michael <i>The Space-filling Pentahedra II</i>, Journal of Combinatorial Theory 17 (1974), 375–378. <a href="http://documentslide.com/documents/the-space-filling-pentahedra-ii.html">PDF</a></li> <li>Goldberg, Michael <i>On the space-filling hexahedra</i> Geom. Dedicata, June 1977, Volume 6, Issue 1, pp 99–108 <a href="https://link.springer.com/article/10.1007/BF00181585">[2]</a> <a href="http://documentslide.com/documents/on-the-space-filling-hexahedra.html">PDF</a></li> <li>Goldberg, Michael <i>On the space-filling heptahedra</i> Geometriae Dedicata, June 1978, Volume 7, Issue 2, pp 175–184 <a href="https://link.springer.com/article/10.1007/BF00181630">[3]</a> <a href="http://documentslide.com/documents/on-the-space-filling-heptahedra.html">PDF</a></li> <li>Goldberg, Michael <i>Convex Polyhedral Space-Fillers of More than Twelve Faces.</i> Geom. Dedicata 8, 491-500, 1979.</li> <li>Goldberg, Michael <i>On the space-filling octahedra</i>, Geometriae Dedicata, January 1981, Volume 10, Issue 1, pp 323–335 <a href="https://link.springer.com/article/10.1007/BF01447431">[4]</a> <a href="http://documents.mx/documents/on-the-space-filling-octahedra.html">PDF</a></li> <li>Goldberg, Michael <i>On the Space-filling Decahedra</i>. Structural Topology, 1982, num. Type 10-II <a href="https://upcommons.upc.edu/handle/2099/990">PDF</a></li> <li>Goldberg, Michael <i>On the space-filling enneahedra</i> Geometriae Dedicata, June 1982, Volume 12, Issue 3, pp 297–306 <a href="https://link.springer.com/article/10.1007/BF00147314">[5]</a> <a href="http://documentslide.com/documents/on-the-space-filling-enneahedra.html">PDF</a></li></ul>