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class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>By symmetries and Bravais lattices</span> </div> </a> <ul id="toc-By_symmetries_and_Bravais_lattices-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_shapes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_shapes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Related shapes</span> </div> </a> <ul id="toc-Related_shapes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Polyhedron that tiles space by translation</div> <p class="mw-empty-elt"> </p> <table class="wikitable" align="right" width="300"> <caption>Five types of parallelohedron </caption> <tbody><tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:Hexahedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Hexahedron.svg/120px-Hexahedron.svg.png" decoding="async" width="100" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Hexahedron.svg/250px-Hexahedron.svg.png 1.5x" data-file-width="540" data-file-height="600" /></a></span><br /><a href="/wiki/Cube" title="Cube">Cube</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Hexagonal_Prism_BC.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Hexagonal_Prism_BC.svg/120px-Hexagonal_Prism_BC.svg.png" decoding="async" width="100" height="82" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Hexagonal_Prism_BC.svg/250px-Hexagonal_Prism_BC.svg.png 1.5x" data-file-width="385" data-file-height="315" /></a></span><br /><a href="/wiki/Hexagonal_prism" title="Hexagonal prism">Hexagonal prism</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Rhombicdodecahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/100px-Rhombicdodecahedron.jpg" decoding="async" width="100" height="89" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/150px-Rhombicdodecahedron.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Rhombicdodecahedron.jpg/200px-Rhombicdodecahedron.jpg 2x" data-file-width="849" data-file-height="754" /></a></span><br /><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">Rhombic dodecahedron</a> </td></tr> <tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:Rhombo-hexagonal_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Rhombo-hexagonal_dodecahedron.png/120px-Rhombo-hexagonal_dodecahedron.png" decoding="async" width="100" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Rhombo-hexagonal_dodecahedron.png/250px-Rhombo-hexagonal_dodecahedron.png 1.5x" data-file-width="673" data-file-height="814" /></a></span><br /><a href="/wiki/Elongated_dodecahedron" title="Elongated dodecahedron">Elongated dodecahedron</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Truncatedoctahedron.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Truncatedoctahedron.jpg/120px-Truncatedoctahedron.jpg" decoding="async" width="120" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Truncatedoctahedron.jpg/250px-Truncatedoctahedron.jpg 1.5x" data-file-width="857" data-file-height="789" /></a></span><br /><a href="/wiki/Truncated_octahedron" title="Truncated octahedron">Truncated octahedron</a> </td></tr></tbody></table> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, a <b>parallelohedron</b> or <b>Fedorov polyhedron</b><sup id="cite_ref-hargittai_1-0" class="reference"><a href="#cite_note-hargittai-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a <a href="/wiki/Convex_polyhedron" class="mw-redirect" title="Convex polyhedron">convex polyhedron</a> that can be <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translated</a> without rotations to fill <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, producing a <a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">honeycomb</a> in which all copies of the polyhedron meet face-to-face. <a href="/wiki/Evgraf_Fedorov" title="Evgraf Fedorov">Evgraf Fedorov</a> identified the five types of parallelohedron in 1885 in his studies of crystallographic systems. They are the <a href="/wiki/Cube" title="Cube">cube</a>, <a href="/wiki/Hexagonal_prism" title="Hexagonal prism">hexagonal prism</a>, <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a>, <a href="/wiki/Elongated_dodecahedron" title="Elongated dodecahedron">elongated dodecahedron</a>, and <a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a>. </p><p>Each parallelohedron is centrally symmetric with symmetric faces, making it a special case of a <a href="/wiki/Zonohedron" title="Zonohedron">zonohedron</a>. Each parallelohedron is also a <a href="/wiki/Stereohedron" title="Stereohedron">stereohedron</a>, a polyhedron that tiles space so that <a href="/wiki/Isohedral_tiling" class="mw-redirect" title="Isohedral tiling">all tiles are symmetric</a>. The centers of the tiles in a tiling of space by parallelohedra form a <a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattice</a>, and every Bravais lattice can be formed in this way. Adjusting the lengths of parallel edges in a parallelohedron, or performing an <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a> of the parallelohedron, results in another parallelohedron of the same combinatorial type. It is possible to choose this adjustment so that the tiling by parallelohedra is the <a href="/wiki/Voronoi_diagram" title="Voronoi diagram">Voronoi diagram</a> of its Bravais lattice, and so that the resulting parallelohedra become special cases of the <a href="/wiki/Plesiohedron" title="Plesiohedron">plesiohedra</a>. </p><p>The three-dimensional parallelohedra are analogous to two-dimensional <a href="/wiki/Parallelogon" title="Parallelogon">parallelogons</a> and higher-dimensional parallelotopes. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_construction">Definition and construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelohedron&action=edit&section=1" title="Edit section: Definition and construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A parallelohedron is defined to be a polyhedron whose translated copies meet face-to-face to fill space, forming a <a href="/wiki/Honeycomb_(geometry)" title="Honeycomb (geometry)">honeycomb</a>.<sup id="cite_ref-alexandrov_2-0" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The resulting honeycomb must be <a href="/wiki/Periodic_tiling" class="mw-redirect" title="Periodic tiling">periodic</a>, having a <a href="/wiki/Space_group" title="Space group">three-dimensional system of global symmetries</a>, because each translation from a copy of the polyhedron to an adjoining copy must apply to all copies, forming a symmetry of the whole tiling.<sup id="cite_ref-engel3_3-0" class="reference"><a href="#cite_note-engel3-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> For the same reason, the honeycomb is uniquely determined by the position of any one parallelohedron in it.<sup id="cite_ref-dolbilin_4-0" class="reference"><a href="#cite_note-dolbilin-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In order to meet face-to-face with another copy, each face of the polyhedron must correspond to a parallel face with the same shape but the opposite orientation. By a result of <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a>, the shape of a parallelohedron is <a href="/wiki/Minkowski_problem_for_polytopes" title="Minkowski problem for polytopes">uniquely determined</a> by the <a href="/wiki/Normal_vector" class="mw-redirect" title="Normal vector">normal vectors</a> and areas of these opposite face pairs. This implies that a parallelohedron must be <a href="/wiki/Point_reflection" title="Point reflection">centrally symmetric</a>, because otherwise a point reflection of the polyhedron would produce a different shape with the same normal vectors and face areas, contradicting Minkowski's uniqueness theorem. Each face of a parallelohedron must also be centrally symmetric, to match its symmetric copy in the adjoining copy of the parallelohedron.<sup id="cite_ref-alexandrov_2-1" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>These two properties of parallelohedra, having central symmetry and centrally symmetric faces, characterize a broader class of polyhedra, the <a href="/wiki/Zonohedron" title="Zonohedron">zonohedra</a>, so every parallelohedron is a zonohedron. In any zonohedron, the edges can be grouped into <i>zones</i>, systems of parallel edges of equal length. If one edge is selected from each zone, the <a href="/wiki/Minkowski_sum" class="mw-redirect" title="Minkowski sum">Minkowski sum</a> of the selected edges gives a translated copy of the zonohedron itself. All Minkowski sums of finite sets of line segments produce zonohedra; the segments forming a zonohedron in this way are called its <i>generators</i>. Unlike some other zonohedra, the parallelohedra can only have from three to six zones and, correspondingly, from three to six generators.<sup id="cite_ref-dienst_5-0" class="reference"><a href="#cite_note-dienst-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Any zonohedron whose faces have the same combinatorial structure as one of the five parallelohedron is itself a parallelohedron. Any <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a> of a parallelohedron will produce another parallelohedron of the same type.<sup id="cite_ref-alexandrov_2-2" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> One way to characterize the parallelohedra among all zonohedra is using <i>belts</i>. <a href="#CITEREFSenechalTaylor2023">Senechal & Taylor (2023)</a> define a belt of a zonohedron to be the cycle of faces that contain all parallel copies of one edge. The number of faces in any belt of any zonohedron must be even, and can be any even number greater than two. As Federov and many others showed, the parallelohedra are exactly the zonohedra all of whose belts consist of only four or six faces.<sup id="cite_ref-sentay_6-0" class="reference"><a href="#cite_note-sentay-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Classification">Classification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelohedron&action=edit&section=2" title="Edit section: Classification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="By_combinatorial_structure">By combinatorial structure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelohedron&action=edit&section=3" title="Edit section: By combinatorial structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The five types of parallelohedron, and their most symmetric forms, are as follows.<sup id="cite_ref-alexandrov_2-3" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ul><li>A <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a> is generated from three line segments that are not all parallel to a common plane. Its most symmetric form is the <a href="/wiki/Cube" title="Cube">cube</a>, generated by three perpendicular unit-length line segments.<sup id="cite_ref-alexandrov_2-4" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The tiling of space by the cube is the <a href="/wiki/Cubic_honeycomb" title="Cubic honeycomb">cubic honeycomb</a>.<sup id="cite_ref-froth_7-0" class="reference"><a href="#cite_note-froth-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>A <a href="/wiki/Hexagonal_prism" title="Hexagonal prism">hexagonal prism</a> is generated from four line segments, three of them parallel to a common plane and the fourth not. Its most symmetric form is the right prism over a regular hexagon.<sup id="cite_ref-alexandrov_2-5" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It tiles space to form the <a href="/wiki/Hexagonal_prismatic_honeycomb" class="mw-redirect" title="Hexagonal prismatic honeycomb">hexagonal prismatic honeycomb</a>.<sup id="cite_ref-onset_8-0" class="reference"><a href="#cite_note-onset-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">rhombic dodecahedron</a> is generated from four line segments, no two of which are parallel to a common plane. Its most symmetric form is generated by the four long diagonals of a cube.<sup id="cite_ref-alexandrov_2-6" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It tiles space to form the <a href="/wiki/Rhombic_dodecahedral_honeycomb" title="Rhombic dodecahedral honeycomb">rhombic dodecahedral honeycomb</a>.<sup id="cite_ref-2orbit_9-0" class="reference"><a href="#cite_note-2orbit-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Bilinski_dodecahedron" title="Bilinski dodecahedron">Bilinski dodecahedron</a>, another less-symmetric form of the rhombic dodecahedron, is notable for (like the symmetric rhombic dodecahedron) having all of its faces congruent; its faces are <a href="/wiki/Golden_rhombus" title="Golden rhombus">golden rhombi</a>.<sup id="cite_ref-bilinski_10-0" class="reference"><a href="#cite_note-bilinski-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Elongated_dodecahedron" title="Elongated dodecahedron">elongated dodecahedron</a> is generated from five line segments, with two triples of coplanar segments. It can be generated by using an edge of the cube and its four long diagonals as generators.<sup id="cite_ref-alexandrov_2-7" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Truncated_octahedron" title="Truncated octahedron">truncated octahedron</a> is generated from six line segments with four triples of coplanar segments. It can be embedded in four-dimensional space as the 4-<a href="/wiki/Permutahedron" class="mw-redirect" title="Permutahedron">permutahedron</a>, whose vertices are all permutations of the counting numbers (1,2,3,4). In three-dimensional space, its most symmetric form is generated from six line segments parallel to the face diagonals of a cube.<sup id="cite_ref-alexandrov_2-8" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The tiling of space generated by its translations has been called the <a href="/wiki/Bitruncated_cubic_honeycomb" title="Bitruncated cubic honeycomb">bitruncated cubic honeycomb</a>.<sup id="cite_ref-self-affine_11-0" class="reference"><a href="#cite_note-self-affine-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul> <table class="wikitable" width="450"> <tbody><tr> <th>Name </th> <th><a href="/wiki/Cube" title="Cube">Cube</a><br />(parallelepiped) </th> <th><a href="/wiki/Hexagonal_prism" title="Hexagonal prism">Hexagonal prism</a><br />Elongated cube </th> <th><a href="/wiki/Rhombic_dodecahedron" title="Rhombic dodecahedron">Rhombic dodecahedron</a> </th> <th><a href="/wiki/Elongated_dodecahedron" title="Elongated dodecahedron">Elongated dodecahedron</a> </th> <th><a href="/wiki/Truncated_octahedron" title="Truncated octahedron">Truncated octahedron</a> </th></tr> <tr> <th>Images (colors indicate parallel edges) </th> <td><span typeof="mw:File"><a href="/wiki/File:Parallelohedron_edges_cube.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Parallelohedron_edges_cube.png/120px-Parallelohedron_edges_cube.png" decoding="async" width="120" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Parallelohedron_edges_cube.png/180px-Parallelohedron_edges_cube.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Parallelohedron_edges_cube.png/240px-Parallelohedron_edges_cube.png 2x" data-file-width="833" data-file-height="821" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Parallelohedron_edges_hexagonal_prism.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Parallelohedron_edges_hexagonal_prism.png/120px-Parallelohedron_edges_hexagonal_prism.png" decoding="async" width="120" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Parallelohedron_edges_hexagonal_prism.png/180px-Parallelohedron_edges_hexagonal_prism.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Parallelohedron_edges_hexagonal_prism.png/240px-Parallelohedron_edges_hexagonal_prism.png 2x" data-file-width="761" data-file-height="841" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Parallelohedron_edges_rhombic_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Parallelohedron_edges_rhombic_dodecahedron.png/120px-Parallelohedron_edges_rhombic_dodecahedron.png" decoding="async" width="120" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Parallelohedron_edges_rhombic_dodecahedron.png/250px-Parallelohedron_edges_rhombic_dodecahedron.png 1.5x" data-file-width="770" data-file-height="811" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Parallelohedron_edges_elongated_rhombic_dodecahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Parallelohedron_edges_elongated_rhombic_dodecahedron.png/120px-Parallelohedron_edges_elongated_rhombic_dodecahedron.png" decoding="async" width="120" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Parallelohedron_edges_elongated_rhombic_dodecahedron.png/180px-Parallelohedron_edges_elongated_rhombic_dodecahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/Parallelohedron_edges_elongated_rhombic_dodecahedron.png/240px-Parallelohedron_edges_elongated_rhombic_dodecahedron.png 2x" data-file-width="675" data-file-height="877" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Parallelohedron_edge_truncated_octahedron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Parallelohedron_edge_truncated_octahedron.png/140px-Parallelohedron_edge_truncated_octahedron.png" decoding="async" width="140" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Parallelohedron_edge_truncated_octahedron.png/210px-Parallelohedron_edge_truncated_octahedron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Parallelohedron_edge_truncated_octahedron.png/280px-Parallelohedron_edge_truncated_octahedron.png 2x" data-file-width="891" data-file-height="798" /></a></span> </td></tr> <tr> <th>Number of generators </th> <th>3 </th> <th>4 </th> <th>4 </th> <th>5 </th> <th>6 </th></tr> <tr> <th>Vertices </th> <th>8 </th> <th>12 </th> <th>14 </th> <th>18 </th> <th>24 </th></tr> <tr> <th>Edges </th> <th>12 </th> <th>18 </th> <th>24 </th> <th>28 </th> <th>36 </th></tr> <tr> <th>Faces </th> <th>6 </th> <th>8 </th> <th>12 </th> <th>12 </th> <th>14 </th></tr> <tr valign="top"> <th>Tiling </th> <td><span typeof="mw:File"><a href="/wiki/File:Partial_cubic_honeycomb.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Partial_cubic_honeycomb.png/250px-Partial_cubic_honeycomb.png" decoding="async" width="160" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Partial_cubic_honeycomb.png/330px-Partial_cubic_honeycomb.png 2x" data-file-width="873" data-file-height="866" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Hexagonal_prismatic_honeycomb.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Hexagonal_prismatic_honeycomb.png/160px-Hexagonal_prismatic_honeycomb.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Hexagonal_prismatic_honeycomb.png/240px-Hexagonal_prismatic_honeycomb.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Hexagonal_prismatic_honeycomb.png/320px-Hexagonal_prismatic_honeycomb.png 2x" data-file-width="860" data-file-height="858" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:HC_R1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/HC_R1.png/120px-HC_R1.png" decoding="async" width="120" height="158" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/HC_R1.png/250px-HC_R1.png 1.5x" data-file-width="640" data-file-height="840" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Rhombo-hexagonal_dodecahedron_tessellation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Rhombo-hexagonal_dodecahedron_tessellation.png/160px-Rhombo-hexagonal_dodecahedron_tessellation.png" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Rhombo-hexagonal_dodecahedron_tessellation.png/240px-Rhombo-hexagonal_dodecahedron_tessellation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Rhombo-hexagonal_dodecahedron_tessellation.png/320px-Rhombo-hexagonal_dodecahedron_tessellation.png 2x" data-file-width="1294" data-file-height="971" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:HC-A4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/HC-A4.png/250px-HC-A4.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/HC-A4.png/330px-HC-A4.png 2x" data-file-width="1024" data-file-height="1024" /></a></span> </td></tr> <tr> <th>Tiling name and <a href="/wiki/Coxeter%E2%80%93Dynkin_diagram" title="Coxeter–Dynkin diagram">Coxeter–Dynkin diagram</a> </th> <th><a href="/wiki/Cubic_honeycomb" title="Cubic honeycomb">Cubic</a><br /><span style="display:inline-block;"><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span><br /><span style="display:inline-block;"><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/be/CDel_infin.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/be/CDel_infin.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/be/CDel_infin.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </th> <th><a href="/wiki/Hexagonal_prismatic_honeycomb" class="mw-redirect" title="Hexagonal prismatic honeycomb">Hexagonal prismatic</a><br /><span style="display:inline-block;"><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/be/CDel_infin.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </th> <th><a href="/wiki/Rhombic_dodecahedral_honeycomb" title="Rhombic dodecahedral honeycomb">Rhombic dodecahedral</a><br /><span style="display:inline-block;"><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/4/4b/CDel_split1-43.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/1f/CDel_nodes.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </th> <th><a href="/wiki/Elongated_dodecahedral_honeycomb" class="mw-redirect" title="Elongated dodecahedral honeycomb">Elongated dodecahedral</a> </th> <th><a href="/wiki/Bitruncated_cubic_honeycomb" title="Bitruncated cubic honeycomb">Bitruncated cubic</a><br /><span style="display:inline-block;"><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </th></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="By_symmetries_and_Bravais_lattices">By symmetries and Bravais lattices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelohedron&action=edit&section=4" title="Edit section: By symmetries and Bravais lattices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The lengths of the segments within each zone can be adjusted arbitrarily, independently of the other zones. Doing so extends or shrinks the corresponding edges of the parallelohedron, without changing its combinatorial type or its property of tiling space. As a limiting case, for a parallelohedron with more than three parallel classes of edges, the length of any one of these classes can be adjusted to zero, producing a different parallelohedron of a simpler form, with one fewer zone. Beyond the central symmetry common to all zonohedra and all parallelohedra, additional symmetries are possible with an appropriate choice of the generating segments.<sup id="cite_ref-tutton_12-0" class="reference"><a href="#cite_note-tutton-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>When further subdivided according to their symmetry groups, there are 22 forms of the parallelohedra. For each form, the centers of its copies in its honeycomb form the points of one of the 14 <a href="/wiki/Bravais_lattice" title="Bravais lattice">Bravais lattices</a>. Because there are fewer Bravais lattices than symmetric forms of parallelohedra, certain pairs of parallelohedra map to the same Bravais lattice.<sup id="cite_ref-tutton_12-1" class="reference"><a href="#cite_note-tutton-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>By placing one endpoint of each generating line segment of a parallelohedron at the origin of three-dimensional space, the generators may be represented as three-dimensional <a href="/wiki/Vector_(mathematics)" class="mw-redirect" title="Vector (mathematics)">vectors</a>, the positions of their opposite endpoints. For this placement of the segments, one vertex of the parallelohedron will itself be at the origin, and the rest will be at positions given by sums of certain subsets of these vectors. A parallelohedron with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" /></span> vectors can in this way be parameterized by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf460b673a8c275b9e3942a23b78c102dea145d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.278ex; height:2.509ex;" alt="{\displaystyle 3g}" /></span> coordinates, three for each vector, but only some of these combinations are valid (because of the requirement that certain triples of segments lie in parallel planes, or equivalently that certain triples of vectors are coplanar) and different combinations may lead to parallelohedra that differ only by a rotation, scaling transformation, or more generally by an <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a>. When affine transformations are factored out, the number of free parameters that describe the shape of a parallelohedron is zero for a parallelepiped (all parallelepipeds are equivalent to each other under affine transformations), two for a hexagonal prism, three for a rhombic dodecahedron, four for an elongated dodecahedron, and five for a truncated octahedron.<sup id="cite_ref-din_13-0" class="reference"><a href="#cite_note-din-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelohedron&action=edit&section=5" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The classification of parallelohedra into five types was first made by Russian crystallographer <a href="/wiki/Evgraf_Fedorov" title="Evgraf Fedorov">Evgraf Fedorov</a>, as chapter 13 of a Russian-language book first published in 1885, whose title has been translated into English as <i>An Introduction to the Theory of Figures</i>.<sup id="cite_ref-fed_14-0" class="reference"><a href="#cite_note-fed-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Federov was working under an incorrect theory of the structure of <a href="/wiki/Crystal" title="Crystal">crystals</a>, according to which every crystal has a repeating structure in the shape of a parallelohedron, which in turn is formed from one or more <a href="/wiki/Molecule" title="Molecule">molecules</a> that all take the same shape (a <a href="/wiki/Stereohedron" title="Stereohedron">stereohedron</a>). This theory was falsified by the 1913 discovery of the structure of <a href="/wiki/Halite" title="Halite">halite</a> (table salt) which is not partitioned into separate molecules, and more strongly by the much later discovery of <a href="/wiki/Quasicrystal" title="Quasicrystal">quasicrystals</a>.<sup id="cite_ref-sentay_6-1" class="reference"><a href="#cite_note-sentay-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Some of the mathematics in Federov's book is faulty; for instance it includes an incorrect proof of a lemma stating that every monohedral tiling of the plane is periodic,<sup id="cite_ref-senechal_15-0" class="reference"><a href="#cite_note-senechal-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> proven to be false in 2023 as part of the solution to the <a href="/wiki/Einstein_problem" title="Einstein problem">einstein problem</a>.<sup id="cite_ref-roberts_16-0" class="reference"><a href="#cite_note-roberts-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In the case of parallelohedra, Fedorov assumed without proof that every parallelohedron is centrally symmetric, and used this assumption to prove his classification. The classification of parallelohedra was later placed on a firmer footing by <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a>, who used his <a href="/wiki/Minkowski_problem_for_polytopes" title="Minkowski problem for polytopes">uniqueness theorem for polyhedra with given face normals and areas</a> to prove that parallelohedra are centrally symmetric.<sup id="cite_ref-alexandrov_2-9" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Related_shapes">Related shapes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelohedron&action=edit&section=6" title="Edit section: Related shapes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-mathematics">Unsolved problem in mathematics</span>:</div> <div class="unsolved-body">Can every spherical non-convex polyhedron that tiles space by translation have its faces grouped into patches with the same combinatorial structure as a parallelohedron?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1233989161" /> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-mathematics">Unsolved problem in mathematics</span>:</div> <div class="unsolved-body">Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div> <p>In two dimensions the analogous figure to a parallelohedron is a <a href="/wiki/Parallelogon" title="Parallelogon">parallelogon</a>, a polygon that can tile the plane edge-to-edge by translation. There are two kinds of parallelogons: the <a href="/wiki/Parallelogram" title="Parallelogram">parallelograms</a> and the <a href="/wiki/Hexagon" title="Hexagon">hexagons</a> in which each pair of opposite sides is parallel and of equal length.<sup id="cite_ref-gs_17-0" class="reference"><a href="#cite_note-gs-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>There are multiple non-convex polyhedra that tile space by translation, beyond the five Federov parallelohedra.<sup id="cite_ref-sentay_6-2" class="reference"><a href="#cite_note-sentay-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ranlor_18-0" class="reference"><a href="#cite_note-ranlor-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> These are not zonohedra and need not be centrally symmetric. For instance, some of these can be obtained from a <a href="/wiki/Rhombic_triacontahedron" title="Rhombic triacontahedron">rhombic triacontahedron</a> by replacing certain triples of faces by indentations. According to a conjecture of <a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Branko Grünbaum</a>, for every polyhedron that is topologically a sphere and can tile space by translation, it is possible to group its faces into patches (unions of connected subsets of faces) so that the combinatorial structure of these patches is the same as the combinatorial structure of the faces of one of the five Federov parallelohedra. This conjecture remains unproven.<sup id="cite_ref-sentay_6-3" class="reference"><a href="#cite_note-sentay-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In higher dimensions a <a href="/wiki/Convex_polytope" title="Convex polytope">convex polytope</a> that tiles space by translation is called a <i>parallelotope</i>. There are 52 different four-dimensional parallelotopes, first enumerated by <a href="/wiki/Boris_Delaunay" title="Boris Delaunay">Boris Delaunay</a> (with one missing parallelotope, later discovered by Mikhail Shtogrin),<sup id="cite_ref-engel_19-0" class="reference"><a href="#cite_note-engel-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> and exactly 110,244 types in five dimensions.<sup id="cite_ref-garber_20-0" class="reference"><a href="#cite_note-garber-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Unlike the case for three dimensions, not all of them are <a href="/wiki/Zonotope" class="mw-redirect" title="Zonotope">zonotopes</a>. 17 of the four-dimensional parallelotopes are zonotopes, one is the regular <a href="/wiki/24-cell" title="24-cell">24-cell</a>, and the remaining 34 of these shapes are <a href="/wiki/Minkowski_sum" class="mw-redirect" title="Minkowski sum">Minkowski sums</a> of zonotopes with the 24-cell.<sup id="cite_ref-dg_21-0" class="reference"><a href="#cite_note-dg-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> A <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span>-dimensional parallelotope can have at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{d+1}-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{d+1}-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/607ec0e9bd39a7036088ea300c4db7b3b12a329d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.358ex; height:2.843ex;" alt="{\displaystyle 2^{d+1}-2}" /></span> facets, with the <a href="/wiki/Permutohedron" title="Permutohedron">permutohedron</a> achieving this maximum.<sup id="cite_ref-dienst_5-1" class="reference"><a href="#cite_note-dienst-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Every parallelohedron is a <a href="/wiki/Stereohedron" title="Stereohedron">stereohedron</a>, a convex polyhedron that tiles space in such a way that there exist symmetries of the tiling that take any tile to any other tile. A <a href="/wiki/Plesiohedron" title="Plesiohedron">plesiohedron</a> is a related class of three-dimensional space-filling polyhedra, formed from the <a href="/wiki/Voronoi_diagram" title="Voronoi diagram">Voronoi diagrams</a> of periodic sets of points (of a more general type than the lattices).<sup id="cite_ref-gs_17-1" class="reference"><a href="#cite_note-gs-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The Voronoi diagram of a lattice produces a tiling of space by parallelohedra,<sup id="cite_ref-garber_20-1" class="reference"><a href="#cite_note-garber-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> but not every parallelohedron and its tiling can be generated in this way: for a parallelohedron to be a plesiohedron, it is required that each vector from the center of the parallelohedron to the center of a face be perpendicular to the face.<sup id="cite_ref-alexandrov_2-10" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> However, as <a href="/wiki/Boris_Delaunay" title="Boris Delaunay">Boris Delaunay</a> proved in 1929,<sup id="cite_ref-austin_22-0" class="reference"><a href="#cite_note-austin-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> every parallelohedron can be made into a plesiohedron by an affine transformation.<sup id="cite_ref-alexandrov_2-11" class="reference"><a href="#cite_note-alexandrov-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> He also proved the same fact for four-dimensional parallelohedra,<sup id="cite_ref-gs_17-2" class="reference"><a href="#cite_note-gs-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> and it has been proven as well for five dimensions,<sup id="cite_ref-garber_20-2" class="reference"><a href="#cite_note-garber-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> but this remains open in higher dimensions.<sup id="cite_ref-dienst_5-2" class="reference"><a href="#cite_note-dienst-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gs_17-3" class="reference"><a href="#cite_note-gs-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-garber_20-3" class="reference"><a href="#cite_note-garber-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Some other three-dimensional plesiohedra are not parallelohedra. The tilings of space by plesiohedra have symmetries taking any cell to any other cell, but unlike for the parallelohedra, these symmetries may involve rotations, not just translations.<sup id="cite_ref-gs_17-4" class="reference"><a href="#cite_note-gs-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelohedron&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Keller%27s_conjecture" title="Keller's conjecture">Keller's conjecture</a>, on tilings by translated copies of cubes and hypercubes that are not required to be face-to-face</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelohedron&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-hargittai-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-hargittai_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration 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(September 2008). <a rel="nofollow" class="external text" href="https://scholar.archive.org/work/fnig6aahfbexfcvn5aaajhm25a">"Parallelohedra and topological transitions in cellular structures"</a>. <i>Philosophical Magazine Letters</i>. <b>88</b> (<span class="nowrap">9–</span>10): <span class="nowrap">703–</span>713. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008PMagL..88..703R">2008PMagL..88..703R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F09500830802112173">10.1080/09500830802112173</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine+Letters&rft.atitle=Parallelohedra+and+topological+transitions+in+cellular+structures&rft.volume=88&rft.issue=%3Cspan+class%3D%22nowrap%22%3E9%E2%80%93%3C%2Fspan%3E10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E703-%3C%2Fspan%3E713&rft.date=2008-09&rft_id=info%3Adoi%2F10.1080%2F09500830802112173&rft_id=info%3Abibcode%2F2008PMagL..88..703R&rft.aulast=Ranganathan&rft.aufirst=S.&rft.au=Lord%2C+E.+A.&rft_id=https%3A%2F%2Fscholar.archive.org%2Fwork%2Ffnig6aahfbexfcvn5aaajhm25a&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelohedron" class="Z3988"></span></span> </li> <li id="cite_note-engel-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-engel_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEngel1988" class="citation journal cs1">Engel, P. (1988). Hargittai, I.; Vainshtein, B. K. (eds.). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0898-1221%2888%2990232-5">"Mathematical problems in modern crystallography"</a>. Crystal Symmetries: Shubnikov Centennial Papers. <i>Computers & Mathematics with Applications</i>. <b>16</b> (<span class="nowrap">5–</span>8): <span class="nowrap">425–</span>436. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0898-1221%2888%2990232-5">10.1016/0898-1221(88)90232-5</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0991578">0991578</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computers+%26+Mathematics+with+Applications&rft.atitle=Mathematical+problems+in+modern+crystallography&rft.volume=16&rft.issue=%3Cspan+class%3D%22nowrap%22%3E5%E2%80%93%3C%2Fspan%3E8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E425-%3C%2Fspan%3E436&rft.date=1988&rft_id=info%3Adoi%2F10.1016%2F0898-1221%2888%2990232-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D991578%23id-name%3DMR&rft.aulast=Engel&rft.aufirst=P.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0898-1221%252888%252990232-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelohedron" class="Z3988"></span> See in particular <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UdnSBQAAQBAJ&lpg=PA435">p. 435</a>.</span> </li> <li id="cite_note-garber-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-garber_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-garber_20-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-garber_20-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-garber_20-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGarber2025" class="citation journal cs1">Garber, Alexey (February 2025). "Voronoi conjecture for five-dimensional parallelohedra". <i>Inventiones Mathematicae</i>. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1906.05193">1906.05193</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00222-025-01325-0">10.1007/s00222-025-01325-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Inventiones+Mathematicae&rft.atitle=Voronoi+conjecture+for+five-dimensional+parallelohedra&rft.date=2025-02&rft_id=info%3Aarxiv%2F1906.05193&rft_id=info%3Adoi%2F10.1007%2Fs00222-025-01325-0&rft.aulast=Garber&rft.aufirst=Alexey&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelohedron" class="Z3988"></span></span> </li> <li id="cite_note-dg-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-dg_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDezaGrishukhin2008" class="citation journal cs1"><a href="/wiki/Michel_Deza" title="Michel Deza">Deza, Michel</a>; Grishukhin, Viacheslav P. (2008). "More about the 52 four-dimensional parallelotopes". <i>Taiwanese Journal of Mathematics</i>. <b>12</b> (4): <span class="nowrap">901–</span>916. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0307171">math/0307171</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.11650%2Ftwjm%2F1500404985">10.11650/twjm/1500404985</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2426535">2426535</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Taiwanese+Journal+of+Mathematics&rft.atitle=More+about+the+52+four-dimensional+parallelotopes&rft.volume=12&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E901-%3C%2Fspan%3E916&rft.date=2008&rft_id=info%3Aarxiv%2Fmath%2F0307171&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2426535%23id-name%3DMR&rft_id=info%3Adoi%2F10.11650%2Ftwjm%2F1500404985&rft.aulast=Deza&rft.aufirst=Michel&rft.au=Grishukhin%2C+Viacheslav+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelohedron" class="Z3988"></span></span> </li> <li id="cite_note-austin-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-austin_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAustin2013" class="citation web cs1">Austin, David (November 2013). <a rel="nofollow" class="external text" href="https://www.ams.org/samplings/feature-column/fc-2013-11">"Fedorov's five parallelohedra"</a>. <i>AMS Feature Column</i>. American Mathematical Society.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=AMS+Feature+Column&rft.atitle=Fedorov%27s+five+parallelohedra&rft.date=2013-11&rft.aulast=Austin&rft.aufirst=David&rft_id=https%3A%2F%2Fwww.ams.org%2Fsamplings%2Ffeature-column%2Ffc-2013-11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelohedron" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Parallelohedron&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Parallelohedron"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Parallelohedron.html">"Parallelohedron"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Parallelohedron&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FParallelohedron.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AParallelohedron" class="Z3988"></span></span></li></ul>    </div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Parallelohedron&oldid=1283383815">https://en.wikipedia.org/w/index.php?title=Parallelohedron&oldid=1283383815</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Space-filling_polyhedra" title="Category:Space-filling polyhedra">Space-filling polyhedra</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_Russian-language_sources_(ru)" title="Category:CS1 Russian-language sources (ru)">CS1 Russian-language sources (ru)</a></li><li><a href="/wiki/Category:CS1_French-language_sources_(fr)" title="Category:CS1 French-language sources (fr)">CS1 French-language sources (fr)</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_mdy_dates_from_October_2020" title="Category:Use mdy dates from October 2020">Use mdy dates from October 2020</a></li><li><a href="/wiki/Category:Use_list-defined_references_from_March_2025" title="Category:Use list-defined references from March 2025">Use list-defined references from March 2025</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 1 April 2025, at 05:07<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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