Isogonal figure

<h2 id="isogonal-polygons-and-apeirogons">Isogonal polygons and apeirogons</h2>
<table><tbody><tr><td></td></tr><tr><td></td></tr><tr><th>Isogonal <a href="/facts/Apeirogon/r0V9hleb">apeirogons</a></th></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><th>Isogonal <a href="/facts/Skew_apeirogon/l0Y1Kg6c">skew apeirogons</a></th></tr></tbody></table>
<p>All <a href="/facts/Regular_polygons/ws89Nhpt">regular polygons</a>, <a href="/facts/Apeirogon/r0V9hleb">apeirogons</a> and <a href="/facts/Regular_star_polygon/ws89Nhpt">regular star polygons</a> are <i>isogonal</i>. The <a href="/facts/Dual_polygon/mbR8QK0n">dual</a> of an isogonal polygon is an <a href="/facts/Isotoxal_polygon/p5j7GhwX">isotoxal polygon</a>.
</p><p>Some even-sided polygons and <a href="/facts/Apeirogon/r0V9hleb">apeirogons</a> which alternate two edge lengths, for example a <a href="/facts/Rectangle/mdLfIGKA">rectangle</a>, are <i>isogonal</i>.
</p><p>All planar isogonal 2<i>n</i>-gons have <a href="/facts/Dihedral_symmetry/1PUlRh4M">dihedral symmetry</a> (D<i>n</i>, <i>n</i> = 2, 3, ...) with reflection lines across the mid-edge points.
</p>
<table><tbody><tr><th>D2</th><th>D3</th><th>D4</th><th>D7</th></tr><tr><td>Isogonal <a href="/facts/Rectangle/mdLfIGKA">rectangles</a> and <a href="/facts/Crossed_rectangle/mdLfIGKA">crossed rectangles</a> sharing the same <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a></td><td>Isogonal <a href="/facts/Hexagram/IF4PEJbL">hexagram</a> with 6 identical vertices and 2 edge lengths.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></td><td>Isogonal convex <a href="/facts/Octagon/H5PuP3wP">octagon</a> with blue and red radial lines of reflection</td><td>Isogonal "star" <a href="/facts/Tetradecagon/oWPEMAly">tetradecagon</a> with one vertex type, and two edge types<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></td></tr></tbody></table>

<h2 id="isogonal-polyhedra-and-2d-tilings">Isogonal polyhedra and 2D tilings</h2>
Isogonal tilings<table><tbody><tr><td></td></tr><tr><td>Distorted <a href="/facts/Square_tiling/3ym86mvv">square tiling</a></td></tr><tr><td></td></tr><tr><td>A distorted<a href="/facts/Truncated_square_tiling/Mbj1fVzm">truncated square tiling</a></td></tr></tbody></table>
<p>An isogonal polyhedron and 2D tiling has a single kind of vertex. An isogonal polyhedron with all regular faces is also a <a href="/facts/Uniform_polyhedron/PHnprqRC">uniform polyhedron</a> and can be represented by a <a href="/facts/Vertex_configuration/wWaBS6t0">vertex configuration</a> notation sequencing the faces around each vertex. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration.
</p>
Isogonal polyhedra<table><tbody><tr><th>D3d, order 12</th><th><a href="/facts/Pyritohedral_symmetry/hIVlf9pu">Th</a>, order 24</th><th colspan="2"><a href="/facts/Octahedral_symmetry/qLqhYxgm">Oh</a>, order 48</th></tr><tr><th>4.4.6</th><th>3.4.4.4</th><th>4.6.8</th><th>3.8.8</th></tr><tr><td>A distorted <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a> (ditrigonal trapezoprism)</td><td>A distorted <a href="/facts/Rhombicuboctahedron/TmxUmEnl">rhombicuboctahedron</a></td><td>A shallow <a href="/facts/Truncated_cuboctahedron/nd2ltxcm">truncated cuboctahedron</a></td><td>A hyper-truncated cube</td></tr></tbody></table>
<p>Isogonal polyhedra and 2D tilings may be further classified:
</p>
<ul><li><i><a href="/facts/Regular_polyhedron/HJ6oGyug">Regular</a></i> if it is also <a href="/facts/Isohedral/ukLVQlK2">isohedral</a> (face-transitive) and <a href="/facts/Isotoxal/p5j7GhwX">isotoxal</a> (edge-transitive); this implies that every face is the same kind of <a href="/facts/Regular_polygon/ws89Nhpt">regular polygon</a>.</li>
<li><i><a href="/facts/Quasiregular_polyhedron/1ukrFhec">Quasi-regular</a></i> if it is also <a href="/facts/Isotoxal/p5j7GhwX">isotoxal</a> (edge-transitive) but not <a href="/facts/Isohedral/ukLVQlK2">isohedral</a> (face-transitive).</li>
<li><i><a href="/facts/Semiregular_polyhedron/fHygpjHu">Semi-regular</a></i> if every face is a regular polygon but it is not  <a href="/facts/Isohedral/ukLVQlK2">isohedral</a> (face-transitive) or <a href="/facts/Isotoxal/p5j7GhwX">isotoxal</a> (edge-transitive).  (Definition varies among authors; e.g. some exclude solids with dihedral symmetry, or nonconvex solids.)</li>
<li><i><a href="/facts/Uniform_polyhedron/PHnprqRC">Uniform</a></i> if every face is a regular polygon, i.e. it is regular, quasiregular or semi-regular.</li>
<li><i>Semi-uniform</i> if its elements are also isogonal.</li>
<li><i>Scaliform</i> if all the edges are the same length.</li>
<li><i><a href="/facts/Noble_polyhedron/AkbkkA02">Noble</a></i> if it is also <a href="/facts/Isohedral/ukLVQlK2">isohedral</a> (face-transitive).</li></ul>
<h2 id="ndimensions-isogonal-polytopes-and-tessellations"><i>N</i> dimensions: Isogonal polytopes and tessellations</h2>
<p>These definitions can be extended to higher-dimensional <a href="/facts/Polytope/a76nXrL0">polytopes</a> and <a href="/facts/Honeycomb_(geometry)/seyRBcel">tessellations</a>.  All <a href="/facts/Uniform_polytope/nFcNAzLy">uniform polytopes</a> are <i>isogonal</i>, for example, the <a href="/facts/Uniform_4-polytope/NHxomMxC">uniform 4-polytopes</a> and <a href="/facts/Convex_uniform_honeycomb/78IkTl7A">convex uniform honeycombs</a>.
</p><p>The <a href="/facts/Dual_polytope/rkIHbmAf">dual</a> of an isogonal polytope is an <a href="/facts/Isohedral_figure/ukLVQlK2">isohedral figure</a>, which is transitive on its <a href="/facts/Facet_(geometry)/IreRHGJ7">facets</a>.
</p>
<h2 id="k-isogonal-and-k-uniform-figures"><i>k</i>-isogonal and <i>k</i>-uniform figures</h2>
<p>A polytope or tiling may be called <i>k</i>-isogonal if its vertices form <i>k</i> transitivity classes. A more restrictive term, <i>k</i>-uniform is defined as a <i>k-isogonal figure</i> constructed only from <a href="/facts/Regular_polygon/ws89Nhpt">regular polygons</a>. They can be represented visually with colors by different <a href="/facts/Uniform_coloring/bNglRNDS">uniform colorings</a>.
</p>
<table><tbody><tr><td>This <a href="/facts/Truncated_rhombic_dodecahedron/hQEGYQpl">truncated rhombic dodecahedron</a> is 2-isogonal because it contains two transitivity classes of vertices. This polyhedron is made of <a href="/facts/Square_(geometry)/JHJeMvmL">squares</a> and flattened <a href="/facts/Hexagon/DPzuCzWE">hexagons</a>.</td><td>This <a href="/facts/Euclidean_tilings_of_regular_polygons/tR5CQACy">demiregular tiling</a> is also 2-isogonal (and 2-uniform). This tiling is made of <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a> and regular <a href="/facts/Hexagon/DPzuCzWE">hexagonal</a> faces.</td><td>2-isogonal 9/4 <a href="/facts/Enneagram_(geometry)/XLmA6xSX">enneagram</a> (face of the <a href="/facts/Final_stellation_of_the_icosahedron/UMKwLhvr">final stellation of the icosahedron</a>)</td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Edge-transitive/p5j7GhwX">Edge-transitive</a> (Isotoxal figure)</li>
<li><a href="/facts/Face-transitive/ukLVQlK2">Face-transitive</a> (Isohedral figure)</li></ul>

<ul><li>Peter R. Cromwell, <i>Polyhedra</i>, Cambridge University Press 1997, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-55432-2, p. 369 Transitivity</li>
<li><a href="/facts/Branko_Gr%25C3%25BCnbaum/taVzX1MG">Grünbaum, Branko</a>; <a href="/facts/G.C._Shephard/5AtJ6sMW">Shephard, G. C.</a> (1987). <a href="https://archive.org/details/isbn_0716711931"><i>Tilings and Patterns</i></a>. W. H. Freeman and Company. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7167-1193-1. (p. 33 <i>k-isogonal</i> tiling, p. 65 <i>k-uniform tilings</i>)</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Vertex-TransitiveGraph.html">"Vertex-transitive graph"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li><a href="http://bulatov.org/polyhedra/mosaic2000/">Isogonal Kaleidoscopical Polyhedra</a> Vladimir L. Bulatov, Physics Department, Oregon State University, Corvallis, Presented at Mosaic2000, Millennial Open Symposium on the Arts and Interdisciplinary Computing, 21–24 August 2000, Seattle, WA <a href="http://www.bulatov.org/polyhedra//mosaic2000/kaleido_poly/kaleido_frames.html">VRML models</a></li>
<li><a href="https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm">Steven Dutch uses the term k-uniform for enumerating k-isogonal tilings</a></li>
<li><a href="http://probabilitysports.com/tilings.html">List of n-uniform tilings</a></li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/DemiregularTessellation.html">"Demiregular tessellations"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>. (Also uses term k-uniform for k-isogonal)</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Grünbaum, Branko (1997), "Isogonal prismatoids", Discrete & Computational Geometry, 18 (1): 13–52, doi:10.1007/PL00009307, MR 1453440 <a href="/wiki/Branko_Gr%C3%BCnbaum" target="_blank">/wiki/Branko_Gr%C3%BCnbaum</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Coxeter, The Densities of the Regular Polytopes II, p54-55, "hexagram" vertex figure of h{5/2,5}. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum, Figure 1. Parameter t=2.0 <a href="/wiki/Branko_Gr%C3%BCnbaum" target="_blank">/wiki/Branko_Gr%C3%BCnbaum</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Isogonal figure open-in-new

Isogonal figure