Vertex configuration

<h2 id="notation">Notation</h2>
Regular vertex figure nets, {p,q} = pq<table><tbody><tr><td><a href="/facts/Tetrahedron/7mkSrl6j">{3,3}</a> = 33Defect 180°</td><td><a href="/facts/Octahedron/G38q92xW">{3,4}</a> = 34Defect 120°</td><td><a href="/facts/Icosahedron/wWbpnDYp">{3,5}</a> = 35Defect 60°</td><td><a href="/facts/Triangular_tiling/Jx8f8Krh">{3,6}</a> =36Defect 0°</td></tr><tr><td><a href="/facts/Cube/65lUaAqN">{4,3}</a>Defect 90°</td><td><a href="/facts/Square_tiling/3ym86mvv">{4,4}</a> =44Defect 0°</td><td><a href="/facts/Dodecahedron/7r519eCl">{5,3}</a> = 53Defect 36°</td><td><a href="/facts/Hexagonal_tiling/HwG5F0YH">{6,3}</a> =63Defect 0°</td></tr><tr><td colspan="4">A vertex needs at least 3 faces, and an <a href="/facts/Angle_defect/FMHKeUVS">angle defect</a>.A 0° angle defect will fill the Euclidean plane with regular tiling.By <a href="/facts/Angular_defect/FMHKeUVS">Descartes' theorem</a>, the number of vertices is 720°/defect (4π radians/defect).</td></tr></tbody></table>
A vertex configuration is written as one or more numbers separated by either dots or commas. Each number represents the number of sides in each face that meets at each vertex.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> An <a href="/facts/Icosidodecahedron/P0JluaVA">icosidodecahedron</a> is denoted as 
 
 
 
 3.5.3.5
 
 
 {\displaystyle 3.5.3.5}
 
 because there are four faces at each vertex, alternating between <a href="/facts/Triangle/cRXUwsMn">triangles</a> (with 3 sides) and <a href="/facts/Pentagon/xoq5oJeQ">pentagons</a> (with 5 sides). This can also be written as 
 
 
 
 (
 3.5
 
 )
 
 2
 
 
 
 
 {\displaystyle (3.5)^{2}}
 
.
The vertex configuration can also be considered an expansive form of the simple <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> for <a href="/facts/Platonic_solid/jraREXFD">regular polyhedra</a>. The Schläfli notation has the form 
 
 
 
 {
 p
 ,
 q
 }
 
 
 {\displaystyle \{p,q\}}
 
, where 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is the number of sides in each face and 
 
 
 
 q
 
 
 {\displaystyle q}
 
 is the number of faces that meet at each vertex. Hence, the Schläfli notation 
 
 
 
 {
 p
 ,
 q
 }
 
 
 {\displaystyle \{p,q\}}
 
 can be written as 
 
 
 
 p
 .
 p
 .
 p
 ⋯
 
 
 {\displaystyle p.p.p\cdots }
 
 (where 
 
 
 
 p
 
 
 {\displaystyle p}
 
 appears 
 
 
 
 q
 
 
 {\displaystyle q}
 
 times), or simply 
 
 
 
 
 p
 
 q
 
 
 
 
 {\displaystyle p^{q}}
 
.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>
This notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron.
The notation is ambiguous for <a href="/facts/Chirality_(mathematics)/FtKNlHTb">chiral</a> forms. For example, the <a href="/facts/Snub_cube/pbnr40gk">snub cube</a> has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration.

<h2 id="star-polygons">Star polygons</h2>
The notation also applies for nonconvex regular faces, the <a href="/facts/Star_polygon/mDfuIe81">star polygons</a>. For example, a <a href="/facts/Pentagram/cN0ke8tp">pentagram</a> has the symbol {5/2}, meaning it has 5 sides going around the centre twice.
For example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The <a href="/facts/Small_stellated_dodecahedron/lX1zlo6C">small stellated dodecahedron</a> has the <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> of {5/2,5} which expands to an explicit vertex configuration 5/2.5/2.5/2.5/2.5/2 or combined as (5/2)5. The <a href="/facts/Great_stellated_dodecahedron/7efrgSub">great stellated dodecahedron</a>, {5/2,3} has a triangular vertex figure and configuration (5/2.5/2.5/2) or (5/2)3. The <a href="/facts/Great_dodecahedron/LWt3Hgg4">great dodecahedron</a>, {5,5/2} has a pentagrammic vertex figure, with vertex configuration is (5.5.5.5.5)/2 or (55)/2. A <a href="/facts/Great_icosahedron/jaAk3a5Q">great icosahedron</a>, {3,5/2} also has a pentagrammic vertex figure, with vertex configuration (3.3.3.3.3)/2 or (35)/2.

<table><tbody><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Small_stellated_dodecahedron/lX1zlo6C">{5/2,5}</a> = (5/2)5</th><th><a href="/facts/Great_stellated_dodecahedron/7efrgSub">{5/2,3}</a> = (5/2)3</th><th><a href="/facts/Great_snub_icosidodecahedron/n0vFdDWt">34.5/2</a></th><th><a href="/facts/Small_retrosnub_icosicosidodecahedron/SItHvRB2">34.5/3</a></th><th><a href="/facts/Great_retrosnub_icosidodecahedron/ofggBlnC">(34.5/2)/2</a></th></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Great_dodecahedron/LWt3Hgg4">{5,5/2}</a> = (55)/2</th><th><a href="/facts/Great_icosahedron/jaAk3a5Q">{3,5/2}</a> = (35)/2</th><th><a href="/facts/Great_pentagonal_hexecontahedron/n0vFdDWt">V.34.5/2</a></th><th><a href="/facts/Small_hexagrammic_hexecontahedron/BX0V4070">V34.5/3</a></th><th><a href="/facts/Great_pentagrammic_hexecontahedron/xCqBazQH">V(34.5/2)/2</a></th></tr></tbody></table>
<h2 id="inverted-polygons">Inverted polygons</h2>
Faces on a vertex figure are considered to progress in one direction. Some uniform polyhedra have vertex figures with inversions where the faces progress retrograde. A vertex figure represents this in the <a href="/facts/Star_polygon/mDfuIe81">star polygon</a> notation of sides p/q such that p<2q, where p is the number of sides and q the number of turns around a circle. For example, "3/2" means a triangle that has vertices that go around twice, which is the same as backwards once. Similarly "5/3" is a backwards pentagram 5/2.

<h2 id="all-uniform-vertex-configurations-of-regular-convex-polygons">All uniform vertex configurations of regular convex polygons</h2>
See also: <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean_solid § Classification</a>, <a href="/facts/Tiling_by_regular_polygons/tR5CQACy">Tiling by regular polygons § Combinations of regular polygons that can meet at a vertex</a>, and <a href="/facts/Uniform_tiling/QVK02ksI">Uniform tiling § Expanded lists of uniform tilings</a>
<a href="/facts/Semiregular_polyhedron/fHygpjHu">Semiregular polyhedra</a> have vertex configurations with positive <a href="/facts/Defect_(geometry)/FMHKeUVS">angle defect</a>.
NOTE: The vertex figure can represent a regular or semiregular tiling on the plane if its defect is zero. It can represent a tiling of the hyperbolic plane if its defect is negative.
For uniform polyhedra, the angle defect can be used to compute the number of vertices. Descartes' theorem states that all the angle defects in a topological sphere must sum to 4π radians or
720 degrees.
Since uniform polyhedra have all identical vertices, this relation allows us to compute the number of vertices, which is 4π/defect or
720/defect.
Example: A <a href="/facts/Truncated_cube/XVAUG4fD">truncated cube</a> 3.8.8 has an angle defect of 30 degrees. Therefore, it has
720/30 = 24 vertices.
In particular it follows that {a,b} has 4 / (2 − b(1 − 2/a)) vertices.
Every enumerated vertex configuration potentially uniquely defines a semiregular polyhedron. However, not all configurations are possible.
Topological requirements limit existence. Specifically p.q.r implies that a p-gon is surrounded by alternating q-gons and r-gons, so either p is even or q equals r. Similarly q is even or p equals r, and r is even or p equals q. Therefore, potentially possible triples are 3.3.3, 3.4.4, 3.6.6, 3.8.8, 3.10.10, 3.12.12, 4.4.n (for any n>2), 4.6.6, 4.6.8, 4.6.10, 4.6.12, 4.8.8, 5.5.5, 5.6.6, 6.6.6. In fact, all these configurations with three faces meeting at each vertex turn out to exist.
The number in parentheses is the number of vertices, determined by the angle defect.

Triples
<ul><li>Platonic solids <a href="/facts/Tetrahedron/7mkSrl6j">3.3.3</a> (4), <a href="/facts/Cube/65lUaAqN">4.4.4</a> (8), <a href="/facts/Dodecahedron/7r519eCl">5.5.5</a> (20)</li>
<li><a href="/facts/Prism_(geometry)/baIRRkCy">prisms</a> 4.4.n (2n)</li>
<li>Archimedean solids <a href="/facts/Truncated_tetrahedron/K9ItvmUK">3.6.6</a> (12), <a href="/facts/Truncated_cube/XVAUG4fD">3.8.8</a> (24), <a href="/facts/Truncated_dodecahedron/PVTAsUWQ">3.10.10</a> (60), <a href="/facts/Truncated_octahedron/GA98b4TH">4.6.6</a> (24), <a href="/facts/Truncated_cuboctahedron/nd2ltxcm">4.6.8</a> (48), <a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">4.6.10</a> (120), <a href="/facts/Truncated_icosahedron/LR8ugaY9">5.6.6</a> (60).</li>
<li>regular tiling <a href="/facts/Hexagonal_tiling/HwG5F0YH">6.6.6</a></li>
<li>semiregular tilings <a href="/facts/Truncated_hexagonal_tiling/c9dRoGFS">3.12.12</a>, <a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">4.6.12</a>, <a href="/facts/Truncated_square_tiling/Mbj1fVzm">4.8.8</a></li></ul>
Quadruples
<ul><li>Platonic solid <a href="/facts/Octahedron/G38q92xW">3.3.3.3</a> (6)</li>
<li><a href="/facts/Antiprism/FgP5jIVd">antiprisms</a> 3.3.3.n (2n)</li>
<li>Archimedean solids <a href="/facts/Cuboctahedron/hD7p0Mem">3.4.3.4</a> (12), <a href="/facts/Icosidodecahedron/P0JluaVA">3.5.3.5</a> (30), <a href="/facts/Rhombicuboctahedron/TmxUmEnl">3.4.4.4</a> (24), <a href="/facts/Rhombicosidodecahedron/AaDckLGP">3.4.5.4</a> (60)</li>
<li>regular tiling <a href="/facts/Square_tiling/3ym86mvv">4.4.4.4</a></li>
<li>semiregular tilings <a href="/facts/Trihexagonal_tiling/spAKhDvd">3.6.3.6</a>, <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">3.4.6.4</a></li></ul>
Quintuples
<ul><li>Platonic solid <a href="/facts/Icosahedron/wWbpnDYp">3.3.3.3.3</a> (12)</li>
<li>Archimedean solids <a href="/facts/Snub_cube/pbnr40gk">3.3.3.3.4</a> (24), <a href="/facts/Snub_dodecahedron/oouEPVxD">3.3.3.3.5</a> (60) (both <a href="/facts/Chirality_(mathematics)/FtKNlHTb">chiral</a>)</li>
<li>semiregular tilings <a href="/facts/Snub_hexagonal_tiling/0Km9nP9l">3.3.3.3.6</a> (chiral), <a href="/facts/Elongated_triangular_tiling/mH6cUVHj">3.3.3.4.4</a>, <a href="/facts/Snub_square_tiling/hNu3KtEA">3.3.4.3.4</a> (note that the two different orders of the same numbers give two different patterns)</li></ul>
Sextuples
<ul><li>regular tiling <a href="/facts/Triangular_tiling/Jx8f8Krh">3.3.3.3.3.3</a></li></ul>
<h2 id="face-configuration">Face configuration</h2>

The uniform dual or <a href="/facts/Catalan_solid/R8D2cP7D">Catalan solids</a>, including the <a href="/facts/Bipyramid/MkzrCA0W">bipyramids</a> and <a href="/facts/Trapezohedra/8QpjlnXe">trapezohedra</a>, are vertically-regular (<a href="/facts/Face-transitive/ukLVQlK2">face-transitive</a>) and so they can be identified by a similar notation which is sometimes called face configuration.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> Cundy and Rollett prefixed these dual symbols by a V. In contrast, <a href="/facts/Tilings_and_patterns/bX261kQS">Tilings and patterns</a> uses square brackets around the symbol for isohedral tilings.
This notation represents a sequential count of the number of faces that exist at each <a href="/facts/Vertex_(geometry)/ID3cea9z">vertex</a> around a <a href="/facts/Face_(geometry)/T2E6BTU4">face</a>.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> For example, V3.4.3.4 or V(3.4)2 represents the <a href="/facts/Rhombic_dodecahedron/X3npAlmr">rhombic dodecahedron</a> which is face-transitive: every face is a <a href="/facts/Rhombus/T4pU50GF">rhombus</a>, and alternating vertices of the rhombus contain 3 or 4 faces each.

<ul><li><a href="/facts/Robert_Williams_(geometer)/1FJDMSJ2">Williams, Robert</a> (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-23729-X. Uses Cundy-Rollett symbol.</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">Tilings and Patterns</a>. W. H. Freeman and Company. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7167-1193-1. Pp. 58–64, Tilings of regular polygons a.b.c.... (Tilings by regular polygons and star polygons) pp. 95–97, 176, 283, 614–620, Monohedral tiling symbol [v1.v2. ... .vr]. pp. 632–642 hollow tilings.</li>
<li>The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-220-5 (p. 289 Vertex figures, uses comma separator, for Archimedean solids and tilings).</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.software3d.com/VertexDesc.php">Consistent Vertex Descriptions</a> <a href="/facts/Stella_(software)/MDH2P21V">Stella (software)</a>, Robert Webb</li></ul>

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

<ol>
<li id="fn:1">Archimedean Polyhedra Archived 2017-07-05 at the Wayback Machine Steven Dutch <a href="https://www.uwgb.edu/dutchs/symmetry/archpol.htm" target="_blank">https://www.uwgb.edu/dutchs/symmetry/archpol.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Uniform Polyhedra Jim McNeill <a href="http://www.orchidpalms.com/polyhedra/uniform/uniform.html" target="_blank">http://www.orchidpalms.com/polyhedra/uniform/uniform.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Uniform Polyhedra and their Duals Robert Webb <a href="http://www.software3d.com/Uniform.php" target="_blank">http://www.software3d.com/Uniform.php</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Symmetry-type graphs of Platonic and Archimedean solids, Jurij Kovič, (2011) <a href="http://hrcak.srce.hr/file/111177/" target="_blank">http://hrcak.srce.hr/file/111177/</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">3. General Theorems: Regular and Semi-Regular Tilings Kevin Mitchell, 1995 <a href="http://people.hws.edu/mitchell/tilings/part3.html" target="_blank">http://people.hws.edu/mitchell/tilings/part3.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Resources for Teaching Discrete Mathematics: Classroom Projects, History, modules, and articles, edited by Brian Hopkins <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Vertex Symbol Robert Whittaker <a href="http://polyhedra.mathmos.net/entry/vertexsymbol.html" target="_blank">http://polyhedra.mathmos.net/entry/vertexsymbol.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Structure and Form in Design: Critical Ideas for Creative Practice
 By Michael Hann <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Symmetry-type graphs of Platonic and Archimedean solids Jurij Kovič <a href="https://www.researchgate.net/publication/264848030_Symmetry-type_graphs_of_Platonic_and_Archimedean_solids" target="_blank">https://www.researchgate.net/publication/264848030_Symmetry-type_graphs_of_Platonic_and_Archimedean_solids</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Deza, Michel; Shtogrin, Mikhail (2000), "Uniform partitions of 3-space, their relatives and embedding", European Journal of Combinatorics, 21 (6): 807–814, arXiv:math/9906034, doi:10.1006/eujc.1999.0385, MR 1791208 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Boag, Tom; Boberg, Charles; Hughes, Lyn (1979). "On Archimedean Solids". The Mathematics Teacher. 72 (5): 371–376. doi:10.5951/MT.72.5.0371. ISSN 0025-5769. JSTOR 27961672. <a href="http://www.jstor.org/stable/27961672" target="_blank">http://www.jstor.org/stable/27961672</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Cundy, Martyn; Rollett, A. (1989). Mathematical Models (3rd ed.). Stradbroke, England: Tarquin Pub. See 3.7 The Archimedean Polyhedra, pp. 101–115, pp. 118–119 in Table I, Nets of Archimedean Duals, 
 
 
 
 
 V
 
 .
 a
 ,
 b
 ,
 c
 ,
 …
 
 
 {\displaystyle \mathrm {V} .a,b,c,\dots }
 
 as vertically-regular symbols. <a href="/wiki/Martyn_Cundy" target="_blank">/wiki/Martyn_Cundy</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere 6.4.1 Cundy-Rollett symbol, p. 164 <a href="https://books.google.com/books?id=HjTSBQAAQBAJ&pg=PA164" target="_blank">https://books.google.com/books?id=HjTSBQAAQBAJ&pg=PA164</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Laughlin (2014), p. 16 <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Walter, Steurer; Deloudi, Sofia (2009). Crystallography of Quasicrystals: Concepts, Methods and Structures. Springer Series in Materials Science. Vol. 126. p. 50. doi:10.1007/978-3-642-01899-2. ISBN 978-3-642-01898-5. <a href="978-3-642-01898-5" target="_blank">978-3-642-01898-5</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Walter, Steurer; Deloudi, Sofia (2009). Crystallography of Quasicrystals: Concepts, Methods and Structures. Springer Series in Materials Science. Vol. 126. p. 50. doi:10.1007/978-3-642-01899-2. ISBN 978-3-642-01898-5. <a href="978-3-642-01898-5" target="_blank">978-3-642-01898-5</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Crystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi, (2009) pp. 18–20 and 51–53 <a href="https://books.google.com/books?id=nVx-tu596twC&pg=PA18" target="_blank">https://books.google.com/books?id=nVx-tu596twC&pg=PA18</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Cundy, Martyn; Rollett, A. (1989). Mathematical Models (3rd ed.). Stradbroke, England: Tarquin Pub. See 3.7 The Archimedean Polyhedra, pp. 101–115, pp. 118–119 in Table I, Nets of Archimedean Duals, 
 
 
 
 
 V
 
 .
 a
 ,
 b
 ,
 c
 ,
 …
 
 
 {\displaystyle \mathrm {V} .a,b,c,\dots }
 
 as vertically-regular symbols. <a href="/wiki/Martyn_Cundy" target="_blank">/wiki/Martyn_Cundy</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
</ol>

Vertex configuration open-in-new

Vertex configuration