View source for Goldberg–Coxeter construction

{{multiple image
 | width = 150
 | footer = Goldberg polyhedron (3,1) and geodesic polyhedron (3,1). The Goldberg polyhedra and geodesic polyhedra were precursors to the Goldberg–Coxeter operation.
 | image1 = Géode V 3 1 duale.gif
 | alt1 = Goldberg polyhedron (3,1)
 | image2 = Géode V 3 1.gif
 | alt2 = Geodesic polyhedron (3,1)
 }}
The '''Goldberg–Coxeter construction''' or '''Goldberg–Coxeter operation''' ('''GC construction''' or '''GC operation''') is a [[graph operation]] defined on [[regular graph|regular]] [[polyhedral graph]]s with [[Degree (graph theory)|degree]] 3 or 4.<ref name="Deza2004">{{cite journal |author1=Deza, M.|author2=Dutour, M|title=Goldberg–Coxeter constructions for 3-and 4-valent plane graphs|journal=The Electronic Journal of Combinatorics |volume=11|year=2004|pages=#R20|doi=10.37236/1773|url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1r20|doi-access=free}}</ref><ref name=Deza2015>{{cite book|author1=Deza, M.-M.|author2=Sikirić, M. D.|author3=Shtogrin, M. I. | year=2015 |title=Geometric Structure of Chemistry-Relevant Graphs: Zigzags and Central Circuits|publisher=Springer|pages=131–148 |chapter=Goldberg–Coxeter Construction and Parameterization|chapter-url=https://books.google.com/books?id=HLi4CQAAQBAJ&q=goldberg-coxeter&pg=PA130   |isbn=9788132224495}}</ref> It also applies to the [[dual graph]] of these graphs, i.e. graphs with triangular or quadrilateral "faces". The GC construction can be thought of as subdividing the faces of a polyhedron with a lattice of triangular, square, or hexagonal polygons, possibly skewed with regards to the original face: it is an extension of concepts introduced by the [[Goldberg polyhedra]] and [[geodesic polyhedra]]. The GC construction is primarily studied in [[organic chemistry]] for its application to [[fullerene]]s,<ref name=Deza2004 /><ref name=Deza2015 /> but it has been applied to [[nanoparticles]],<ref>{{cite journal|doi=10.1098/rsos.161092|pmid=28484626|pmc=5414263|title=Classification of self-assembling protein nanoparticle architectures for applications in vaccine design|journal=Royal Society Open Science|volume=4|issue=4|pages=161092|year=2017|last1=Indelicato|first1=G|last2=Burkhard|first2=P|last3=Twarock|first3=R|bibcode=2017RSOS....461092I}}</ref> [[computer-aided design]],<ref>{{cite journal|doi=10.1016/j.cagd.2017.09.002|title=A discrete surface theory|journal=Computer Aided Geometric Design|volume=58|pages=24–54|year=2017|last1=Kotani|first1=M|last2=Naito|first2=H|last3=Omori|first3=T|doi-access=free}}</ref> [[basket weaving]],<ref>{{cite conference|author=Tarnai, T.|year=2006|title=Baskets|conference=IASS-APCS 2006 Int. Symp. New Olympics New Shell and Spatial Structures| location=Beijing University of Technology|publisher=International Assoc. for Shell and Spatial Structures|pages=IL09|url=http://www.me.bme.hu/sites/default/files/flori/Baskets.pdf}}</ref><ref>{{cite journal |author1=Tarnai, T.|author2=Kovacs, F. |author3=Fowler, P.W. |author4=Guest, S.D. |title=Wrapping the cube and other polyhedra|journal=Proceedings of the Royal Society A |volume=468 |issue=2145 |pages=2652 |year=2012|doi=10.1098/rspa.2012.0116 |bibcode=2012RSPSA.468.2652T|doi-access=free }}</ref> and the general study of [[graph theory]] and [[polyhedra]].<ref>{{cite journal|doi=10.1016/j.ejc.2017.08.004|title=Packing and covering odd cycles in cubic plane graphs with small faces|journal=European Journal of Combinatorics|volume=67|pages=208–221|year=2018|last1=Nicodemos|first1=Diego|last2=Stehlík|first2=Matěj|arxiv=1701.07748|s2cid=27137740}}</ref>

The Goldberg–Coxeter construction may be denoted as <math>GC_{k,\ell}(G_0)</math>, where <math>G_0</math> is the graph being operated on, <math>k</math> and <math>l</math> are integers, <math>k >0</math>, and <math>\ell \ge 0</math>.

== History ==
Michael Goldberg introduced the [[Goldberg polyhedron]] in 1937.<ref>{{cite journal |author=Goldberg, M. |year=1937|title=A class of multi-symmetric polyhedra|journal=Tohoku Mathematical Journal}}</ref> [[Buckminster Fuller]] coined the term "[[geodesic dome]]" in the 1940s, although he largely kept the mathematics behind the domes a trade secret.<ref>{{cite book|author=Kenner, H. |year=1976|title=Geodesic Math and How to Use It|publisher=University of California Press}}</ref> Geodesic domes are the geometric dual of (a section of) a Goldberg polyhedron: a full geodesic dome can be thought of as a [[geodesic polyhedron]], dual to the Goldberg polyhedron. In 1962, [[Donald Caspar]] and [[Aaron Klug]] published an article on the geometry of [[viral capsid]]s that applied and expanded upon concepts from Goldberg and Fuller.<ref>{{cite journal |year=1962 |title=Physical Principles in the Construction of Regular Viruses |journal=Cold Spring Harb. Symp. Quant. Biol. |volume=27 |pages=1–24 |pmid=14019094 |author1=Caspar, D. L. D.  |author2=Klug, A. |doi=10.1101/sqb.1962.027.001.005}}</ref> [[H.S.M. Coxeter]] published an article in 1971 covering much of the same information.<ref>{{cite book|author=Coxeter, H.S.M. |chapter=Virus macromolecules and geodesic domes.|editor=Butcher, J.C. |title=A spectrum of mathematics|pages=98–107|publisher=Oxford University Press|year=1971}}</ref> Caspar and Klug were the first to publish the most general correct construction of a geodesic polyhedron, making the name "Goldberg–Coxeter construction" an instance of [[Stigler's law of eponymy]].<ref name=Brinkmann>{{Cite journal|year=2017|title=Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations|journal=Proceedings of the Royal Society A|volume=473|issue=2206|pages=20170267|arxiv=1705.02848|last1=Brinkmann|first1=G.|last2=Goetschalckx|first2=P.|last3=Schein|first3=S.|doi=10.1098/rspa.2017.0267|bibcode=2017RSPSA.47370267B|s2cid=119171258}}</ref>

The discovery of [[Buckminsterfullerene]] in 1985 motivated research into other molecules with the structure of a Goldberg polyhedron. The terms "Goldberg–Coxeter fullerene" and "Goldberg–Coxeter construction" were introduced by [[Michel Deza]] in 2000.<ref>{{cite journal|doi=10.1021/ci990066h|pmid=10850758|title=Fullerenes as Tilings of Surfaces|journal=Journal of Chemical Information and Computer Sciences|volume=40|issue=3|pages=550–8|year=2000|last1=Deza|first1=M|last2=Fowler|first2=P. W|last3=Rassat|first3=A|last4=Rogers|first4=K. M|citeseerx=10.1.1.105.5973}}</ref><ref>{{cite journal|doi=10.1002/1521-3773(20001103)39:21<3915::AID-ANIE3915>3.0.CO;2-O|title=Spherical Aromaticity in ''Ih'' Symmetrical Fullerenes: The 2(N+1)2 Rule|journal=Angewandte Chemie|volume=39|issue=21|pages=3915–3917|year=2000|last1=Hirsch|first1=Andreas|last2=Chen|first2=Zhongfang|last3=Jiao|first3=Haijun|pmid=29711706}}</ref> This is also the first time the degree 4 case was considered.

==Construction==
This section largely follows Deza et al.'s two articles.<ref name=Deza2004 /><ref name=Deza2015 />

===Master polygons===
{| class="wikitable" align="right" 
|+ Lattices
|-
! n-Regular !! 3 !! 4
|-
| [[Euclidean domain|Domain]] || Eisenstein<br><math>a+bu</math> || Gaussian<br><math>a + bi</math>
|-
| Adjoined<br>unit
| <math>\begin{align}
  u &= \frac{1}{2}\left(1 + i\sqrt{3}\right) \\
    &= e^{\frac{1}{3}\pi i} = \omega + 1
\end{align}</math>
| <math>i = \sqrt{-1}</math>
|-
| Norm <math>t(a, b)</math>
| <math>a^2 + ab + b^2</math>
| <math>a^2 + b^2</math>.
|-
| Master polygon
| <math>(0, x, xu)</math>
| <math>(0, x, x(1 + i), xi)</math>
|}
Regular lattices over the [[complex plane]] can be used to create "master polygons". In geodesic dome terminology, this is the "breakdown structure" or "principal polyhedral triangle" (PPT). The 4-regular case uses the square lattice over the [[Gaussian integers]], and the 3-regular case uses triangular lattice over the [[Eisenstein integers]]. For convenience, an alternate parameterization of the Eisenstein integers is used, based on the sixth root of unity instead of the third.{{efn|This simplifies the definition of the equivalence class, makes the class definition the same for 3- and 4-regular graphs, and corresponds with the parameterization traditionally used for geodesic domes and Goldberg polyhedra.}} The usual definition of Eisenstein integers uses the element <math display="inline">\omega = \frac{1}{2}\left(-1 + i\sqrt{3}\right) = e^{\frac{2}{3}\pi i} = u - 1</math>.  A norm, <math>t(k, \ell)</math>, is defined as the square of the absolute value of the [[complex number]]. For 3-regular graphs this norm is the [[Capsid#Triangulation number (T-number)|T-number or triangulation number]] used in virology.

The master polygon is an equilateral triangle or square laid over the lattice. The table to the right gives formulas for the vertices of the master polygons in the complex plane, and the gallery below shows the (3,2) master triangle and square. So that the polygon can be described by a single complex number, one vertex is fixed at 0. There are multiple numbers that can describe the same polygon: these are [[Integral domain#Divisibility, prime and irreducible elements|associate]]s of each other: if <math>x</math> and <math>y</math> are associates, then <math>x=u^n y</math> in the Eisensteins or <math>x=i^n y</math> in the Gaussians for some integer <math>n</math>. The set of elements that are associates of each other is an [[equivalence class]], and the element of each equivalence class that has <math>a>0</math> and <math>b\ge0</math> is the [[Normal form (mathematics)|normal form]].
<gallery widths=400 heights=400>
File:Breakdown 3 2 tri.svg|(3,2) master triangle over triangular grid
File:Subdivided square 3 2 with explanation.svg|(3,2) master square over square grid
</gallery>

Master polygons, and the operator <math>GC_{k,\ell}\left(G_0\right)</math>, can be classified as follows:
* Class I: <math>\ell = 0.</math>
* Class II: <math>k = \ell.</math>
* Class III: all other. Class III operators exist in chiral pairs: <math>GC_{k,\ell}\left(G_0\right)</math> is the chiral pair of <math>GC_{\ell,k}(G_0)</math>.
Below are tables of master triangles and squares. Class I corresponds to the first column, and Class II corresponds to the diagonal with a slightly darker background.
====Master polygons for triangles ====
{| class="wikitable collapsible collapsed" style="border:1px solid #aaa;"
! colspan="10" bgcolor="#ddd" | Master triangles through (8,8)
|-
!
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
|-
! 1
| [[File:Subdivided triangle 01 00.svg|100px]]
! [[File:Subdivided triangle 01 01.svg|100px]]
| [[File:Subdivided triangle 01 02.svg|100px]]
| [[File:Subdivided triangle 01 03.svg|100px]]
| [[File:Subdivided triangle 01 04.svg|100px]]
| [[File:Subdivided triangle 01 05.svg|100px]]
| [[File:Subdivided triangle 01 06.svg|100px]]
| [[File:Subdivided triangle 01 07.svg|100px]]
| [[File:Subdivided triangle 01 08.svg|100px]]

|-
! 2
| [[File:Subdivided triangle 02 00.svg|100px]]
| [[File:Subdivided triangle 02 01.svg|100px]]
! [[File:Subdivided triangle 02 02.svg|100px]]
| [[File:Subdivided triangle 02 03.svg|100px]]
| [[File:Subdivided triangle 02 04.svg|100px]]
| [[File:Subdivided triangle 02 05.svg|100px]]
| [[File:Subdivided triangle 02 06.svg|100px]]
| [[File:Subdivided triangle 02 07.svg|100px]]
| [[File:Subdivided triangle 02 08.svg|100px]]
|-
! 3
| [[File:Subdivided triangle 03 00.svg|100px]]
| [[File:Subdivided triangle 03 01.svg|100px]]
| [[File:Subdivided triangle 03 02.svg|100px]]
! [[File:Subdivided triangle 03 03.svg|100px]]
| [[File:Subdivided triangle 03 04.svg|100px]]
| [[File:Subdivided triangle 03 05.svg|100px]]
| [[File:Subdivided triangle 03 06.svg|100px]]
| [[File:Subdivided triangle 03 07.svg|100px]]
| [[File:Subdivided triangle 03 08.svg|100px]]
|-
! 4
| [[File:Subdivided triangle 04 00.svg|100px]]
| [[File:Subdivided triangle 04 01.svg|100px]]
| [[File:Subdivided triangle 04 02.svg|100px]]
| [[File:Subdivided triangle 04 03.svg|100px]]
! [[File:Subdivided triangle 04 04.svg|100px]]
| [[File:Subdivided triangle 04 05.svg|100px]]
| [[File:Subdivided triangle 04 06.svg|100px]]
| [[File:Subdivided triangle 04 07.svg|100px]]
| [[File:Subdivided triangle 04 08.svg|100px]]
|-
! 5
| [[File:Subdivided triangle 05 00.svg|100px]]
| [[File:Subdivided triangle 05 01.svg|100px]]
| [[File:Subdivided triangle 05 02.svg|100px]]
| [[File:Subdivided triangle 05 03.svg|100px]]
| [[File:Subdivided triangle 05 04.svg|100px]]
! [[File:Subdivided triangle 05 05.svg|100px]]
| [[File:Subdivided triangle 05 06.svg|100px]]
| [[File:Subdivided triangle 05 07.svg|100px]]
| [[File:Subdivided triangle 05 08.svg|100px]]
|-
! 6
| [[File:Subdivided triangle 06 00.svg|100px]]
| [[File:Subdivided triangle 06 01.svg|100px]]
| [[File:Subdivided triangle 06 02.svg|100px]]
| [[File:Subdivided triangle 06 03.svg|100px]]
| [[File:Subdivided triangle 06 04.svg|100px]]
| [[File:Subdivided triangle 06 05.svg|100px]]
! [[File:Subdivided triangle 06 06.svg|100px]]
| [[File:Subdivided triangle 06 07.svg|100px]]
| [[File:Subdivided triangle 06 08.svg|100px]]
|-
! 7
| [[File:Subdivided triangle 07 00.svg|100px]]
| [[File:Subdivided triangle 07 01.svg|100px]]
| [[File:Subdivided triangle 07 02.svg|100px]]
| [[File:Subdivided triangle 07 03.svg|100px]]
| [[File:Subdivided triangle 07 04.svg|100px]]
| [[File:Subdivided triangle 07 05.svg|100px]]
| [[File:Subdivided triangle 07 06.svg|100px]]
! [[File:Subdivided triangle 07 07.svg|100px]]
| [[File:Subdivided triangle 07 08.svg|100px]]
|-
! 8
| [[File:Subdivided triangle 08 00.svg|100px]]
| [[File:Subdivided triangle 08 01.svg|100px]]
| [[File:Subdivided triangle 08 02.svg|100px]]
| [[File:Subdivided triangle 08 03.svg|100px]]
| [[File:Subdivided triangle 08 04.svg|100px]]
| [[File:Subdivided triangle 08 05.svg|100px]]
| [[File:Subdivided triangle 08 06.svg|100px]]
| [[File:Subdivided triangle 08 07.svg|100px]]
! [[File:Subdivided triangle 08 08.svg|100px]]
|}
====Master polygons for squares ====
{| class="wikitable collapsible collapsed" style="border: 1px solid #aaa;"
! colspan="10" bgcolor="#ddd" |Master squares through (8,8)
|-
!
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
|-
! 1
| [[File:Subdivided square 01 00.svg|100px]]
! [[File:Subdivided square 01 01.svg|100px]]
| [[File:Subdivided square 01 02.svg|100px]]
| [[File:Subdivided square 01 03.svg|100px]]
| [[File:Subdivided square 01 04.svg|100px]]
| [[File:Subdivided square 01 05.svg|100px]]
| [[File:Subdivided square 01 06.svg|100px]]
| [[File:Subdivided square 01 07.svg|100px]]
| [[File:Subdivided square 01 08.svg|100px]]

|-
! 2
| [[File:Subdivided square 02 00.svg|100px]]
| [[File:Subdivided square 02 01.svg|100px]]
! [[File:Subdivided square 02 02.svg|100px]]
| [[File:Subdivided square 02 03.svg|100px]]
| [[File:Subdivided square 02 04.svg|100px]]
| [[File:Subdivided square 02 05.svg|100px]]
| [[File:Subdivided square 02 06.svg|100px]]
| [[File:Subdivided square 02 07.svg|100px]]
| [[File:Subdivided square 02 08.svg|100px]]
|-
! 3
| [[File:Subdivided square 03 00.svg|100px]]
| [[File:Subdivided square 03 01.svg|100px]]
| [[File:Subdivided square 03 02.svg|100px]]
! [[File:Subdivided square 03 03.svg|100px]]
| [[File:Subdivided square 03 04.svg|100px]]
| [[File:Subdivided square 03 05.svg|100px]]
| [[File:Subdivided square 03 06.svg|100px]]
| [[File:Subdivided square 03 07.svg|100px]]
| [[File:Subdivided square 03 08.svg|100px]]
|-
! 4
| [[File:Subdivided square 04 00.svg|100px]]
| [[File:Subdivided square 04 01.svg|100px]]
| [[File:Subdivided square 04 02.svg|100px]]
| [[File:Subdivided square 04 03.svg|100px]]
! [[File:Subdivided square 04 04.svg|100px]]
| [[File:Subdivided square 04 05.svg|100px]]
| [[File:Subdivided square 04 06.svg|100px]]
| [[File:Subdivided square 04 07.svg|100px]]
| [[File:Subdivided square 04 08.svg|100px]]
|-
! 5
| [[File:Subdivided square 05 00.svg|100px]]
| [[File:Subdivided square 05 01.svg|100px]]
| [[File:Subdivided square 05 02.svg|100px]]
| [[File:Subdivided square 05 03.svg|100px]]
| [[File:Subdivided square 05 04.svg|100px]]
! [[File:Subdivided square 05 05.svg|100px]]
| [[File:Subdivided square 05 06.svg|100px]]
| [[File:Subdivided square 05 07.svg|100px]]
| [[File:Subdivided square 05 08.svg|100px]]
|-
! 6
| [[File:Subdivided square 06 00.svg|100px]]
| [[File:Subdivided square 06 01.svg|100px]]
| [[File:Subdivided square 06 02.svg|100px]]
| [[File:Subdivided square 06 03.svg|100px]]
| [[File:Subdivided square 06 04.svg|100px]]
| [[File:Subdivided square 06 05.svg|100px]]
! [[File:Subdivided square 06 06.svg|100px]]
| [[File:Subdivided square 06 07.svg|100px]]
| [[File:Subdivided square 06 08.svg|100px]]
|-
! 7
| [[File:Subdivided square 07 00.svg|100px]]
| [[File:Subdivided square 07 01.svg|100px]]
| [[File:Subdivided square 07 02.svg|100px]]
| [[File:Subdivided square 07 03.svg|100px]]
| [[File:Subdivided square 07 04.svg|100px]]
| [[File:Subdivided square 07 05.svg|100px]]
| [[File:Subdivided square 07 06.svg|100px]]
! [[File:Subdivided square 07 07.svg|100px]]
| [[File:Subdivided square 07 08.svg|100px]]
|-
! 8
| [[File:Subdivided square 08 00.svg|100px]]
| [[File:Subdivided square 08 01.svg|100px]]
| [[File:Subdivided square 08 02.svg|100px]]
| [[File:Subdivided square 08 03.svg|100px]]
| [[File:Subdivided square 08 04.svg|100px]]
| [[File:Subdivided square 08 05.svg|100px]]
| [[File:Subdivided square 08 06.svg|100px]]
| [[File:Subdivided square 08 07.svg|100px]]
! [[File:Subdivided square 08 08.svg|100px]]
|}

Composition of Goldberg–Coxeter operations corresponds to multiplication of complex numbers. If and only if <math>GC_{a,b}\left(GC_{c,d}\left(G_0\right)\right) = GC_{e,f}\left(G_0\right)</math> (i.e. the series of operations on the left produces a graph isomorphic to the one on the right), then for a 3-regular graph <math>(a + bu)(c + du)</math> is in the equivalence class of <math>(e + fu)</math>, and for a 4-regular graph <math>(a + bi)(c + di)</math> is in the equivalence class of <math>(e + fi)</math>. There are some useful consequences of this:
* The application of repeated Goldberg–Coxeter operations is [[commutative]] and [[associative]].
* Complex conjugation of the element <math>a + bi</math> or <math>a + bu</math> corresponds to reflection of the constructed graph.
* Since the Gaussian integers and Euclidean integers are both [[Euclidean domain]]s, elements of those domains can be uniquely factored into prime elements. Therefore, it also makes sense to decompose a Goldberg–Coxeter operator into a sequence of "prime" Goldberg–Coxeter operators, and this sequence is unique (up to rearrangement).

===Performing the GC construction===
The steps of performing the GC construction <math>GC_{k,\ell}(G_0)</math>are as follows:
# Determine the master polygon, based on <math>k</math>, <math>\ell</math>, and <math>G_0</math>
# If operating on a 3- or 4-regular graph (instead of a graph with triangular/quadrilateral faces), take its [[dual graph]]. This dual graph will have triangular or quadrilateral faces.
# Replace the faces of the triangulated/quadrangulated graph with the master polygon. Be aware that planar graphs have an "external" face that must be replaced as well. In the below example, this is done by attaching it to one side of the graph and connecting other sides as necessary. This temporarily introduces overlapping edges into the graph, but the resulting graph is planar. The vertices can be rearranged so that there are no overlapping edges.
# If the original graph was a 3- or 4-regular graph, take the dual of the result of step 3. Otherwise, the result of step 3 is the GC construction.

Below is an example, where <math>GC_{1,1}(G_0)</math> is constructed on the [[n-skeleton|skeleton]] of a [[cube]]. In the last two graphs, blue lines are edges of <math>G_0</math>, while black lines are edges of <math>GC_{1,1}(G_0)</math>. (Dotted lines are normal graph edges, just drawn differently to make overlapping graph edges more visible.) Red vertices are in <math>G_0</math> and remain in <math>GC_{1,1}(G_0)</math>, while blue vertices are newly created by the construction and are only in <math>GC_{1,1}(G_0)</math>.
<gallery widths=200 heights=200>File:Subdivided square 01 01.svg|Master square (1,1)
File:Hexahedron.svg|Starting polyhedron (Cube)
File:Cube skeleton.svg|<math>G_0</math>, the skeleton of the cube
File:GC_1,1_Cube_faces.svg|Intermediate step of constructing <math>GC_{1,1}(G_0)</math>. 
File:Rhombic_dodecahedron_skeleton.svg|The result <math>GC_{1,1}(G_0)</math>, after rearrangement
File:Rhombicdodecahedron.jpg|Embedding of the result ([[rhombic dodecahedron]])</gallery>

==Extensions==
The Goldberg–Coxeter construction can be easily extended to some non-planar graphs, such as [[toroidal graph]]s.<ref>{{cite journal|doi=10.1007/s10910-016-0706-8|title=Lego-like spheres and tori|journal=Journal of Mathematical Chemistry|volume=55|issue=3|pages=752|year=2016|last1=Deza|first1=Michel-Marie|last2=Sikirić|first2=Mathieu Dutour|s2cid=125087322}}</ref> Class III operators, because of their chirality, require a graph that can be [[graph embedding|embedded]] on an [[orientable surface]], but class I and II operators can be used on non-orientable graphs.

==See also==
{{commons}}
{{div col}}
*[[Geodesic grid]]
*[[Quadrilateralized spherical cube]]
*[[Loop subdivision surface]]
*[[Catmull–Clark subdivision surface]]
*[[Conway polyhedron notation]]
{{div col end}}

==Footnotes==
{{notelist}}

==References==
{{reflist}}

{{DEFAULTSORT:Goldberg-Coxeter construction}}
[[Category:Graph operations]]
[[Category:Polyhedra]]