In mathematics, and more particularly in
polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.
Research in polyhedral c ...
, Eberhard's theorem partially characterizes the
multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...
s of
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...
s that can form the faces of
simple
Simple or SIMPLE may refer to:
* Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by John ...
convex polyhedra
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wor ...
. It states that, for given numbers of triangles, quadrilaterals, pentagons, heptagons, and other polygons other than hexagons,
there exists a convex polyhedron with those given numbers of faces of each type (and an unspecified number of hexagonal faces) if and only if those numbers of polygons obey a linear equation derived from
Euler's polyhedral formula
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
.
The theorem is named after
Victor Eberhard, a blind German mathematician, who published it in 1888 in his
habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including ...
thesis and in expanded form in an 1891 book on polyhedra.
Definitions and statement
For an arbitrary convex polyhedron, one can define numbers
,
,
, etc., where
counts the faces of the polyhedron that have exactly
sides. A three-dimensional convex polyhedron is defined to be simple when every
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
*Vertex (computer graphics), a data structure that describes the position ...
of the polyhedron is incident to exactly three edges. In a simple polygon, every vertex is incident to three angles of faces, and every edge is incident to two sides of faces. Since the numbers of angles and sides of the faces are given, one can calculate the three numbers
(the total number of vertices),
(the total number of edges), and
(the total number of faces), by summing over all faces and multiplying by an appropriate factor:
:
:
and
:
Plugging these values into
Euler's polyhedral formula
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
and clearing denominators leads to the equation
:
which must be satisfied by the face counts of every simple polyhedron. However,
this equation is not affected by the value of
(as its multiplier
is zero), and, for some choices of the other face counts, changing
can change whether or not a polyhedron with those face counts exists. That is, obeying this equation on the face counts is a necessary condition for the existence of a polyhedron, but not a sufficient condition, and a complete characterization of which face counts are realizable would need to take into account the value of
.
Eberhard's theorem implies that the equation above is the only necessary condition that does not depend on
. It states that, if an assignment of numbers to
(omitting
) obeys the equation
:
then there exists a value of
and a simple convex polyhedron with exactly
-sided faces for all
.
Examples
There are three simple
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s, the
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
,
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the on ...
, and
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentag ...
. The tetrahedron has
, the cube has
, and the dodecahedron has
, with all other values of
being zero. These three assignments of numbers to
all obey the equation that Eberhard's theorem requires them to obey. The existence of these polyhedra shows that, for these three assignments of numbers to
, there exists a polyhedron with
. The case of the dodecahedron, with
and all others except
zero, describes more generally the
fullerene
A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...
s. There is no fullerene with
but these graphs are realizable for any other value of
; see for instance, the
26-fullerene graph
In the mathematical field of graph theory, the 26-fullerene graph is a polyhedral graph with ''V'' = 26 vertices and ''E'' = 39 edges. Its planar embedding has three hexagonal faces (including the one shown as the external face ...
, with
.

There is no simple convex polyhedron with three triangle faces, three pentagon faces, and no other faces. That is, it is impossible to have a simple convex polyhedron with
, and
for
. However, Eberhard's theorem states that it should be possible to form a simple polyhedron by adding some number of hexagons, and in this case one hexagon suffices: bisecting a cube on a regular hexagon passing through six of its faces produces two copies of a simple
roofless polyhedron with three triangle faces, three pentagon faces, and one hexagon face. That is, setting
suffices in this case to produce a realizable combination of face counts.
[For both the nonexistence of a polyhedron with three triangles, three pentagons, and no hexagons, and the existence with one hexagon, see , third row of Table 13.3.1, page 268.]
Related results
An analogous result to Eberhard's theorem holds for the existence of polyhedra in which all vertices are incident to exactly four edges. In this case the equation derived from Euler's formula is not affected by the number
of quadrilaterals, and for every assignment to the numbers of faces of other types that obeys this equation it is possible to choose a number of quadrilaterals that allows a 4-regular polyhedron to be realized.
A strengthened version of Eberhard's theorem states that, under the same conditions as the original theorem, there exists a number
such that all choices of
that are greater than equal to
and have the same parity as
are realizable by simple convex polyhedra.
A theorem of
David W. Barnette
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". w ...
provides a
lower bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an ele ...
on the number of hexagons that are needed, whenever the number of faces of order seven or higher is at least three. It states that, in these cases,
:
For polygons with few pentagons and many high-order faces, this inequality can force the number of hexagons to be arbitrarily large. More strongly, it can be used to find assignments to the numbers of faces for which the required number of hexagons cannot be bounded by any function of the maximum number of sides of a face.
Analogues of Eberhard's theorem have also been studied for other systems of faces and face counts than simple convex polyhedra, for instance for
toroidal graph
In the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross.
Examples
Any graph that can be embedded in a plane ...
s and for
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ge ...
s.
See also
*
Erdős–Gallai theorem
The Erdős–Gallai theorem is a result in graph theory, a branch of combinatorial mathematics. It provides one of two known approaches to solving the graph realization problem, i.e. it gives a necessary and sufficient condition for a finite se ...
*
Grinberg's theorem
In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. If a graph does not meet this condition, it is not Hamiltonian. The result has been widely use ...
References
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[{{citation, url=https://www.catalogus-professorum-halensis.de/eberhardviktor.html, title=Viktor Eberhard, work=Catalogus Professorum Halensis, publisher=Martin Luther University of Halle-Wittenberg, accessdate=2020-09-02, language=German]
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Polyhedral combinatorics