In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a dessin d'enfant is a type of
graph embedding
In topological graph theory, an embedding (also spelled imbedding) of a Graph (discrete mathematics), graph G on a surface (mathematics), surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with graph the ...
used to study
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s and to provide combinatorial
invariants for the action of the
absolute Galois group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' tha ...
of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s. The name of these embeddings is
French for a "child's drawing"; its plural is either ''dessins d'enfant'', "child's drawings", or ''dessins d'enfants'', "children's drawings".
A dessin d'enfant is a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
, with its
vertices colored alternately black and white,
embedded in an
oriented surface
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
that, in many cases, is simply a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
. For the coloring to exist, the graph must be
bipartite. The faces of the embedding are required be topological disks. The surface and the embedding may be described combinatorially using a
rotation system In combinatorial mathematics, rotation systems (also called combinatorial embeddings or combinatorial maps) encode embeddings of graphs onto orientable surfaces by describing the circular ordering of a graph's edges around each vertex.
A more for ...
, a
cyclic order
In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. In ...
of the edges surrounding each vertex of the graph that describes the order in which the edges would be crossed by a path that travels clockwise on the surface in a small loop around the vertex.
Any dessin can provide the surface it is embedded in with a structure as a Riemann surface. It is natural to ask which Riemann surfaces arise in this way. The answer is provided by
Belyi's theorem
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified a ...
, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s over the field of
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s. The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins.
For a more detailed treatment of this subject, see or .
History
19th century
Early proto-forms of dessins d'enfants appeared as early as 1856 in the
icosian calculus
The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856.
In modern terms, he gave a group presentation of the icosahedral rotation group by generators and relations.
Ham ...
of
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
; in modern terms, these are
Hamiltonian path
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...
s on the icosahedral graph.
Recognizable modern dessins d'enfants and
Belyi function
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at t ...
s were used by . Klein called these diagrams ''Linienzüge'' (German, plural of ''Linienzug'' "line-track", also used as a term for
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 toge ...
); he used a white circle for the preimage of 0 and a '+' for the preimage of 1, rather than a black circle for 0 and white circle for 1 as in modern notation. He used these diagrams to construct an 11-fold cover of the Riemann sphere by itself, with
monodromy group
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
, following earlier constructions of a 7-fold cover with monodromy
connected to the
Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms ...
in . These were all related to his investigations of the geometry of the quintic equation and the group collected in his famous 1884/88 ''Lectures on the Icosahedron''. The three surfaces constructed in this way from these three groups were much later shown to be closely related through the phenomenon of
trinity
The Christian doctrine of the Trinity (, from 'threefold') is the central dogma concerning the nature of God in most Christian churches, which defines one God existing in three coequal, coeternal, consubstantial divine persons: God the F ...
.
20th century
Dessins d'enfant in their modern form were then rediscovered over a century later and named by
Alexander Grothendieck in 1984 in his ''
Esquisse d'un Programme
"Esquisse d'un Programme" (Sketch of a Programme) is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the sequence of logically linked ideas in his impor ...
''. quotes Grothendieck regarding his discovery of the Galois action on dessins d'enfants:
Part of the theory had already been developed independently by some time before Grothendieck. They outline the correspondence between maps on topological surfaces, maps on Riemann surfaces, and groups with certain distinguished generators, but do not consider the Galois action. Their notion of a map corresponds to a particular instance of a dessin d'enfant. Later work by extends the treatment to surfaces with a boundary.
Riemann surfaces and Belyi pairs
The
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, together with a special point designated as
, form a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
known as the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
. Any
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
, and more generally any
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
where
and
are polynomials, transforms the Riemann sphere by mapping it to itself.
Consider, for example, the
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
At most points of the Riemann sphere, this transformation is a
local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If f : X \to Y is a local homeomorphism, X is said to be an é ...
: it maps a small disk centered at any point in a one-to-one way into another disk. However, at certain
critical points, the mapping is more complicated, and maps a disk centered at the point in a
-to-one way onto its image. The number
is known as the ''degree'' of the critical point and the transformed image of a critical point is known as a
critical value
Critical value may refer to:
*In differential topology, a critical value of a differentiable function between differentiable manifolds is the image (value of) ƒ(''x'') in ''N'' of a critical point ''x'' in ''M''.
*In statistical hypothesis ...
.
The example given above,
, has the following critical points and critical values. (Some points of the Riemann sphere that, while not themselves critical, map to one of the critical values, are also included; these are indicated by having degree one.)
One may form a dessin d'enfant from
by placing black points at the
preimage
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
s of 0 (that is, at 1 and 9), white points at the preimages of 1 (that is, at
), and arcs at the preimages of the
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
, 1 This line segment has four preimages, two along the line segment from 1 to 9 and two forming a
simple closed curve
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
that loops from 1 to itself, surrounding 0; the resulting dessin is shown in the figure.
In the other direction, from this dessin, described as a combinatorial object without specifying the locations of the critical points, one may form a
compact Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
, and a map from that surface to the Riemann sphere, equivalent to the map from which the dessin was originally constructed. To do so, place a point labeled
within each region of the dessin (shown as the red points in the second figure), and
triangulate
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Applications
In surveying
Specifically in surveying, triangulation involves only angle me ...
each region by connecting this point to the black and white points forming the boundary of the region, connecting multiple times to the same black or white point if it appears multiple times on the boundary of the region. Each triangle in the triangulation has three vertices labeled 0 (for the black points), 1 (for the white points), or
. For each triangle, substitute a
half-plane, either the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...
for a triangle that has 0, 1, and
in counterclockwise order or the lower half-plane for a triangle that has them in clockwise order, and for every adjacent pair of triangles glue the corresponding half-planes together along the portion of their boundaries indicated by the vertex labels. The resulting Riemann surface can be mapped to the Riemann sphere by using the identity map within each half-plane. Thus, the dessin d'enfant formed from
is sufficient to describe
itself up to
biholomorphism
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.
Formal definit ...
. However, this construction identifies the Riemann surface only as a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
with complex structure; it does not construct an embedding of this manifold as an
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
in the
complex projective plane
In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates
:(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...
, although such an embedding always exists.
The same construction applies more generally when
is any Riemann surface and
is a
Belyi function
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at t ...
; that is, a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
from
to the Riemann sphere having only 0, 1, and
as critical values. A pair
of this type is known as a ''Belyi pair''. From any Belyi pair
one can form a dessin d'enfant, drawn on the that has its black points at the preimages
of 0, its white points at the preimages
of 1, and its edges placed along the preimages
of the line segment