In geometry, the Hesse configuration, introduced by

Colin Maclaurin Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for be ...

and studied by , is a

configuration Configuration or configurations may refer to: Computing * Computer configuration or system configuration * Configuration file, a software file used to configure the initial settings for a computer program * Configurator, also known as choice board ...

of 9 points and 12 lines with three points per line and four lines through each point. It can be realized 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 ...

as the set of

inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case o ...

s of an

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

, but it has no realization in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

Description

The Hesse configuration has the same incidence relations as the lines and points of the

affine plane In geometry, an affine plane is a two-dimensional affine space. Examples Typical examples of affine planes are *Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance. In other words, an affine pl ...

over the field of 3 elements. That is, the points of the Hesse configuration may be identified with

ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...

s of numbers modulo 3, and the lines of the configuration may correspondingly be identified with the triples of points satisfying a linear equation . Alternatively, the points of the configuration may be identified by the squares of a

tic-tac-toe Tic-tac-toe (American English), noughts and crosses ( Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with ''X'' or ''O''. ...

board, and the lines may be identified with the lines and broken diagonals of the board. Each point belongs to four lines: in the tic tac toe interpretation of the configuration, one line is horizontal, one vertical, and two are diagonals or broken diagonals. Each line contains three points. In the language of configurations the Hesse configuration has the notation 9₄12₃, meaning that there are 9 points, 4 lines per point, 12 lines, and 3 points per line. The Hesse configuration has 216 symmetries (its

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

has order 216). The group of its symmetries is known as the

Hessian group In mathematics, the Hessian group is a finite group of order 216, introduced by who named it for Otto Hesse. It may be represented as the group of affine transformations with determinant 1 of the affine plane over the field of 3 elements.Hessian gr ...

Related configurations

Removing any one point and its four incident lines from the Hesse configuration produces another configuration of type 8₃8₃, the

Möbius–Kantor configuration In geometry, the Möbius–Kantor configuration is a configuration consisting of eight points and eight lines, with three points on each line and three lines through each point. It is not possible to draw points and lines having this pattern of ...

.. In the Hesse configuration, the 12 lines may be grouped into four triples of parallel (non-intersecting) lines. Removing from the Hesse configuration the three lines belonging to a single triple produces a configuration of type 9₃9₃, the

Pappus configuration In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point. History and construction This configuration is named after Pappus of A ...

.. The Hesse configuration may in turn be augmented by adding four points, one for each triple of non-intersecting lines, and one line containing the four new points, to form a configuration of type 13₄13₄, the set of points and lines of the

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...

over the three-element field.

Realizability

The Hesse configuration can be realized in the

as the 9

s of an

and the 12 lines through triples of inflection points. If a given set of nine points in the complex plane is the set of inflections of an elliptic curve ''C'', it is also the set of inflections of every curve in a

pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion (mechanical), abra ...

of curves generated by ''C'' and by the Hessian curve of ''C'', the

Hesse pencil In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation :\lambda(x^3+y^3+z^3) + \mu xyz =0. Each curve ...

. The

Hessian polyhedron In geometry, the Hessian polyhedron is a regular complex polyhedron 333, , in \mathbb^3. It has 27 vertices, 72 3 edges, and 27 33 faces. It is self-dual. Coxeter named it after Ludwig Otto Hesse for sharing the '' Hessian configuration'' \l ...

is a representation of the Hesse configuration in the complex plane. The Hesse configuration shares with the Möbius–Kantor configuration the property of having a complex realization but not being realizable by points and straight lines in the

. In the Hesse configuration, every two points are connected by a line of the configuration (the defining property of the

Sylvester–Gallai configuration In geometry, a Sylvester–Gallai configuration consists of a finite subset of the points of a projective space with the property that the line through any two of the points in the subset also passes through at least one other point of the subset. ...

s) and therefore every line through two of its points contains a third point. But in the Euclidean plane, every finite set of points is either collinear, or includes a pair of points whose line does not contain any other points of the set; this is the

Sylvester–Gallai theorem The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester ...

. Because the Hesse configuration disobeys the Sylvester–Gallai theorem, it has no Euclidean realization. This example also shows that the Sylvester–Gallai theorem cannot be generalized to the complex projective plane. However, in complex spaces, the Hesse configuration and all Sylvester–Gallai configurations must lie within a two-dimensional flat subspace..

References

{{Incidence structures Configurations (geometry) Elliptic curves