In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization has at least one irrational number as one of its coordinates. It can be constructed from some of the diagonals and symmetry lines of a regular pentagon, and their crossing points. All of the realizations of the Perles configuration in the projective plane are equivalent to each other under projective transformations.

The Perles configuration is the smallest configuration of points and lines that cannot be realized with rational coordinates. It is named after Micha Perles, who used it to construct an eight-dimensional convex polytope that cannot be given rational number coordinates and that has the fewest vertices (twelve) of any known irrational polytope. It has additional applications as a counterexample in the theory of visibility graphs and in graph drawing.

One way of constructing the Perles configuration is to start with a regular pentagon and its five diagonals. These diagonals form the sides of a smaller inner pentagon nested inside the outer pentagon. Each vertex of the outer pentagon is situated opposite from a vertex of the inner pentagon. The nine points of the configuration consist of four out of the five vertices of each pentagon and the shared center of the two pentagons. Two opposite vertices are omitted, one from each pentagon.

The nine lines of the configuration consist of the five lines that are diagonals of the outer pentagon and sides of the inner pentagon, and the four lines that pass through the center and through opposite pairs of vertices from the two pentagons.

A realization of the Perles configuration is defined to consist of any nine points and nine lines with the same intersection pattern. That means that a point and line intersect each other in the realization, if and only if they intersect in the configuration constructed from the regular pentagon. Every realization of this configuration in the Euclidean plane or, more generally, in the real projective plane is equivalent, under a projective transformation, to a realization constructed from a regular pentagon. One proof of this fact assigns arbitrary projective coordinates to the two outer points on the four-point line, the center point of the configuration, and one of the remaining two outer points. These points determine the position of one middle point on the four-point line. One then defines a parameter specifying, in terms of these coordinates, the position of the fourth point on this line. This parameter can be calculated in terms of the projective coordinates of the remaining points. The requirement that these points be collinear translates into a constraint on the parameter: it must obey the quadratic equation satisfied by the golden ratio. The two solutions to this equation both produce configurations of the same type, with rearranged points. Therefore, when this parameterization is applied both to the regular pentagon realization and any other realization, the two realizations end up with the same projective coordinates, meaning that they are projectively equivalent.

The cross-ratio is a number defined from any four collinear points. In the realization derived from the regular pentagon, the four collinear points have cross-ratio ${\displaystyle 1+\varphi }$, where ${\displaystyle \varphi }$ is the golden ratio, an irrational number. The cross-ratio does not change under projective transformations, so it remains irrational for every realization. However, every four collinear points with rational coordinates have a rational cross ratio, so the Perles configuration cannot be realized by rational points.

Answering a question of Branko Grünbaum, József Solymosi has proved that the Perles configuration is the smallest possible irrational configuration of points and lines: every configuration of eight or fewer points in the Euclidean plane, and lines through subsets of these points, has a combinatorially equivalent configuration with points that have rational numbers as their coordinates.

Perles used his configuration to construct the Perles polytope, an eight-dimensional convex polytope with twelve vertices that can similarly be realized with real coordinates but not with rational coordinates. The points of the configuration, three of them repeated and with positive, negative, or zero signs assigned to each point, form the affine Gale diagram of this polytope, from which the polytope itself can be recovered.