View source for Configuration

A finite set of points, lines and planes related by mutual incidences (cf. [[Incidence|Incidence]]) between them. Configurations can be both located in a place and in space.

A plane configuration is a finite system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c0246701.png" /> points and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c0246702.png" /> (straight) lines in the plane arranged in such a way that every point is incident to a fixed number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c0246703.png" /> of lines, while every line is incident to a fixed number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c0246704.png" /> of points. A minimal system of points of a given configuration, from which the whole configuration can be obtained from incidence of pairs of points with lines and intersections of pairs of lines, is called a system of generators of the given configuration. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c0246705.png" /> are connected by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c0246706.png" />, and the configuration is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c0246707.png" />. A configuration containing the same number of points and lines is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c0246708.png" />.

Examples of plane configurations. 1) A single point and a single line incident to each other form the configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c0246709.png" />. 2) Three points not lying on a line and three lines incident with each pair forms the configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467010.png" />, this figure is a triangle or trigon in the plane. 3) The complete quadrilateral consisting of four lines and the six points of their pairwise intersections forms the configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467011.png" />. However, here not all the lines joining pairs of points of the configuration are lines of the configuration; the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467012.png" /> and the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467013.png" /> do not belong to it (Fig. a).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670a.gif" />

Figure: c024670a

An automorphism of a configuration is a mapping of this configuration onto itself under which points are taken to points, lines to lines and, furthermore, no incidence is lost and no new incidences are generated. A configuration is called regular if its automorphism group is transitive on the points and on the lines of the configuration.

For a given configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467014.png" /> the configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467015.png" /> is called its dual. The configurations of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467016.png" />, which are isomorphic to their dual, are called self-dual.

A configuration is called projective if the incidence of its elements is preserved under projective transformations. For example, the incidence of elements of some configurations in the projective plane satisfies the axioms for the latter and therefore the incidence of points and lines is preserved under projective transformations of this plane, so that the configuration is projective. All the incidences and elements of such a plane configuration can be drawn in a diagram by means of just a ruler. A plane configuration always has a dual, as a consequence of the duality principle.

A configuration can be defined as a finite partial plane. The possibility of the existence of a certain configuration is determined from geometrical and combinatorial relations between the collection of points and lines and the collection of incidences between them. A configuration can also be defined by means of an abstract scheme, for example, in the table (Fig. b) the incidences are shown (denoted by crosses) of four points and four planes, the vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467017.png" /> and the faces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467018.png" /> of a tetrahedron.''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467019.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467020.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467021.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467022.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467023.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467024.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467025.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467026.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467027.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467028.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467029.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467030.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467031.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467032.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467033.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467034.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467035.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467036.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467037.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467038.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467039.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467040.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467041.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467042.png" /></td> </tr> </tbody> </table>

</td></tr> </table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670b.gif" />

Figure: c024670b

Once a certain configuration has been abstractly defined, the question of its realization arises, that is, the possibility of constructing all incidences in terms of a given systems of generators. The realizability of a configuration as a finite partial plane entails the possibility of isomorphically mapping it onto a certain subplane of some plane.

A configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467044.png" />, can be realized in the form of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467045.png" />-gon such that its vertices and sides are pairwise incident. The abstract scheme of a configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467046.png" /> can be constructed, for example, by a table as in Fig. b. Plane configurations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467047.png" /> are only possible for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467048.png" />, since through every point of the configuration three lines must pass and each of them must contain two further points of the configuration (the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467049.png" /> must satisfy the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467050.png" />). In general there will be several non-isomorphic configurations of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467051.png" />, the number of which depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467052.png" />. The (abstractly speaking unique) configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467053.png" /> is represented by the scheme of Fig. c, where the number to the right of the bracket denotes the point of intersection of the lines passing through the pairs of points written to the left of the bracket, and the numbers beyond the vertical line denote those points that must be collinear for the validity of all the incidences indicated on the left.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467054.png" /></td> </tr></table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670c.gif" />

Figure: c024670c

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670d.gif" />

Figure: c024670d

In the real projective plane all the incidences are realized by the complete quadrangle (Fig. d), however, the last incidence (collinearity of three points) does not hold here. There are three non-isomorphic configurations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467055.png" />, one of which, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467056.png" />, is called the Brianchon–Pascal configuration (Fig. e): Each line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467057.png" /> is incident with three distinct points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467058.png" /> and each point with three distinct lines.''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467059.png" /></td> <td colname="2" style="background-color:white;" colspan="1">(1)</td> <td colname="3" style="background-color:white;" colspan="1">(2)</td> <td colname="4" style="background-color:white;" colspan="1">(3)</td> <td colname="5" style="background-color:white;" colspan="1">(4)</td> <td colname="6" style="background-color:white;" colspan="1">(5)</td> <td colname="7" style="background-color:white;" colspan="1">(6)</td> <td colname="8" style="background-color:white;" colspan="1">(7)</td> <td colname="9" style="background-color:white;" colspan="1">(8)</td> <td colname="10" style="background-color:white;" colspan="1">(9)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467060.png" /></td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1">1</td> <td colname="5" style="background-color:white;" colspan="1">2</td> <td colname="6" style="background-color:white;" colspan="1">2</td> <td colname="7" style="background-color:white;" colspan="1">3</td> <td colname="8" style="background-color:white;" colspan="1">3</td> <td colname="9" style="background-color:white;" colspan="1">4</td> <td colname="10" style="background-color:white;" colspan="1">5</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1">2</td> <td colname="3" style="background-color:white;" colspan="1">4</td> <td colname="4" style="background-color:white;" colspan="1">6</td> <td colname="5" style="background-color:white;" colspan="1">4</td> <td colname="6" style="background-color:white;" colspan="1">7</td> <td colname="7" style="background-color:white;" colspan="1">6</td> <td colname="8" style="background-color:white;" colspan="1">5</td> <td colname="9" style="background-color:white;" colspan="1">6</td> <td colname="10" style="background-color:white;" colspan="1">7</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1">3</td> <td colname="3" style="background-color:white;" colspan="1">5</td> <td colname="4" style="background-color:white;" colspan="1">7</td> <td colname="5" style="background-color:white;" colspan="1">8</td> <td colname="6" style="background-color:white;" colspan="1">9</td> <td colname="7" style="background-color:white;" colspan="1">8</td> <td colname="8" style="background-color:white;" colspan="1">9</td> <td colname="9" style="background-color:white;" colspan="1">9</td> <td colname="10" style="background-color:white;" colspan="1">8</td> </tr> </tbody> </table>

</td></tr> </table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670e.gif" />

Figure: c024670e

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670f.gif" />

Figure: c024670f

The configuration can be realized in the projective plane (Fig. f); it is projective, regular and self-dual (see [[Brianchon theorem|Brianchon theorem]]; [[Pascal theorem|Pascal theorem]]). The two other realizations of a configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467063.png" />, (Fig. g) are essentially different from the Brianchon–Pascal configuration.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670g.gif" />

Figure: c024670g

For example, the configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467064.png" /> is not regular, while the construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467065.png" /> requires an auxiliary second-order curve. The configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467066.png" /> already has ten distinct non-isomorphic realizations, the most important of which is the Desarguesian configuration (Fig. h).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670h.gif" />

Figure: c024670h

It can be realized in the real projective plane, and is projective, regular, and self-dual. The other nine realizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467067.png" /> do not express any general geometrical theorem, and only eight of them can be realized in the real projective plane, but for their construction one requires a special arrangement of the system of generating points (in particular, a configuration of such a type can be realized in the form of regular polygons (Fig. i)).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670i.gif" />

Figure: c024670i

In the one given in the diagram, the sequence of vertices is also obtained as a simultaneously inscribed and circumscribed decagon. The configurations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467069.png" /> also admit of a geometric construction by means of mutually inscribed and circumscribed polygons; thus, the configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467070.png" /> is represented in the form of an enneagon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467071.png" /> (Fig. f), the Desarguesian configuration in the form of a decagon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467072.png" /> (Fig. h). Such representations of these configurations are unique up to automorphisms. In general, the construction of mutually inscribed and circumscribed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467073.png" />-gons leads to a configuration of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467074.png" />. There also exists representations of configurations of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467075.png" /> in the form of certain mutually inscribed and circumscribed polygons. For example, the Desarguesian configuration admits of a unique way (up to an automorphism) of representing a pair of mutually inscribed and circumscribed pentagons <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467077.png" /> (Fig. h). As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467078.png" /> increases, the number of non-isomorphic configurations of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467079.png" /> rapidly increases.

A spatial configuration is a finite system of points and planes such that each point is incident to a fixed number of planes and each plane is incident to a fixed number of points. Along with configurations consisting of points and planes in space, one can also consider configurations consisting of points and lines. Thus, the Desarguesian configuration considered above, consisting of points and lines, is also a spatial configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467080.png" /> (Fig. h) if the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467081.png" />-gons lie in different planes. At the same time it can be regarded as the configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467082.png" /> consisting of points and planes. The six vertices of corresponding triangles, the centre of perspectivity and the three points on the axis of perspectivity at which the corresponding sides of the trigons meet, given in total ten points. The three points formed by the respective sides of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467083.png" />-gons, and the two planes of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467084.png" />-gons themselves give in total five planes. In each plane there are six points of the configuration and each point is incident to three distinct planes.

A configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467085.png" /> is a trivial spatial configuration of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467086.png" />, its scheme is given in Fig. b; it is depicted by a tetrahedron. A configuration of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467087.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467088.png" /> is not possible, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467089.png" /> there are three distinct schemes of the configuration, one of which is the so-called Möbius configuration consisting of two mutually inscribed and circumscribed tetrahedra. Each of the eight points, the vertices of the tetrahedra, is incident to four planes of the faces of the tetrahedra, and each of the four planes is incident of four vertices. For configurations of higher order, the number of distinct schemes rapidly increases; for example, a configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467090.png" /> already has 26 distinct geometric realizations.

The most remarkable of the spatial configurations of higher order are those of Reye and Schläfli. The so-called Reye configuration consists of the points and the planes of a configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467091.png" />. In real projective space this can be constructed, for example, from the vertices of a cube, its centre and the three points (at infinity) at which the parallel edges of the cube meet, while the planes of this configurations are the six faces of the cube and the six diagonal planes passing through pairs of opposite edges (Fig. j).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670j.gif" />

Figure: c024670j

The Reye configuration is projective, regular and self-dual, and a representation of this configuration can be constructed (according to the great duality principle) using an octahedron instead of a cube (Fig. k).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670k.gif" />

Figure: c024670k

The Reye configuration can also be regarded as the spatial configuration of points and lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467092.png" />.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670l.gif" />

Figure: c024670l

The configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467093.png" />, consisting of points and lines, is called the Schläfli double-six. It is depicted in space, for example, in the form of symmetrically-placed lines and points on each face of cube (with respect to one of the two sixes) (Fig. l).

Not every planar configuration can be realized in the real projective plane. For example, configurations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467095.png" /> cannot be so realized, and in this connection the problem arises to realize the configuration in other projective planes. Each hypothesis asserting that the realizability in a given plane of some configuration with all the incidences except, perhaps, one, implies the realizability of the configuration with all its incidences, is called a configuration hypothesis of the given plane. Thus, if all the incidences of some configuration can be realized in some plane as a result of the geometric properties of this plane, then the configuration can be realized in this plane, i.e. is planar. For example, the configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467096.png" /> can always be realized in the real projective plane, since, as a result of the theorems of Brianchon and Pascal, the realizability of all its incidences except one will imply that the last incidence will hold. In other words, these theorems are configuration hypotheses. The fulfillment of the last incidence in a configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467097.png" /> (Fig. d) does not hold in such a plane (since the diagonal points of a complete quadrangle do not lie on a line). Therefore such an assertion is not a configuration hypothesis. However, it becomes one in the complex projective plane. In a similar way one may consider the realizability of other configurations; in particular, a configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467098.png" /> can be realized in the finite Pappus planes constructed over the Galois fields with three and four elements.

Configuration hypothesis that are valid in a given plane organize in a certain way the elements of this plane and therefore play a role in the axiomatic construction of the geometry of this plane. For example, if in a given plane all the axioms of a projective plane hold, but the Desarguesian configuration is not realized, then the given plane is a so-called non-Desarguesian geometry. A configuration hypothesis can also be described in algebraic form as a certain algebraic identity. In an arbitrary projective plane configuration hypotheses can only be introduced as new axioms, where the postulation of one configuration hypothesis may lead to the validity of other configuration hypotheses in this plane. Generally speaking, to every algebraic identity corresponds a configuration hypothesis, called the geometric representation of this identity in the plane in which the configuration is realized. (However, it is not established whether each configuration hypothesis is the geometric representation of a definite type of identity in the planar ternary ring or skew-field coordinatizing the projective plane.) The consideration of similar algebraic equivalents enables one to study both the properties of a configuration and the possibility of their realizability as a finite partial plane in a specific projective plane, and also the logical connection between different configuration hypotheses.

The theory of configurations finds application in the solution of a number of geometric problems. Thus, a plane configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c02467099.png" /> plays an important role in the theory of plane cubic curves without double points, in particular in the study of points of inflection. Reye configurations are applied in the study of regular polyhedra in a four-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670100.png" />. A regular polyhedron in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670101.png" /> is bounded by regions that are regular three-dimensional polyhedra. A regular polyhedron in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670102.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670104.png" />-cell if it is bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670105.png" /> regular polyhedra. For example, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670106.png" />-cell is bounded by five three-dimensional tetrahedra, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670107.png" />-cell by eight cubes, etc., the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670108.png" />-cell and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670109.png" />-cell being duals of each other (the points correspond to spaces and lines to planes). The study of regular cells is carried out by investigating their projections into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670110.png" />. If for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670111.png" />-cell one chooses as the projection space some three-dimensional face, then the decomposition of the space into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670112.png" /> octahedra is obtained; all of these, except the central one, extend to infinity, which leads to the kind of projection that is realized by the Reye configuration (Fig. j). If the projection space is chosen to be a three-dimensional space passing through a vertex of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670113.png" />-cell, then one also obtains the Reye configuration (Fig. k). (The projection of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670114.png" />-cell in three-dimensional space is depicted in Fig.13.)

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024670m.gif" />

Figure: c024670m

The Reye configuration also arises in the system of points and axes of symmetry of four balls whose centres do not lie in a plane. In this system each axis is incident to three points and each point to four axes, and the spatial configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670115.png" /> of points and lines is obtained. Each triple of points of symmetry defines the same plane, every two axes incident with a point of similarity form eight more distinct planes, in total twelve, each of which is incident with six points of similarity and each of the twelve points of similarity with six planes; that is, a configuration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670116.png" /> is obtained. The configuration represented by the Schläfli double-six is applied in the study of the properties of cubic algebraic surfaces. Such a cubic surface is defined by nineteen points and always passes through some Schläfli double-six. Here it is essential that any four lines of it have a hyperboloidal arrangement.

Configuration hypotheses on the realization of incidences in a plane are applied to the study of polyhedra (or polyangles) and to the solution of problems concerning the construction under various restrictions (structures with inaccessible elements, structures with a single ruler, etc.). The theory of configurations of polyhedra finds application in kinematics and graphical statics.

Configurations consisting of other geometric elements are possible, for example, consisting of discs of arbitrary dimension and unit radius in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670117.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,   S.E. Cohn-Vossen,   "Anschauliche Geometrie" , Springer  (1932)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,   "Projective planes"  ''Uspekhi Mat. Nauk'' , '''6''' :  6  (1951)  pp. 112–154  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.I. Argunov,   "Configuration hypotheses and their algebraic equivalents"  ''Mat. Sb.'' , '''26''' :  3  (1950)  pp. 425–456  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Levi,   "Geometrische Konfigurationen (mit einer Einführung in die kombinatorische Flächentopologie)" , S. Hirzel , Leipzig  (1929)</TD></TR></table>



====Comments====
A planar configuration is a configuration that can be imbedded into a plane, perhaps in more than one way, whereas a plane configuration (also defined above) is a configuration in a plane.

If there exists a one-to-one correspondence between the sides of a polygon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670118.png" /> and the vertices of a polygon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670119.png" /> such that every side of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670120.png" /> passes through the vertex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670121.png" /> corresponding to it, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670122.png" /> is said to be a circumscribed polygon with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670123.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670124.png" /> is said to be an inscribed polygon with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024670/c024670125.png" />. Thus, in Fig. ga above the triangle 147 is circumscribed with respect to 369, which in turn is circumscribed with respect to 258, which in its turn is circumscribed with respect to 147.

A modern reference on configurations is [[#References|[a3]]], Chapt. 3. For some material on the Schläfli six-fold in cubic surfaces cf. [[#References|[a4]]], Chapt. 4.

This subject as described is no longer popular. On the one hand one has (finite) group theory, especially the theory of permutation groups (cf. [[Group|Group]]; [[Finite group|Finite group]]; [[Permutation group|Permutation group]]), where the geometrical nature of the objects being permuted usually is irrelevant. On the other hand, if one is interested in the configurations themselves and not in their group of automorphisms, then there are subjects like finite geometry (with objects of geometric nature) or the theory of block designs (cf. [[Block design|Block design]]) (where only the [[Incidence|incidence]] relation is relevant).

A reference for these matters is [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Steinitz,   "Konfigurationen der projektiven Geometrie"  W.F. Meyer (ed.)  H. Mohrmann (ed.) , ''Enzyklopaedie der math. Wissenschaften'' , '''3''' , Teubner  (1907–1910)  pp. 481–516</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Dembowski,   "Finite geometries" , Springer  (1968)  pp. 254</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Kärteszi,   "Introduction to finite geometries" , North-Holland  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Yu.I. Manin,   "Cubic forms. Algebra, geometry, arithmetic" , North-Holland  (1984)  (Translated from Russian)</TD></TR></table>