View source for Net (in finite geometry)

A net is an incidence structure of points and lines, i.e. a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n0663501.png" /> consisting of a set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n0663502.png" />, a set of lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n0663503.png" /> and an incidence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n0663504.png" />, such that

1) there exist points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n0663505.png" />, and lines, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n0663506.png" />, and to every point (line) there exist two lines (points) not incident with it;

2) there is at most one line through every two distinct points;

3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n0663507.png" /> is not incident with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n0663508.png" />, then there exist one and only one line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n0663509.png" /> incident with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n06635010.png" /> that does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n06635011.png" />.

Thus, a net is a partial plane satisfying the parallelism axiom.

A more elaborate structure in finite geometry also goes by the name net, or geometrical net. It consists of a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n06635012.png" /> elements and three families of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n06635013.png" /> lines, with each line containing precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n06635014.png" /> points. This structure of lines and points is required to satisfy:

a) any two lines of two different families have precisely one point in common;

b) two lines from the same family have no point in common;

c) for each point there is exactly one line from each family passing through that point.

This notion relates to that of a web in differential geometry (cf. [[Webs, geometry of|Webs, geometry of]]). Pictorially a geometrical net can be seen as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n06635015.png" />-array of points (squares). The first two families of lines form the columns and the rows, and the lines of the third family are subsets with precisely one point in each row and in each column. Giving the squares belonging to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n06635016.png" />-th line of the third family the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n06635017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066350/n06635018.png" />, results in a [[Latin square|Latin square]] (and vice versa).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Dembowski,   "Finite geometries" , Springer  (1968)  pp. 254</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Kårteszi,   "Introduction to finite geometries" , North-Holland  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.M. Batten,   "Combinatorics of finite geometries" , Cambridge Univ. Press  (1986)</TD></TR></table>