(n_{3}) Configurations Last update=4 Sept, 2021 A configuration (n_{3}) is an incidence structure consisting of n "points" and n "lines" such that each point is contained in 3 lines, each line contains 3 points, and each pair of points is contained in at most one line. Two recommended books on configurations are
B. Grünbaum, Configurations of Points and Lines, American Mathematical Society, 2009. The smallest (n_{3}) configuration has n=7, shown below.
One line is drawn as a circle, because it is not possible to represent each L_{i} as a straight line. There is exactly one configuration (7_{3}), and one configuration (8_{3}), which is known as the MöbiusKantor configuration. It also cannot be drawn with straight lines, but can be drawn with one line represented as a circle.
When n=9, there are three configurations (9_{3}), and they can all be drawn such that every "line" is a straight line. When this is possible, the configuration is said to be geometric or realizable.
There are ten configurations (10_{3}), of which nine are geometric. One is nongeometric, it is known as the antiPappian. A diagram of it follows !
There are 31 configurations (11_{3}), originally found by Martinetti (1887). And there are 229 configurations (12_{3}), originally found by Daublebsky von Sterneck (1888). The (11_{3}) and (12_{3}) configurations are all geometric. The numbers of configurations are known up to n=19, found by computer by Betten, Brinkman and Pisanski. Grünbaum has conjectured that if a configuration is geometric, then it has a drawing in the plane such that all point and line coordinates are rational. Bokowski, Sturmfels and White have shown that this is true when n ≤ 12. Recent work (see below) shows that all (13_{3}) configurations are also geometric and rational. It is not yet known whether there are any other nongeometric configurations that are nontrivial. Note that a configuration must be connected, and must satisfy certain other basic properties described in Grünbaum's book. Recently an algorithm (see W. Kocay, "Onepoint extensions in (n_{3}) configurations", Ars Math. Contemp. 10 (2016), 291322 and "Coordinatizing (n_{3}) configurations", Ars Math. Contemp. to appear) shows how to construct extensions of a configuration (n_{3}) to ((n+1)_{3}). If the (n_{3}) has an integer coordinatization, then the algorithm can usually find an integer coordinatization of the ((n+1)_{3}) configuration, extended from the (n_{3}), when one exists. Using this algorithm, integer coordinatizations of all 2036 configurations (13_{3}) have now been constructed. Thus all (13_{3})'s are geometric and all are rational. One noticeable feature of the integer coordinatizations is that the number of digits in the point and line coordinates grows very rapidly with increasing n.
CoordinatesIt is convenient to use homogeneous coordinates in the plane for the points and lines of a geometric configuration. Then point i with coordinates P_{i} and line j with coordinates L_{j} are incident if and only ifP_{i} ⋅ L_{j} = 0. For example, the Pappus configuration, a (9_{3}), is given by the table following.
A point (x, y, z) with z = 0 is a point at infinity.
Text FilesSome text files of the small configurations can be downloaded here. They are in the Haiku G&G format. The naming of the (10_{3})'s corresponds to that in Grünbaum's book. The naming of the others is based on the order in which they were generated by the software. The order of the collineation group is shown in square brackets as part of the configuration's name, eg. (10,3)#10[120sd] means that configuration (10,3)#10 has a collineation group of order 120, and that it is selfdual.Download:
The AntiPappianThe antiPappian has a very interesting construction. Choose four points P_{1}, P_{2}, P_{3}, P_{4} in general position. They determine six lines P_{i}P_{j}, where 1 ≤ i < j ≤ 4. Choose four lines L_{1}, L_{2}, L_{3}, L_{4} in general position, such that no point P_{i} is on a line L_{j}. The four lines determine six points of intersection L_{i}L_{j}, where 1 ≤ i < j ≤ 4. Each line P_{i}P_{j} currently contains two points, and each point L_{i}L_{j} is currently on two lines. Place point L_{i}L_{j} on line P_{i}P_{j}. The result is the configuration (10_{3}) known as the antiPappian. Clearly the symmetric group S_{4} permutes the points and lines as collineations.Back to the Groups & Graphs home page.
