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Grünbaum-Rigby Configuration


GruenbaumRigbyConfiguration

The Grünbaum-Rigby configuration is a 21_4 configuration consisting of 21 points and 21 lines such that four points lie on each line and four lines pass through each point (left figure above). It was first described by Klein (1879), but a geometric realization in the Euclidean plane was not found until Grünbaum and Rigby (1990) constructed one by overlaying three regular heptagrams. Note that the points also determine seven additional lines through three points (illustrated in red in the right figure above) which are not however part of the configuration.

The Levi graph of the Grünbaum-Rigby configuration may be termed the Grünbaum-Rigby graph.

GruenbaumRigbyConfigurationOrchardPlanting

The best known solution to the orchard-planting problem for 22 points with 4 points per line is obtained by a adding a point at the center of the configuration through which opposite vertices of the configuration lie, yielding 28 lines (E. Pegg, Jr., pers. comm., Dec. 11, 2023). Note that this arrangement is not a configuration since while five lines pass through 21 of the points, seven lines pass through the central point.


See also

Grünbaum-Rigby Graph, Orchard-Planting Problem

Portions of this entry contributed by Ed Pegg, Jr. (author's link)

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References

Burnside, W. "On the Hessian Configuration and Its Connection With the Group of 360 Plane Collineations." Proc. London Math. Soc. 4, 54-71, 1907.Coxeter, H. S. M. "My Graph." Proc. London Math. Soc. 46, 117-136, 1983.Grünbaum, B. and Rigby, J. F. "the Real Configuration (21_4)." J. London Math. Soc. 41, 336-348, 1990.Grünbaum, B. Configurations of Points and Lines. Providence, RI: Amer. Math. Soc., p. 311, 2009.Klein, F. "Über die Transformation siebenter Ordnung der elliptischen Functionen." Math. Ann. 14, 428-471, 1879.

Cite this as:

Pegg, Ed Jr. and Weisstein, Eric W. "Grünbaum-Rigby Configuration." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Gruenbaum-RigbyConfiguration.html

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