Department of Mathematics
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It is well-known that if a finite configuration is representable in the complex projective plane, then it is also representable in the projective plane over a finite algebraic extension of the rationals. A fundamental theorem of Saunders Mac Lane says that given a finite extension k of the field of rational numbers, there exists a configuration C which can be faithfully represented in PG(2,k) and moreover, any other representation of C in PG(2, K) requires that the field K contains k. In particular, one can capture the algebraic essence of an irrational number, say "square root of 5", by means of a block design expressed purely in the language of geometry. We will give some typical examples illustrating these theorems and relate this to a conjecture of Gruenbaum which states that every self- dual (n,3)-configuration in the real plane can also be realized in the rational plane.

References

[1] B. Gruenbaum, Notes on Configurations, Lecture Notes, University of Washington, 1986.
[2] S. Mac Lane, Abstract linear independence in terms of projective geometry, American J of Mathematics, 58 (1936) 236-240.