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# Birkhoff

Birkhoff, Garrett, 1911-, American mathematician, b. Princeton, N.J.; son of George David Birkhoff. He was educated at Harvard (B.A., 1932) where he was elected a fellow in 1933 and taught until his retirement in 1981. Birkoff has made several important contributions to abstract mathematics, the teaching of mathematics, and mathematical physics. From 1934 on he developed the concept of a lattice, a generalized algebra with two operators, and showed how a number of subjects, e.g., Boolean algebra, projective geometry, and affine geometry, could be treated as special types of lattices. His text A Survey of Modern Algebra (with Saunders MacLane, 1941) became a standard undergraduate textbook.

See his Lattice Theory (1940, 3d ed. 1967).

Birkhoff, George David, 1884-1944, American mathematician, b. Overisel, Mich.; father of Garrett Birkhoff. The son of a physician, he was educated at Harvard (B.A., 1905) and the Univ. of Chicago (Ph.D., 1907) After teaching shortly at Chicago and Princeton, he joined the faculty at Harvard (1912) where he taught until his death. Birkhoff, perhaps the first American mathematician of international repute, is known for his work on linear differential equations and difference equations. He was also deeply interested in and made contributions to the analysis of dynamical systems, celestial mechanics, the four-color map problem, and function spaces. In addition he wrote on the foundations of relativity and quantum mechanics and on art and music, e.g., Aesthetic Measure (1933).

See his Collected Mathematical Papers (3 vol., 1950).

In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry. Other often-used axiomizations of plane geometry are Hilbert's axioms and Tarski's axioms.

## Postulates

Postulate I: Postulate of Line Measure. A set of points {A, B...} on any line can be put into a 1:1 correspondence with the real numbers {a, b...} so that |b-a| = d(A,B) for all points A and B.

Postulate II: Point-Line Postulate. There is one and only one line, l, that contains any two given distinct points P and Q.

Postulate III: Postulate of Angle Measure. A set of rays {l, m, n...} through any point O can be put into 1:1 correspondence with the real numbers a(mod 2π) so that if A and B are points (not equal to O) of l and m, respectively, the difference am - al (mod 2π) of the numbers associated with the lines l and m is $angle$AOB.

Postulate IV: Postulate of Similarity. Given two triangles ABC and A'B'C' and some constant k>0, d(A', B') = kd(A, B), d(A', C')=kd(A, C) and $angle$B'A'C'$angle$BAC, then d(B', C')=kd(B,C), $angle$C'B'A'$angle$CBA, and $angle$A'C'B'$angle$ACB

## References

• Birkhoff, G. D., 1932, "A Set of Postulates for Plane Geometry (Based on Scale and Protractors)," Annals of Mathematics 33.

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