Sources page biographical material

6.H. PICK'S THEOREM
Georg Pick. Geometrisches zur Zahlenlehre. Sitzungsberichte des deutschen naturwissenschaftlich medicinischen Vereines für Böhmen "Lotos" in Prag (NS) 19 (1899) 311 319. Pp. 311 314 gives the proof, for an oblique lattice. Pp. 318 319 gives the extension to multiply connected and separated regions. Rest relates to number theory. [I have made a translation of the material on Pick's Theorem.]

Charles Howard Hinton. The Fourth Dimension. Swan Sonnenschein & Co., London, 1906. Metageometry, pp. 46-60. [This material is in Speculations on the Fourth Dimension, ed. by R. v. B. Rucker; Dover, 1980, pp. 130-141. Rucker says the book was published in 1904, so my copy may be a reprint??] In the beginning of this section, he draws quadrilateral shapes on the square lattice and determines the area by counting points, but he counts I + E/2 + C/4, which works for quadrilaterals but is not valid in general.

H. Steinhaus. O mierzeniu pól płaskich. Przegląd Matematyczno Fizyczny 2 (1924) 24 29. Gives a version of Pick's theorem, but doesn't cite Pick. (My thanks to A. Mąkowski for an English summary of this.)

H. Steinhaus. Mathematical Snapshots. Stechert, NY, 1938, pp. 16-17 & 132. OUP, NY: 1950: pp. 76 77 & 260 (note 77); 1960: pp. 99 100 & 324 (note 95); 1969 (1983): pp. 96 97 & 301 (note 107). In 1938 he simply notes the theorem and gives one example. In 1950, he outlines Pick's argument. He refers to Pick's paper, but in "Ztschr. d. Vereins 'Lotos' in Prag". Steinhaus also cites his own paper, above.

J. F. Reeve. On the volume of lattice polyhedra. Proc. London Math. Soc. 7 (1957) 378 395. Deals with the failure of the obvious form of Pick's theorem in 3 D and finds a valid generalization.

Ivan Niven & H. S. Zuckerman. Lattice points and polygonal area. AMM 74 (1967) 1195 1200. Straightforward proof. Mention failure for tetrahedra.

D. W. De Temple & J. M. Robertson. The equivalence of Euler's and Pick's theorems. MTr 67 (1974) 222 226. ??NYS.

W. W. Funkenbusch. From Euler's formula to Pick's formula using an edge theorem. AMM 81 (1974) 647 648. Easy proof though it could be easier.

R. W. Gaskell, M. S. Klamkin & P. Watson. Triangulations and Pick's theorem. MM 49 (1976) 35 37. A bit roundabout.

Richard A. Gibbs. Pick iff Euler. MM 49 (1976) 158. Cites DeTemple & Robertson and observes that both Pick and Euler can be proven from a result on triangulations.

John Reay. Areas of hex-lattice polygons, with short sides. Abstracts Amer. Math. Soc. 8:2 (1987) 174, #832-51-55. Gives a formula for the area in terms of the boundary and interior points and the characteristic of the boundary, but it is an open question to determine when this formula gives the actual area.