Sources page biographical material

6.BI. VENN DIAGRAMS FOR N SETS
New topic. I think I have seen more papers on this and Anthony Edwards has recently sent several more papers.
Martin Gardner. Logic diagrams. IN: Logic Machines and Diagrams; McGraw Hill, NY, 1958, pp. 28-59. Slightly amended in the 2nd ed., Univ. of Chicago Press, 1982, and Harvester Press, Brighton, 1983, pp. 28-59. This surveys the history of all types of diagrams. John Venn [Symbolic Logic, 2nd ed., ??NYS] already gave Venn diagrams with 4 ovals and with 4 ovals and a disconnected set. Gardner describes various binary diagrams from 1881 onward, but generalised Venn diagrams seem to first occur in 1909 and then in 1938-1939, before a surge of interest from 1959. His references are much expanded in the 2nd ed. and he cites most of the following items.

John Venn. On the diagrammatic and mechanical representation of propositions and reasonings. London, Edinburgh and Dublin Philos. Mag. 10 (1880) 1-18. ??NYS -- cited by Henderson.

John Venn. Symbolic Logic. 2nd ed., Macmillan, 1894. ??NYS. Gardner, p. 105, reproduces a four ellipse diagram.

Lewis Carroll. Symbolic Logic, Part I. 4th ed., Macmillan, 1897; reprinted by Dover, 1958. Appendix -- Addressed to Teachers, sections 5 - 7: Euler's method of diagrams; Venn's method of diagrams; My method of diagrams, pp. 173-179. Describes Euler's simple approach and Venn's thorough approach. Reproduces Venn's four-ellipse diagram and his diagram for five sets using four ellipses and a disconnected region. He notes that Venn suggests using two five-set diagrams to deal with six sets and does not go further. He then describes his own method, which easily does up to eight sets. The diagram for four sets is the same as the common Karnaugh diagram used by electrical engineers. For more than four sets, the regions become disconnected with the cells of the four-set case being subdivided, using a simple diagonal, then his 2-set, 3-set and 4-set diagrams within each cell of the 4-set case. ?? -- is this in the 1st ed. -- ??NYS date??

Carroll-Gardner, p. 61, says this is in the first ed. of 1896.

William E. Hocking. Two extensions of the use of graphs in elementary logic. University of California Publications in Philosophy 2:2 (1909) 31(-??). ??NYS -- cited by Gardner who says Hocking uses nonconvex regions to get a solution for any n.

Edmund C. Berkeley. Boolean algebra and applications to insurance. Record of the Amer. Inst. of Actuaries 26:2 (Oct 1937) & 27:1 (Jun 1938). Reprinted as a booklet by Berkeley and Associates, 1952. ??NYS -- cited by Gardner. Uses nonconvex sets.

Trenchard More Jr. On the construction of Venn diagrams. J. Symbolic Logic 24 (Dec 1959) 303-304. ??NYS -- cited by Gardner. Uses nonconvex sets.

David W. Henderson. Venn diagrams for more than four classes. AMM 70:4 (1963) 424-426. Gives diagrams with 5 congruent irregular pentagons and with 5 congruent quadrilaterals. Considers problem of finding diagrams that have n-fold rotational symmetry and shows that then n must be a prime. Says he has found an example for n = 7, but doesn't know if examples can be found for all prime n.

Margaret E. Baron. A note on the historical development of logic diagrams: Leibniz, Euler and Venn. MG 53 (No. 384) (May 1969) 113 125. She notes Venn's solutions for n = 4, 5. She gives toothed rectangles for n = 5, 6.

K. M. Caldwell. Multiple set Venn diagrams. MTg 53 (1970) 29. Does n = 4 with rectangles and then uses indents.

A. K. Austin, proposer; Heiko Harborth, solver. Problem E2314 -- Venn again. AMM 78:8 (Oct 1971) 904 & 79:8 (Oct 1972) 907-908. Shows that a diagram for 4 or more sets cannot be formed with translates of a convex set, using simple counting and Euler's formula. (The case of circles is in Yaglom & Yaglom I, pp. 103-104.) Editor gives a solution of G. A. Heuer with 4 congruent rectangles and more complex examples yielding disconnected subsets.

Lynette J. Bowles. Logic diagrams for up to n classes. MG 55 (No. 394) (Dec 1971) 370 373. Following Baron's note, she gives a binary tooth like structure with examples for n = 7, 8.

Vern S. Poythress & Hugo S. Sun. A method to construct convex connected Venn diagrams for any finite number of sets. Pentagon (Spring 1972) 80-83. ??NYS -- cited by Gardner.

S. N. Collings. Further logic diagrams in various dimensions. MG 56 (No. 398) (Dec 1972) 309 310. Extends Bowles.

Branko Grünbaum. Venn diagrams and independent families of sets. MM 48 (1975) 12 22. Considers general case. Substantial survey of different ways to consider the problem. References to earlier literature. Shows one can use 5 identical ellipses, but one cannot use ellipses for n > 5.

B. Grünbaum. The construction of Venn diagrams. CMJ 15 (1984) 238 247. ??NYS.

Allen J. Schwenk. Venn diagram for five sets. MM 57 (1984) 297. Five ovals in a pentagram shape.

A. V. Boyd. Letter: Venn diagram of rectangles. MM 58 (1985) 251. Does n = 5 with rectangles.

W. O. J. Moser & J. Pach. Research Problems in Discrete Geometry. Op. cit. in 6.T. 1986. Prob. 27: On the extension of Venn diagrams. Considers whether a diagram for n classes can be extended to one for n+1 classes.

Mike Humphries. Note 71.11: Venn diagrams using convex sets. MG 71 (No. 455) (Mar 1987) 59. His fourth set is a square; fifth is an octagon.

J. Chris Fisher, E. L. Koh & Branko Grünbaum. Diagrams Venn and how. MM 61 (1988) 36 40. General case done with zig zag lines. References.

Anthony W. F. Edwards. Venn diagrams for many sets. New Scientist 121 (No. 1646) (7 Jan 1989) 51-56. Discusses history, particularly Venn and Carroll, the four set version with ovals and Carroll's four set version where the third and fourth sets are rectangles. Edwards' diagram starts with a square divided into quadrants, then a circle. Fourth set is a two-tooth 'cogwheel' which he relates to a Hamiltonian circuit on the 3-cube. The fifth set is a four-tooth cogwheel, etc. The result is rather pretty. Edwards notes that the circle in the n set diagram meets the 2^n-1 subsets of the n-1 sets other than that given by the circle, hence travelling around the circle gives a sequence of the subsets of n-1 objects and this is the Gray code (though he attributes this to Elisha Gray, the 19C American telephone engineer -- cf 7.M.3). The relationship with the n-cube leads to a partial connection between Edwards' diagram and the lattice of subsets of a set of n things.

New Scientist (11 Feb 1989) 77 has: Drawing the lines -- letters from Michael Lockwood -- describing a version with indented rectangles -- and from Anthony Edwards -- noting some errors in the article.

Ian Stewart. Visions mathématiques: Les dentelures de l'esprit. Pour la Science No. 138 (Apr 1989) 104-109. c= Cogwheels of the mind, IN: Ian Stewart; Another Fine Math You've Got Me Into; Freeman, NY, 1992, chap. 4, pp. 51-64. Exposits Edwards' work with a little more detail about the connection with the Gray code.

A. W. F. Edwards & C. A. B. Smith. New 3-set Venn diagram. Nature 339 (25 May 1989) 263. Notes connection with the family of cosine curves, y = 2^-n cos 2ⁿx on [0, π] and Gray codes and with the family of sine curves, y = 2^-n sin 2ⁿx on [0, 2π] and ordinary binary codes. Applying a similar phase shift to Edwards' diagram leads to diagrams where more than two set boundaries are allowed to meet at a point.

A. W. F. Edwards. Venn diagrams for many sets. Bull. Intern. Statistical Inst., 47th Session, Paris, 1989. Contrib. Papers, Book 1, pp. 311-312.

A. W. F. Edwards. To make a rotatable Edwards map of a Venn diagram. 4pp of instructions and cut-out figures. The author, Gonville and Caius College, Cambridge, CB2 1TA, 21 Feb 1991.

A. W. F. Edwards. Note 75.39: How to iron a hypercube. MG 75 (No. 474) (1991) 433-436. Discusses his diagram and its connection with the n-cube.

Anthony Edwards. Rotatable Venn diagrams. Mathematics Review 2:3 (Feb 1992) 19-21. +  Letter: Venn revisited. Ibid. 3:2 (Nov 1992) 29.