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David McCandless of "Information is Beautiful" recently posted an image of the 7-set Venn diagram construction.

What rules generate Venn diagrams for an arbitrary number of sets $k$?

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Did you check the references in the Wikipedia page? The first link has a section called A General Construction for Symmetric Venn Diagrams. – Rahul Jul 15 '12 at 5:11

Since this is an interesting topic, I thought that it might be useful to collect interesting references on this topic. I made the post CW, feel free to edit it.

First of all the references given in Wikipedia article might be useful.

Maybe some information on the talk page of that article could be useful, too. E.g. the link to review of Cogwheels of Mind given here.

Here is an alphabetical list of some more interesting papers related to this topic.

  • Branko Grünbaum: The search for symmetric Venn diagrams; Geombinatorics 8(1999), pp. 104 - 109 author's homepage. Note: The same author has several other papers on the same topic.

  • Branko Grünbaum: The Construction of Venn Diagrams, The College Mathematics Journal, Vol. 15, No. 3 (Jun., 1984), pp. 238-247; jstor.

  • Peter Hamburger, Raymond E. Pippert: Venn Said It Couldn't Be Done, Mathematics Magazine, Vol. 73, No. 2 (Apr., 2000), pp. 105-110; jstor

  • Ruskey, Frank; Savage, Carla D.; Wagon, Stan. The search for simple symmetric Venn diagrams. Notices Amer. Math. Soc. 53 (2006), no. 11, 1304–1312. link

  • Allen J. Schwenk: Venn Diagram for Five Sets, Mathematics Magazine, Vol. 57, No. 5 (Nov., 1984), p. 297; jstor

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Using ellipses, you can construct a symmetrical Venn diagram for all orders $k$:

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