The murmur of the snarkmatrix…

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Beyond the Venn diagram

Edwards-Venn Cogwheel

Joshua Glenn buried this nugget in a comment at HiLobrow:

Anyone who has ever spent time in a conference room equipped with an overhead projector is familiar with the basic Venn diagram — three overlapping circles whose eight regions represent every possible intersection of three given sets, the eighth region being the space around the diagram. Although it resembled the intertwined rings already familiar in Christian (and later Led Zeppelin) iconography, when Venn devised the diagram in 1880, it was hailed as a conceptually innovative way to represent complex logical problems in two dimensions.

There was just one problem with it, according to British statistician, geneticist, and Venn diagram expert A.W.F. Edwards, author of the entertaining book “Cogwheels of the Mind” (Johns Hopkins): It didn’t scale up. With four sets, it turns out, circles are no use — they don’t have enough possible combinations of overlaps. Ovals work for four sets, Venn found, but after that one winds up drawing spaghetti-like messes — and, as he put it, “the visual aid for which mainly such diagrams exist is soon lost.” What to do?

A rival lecturer in mathematics at Oxford by the name of Charles Dodgson — Lewis Carroll — tried to come up with a better logical diagram by using rectangles instead of circles, but failed (though not before producing an 1887 board game based on his “triliteral” design). In fact, it wasn’t until a century later that the problem of drawing visually appealing Venn diagrams for arbitrary numbers of sets was solved — by Edwards, it turns out. In 1988 Edwards came up with a six-set diagram that was nicknamed the “Edwards-Venn cogwheel.”

The original post, which extends Glenn’s ongoing remapping of generational lines past the mid-nineteenth century — Decadents! Pragmatists! Industrial Tyrants! Mark Twain AND Henry James! — is pretty sweet too. (For what it’s worth, one big thing Decadents and Pragmatists had in common was that they were both obsessed with generational changes.)

Now we just need a generational map that’s ALSO an Edwards-Venn cogwheel!


FWIW, back in my Discrete Math II (The Revenge of Discrete Math) class, I learned that you can use four spheres to represent four sets and build a three-dimensional Venn Diagram by putting each of the four spheres in a corner of an imaginary tetrahedron. That way, they all intersect in just the right places to form a Venn Diagram.

So, there’s that. Can’t do any more than four sets, though. Not without a few more dimensions, and those things don’t just grow on trees.

Tim Carmody says…

That reminds me of the joke from Futurama: “[Shrinking you] would require very tiny atoms. And have you priced those lately?”

The snarkmatrix awaits you

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