Create a fully proportional chart of 3 sets and their areas of overlap (2007)

Hi Everyone,

Perhaps there is a semantic or conceptual issue that is preventing me from presenting a cogent problem...

I believe that:

1) A proportional Venn Diagram made with perfect (ie. un-deformed shape) circles (or any other perfect shape) cannot represent all the possible, legitimate relationships between 3 sets. Yes -- you can fill a classic Venn diagram with Roman numerals, and it can communicate any combination of relationships between 3 sets. But if you want the *area* of each sector to be proportional to the number of members that it contains, there are specific relationship combos (as described in my earlier post) that cannot be described with a Venn diagram using perfect shapes.

If that's true... what's the next best thing?

The answer -- perhaps deceptively "easy" seeming to me -- is to throw out the requirement that the totality of each set be represented by a perfect shape. A more flexible approach might be to assign a perfect shape (square or triangle or pentagon -- not circle) to each *set member* Then, by assembling these shapes into a "mosaic", you can enlarge and reduce overlap sectors *independently* of each other, and all possible set relationships can be represented.

To that end, I have seen this type of "mosaic chart" before. It appears most frequently (in the US) as a map of the 50 states made from 437 identical little boxes, with each box representing one vote in the US House of Representatives. On this map, New York state appears super-sized because it has a lot of votes in Congress, and North Dakota, with just one tiny box, practically disappears into oblivion. Now, the reason I think of this map is that it has inherent ability to support overlap sets very nicely (though that ability is not used in the specific implementation regarding votes in Congress). Imagine for a moment that, for whatever silly reason, Michigan and Ohio are granted one additional vote to share between them; i.e. an additional representative representing both states. This new, extra vote is effectively a set overlap, and this new vote could be represented on the map, as a lone yellow box (the same size as all the other boxes, but a different color) that is contained within the boundaries of both Michigan and Ohio.

Anyway... there it is. That's the perceived problem and potential solution presented as clearly as I can think to present them.

Input or ideas, anyone?

Cheers,

Jay

SHG,

Yes, that's it!! Thanks for the great articles. Now I just have to create one with overlapping sets --- thanks for the pointers.

Cheers,

Jay

More progress!

It turns out that the mathematical tool that I was looking for is called... Venn Polyominos!

See: http://home.flash.net/~markthom/html...olyominos.html

At the name "Venn "suggests, Venn polyominos support overlapping sets, and yet each polyomino is proportional to the number of members that it contains (in math lingo, each is "equal density").

A more common, sometimes-more-specific implementation of the same concept is something called a "tile chart". It is sometimes more specific if it adds the requirement that each polyonimo have the same shape as it's constituent tiles (like here: http://bit.ly/bjjzR5 ).

[One major problem is that there is another kind of "tile chart" that has a completely different appearance and purpose.]

Now, what do you suppose is the simplest, fastest way create a series of Venn Polyomino (or Tile) charts representing the areas and intersections of 3 sets?

Hmmmm....

Cheers,

Jay

PS: It turns out that the term "cartogram" is very specifically linked to maps, and that is a slightly more precise and applied case of a problem that I conceive as more generalized.