# Make a Venn diagram with five triangles

A Venn diagram consists of a set of partially overlapping shapes on a plane, arranged in a particular way: for any particular labelling of each of the shapes as "inside" or "outside", it is possible to find a point that is inside all the "inside" shapes and outside all the "outside" shapes.

The most famous Venn diagram is formed out of three circles, as seen in this question. Circles don't work very well for more than three regions, but by using ellipses, it is possible to go up to five shapes, as seen in this question – there are 32 possible ways to choose "inside" and "outside" for each of the ellipses, and for all 32 combinations, you can find a point that's inside or outside each of the ellipses as appropriate.

So my puzzle for you is: do the same thing, but using triangles rather than ellipses. You need to find a way to arrange five triangles, such that for all 32 possible ways to choose "inside" or "outside" for each of the triangles, there is some point that's inside or outside each of the triangles as appropriate (meaning that the triangles need to split the plane into at least 32 regions). The triangles don't need to be any particular shape or size (e.g. they don't have to be equilateral), and they don't have to be the same shape or size as each other; any shape or size of triangle will do.

(Note: the result probably won't be a "perfect" Venn diagram – all the solutions I've found have more than 32 regions, with some of them duplicating each other, but with all 32 combinations present in the diagram somewhere, and the problem merely requires there to be at least one region for all 32 possibilities; it doesn't require there to be only one region per possibility. I don't know the minimum possible number of regions with which it's possible to solve the problem, but suspect it's more than 32.)

• arxiv.org/abs/cs/0512001v1 has a freakish 6-set triangular venn diagram. Just remove one triangle to get 5-set!
– qwr
Commented May 31 at 20:59

Without the constraint of unique regions, I don't think this is especially difficult.

Five overlapping scalene triangles, each rotated 72° from the last, will give you all the regions you need with a minimum of manipulation.

Here is one attempt. I will continue to attempt to find a solution without duplicated regions.

• Hmm, this turned out to be easier than I expected – I should have tried something like that first. (My solution is substantially different from this one and took a lot longer to find – and is probably also less elegant.) Commented May 31 at 20:18
• In the papers in my answer, the duplicated regions form an "independent family'. If all 2^n regions inside and outside the curves are connected, then they call it a Venn Diagram.
– qwr
Commented Jun 1 at 4:31
• I didn't look closely enough at this. Some regions are missing. For example, there is no AE in the above diagram. I think you could still get something that works with a bit of manipulation, though. Commented Jun 4 at 19:53
• @GentlePurpleRain AE isn't labeled, but it is present. It's the little unlabeled triangle in the lower right of the figure, neighboring the ADE and ACE regions. The other unlabeled regions are AB, BC, CD, and DE, but they're all present symmetrically around the diagram. I thought they were missing as well when I first saw your answer, but I found them after another look. Commented Jun 4 at 20:19
• @isaacg Thanks for pointing that out. I thought I remembered carefully checking that all regions were represented, but I guess I just missed some when I was labelling them. Commented Jun 5 at 15:26

My previous answer was flawed, so here is another attempt, which has the additional benefit of not duplicating any regions (and being rather visually pleasing:

Branko Grünbaum - Venn Diagrams and Independent Families of Sets (1975)

Bonus: A freakish 6-Venn Diagram of triangles, though not symmetric, is in Carroll, Ruskey, Weston - Which n-Venn diagrams can be drawn with convex k-gons? (2005)