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Main Triang Relations Discussion
I worked out that if you include another possible relationship (for example, have romantic, platonic, and unknown) the total number of possible triangles is 132. Too bad that that's too much to make for this page.
If you want to check my work, here it is:
(n^6+3*n^3+2*n^2)/6-(n^2+n)/2
which simplifies to
(n^6+3*n^3-n^2-3*n)/6
Totally asymmetric triangles are counted six times in n^6.
Triangles that are only reflection symmetric are counted three times in n^6 and three times in n^3*3.
Triangles that are only rotationally symmetric are counted twice in n^6 and four times in n^2*n (it would be twice in n^2, and I want it to total to six).
Triangles that are reflection symmetric and rotationally symmetric are counted once in n^6, three times in n^3*3, twice in n^2*2.
Now that I counted every triangle six times, I divide the result by six. The problem is, that counts ones where one of the people isn't even in it. (If Alice and Bob are boyfriend/girlfriend, and Carol is a complete stranger, that doesn't qualify as a love triangle.) So now I take those out. Triangles that are asymmetric are counted twice in n^2.
Triangles that are mirror-symmetric are counted once in n^2 and once in n. One of these is also rotationally symmetric, but it's still counted the same number of times.
Then I divide by two and subtract from the original.
Triangles that are only reflection symmetric are counted three times in n^6 and three times in n^3*3.
Triangles that are only rotationally symmetric are counted twice in n^6 and four times in n^2*n (it would be twice in n^2, and I want it to total to six).
Triangles that are reflection symmetric and rotationally symmetric are counted once in n^6, three times in n^3*3, twice in n^2*2.
Now that I counted every triangle six times, I divide the result by six. The problem is, that counts ones where one of the people isn't even in it. (If Alice and Bob are boyfriend/girlfriend, and Carol is a complete stranger, that doesn't qualify as a love triangle.) So now I take those out. Triangles that are asymmetric are counted twice in n^2.
Triangles that are mirror-symmetric are counted once in n^2 and once in n. One of these is also rotationally symmetric, but it's still counted the same number of times.
Then I divide by two and subtract from the original.
The amount of possible relationships between n persons (considering isomorphic cases equal) is the number of connected digraphs with n nodes. If you are curious to know how many there are for other number of people, it's http://oeis.org/A003085
Triang Relations 13 is described as Alice choosing between B and C, as if the attractions between A and B as well as A and C are mutual. However, the arrows are one-sided, as if A likes B and C but they don't feel the same way.
@Fast Eddie: I get the point of trying to trim the page down, but damn if you couldn't have picked a worse way to do it. "Limit of three examples per media type" is a simply horrendous distinction (cue "BAWWWWW MY EXAMPLE IS BETTER THAN YOURS"-style arguments), and once again the scythe that's been taken to the article has been done in a sufficiently careless way to leave hanging references dotted around the page.
Is it just me, or are Type 1 and Type 7 essentially the same thing? Type 1, based on the diagram, implies that A loves two people, and neither of them cares at all. Type 7 would be a much more accurate description of most examples of the Betty And Veronica scenario, although here it's described as an "affair". Which it could be, if you look at it a certain way. But most examples of what you have as Type 1, tend to fall into Type 7, at least in theory. This needs fixed.
Can someone explain to me why the page Love Triangle exists if all the examples can be sorted onto this page?
Hey, could we maybe give the types names in addition to numbers so that non-tropers don't have to tab out to this page whenever a given type is referred to? Also, we REALLY need to merge this with the love triangle page.