Non-euclidean love triangle: Both people who are currently together wish they were with the same third person who lives too far away for a relationship to work.
I'd argue that your best friend whom you are willing to fight and die with or for is someone that you truly love. It may not be romantic, but it's certainly something different than platonic. My concept of love isn't limited to the binary of sexual or non-sexual. How sad that would be if it were.
Technically the triangle just refers to 'relationships' and not 'romantic relationships' so two male friends and one female love interest works as a triangle but one woman with two male strangers doesn't with because that's just a V.
That is the standard of the love triangle. The two love rivals have to be friends with, or at least know, each other or else there is no tension between them for their love of the 2nd party. Or else every person talking to multiple people on dating apps is in a love n-gon. And love should never intersext with n-gons.
There's some Interesting graph theory problem somewhere in here about finding out if a network of N people in M relationships must include at least one gay person.
A relationship graph which requires no gay relationships is called a bigraph (honest, I'm not making this up) or bipartite.
That follows because if you can two color the graph so that edges only connect different colors, you just assign male to one color and female to the other.
This means there's a tone of mathematical identities describing this. Wikipedia has a good introduction.
Very neat. But the fun part of this is that if you're poly/into more than one person, you are now a part of the LGBT group, not the straight one.
Note the above image, where the triangle shape requires that every person be into another person, with the tension being that if it's cishet, they have to choose only one person.
We aren't checking to see if a shape with LGBT people works, we know those do. We are checking to see if a cishet triangle is even possible.
Even numbers are pretty easy to solve at least. They can be straight or lgbt.
I posit that any love shape that includes an odd number of participants requires some form of deception, or at least one LGBT person in order to be valid.
Kalodin, Adolin, and Shalan (spelling is probably wrong, audiobooks of fantasy series) in the storm light archive are the greatest love triangle because each arrow is bidirectional
I remember when I was a young lad seeing a magazine about some drama or another involving a 'love triangle', and, confused, asking my mother if one of the members of the triangle was gay, since I couldn't conceive how a love triangle could work otherwise.
Wait this is great! I love my wife but she doesn't love me back! So all we need to do is find a person who she loves that then loves me so we can have threesomes!
Oh can't wait! I wonder 🤔 who this person will be!
Ok we gotta get this to work. But first questions first. Are you now or have you ever been a hottie? If male, is your penis at least 7.5"? These are very important questions. She's a hottie but it doesn't matter because you're supposed to love me, not her. But she's supposed to love you. Can you sign the backstreet boys songs? Or the songs from Pink? Are you a gambler but you keep your cool and regularly make a ton of money in the stock market? Anyway, as for me, I got the sad bod going. I'm not too tall, not too short, 6" penis. I'm at a good age. And I love to be loved. And I love to love. So we will definitely click. Man, this is going great 😃. So yeah, tell us which end you'll need from me and which end she can get from you! LOL.
If we're just talking math, triangles can be defined in terms of 3-element subsets of all 3 (A)ngles and 3 (S)ides:
SSS - unique
SAS - unique
ASS - may be unique depending on the lengths of the sides
ASA - unique
SAA - unique
AAA - infinite solutions
Maybe someone cleverer than me can figure out how that maps on to love and gender.