One of those thoughts is that I have always liked one of the stained glass windows next to the pew in which my family regularly sits.
In particular, I have always liked the visual representation of the relationships in the trinity.
However, I have never known why I liked it...until now!
The Trinity is an antitransitive relation.
A relation is the formal mathematical notion most closely related to the word I used before, "relationship". In the case of the Trinity relation, each pair of (distinct) words (i.e. God, Father, Jesus, Holy Spirit) is ascribed either "is" or "is not" (so the Trinity relation is a binary relation). (For each pair of identical words, the window does not state the relationship but it is implied that each word is related to itself, thus ascribed an "is" and making it a symmetric relation.)
To understand an antitransitive relation, it helps to know what a transitive relation is. A (binary) relation is transitive if
for all words x, y, and z, if x is related to y and y is related to z, then x is related to z.The most commonly known transitive relation is the equivalence relation. As everyone knows, if x is equal to y and y is equal to z, then x is equal to z.
An antitransitive relation has the exact opposite conclusion about every triple of words. A (binary) relation is antitransitive if
for all words x, y, and z, if x is related to y and y is related to z, then x is NOT related to z.
Although somewhat counterintuitive, I believe it is the mathematical beauty of the Trinity that drew me to this window all those years ago.