The Trinity is Antitransitive

I grew up going to St. James United Methodist Church in Sioux City, Iowa. Every time I return home and attend church there, I rethink many of the thoughts that I have had in that building over the years.

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.

Now, to explain what that means.

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.

(Or maybe it is because it also looks like a planar embedding of K₄...nah.)