The Trinity Functions

The Trinity Functions, also referred to as "Gowan's Trinity Functions," are a set of three geometric functions that each map a triangle to another triangle in a two-dimensional plane. Its name is derived from the apparent relation between the three functions, namely that if all three functions are performed onto a triangle in any order, the result is the starting triangle (i.e. f1 ( f2 ( f3 ( Δ ))) = Δ, where f1, f2, and f3 represent the first, second, and third functions respectively, and Δ is a given triangle).

The topic was first brought forth by mathematician Hunter Gowan in 2016, where he, Tony Pillsbury, and their geometry professor Tyler "Hill" Pickens began to study the functions' properties. The studying of the functions has led to both predictable and unforseeable findings, such as the above relation not appearing to hold true on all triangles.

The first function discussed was originally viewed on its own, as opposed to belonging as a set of three. Following some studies, the idea of the function being one-to-one came into question. Shortly after, Pickens discovered a potential way to find the inverse of the function, namely finding a triangle that when the function is applied to it yields the starting triangle. The way found was to apply two new functions, promplty and originally named function 2 and function 3. By applying the second function, then the third function onto the resultant triangle, the final triangle is such that if function 1 was applied to it, the result would be the original triangle. In function notation, f3 ( f2 ( Δ )) = (f1)-1 ( Δ ). It is also believed that the rearrangement of functions yields the same result, namely f2 ( f3 ( Δ )) = (f1)-1 ( Δ ). From this, it is apparent why the original relation seems to hold true, namely f1 ( f2 ( f3 ( Δ ))) = Δ.

From further studies, it seems that any arrangement of any two of the functions yields the inverse of the remaining third. This, however, does not seem to hold true for every triangle, leading Gowan on a 5+ year quest to try to prove the relations and to find when they work and when they don't.

Function 1 - Outer Equilateral
Function 1, notated as f1, is defined by constructing an equilateral triangle on each side of the given triangle away from the given triangle. Connect the points at each equialteral triangle's vertex not touching the given triangle.

Function 2 - Inner Equilateral
Function 2, notated as f2, is defined by constructing an equilateral triangle on each side of the given triangle towards the given triangle. Connect the points at each equialteral triangle's vertex not touching the given triangle.

Function 3 - Midpoints
Function 3, notated as f3, is defined by finding the midpoint of each side of the given triangle then connecting them.

The Trinity Function Problem
While much of what is discussed about the Trinity Function seems to be known, little to none of it has been proven. Not only is very little proven, but there have been specific cases where the assumed truths of the problem are proven incorrect.

Ultimately, to find the solution of the Trinity Function Problem, one must find the set of triangles T such that for all Δ in T, f1 ( f2 ( f3 ( Δ ))) = Δ for any arrangement of f1, f2, and f3.