If you’re a viewer of Gravity Falls, you’ve likely paused the opening theme near the end in order to get a glimpse of the lovely triangle thingy that appears for a few frames. If you manage to pause correctly, what you will see is this:

While the image in the center, and the symbols in the circle are clearly the thing that draws the eye of the viewer most, take a look in the upper right corner. What do you see?

Matrices. The bane of every high school student who went past the first year of algebra. I was looking at this today (since my life is apparently now consumed by this show), and noticed that the symbols and patterns of the matrices were familiar for some reason.

I looked up how well the page had been interpreted/translated, and the only explanation I found for the matrices were: Geometric Transformation. That’s all well and good, but there was something still not clicking. I determined that I needed a better look at the full matrix equation (you can tell there’s some missing from the top of the previous image, if you look closely).

So, I found an image of the page (from S1E19, taken from the Gravity Falls Wiki) which showed the other portions of the matrices:

Here we have the full matrix equation. It turns out that what we can see in the title card is the final form of the equation itself, which is an expanded form of a column vector with components [hx, hy_z, h, 1]. Suddenly, it hit me:

This is something I learned about in a 6000 level Stat course last semester!

This is not a random “geometric transformation.” This is a projective transformation from 2D to 3D! These transformations preserve the connections between points, but can stretch and warp the connective lines beyond recognition.

If I were to extrapolate this identification into an implication about Bill Cipher, the lovely triangle in the center, I would think it implies three things, one of which is confirmed:

1) Bill can project into minds, and possibly into 3D in general. [Confirmed by S1E19]

2) Bill is trapped in 2D space in general. The summoning described in the ring (which is not fully deciphered/understood in the context of the show) will permanently project him into 3D space/the “real” world (of the show).

3) Bill can alter his form, but will overall stay contained by the form of a triangle (projective transformations can stretch and warp the lines that connect the three points).

I am super, super excited to see this, and was a bit shocked to find a graduate level mathematics reference/clue hidden in a 1-frame image in the opening theme of a children’s television show.

tl;dr - There’s another clue in the title card of Gravity Falls, and it depends on an understanding of a particular topic of graduate-level mathematics.