#Euclidean Geometry

[dat-physics-gal]

Not to be that guy, but you depicted a 4D EUCLIDEAN object, a regular-4-hypercube, or tesseract. If you want to go non Euclidean, may i suggest a hyperbolic crochet? Or a spherical triangle with three right angles?

Any of my followers extradimensional? Can u help this guy out? He's not even supposed to be here.

[pitch-and-moan]

Six Sides of Death

A sequel to Enter the Tetrahedron, in which the kumite between history's greatest geometers continues. This time, they fight both for honor, and to solve the unit distance problem.

[iridescentalchemyst]

Introduction to Sacred Geometry

I have been working on this post and the graphics for over a week now and I am happy to finally hit the publish button! Hope you like it! Let me know what you think in the comments!

“Greek mythology holds that our human capacity for geometric vision is a gift of the divine feminine—energetic sources of wisdom conceptualized as a lineage of goddesses. Born from primal Chaos is Gaia… She gives birth to Mnemosyne, goddess of Memory, from whose name comes “mnemonic.” The daughters of Mnemosyne are the Muses—the arts and sciences. Memory is the legacy of the sacred Earth, and the…

[tagitables]

🤍

[the-barefoot-hatter]

c'mon Sixer, it's like the World's Greatest Playpen! 100% safe!

[megarhyssa-sp]

from this:

@iowntheworldandyou you are so right. he was probably keeping it in his pocket. these two should not be parents 😂

[kaiyax1]

I made a doc of every gravity falls au I could remember, here’s the link if you want to add any or if you know who made one of the au’s I didn’t know who made. gravity falls aus are my hyperfixation.

[giulliadella]

Speculative Biology of Euclydians (and Bill Cipher) part 1

Part 2

We're doing this, babies!

This analysis is based on two assumptions:

Before Bill Cipher became a demigod, he was a biological, living organism and so were the rest of his species.

Even after Bill Cipher became a demigod, he still retained some physical characteristics of his biological form.

I will clearly specify which of his abilities are innate abilities of his species, which ones are definitely his divine abilities and which ones could be both.

This is part one. This analysis became VERY long, so I'll separate it into FIVE parts:

Part 1: What is Euclydia and what are Euclydians?

Part 2: How Euclydians function as animals? (This is where I explain how are they built, what their organs do, how they feed, move, speak etc.)

Part 3: Reproduction NSFW (this one I separated because it's NSFW. It'll be nothing explicit, but I doubt your boss would be thrilled if he found out that you're reading about how triangles fuck in your office)

Part 4: Growth and development (here I will also talk about Bill's deformity and Euclydean society)

Part 5: How Bill Cipher destroyed Euclydia and got his god like powers?

SO, without further ado:

What is Euclydia and what are Euclydians?

I'm gonna drop a bomb first.

Euclydia IS NOT a flat two dimensional plane. Before you load your shotguns, let me explain!

There are many proofs both in the Gravity Falls show and The Book of Bill that Euclydia isn't a flat plane like the imaginary two dimensional world from Flatland by Edwin A. Abbot.

The first one is actually Bill himself. Bill's species has complex camera lens type of eyes. Such eyes are possible in 2D world, but not on the front, like Bill has. He was born like that, so that is proof that Euclydia isn't 2D.

Next, when Bill is talking about his home in Weirdmageddon part 3, he shows an image of his home planet:

This planet has RINGS. That is COMPLETELY impossible in 2D. Even if the planet was completely flat, the rings would go through it. They would never be able to actually encircle this planet. So, if Euclydia was two dimensional, Bill's home planet would not be able to exist.

In the Book of Bill, we see image of Bill as a baby. In that image he's standing on some kind of field with grass and you can clearly see that there's grass in front of him and behind him, and that's impossible in 2D:

(also sorry for the shit quality of this pic)

But the best proof is that image from thisisnotawebsitedotcom.com that you get when you type VALLIS CINERIS in the computer. It shows Bill Cipher as a child with his parents. The parents are holding him in a manner that is completely impossible in 2D:

The image quality sucks, but you can clearly see that his parent's hands are IN FRONT OF him and he is also IN FRONT OF his parents. The position of "in front of" isn't possible in two dimensions and yet on this image the overlap happens many times. (I circled his parents' hands in red where they overlap with Bill and I circled him in blue where he overlaps with his parents. Bill's bow tie is also in front of him.).

With all that being said, what is Euclydia?

Well, just like Bill said, it's a flat world. Not two dimensional, but flat. The third dimension of Euclydia is limited somehow. Basically, in 3D, creatures are defined by 3 axis:

x axis is left and right (width)

y axis is up and down (height)

z axis is towards and away from (depth)

All three dimensional objects have both width, height and depth. Two dimensional objects have just width and height, so just x and y axis. And Bill has depth. It's a very limited depth, but it is depth nonetheless. So he's not really a triangle, more like a very thin pyramid. This is his side profile lmao:

So Euclydeans have some depth, but for whatever reason, they can't move on z axis. They can only move left, right, up and down. They also can't turn around.

This is how thisisnotawebsitedotcom.com explains Euclydian movements:

Two dimension to and fro, you always know which way to go. If you're lost, don't be afraid, in Euclydia you've got it made. Run too far right to right of frame, you'll appear on left again. Jump too high don’t cry or fret, pop up from the ground I bet. In this place, there is no fear, loved ones will be ever near. Roles and rules always flea/clear. Euclydia, we hold you dear.

So, if they move too far left, they'll come from the other side. This is actually something that in possible ONLY in non-euclidean geometry, which means that Euclydia, ironically, is a non-euclidean place. It's actually a sphere (or a similar elliptical body).

In non-euclidean geometry of the sphere, there exists something that sounds paradoxical: a straight line is actually a circle. But it's actually very easy to understand with this example:

Imagine that you're flying a plane in a straight line. You feel like you're going in a straight line, but your plane is actually following the curvature of the Earth. If you manage to fly around the entire Earth, you will appear on the same spot where you started flying. You were flying in a straight line, but because Earth is a non-euclidean sphere, you were actually flying in a circle. And both of those are true!

The plane is very very small compared to the size of the Earth. So, to the plane, Earth's curvature is so negligent that we could say that in a small radius around it the Earth is actually a flat plane. So, for example, houses, neighborhoods, even cities are built relying only on euclidean geometry (the geometry of a flat plane) because the Earth is so goddamn big.

And Euclydia is actually a whole fucking dimension. Let's say that our dimension is our universe. Our universe is approximately 93 billion light-years wide. So let's say that that's the size of Euclydia. How tiny is Earth compared to the Universe? That's why planets and everything else in Euclydia can be treated as a flat plane: every object is so small compared to the size of this giant sphere that the curvature could be completely omitted from the equation.

Now this is my theory, but I imagine that Euclydia looks like a giant soap bubble. Soap bubbles are made when two thin layers of soap molecules trap a thin layer of water:

Euclydia is the water - that thin layer is where all the planets, stars and living beings on them are located. That's why movement on z axis is so limited. The soap molecules are membranes that separate Euclydia from the other dimensions, one inside the bubble and one outside.

Since Euclydians can't move across z axis, they have eyes on their sides that can see only left and right. Their vision is limited to one dimension. But Bill's eye is located in a spot that allows him to see both left and right, but also up and down. He can see two dimensions, just like us! Here's a diagram I made, so you can understand better:

(there are stars outside too, but I didn't want to clutter this image more)

So, now that I've spent SO MUCH TIME explaining what is Euclydia, let me tell you what are Euclydeans.

Euclideans are animals (or their equivalent in their dimension). Animals are defined as multicellular heterotrophic organisms with an internal digestive tract. This basically means multicellular organisms that eat.

Euclydeans have to be multicellular because they have extremely complex structures such as: camera lens eyes, teeth, fingerprints, exoskeleton and so on. These traits cannot be achieved by a unicellular organism. And they definitely eat their food, we've seen Bill do it. So they are (their dimension's equivalent of) animals.

And how they function? What type of animal are they? Well, see you at part two, if this didn't bore you to death already!

Thank you to @forseenconsequences @extremereader and @ok1237 for asking me to do this. Hope you like it, guys!

[chanceofwhat]

@the-barefoot-hatter if you liked my other one you’re gonna love this

I couldn’t stop picturing it and laughing so I drew a little comic for my version of the non-Euclidean geometry au. ¯\_(ツ)_/¯ It’s a bit of a mess, but I enjoyed making it and I think it turned out okay considering I don’t tend to make comics. Long story short, a new player has arrived.

[typhlonectes]

[i-am-harmless]

My ONLY regret is not animating this 😔 (translation may not be accurate)

AU by @the-barefoot-hatter

Art by @the-barefoot-hatter and @megarhyssa-sp

[valyrfia]

this screenshot alone could power sports rpf for months

[paracosmicka]

spiral avatar except it’s just my mechanical engineering professor and he’s saying that these weird shapes are actually letters and that they have value and he just assumes that it makes sense to me

[knotty-et-al]

Pseudosphere and dini surface

^ Visualization about the transformation of a pseudosphere into a dini surface.

v Pseudosphere

[the-barefoot-hatter]

Geometry on a curved surface is considered non-euclidean, where regular rules about lines and angles might not apply-

AKA, the AU where you got wasted on your birthday with your Muse and one thing led to another and you REALLY screwed math and now you gotta deal with the results, FORD. (he didn't know he could that! neither did the other one! and neither remembers what exactly they did to do that!!!)

[megarhyssa-sp]

Lost time | Part 1 | Part 2 | Part 3 | Part 4 (here)

This dad shit gonna be hard

...

i have no idea how @the-barefoot-hatter plans to keep Pyramid Steve from turning into an absolute spoiled brat in the original non-euclidean geometry au that this is based off