#quaternions

[noelements-setempty]

What if we had three square roots of -1? Haha, ijk.

[blake447]

I realized I hadn't shared this here yet! Here's my 3D printed Quaternion Julia Set! See I first learned of these things reading about it from a pretty well known graphics programmer Inigo Quilez. Theirs are a lot prettier! But yeah, if you're familiar with Julia Sets and the Mandelbrot set already, this is basically that but we use 4D complex numbers called quaternions instead of the regular old complex numbers.

The original shaders I used to render them were unity CG/HLSL implementations, but this particular one is from a GLSL implementation over on my shadertoy you can find here: https://www.shadertoy.com/view/tdt3W8 It isn't exactly the same one that I've printed here (I've long lost the exact seed) but it is reasonably close. The way I printed it was I stole some marching cube code for blender and just plugged in the SDF function derived by Inigo Quilez, tweaked the values and eventually got a mesh I can print!

This uh, isn't the one I used lol. It did take a few tries to get one that was both visually interesting and also printable. In fact i wasn't even using the marching cube algorithm at first. I was using Poisson Surface Reconstruction with a python script that casted points to form a point cloud. Basically I was attempting to create a mesh like you would with photogrammetry, just with an abstract object rather than an actual thing or place.

The results were, well not good lol.

*Continues digging through box* I know its around here some where, I should have the one that works. Okay this still isnt it but this one is using the same method, I just wanted to use this for a vrchat world instead of using it for 3D printing. It gets the point across lol

But yeah. 3D printing is really cool if you're into a bit of math

[silverbridge-harbor]

favorite bridge and metal?

bridge: this part in this live version of this song where they take the normal bridge and replace it with the destruction of the universe and slowly reconstruct the void into a wall of sound:

metal: everything clutch has ever done:

hmmmmmm or im the mountain:

...or grim reefer by cloud:

ooouuugghhh i cannot decide on one

as recompense, have this actual answer: broom bridge, in ireland, where william hamilton figured out quaternions and did math graffiti:

i² = j² = k² = ijk = -1

and naquadah from stargate sg1 is pretty sicc

[underwhelmedandoverstimulated]

Hey math Tumblr, how normal is the experience of finally interacting with maths education that actually explains why things are, and then being irrationally angry, because "if you'd just told me that 20 years ago I would have understood!"?

Like, I am 35 and no one every explained that complex numbers are just a perpendicular axis to the real number line. That one piece of information let me understand imaginary numbers and no one told me!

I was randomly watching a video on quaternions, and I'm listening to them explain why it works and I'm nodding and following. And then they show the fucking equation. I'm sitting there in shock, because I know these equations. I learned how to do fucking quaternions and no one even told me that was what I was learning much less how or why it worked; just here's an equation memorize it and use it for these questions. I'm furious.

[dat-physics-gal]

Heyo, I'm also a physicist! I study cosmology and particle physics and I should be doing a phD in cosmology next year! What do you study in physics? 👀

To be honest, not quite sure yet, though i lean heavily toward the theoretical side. Next semester i'll do my Bachelor's degree, so you're ahead of me if you're already at it to get a phD.

I'm fascinated by quaternions as a system though, and would love to make models with them replacing standard 4-vectors, to see where i can leverage their inherent structure and field properties.

I know it's presumptuous of me, but i do have the cliché hope that i may do my part in unifying gravity with quantum field theory. Eventually, and not alone obviously. If even a tiny bit of the groundwork i hope to help lay helps someone in the future to do that, then i'll be satisfied.

[poppiesandpromises]

[wolframcarbide]

[endofunktor]

All numbers are imaginary. Some are useful.

So, @lipshits-continuous (I like your blog very much!) was bringing to my attention someone who was, let's say, a little frustrated about having to learn "imaginary numbers". You know. They're not even real.

I don't want to write about that person specifically, I don't know them at all.

But.

My siblings in crisis.

There are no "real" numbers in this world. The act of counting is already an abstraction. If you have an apple and a pear, then counting them as two pieces of fruit is an act of forgetting all the properties that distinguish them and remembering only those that unite them. Same goes for two apples of course.

So the number 2 is already entirely imaginary. Made up. Nonsense.

This gets worse very quickly, by the way. You can't have negative apples! Sure, you can have positive fractions of apples, but I can assure you: Two halves of an apple do rarely make one apple, except when they are the two halves of the same apple. And even then: Having them cut up makes it impossible to rejoin them. Think about that.

Now you know where this is going, because let's talk about the so-called ""real"" numbers. Nasty little things. Have a look at the irrational numbers. First, there are comparatively tame irrationals: Algebraic numbers. These arise as roots of integer polynomials. You know, like our favorite, the square root of 2, the positive zero of X^2 - 2.

Did you know that the Pythagoreans believed that everything was in integer relation to each other. (For example: the lengths of a rectangular triangle could be 3-4-5, meanging one cathetus is 4/5, the other 3/5 the length of the hypotenuse.) And that when someone found out that sqrt(2) was indeed not in integer relation to, say, 1, they had him murdered? That didn't change the fact that they were wrong though. (This story is not true, of course, like all good stories. You could say: it is imaginary, but nonetheless an interesting tale to tell!)

And now, the non-algebraic, so called transcendent numbers. Like pi. We know (the abstract concept of) pi well enough to calculate the circumference of the observable universe up to the accuracy of a neutrino (or so they say, I'm not a physicist). We don't have to know it any better. We could quit at however many digits we know and that would be pi. Perfectly rational. Because if we are looking for """real""" numbers only, why should we ever even concern ourself with those nasty things?

Did you know that transcendent numbers make the vast majority of real line? Of course you do! Rationals are countable, integer polynomials are countable and thus are their roots. The real line is famously uncountable, so must be the transcendent numbers.

So. Our so called """"real"""" numbers are mostly non-precisely calculable numbers. (Arbitrarily precise, but not precise.)

Do you know what kind of numbers are used in electrical engineering? Me neither because I'm not an electrical engineer. That's right, complex numbers! Actual engineers have to actually work with the actually imaginary square roots of negative one.

Maybe Numbers aren't a thing of this world.

Maybe Abstraction is a thing of our worldview.

And maybe we can learn to cherish abstract math the way we learn to cherish abstract art: Not always a true representation of reality, but a thing of interest and beauty in itself.

[stanwixbuster]

fuck with me

[shwoo]

I figured out what calculations to do to transfer animations from the Bugsnax player rig to an NPC rig, and put a few onto the Filbo model I modified into the Journalist, but I'm not sure what to do with it now. I've got their idle animation, their walking animation, their crouching animation, their crouching and walking animation, some jumping stuff, and their on fire animation, but maybe there's some others I can add? A lot of the tool using animations clip through their body, so they don't look as good from a third person perspective.

Their face almost never moves because the player doesn't have a face to animate, which is also a problem. I guess I could... learn to actually animate. Or steal another character's face animation, like I did with the default expression. This model can do expressions, but that's because expression animations are the same for every character, so I just used the ones from Filbo's animation file.

I think their shoulders are a bit janky still too. I should see if I can fix that. And I'm not totally happy with some of the eye colours.

Heheheh, look at 'em go. They don't know they're trapped in Blender for the foreseeable future.

[vermillioncrown]

long external meeting today w ppl that are ex-lab members (previous supe + a senpai)

i was doing derivations during downtime and my ex-supe joked upon seeing my Chaotic Papers (as denoted by current supe) "it's like a beautiful mind"

but then the topic got into computational fluid dynamics regarding someone's thesis and their use of quaternions and he immediately made a face and declared "that's definitely not a beautiful mind 😬"

[dat-physics-gal]

What would be the major benefit of using quaternions instead of 4-vectors?

The full answer is more involved, but in a nutshell, quaternions form a field. They're not just a vectorspace, they're a field in and of themselves. Which gives you perks like a division algebra.

Also, the multiplication structure of pure vector quaternions with scalar part 0 encodes both the dot and cross product directly.

And spinors arise naturally instead of having to be constructed artificially.

Generally, they have an inherent structure, which has advantages.

Vectors are like bags of numbers, while Quaternions are numbers.

[fastfouriertransfem]

Hi

actual announcement

new tag system !

#real - posts that are the limit of an infinite sequence of rationals

#imaginary - posts that square to a nonpositive real

#complex - posts representable by the sum of a real and an imaginary

#rational - posts representable by the division of an integer by a nonzero integer

#integer - posts that are an equivalence class of ordered pairs of naturals with the equivalence relation being the difference of the two but that's circular so do a thing with addition and stuff to define the relation without using negative numbers or anything yeahg

#natural - something something axiomatic (0 is natural btw)

#quaternion - something something rotation

#octonion - something that is not commutative or associative

#sedenion -

#transfinite - finite but trans (better way of saying infinite)

#art - art is a number post now (just take the binary of the image or whatever) (this reminds me of https://en.wikipedia.org/wiki/Illegal_number)

ok real tag system:

#real - posts that are so true so real (copyright held by Rosemi Lovelock, clipped by RandomVtubers_foreveryone on Aug 18, 2022)

#imaginary - random stuff

#complex - idk

#rational - stuff i wanna look at / be reminded of later

#integer - stuff related to computers and stuff ig

#natural - will come up with this later

#quaternion - will come up with this later

#octonion - ninomae ina'nis or something

#sedenion - idk

#transfinite - /trans(?!gender)/

#art - art (includes visual, auditory, written, combination, etc) (i will retroactively tag this later (i already tagged some) (when i do, i will move these parentheticals to the changelog post))

changelog: https://www.tumblr.com/fastfouriertransfem/734227254388883456/hi-changelog

[livelifelater]

They can also doodle in the 4th dimensional plane #fractal #fractalart #generativeart #generative #digitalart #digitalillustration #4thdimension #quaternions #math #mathart #scienceart #science #abstractart #abstract #trippyart #darkart #psychedelicart #doodle #algorithmicart #experimentalart #glitchart #computerart https://www.instagram.com/p/CnkG60Fukcf/?igshid=NGJjMDIxMWI=

[0utrenoir]

I know i will hate myself for it but i can feel i am on the verge of asking my teacher to explain quaternions to me which would add at least 6 months of lessons when i asked for us to be done in less than a month.