#quaternion-group

[full-metal-furies]

me when i bring a moment of joy to the people around me whether they be strangers or lifelong friends:

nothing that takes me so off guard (in a good way) as when im just running my mouth in public and someone nearby bursts out laughing over what i just said

[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.

[afafmasoud]

Today is the birthday of my beloved niece, Marmar, who has turned four years old. Little Marmar, who always fills our hearts with joy, was eagerly waiting to wear her new dress and cut her favorite birthday cake. However, this time, due to difficult circumstances and rising prices, we couldn’t celebrate as she deserves.

The cost of cakes has surged to more than ten times its usual price, and even the simplest things like dresses and gifts have become out of reach. Yet, little Marmar still dreams of a birthday cake, a new dress, and a gift to make her happy on her special day.

If you care about Marmar and have a place for humanity in your heart, we ask for your help in raising funds to send Marmar a birthday cake, a new dress, and a simple gift to bring a smile to her innocent face. Every contribution, no matter how small, will make a big difference in Marmar’s life on this special day.

Thank you from the bottom of our hearts for your support and kind hearts. Let’s make Marmar’s birthday an unforgettable memory.

https://gofund.me/c3e162db

@fricklefracklefloof @ramshackledtrickster @gasterofficial @pompeii-for-elephants @ponpasta @pomeg-glitch @atlas-of-galaxies @mxwhore @mstormcloud @saintlauhrent @saiwestfield @saintence @samwinchesterblog @sam-the-panda-hero @samwinchesterappreciation

@andvakandi @autisticats @quaternion-group @queenofglitch @electricbluebutterflies @entraptasboobwindow @wadewilsonparker @vialism @memelovingbot @lilliesthings @prinnay @catbushandludicrous @chilewithcarnage @daxdraggon @slutdge @xenodile @xeneric-shrooms @xxang3l-trapxx @froginakettle @fallingtowers @fallingtowers @felix-the-snow-cheetah @sumikatt @ponpasta @turtletoria @unfunnynascarfan @ultimatecryptid @vampyretaemin @zombeesknees @ziilosdiidu @zukoromantic @fierycavalier @pkbunny @jadedisaster

[ostolero]

j and I are doing a roulette challenge from this Nashville hot chicken place called Dave's. 4 mild, 5 medium, and 1 reaper

each bite has me contemplating my life.

each bite has me drinking a whole 12 oz drink

@quaternion-group

[kiiamn]

What you want are non communitive number systems!

Off the top of my head, you can create number systems based on "actions" and have each number be a type of thing you can do (like, say, a series of turns on a rubiks cube) and multiplication gives you another action, which is just, do one thing then the other.

You can imagine that, even with the rubix cube example, a•b ("doing" a move b then "doing" a move a) isnt necessarily the same as b•a ("doing" a move a then "doing", a move b) and there might be reasons you would want to lable some of thease numbers as 1 or 2.

i know the "move" of not doing a move at all usually is called 1, because you get a nice thing where a•1=a, or "doing nothing, then doing something" is the same as "just doing the thing", and similarly 1•a=a

...in this case, no matter what you pick to call 2, 1•2=2=2•1, but you can imagine some reasonable choice of 1 and 2 exists where that doesnt work.

Quaternions are a famous one, where each number corresponds to a rotation and scaling in 3d space.

(its actually like. a 4d object n stuff but it doesn't really matter, i. dont know any great resources for this, wikipedia might be okay? but you can probably keyword "quaternions" and get somewhere)

Its an extension of the complex numbers, where each number is a rotation and scaling in 2d space!

(think about why i might have not been able to use the complex numbers as a non communitive number system, just based off the rotations thing)

And theres just lots of ways besides this im sure. Groups are what we call number systems that are not necessarily communitive, but are necessary associative (a(bc)=(ab)c) if you want to learn more.

Okay math side of Tumblr I need help

So I was reading Broca's Brain by Carl Sagan yesterday, and there's this point where Sagan lists off a few bizarre scientific facts as examples of how true science is often more interesting and whimsical than anything a hack pseudoscientist could make up.

One of these facts was that there's a logically sound arithmetic in which 2 times 1 is not equal to 1 times 2. This piqued my interest, so I took to Firefox and Wikipedia to figure out more, but I searched up a few things and couldn't find anything helpful.

So does anyone here know what he's talking about and how that works?

[keycrash]

semi-daily math post since people asked—

you may have heard about historical arguments in mathematics— irrational numbers, imaginary numbers, even quaternions— but one of the more modern divides is over something called the axiom of choice. an axiom is one of the base assumptions of a system of logic— things that we presume to be true so that we can rely on their logic to create new conclusions. our common system of logic is called zermelo-fraenkel set theory. (if you choose to accept the axiom of choice, it’s abbreviated ZFC to include that.) set theory is extremely foundational and has to do with how we group collections of abstract mathematical objects; one axiom in ZFC, for example, is ‘if we have two sets, there exists a union of the sets.’ for example, the union of {x,y} and {y,z} is {x,y,z}.

the axiom of choice essentially states that given an infinite collection of sets, you can make a new set by choosing one element from each of those sets. kinda abstract. kinda not as abstract as you’d think, too? but once you start thinking about choosing from infinite sets without a ‘rule’ to follow— infinite arbitrary choices— it can get dicey. it was originally controversial because some of its conclusions were kind of counterintuitive; for example, the banach-tarski paradox, which lets you divide an ideal sphere (so, infinitely divisible) into complex parts such that you can manipulate those parts into two identical spheres of the same volume as the original. there’s even a common math joke about it by jerry bona— “the axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about zorn’s lemma?” poking fun at the fact that… those three things are all equivalent to the same thing, the axiom of choice, just presented in different ways that make them seem either very intuitive or very counterintuitive!

these days the axiom of choice is widely used. i wouldn’t say ‘widely accepted,’ exactly, because axioms aren’t exactly ‘true’ or ‘false’; they’re a basis of logic we either decide to use or decide not to use based on whether it’s useful for us. (people study other systems of logic too! look up peano arithmetic). that being said, apparently it’s useful enough to have justified its existence to most mathematicians :-)

[nokmietarchive]

my friends made turtles with me. say hi to Tetsu (first one with the stop sign, belongs to @quaternion-group) and Bietris (second one, belongs to @keeperesque)

they are also humans-turned-turtles and both of them are absolutely fucking lying about having gotten shorter it's just a running bit they commit against Frisbee cuz she's so excited about gaining 4 inches of height post-mutation

[dmdorematrixabc]

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[janmusija]

I disagree; numbers have to be elements of sets with, um, a certain kind of algebraic structure, the exact qualifying details of which I don't have laid out.

Examples of structures, listed by the "numberiness" of their elements:

Most numbery structures: the naturals, the reals, the integers

Complex numbers, quaternions, split complexes, etcetera

ordinals, surreals

combinatorial game values, I guess nimbers

Least numbery things you could still argue are numbers: many other groups, fields, rings...

Non-numbers: Polyhedra, Well-Formed Formulae, graphs, combinatorial games [as opposed to their values]

I'm a maximal inclusivist when it comes to what I consider to be a number. if you can do math with it it's a number. it only makes sense to call something a non-number in math if you're distinguishing between two types of mathematical object and one of them doesn't have a good name besides "number".

[jonathankatwhatever]

It’s now 25 Jan 2024, and I’m still not entirely comfortable with this keyboard, but it is getting better. I find it goes better if I rest my left elbow and keep the right suspended. This keeps the right from getting lazy and not reaching the arm to the keys. With the keyboard tilted the right, the left’s motion is easier. I wonder how many variations this will permute through.

I can now get into some really tight folds and can squat nicely below parallel again.

This was not what I had in mind when I started typing. Caught one: easy to tilt to the right, and that lowers my elbow an inch or two, and that makes the move toward the upper right, which is the delete key, easier. But that’s a lazy position because it makes that one move easier and the others worse. The promise of an undeveloped land versus the reality of the paths it took.

I’m in some scary stuff. Do our paths ever coincide? I have no idea what because I can’t impose a version as though I define how reality occurs. I saw these VR images and they show something interesting, which is the genericization of your image so it fits a blunt sexual fantasy, meaning it is not you but a generated image of viewpoints or perspectives, those multiple takes which are pairs, which all synthesize to very straightforward Attachment of a structure that says looks like this person to any sexual role. Like the I Dream of Jeanie concept in which the issue is the complexity of the story versus the cartoon quality of the fantasy, with that solved through this process, which is Recombinance, because that is how the elements recombine, which is where the associative comes in, meaning this provides a solution to the ‘what fits between you and the mirror’, meaning that if we take you and the mirror then what fits between is the construction which is the solution which fits. That’s how you build toward and away from ideals.

That is, because the structure is within a structure, the old container idea often used in portable or modular programming, so the structure itself is composed or constructed. I keep saying the same thing in slightly different words. Keep right elbow out or the delete key squeaks.

I was looking at Dedekind groups, which led to the quaternion group, which led to the cycle graph, which led to the understanding that this maps to IC and Triangular over gs. It also maps to the Irreducible, which in the elements of the Q8 group are represented as -1 and 1. That answers a minor technical issue I had with the idea, which is why they represent the identity element as they do, typically I see as an e with a bar over it to indicate that it’s the commuter because if you square the other terms, meaning i, j and k, then you get that e bar, which is -1. To be blunt, that means you can switch at -1. This is an elementary example of pairing, isn’t it?

I really and truly forget that I have no idea what happens next in my head. I had no idea this was coming, but it is of course the exact answer needed and the answer which fits to the count of 2 perspective. Don’t have a name for that counting not of 0 to 1 but of 1 to 2, and thus from 2 to 1 rather than 1 to 0. That basic counting shift literally enables almost everything.

One reason is that pushes the count from 1 to 0 off a step, which then allows pairing across that gap to whatever Ends one sees or which occurs if you’re being fatalistic. Remarkably clear insight. This is how Joana thinks, and all I’m doing is accepting that this

I need a break. The illness situation here is not good.

[yoursoldoutforevermore]

Matthew 27:27-31 (Part 1 of 4)

Matthew 27:27-31 Jesus is beaten and mocked (Part 1 of 4). “Gathered the whole garrison around Him.” They only needed a regular group of four soldiers – called a quaternion – to carry out the execution. Yet they “gathered the whole garrison around Him.” It wasn’t to prevent His escape. It wasn’t to prevent a hostile crowd from rescuing Him. It wasn’t to keep the disciples away. “Take heed of…

[msexcelfractal]

This one is the quaternion group. The trick is finding the right initial conditions to showcase the weirdness of the group. Here my top row repeats i j k i j k i j k ...

In the closeup you can see the Cayley table I painstakingly transcribed from wikipedia. I have to code each element of the group as an integer! 0 represents 1, and 1 represents -1... we stay silly XD

steven wolfram generated sierpinski's gasket with a cellular automaton. the rule of action for this automaton is logical xor. I notice loxigal xor is equivalent to addition modulo 2. I make a bunch of art, mostly in addition modulo 2 or 3.

I think to myself, modular addition is the operation of cyclic groups. Why restrict myself to cyclic groups? why restrict myself to abelian groups even? Just because Excel's architecture assumes cell contents are real numbers? Pah. I write a formula to iterate wolfram's sierpinski rule using arbitrarily defined group operation. B11=INDIRECT(ADDRESS(A11+3,B10+3)). This will compute B11 as group operation A11⊕B10, where ⊕ is designated by the cayley table whose upper left corner is C3. Since B10 is above B11 and A10 is to B11's left, the pattern formed by ⊕ cascades down and to the right.

Here's a sierpinski gasket "modulo" [the dihedral group of order 6]. The group D6 is generated by the actions of [horizontal reflection] and [120° rotation] on a plane.

[dat-physics-gal]

Thank you for the answer. While I agree that the properties you described are useful, quaternions have yet to fight for attention with established methods like vector spaces, Pauli matrices and rotation groups in general. Do you think that the switch would be possible and beneficial?

That’s what i’m hoping to find out / prove in the future.

[ostolero]

"seasons beatings" -- @amigarobot

ft @quaternion-group

[jonathankatwhatever]

It’s 18 August 2023 and I’m having a problem with anxiety. It seems to be either coming from you or from a perceived lack of you, and I can’t tell which. It’s intense.

I’ve been glancing at stuff like Yang-Mills Theory and the Solovay Model, and they both make sense. Which is scary because that was only implied before, like I knew they should fit to HC but I didn’t see how. Example is the inaccessible cardinal required in the Solovay Model: I couldn’t get what that meant until I was able to see the 0-1-0 construction and the top down 1 with at least some clarity. Otherwise, it was just a statement that we assume a cardinal we can’t reach.

I looked at glueballs, which I don’t remember even hearing about. They’re gluons, and those generally relate SBE to SBE3, meaning they come in 3’s, and that invokes IC and f1-3 and the entire 13 conception and more.

I say this with more confidence than I feel. I don’t know the underlying calculations, but I know the higher level statements are true. I could not say that before. I couldn’t say I understood why the explanations are groups, like SU(3)*SU(2)*U(1). SU(2) is quaternions, which I see as D3-4//D4-3, while SU(3) is, if I can translate this, the I//I of a gs, meaning D8, and which thus ‘encloses’ SBE3 in the drawings. U(1) is the circle group. So I translate that very loosely as CR of these 2 forms of I//I, if you take say U(1) times SU(2) or SU(3) or both. The both is hard to articulate. I have to let that sit. Electroweak is the first. And the combination is the third. And that all makes sense, though the words aren’t fully there.

This makes me almost queasy. Like I’m having trouble sitting up because it’s twisting something in me hard.

I looked into Vitali sets, which seem to be at the root of the unmeasurable idea because they were the response to Lebesgue measuring, and the notation is annoying but the idea fit is exactly what HC describes.

And now I’m actually shaking. Like me entire body is shaking. I hope this doesn’t continue. Or get worse.

————————

A cross-dimensional algebra. By us.

Is one which would relate algebraic structures. From polynomials of one variable to rings and groups. Each of those forms embodies a perspective with its own sets of problems and solutions, meaning it is constrained, that there are limits to its power. We describe how to cross those constraints using a series of simple mechanisms which generate a dimensional structure, called D-structure. D-structure enables the creation of finite constructions using infinite processes.

Hard to write at that level. Can’t just say a perspective. Have to say has and leave the perspective til later: so each of those forms has its own sets of problems and solutions. The reason this approach works, though, is that it embodies the idea itself, that there is something which generates processes out of which we can identify a certain number, kind, and type, and that this something appears within the processes and within the results. That is itself a manifestation of the basic inner and outer Boundary conception. That means this isn’t a complete explanation.

My intuition at this point gets confused. The usual way is to build up a long explanation, like in a classroom, but I’d rather have people graps the basic concepts, so images. I see, for example, the formation of D1 out of D0, which means from a dimensionless point to an End, then CR to D2, etc. And that has to go with 1Segments and 1-0Segments.

And that means a new way of describing CR is in there. But of course I just realized we are proving stuff like Zorn’s lemma and other stuff, including the Axiom of Choice, which develops because this is finite construction out of the infinite. The puzzle is extremely complicated and I really need help.

So for example, show and describe the basics and then take on individual parts is obvious, but which parts in what order? Say we name Boundary: requires 1-0Segs.

This seems nuts: I’m seeing a 1-0Segment map to other lengths while still mapping to 1. That’s obviously true in some ways, but in the context of a circle or sphere, this means a 1-0Segment’s infinities build dimensional spaces and the circular Boundary. That’s a map of 1 to the scaled 1. I’ve not seen that before, I think. We’re saying a 1-0Segment enables counting 1 to other scales of 1, including Pi. That’s certainly true with e, because this is the Recombinance needed to make 1 + 1. Okay. So if this is true, then maybe this adds to the understanding so this description for others can proceed.

Let me try to think. We need Recombinance. The logic is the same as with quaternions: you need a 4th D to multiply the 3. It’s the same as in LayerView. It requires construction of grid squares, which act as amplitude and other forms of spaces, including vector spaces. So, we’re pointing at a 1-0Segment, and that means suddenly I’m seeing Poisson point processes and distributions. Need gs construction to have those. So I guess this means we are ready to say that gs construction generates a lot of mathematics. And more. Like Physics but also, since we’re constructing finite constructions, the intangible aspects that connect through the Boundary process to the inner core and vice versa, the translation method by which Things interact with themselves and with the external world.

That is no longer an absurdity to say. It never was, but now it’s a basic extension of the provable fact that everything is a construction. So putting D-structure first makes sense to me.

This is exciting, though the cost to me psychically is intense: just had a conversation about how if we can’t get a good price, then I’m not letting the house go to auction, meaning death. Went well. It’s an actual deadline: if HC doesn’t happen, if we don’t get a good offer, then I’m gone. I tell myself I have to see down that road to see down the road I hope to take, because they’re 2 directions on the same highway. Well, the bad End is well defined and the work End is blooming like crazy. They say 2 out of 3 aren’t bad, but they are if one of the three is the Bad End.

There’s some deep truth in there. The same process of the Triangular End, so an fD me to you, etc. Bad End // Good End are rather definitely Irreducible. Where to define Irreducible? I see this as cases: one case is Irreducible when constructed to a plane, so rotational. Another is to higher dimensional space, so flips. Both require a D-count higher, like D2-3 or D3-4 enables D3-2 and D4-3. A flip, for example, of a square diagonally requires CR along the sK or zK. That process is infinite and n-dimensional, but all that is contained because that orthogonal circle form maps to the sK or zK. If we want, then we can switch projections to see the CR, and then switch back. Is this sensible?

Doesn’t seem complete, but it is sensible. So for example, D3 relates to D6 and not just to D4. Wow. That means Recombinance has forms too. One is the LV form: the szK can be read as the value or meaning of the wings as they balance, as grisp flows through the quadrants, with that meaning both generated internally and generated within the larger szK count, meaning it fits or doesn’t fit, both in individual iterations and in lots or even all iterations. This connects to stochastic processes because those happen within a space, meaning within this LV approach because this describes the restraints on the actual processes so they generate random within the constraints.

My thoughts can turn very bad very fast.

Another form is like D3-D6 // D6-D3. That doesn’t mean we construct the soap opera evil twin, but rather that everything which makes D3 can be unwound except it can’t because of the other D3 in the D6. Can I do better than that? I need a break.

[isomorphismes]

Lochlainn O’Raifeartaigh, group structure of gauge theories