For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.

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44
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5answers
1k views

Why are the solutions of polynomial equations so unconstrained over the quaternions?

An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
26
votes
2answers
1k views

Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions? Here a division algebra is an associative algebra where every ...
22
votes
2answers
697 views

What do we lose passing from the reals to the complex numbers?

As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we ...
19
votes
2answers
896 views

What lies beyond the Sedenions

In the construction of types of numbers, we have the following sequence: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$ or: $$2^0 \mathrm{-ions} \subset ...
18
votes
5answers
1k views

Can Euler's identity be extended to quaternions?

Euler's identity is $e^{i \pi} + 1 = 0$, a special case of the Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$, where $\theta = \pi$ (half-turn of the unit circle). It is commonly described ...
17
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2answers
433 views

Is the set of quaternions $\mathbb{H}$ algebraically closed?

A skew field $K$ is said to be algebraically closed if it contains a root for every non-constant polynomial in $K[x]$. I know that this is true for $\mathbb{C}$, which is the algebraic closure of ...
16
votes
9answers
2k views

Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
16
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7answers
1k views

How can one intuitively think about quaternions?

Quaternions came up while I was interning not too long ago and it seemed like no one really know how they worked. While eventually certain people were tracked down and were able to help with the ...
13
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6answers
3k views

Real world uses of Quaternions?

I've recently started reading about Quaternions, and I keep reading that for example they're used in computer graphics and mechanics calculations to calculate movement and rotation, but without real ...
13
votes
1answer
188 views

Why is the dimension of $SL(2,\mathbb{H})$ equal to $15$?

Let me ask a very basic question which is inspired by reading M. Atiyah's "Geometry and physics of knots". Could you explain me (or give a reference to) the definition of the special linear group ...
12
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1answer
249 views

Quaternionic veronese Embedding

I know that the complex projective line $\mathbb{C}P^1$ can be embedded in the complex projective space $\mathbb{C}P^n$ (Veronese embedding). For example, $\mathbb{C}P^1\rightarrow\mathbb{C}P^3$ is ...
11
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4answers
761 views

Quaternions: why does ijk = -1 and ij=k and -ji=k

Currently i am studying quaternions. I do understand that i, j and k are imaginairy numbers. so $i^2 = j^2 =k^2 = -1$. But I could not understand this: ...
11
votes
1answer
219 views

The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
11
votes
1answer
104 views

How to raise a number to a quaternion power

Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2ln(z_1)}$ and $ln(z_1$) can just be found using: ...
10
votes
2answers
597 views

How do quaternions represent rotations?

I wonder how $qvq^{-1}$ gives the rotated vector of $v$. Is there any easy-to-understand proof for it? I was on Wikipedia, but I could not understand the proof there because of the conversions. Why ...
8
votes
3answers
103 views

Quaternions as an Algebra

I'm lacking some vital understanding about quaternions and algebras in general. If we first define $V=\{a+bi+cj+dk|a,b,c,d\in\mathbb{R}\}$. Then we define scalar multiplication, vector multiplication, ...
8
votes
2answers
179 views

Why is $\mathbb{H}\otimes\mathbb{H}\cong\text{End}_\mathbb{R}\mathbb{H}$?

When I first learned of the quaternion algebra $\mathbb{H}$, the most concrete way to get a grip on the ring of its endomorphisms $\operatorname{End}_\mathbb{R}(\mathbb{H})$ was to view them as ...
8
votes
1answer
440 views

Why are properties lost in the the Cayley-Dickson construction?

Motivating question: What lies beyond the Sedenions? I'm aware that one can construct a hierarchy of number systems via the Cayley-Dickson process: $$\mathbb{R} \subset \mathbb{C} \subset ...
8
votes
2answers
242 views

Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$

The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup $$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
8
votes
1answer
291 views

Splitting of quaternion algebras

A rational (definite) quaternion algebra is an algebra of the form $$ \mathcal{K} = \mathbb{Q} + \mathbb{Q}\alpha + \mathbb{Q}\beta + \mathbb{Q}\alpha \beta $$ with $\alpha^2,\beta^2 \in ...
7
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1answer
3k views

Quaternion distance

I am using quaternions to represent orientation as a rotational offset from a global coordinate frame. Is it correct in thinking that quaternion distance gives a metric that defines the closeness of ...
7
votes
2answers
13k views

How do you rotate a vector by a unit quaternion?

Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to ...
7
votes
1answer
144 views

Quaternion group associativity

Consider the eight objects $\pm 1, \pm i, \pm j, \pm k$ with multiplication rules: $ij=k,jk=i,ki=j,ji=-k,kj=-i,ik=-j,i^2=j^2=k^2=-1$, where the minus signs behave as expected and $1$ and $-1$ ...
7
votes
2answers
95 views

Is it possible to use the imaginary components of quaternions to facilitate calculation of vector cross products?

It has come to my attention that the cross products of the vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are almost identical to the products of the imaginary components of quaternions $i$, ...
6
votes
3answers
203 views

4 dimensional numbers

I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the ...
6
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1answer
206 views

How to identify $SL(2,\mathbb{C})/SU(2)$ and the hyperbolic 3-space?

I know that every coset representative $g\in SL(2,\mathbb{C})$ for $SL(2,\mathbb{C})/SU(2)$ can be chosen of the form $$ g = \left( \begin{array}{cc} \sqrt{t} & \frac{z}{\sqrt{t}}\\ 0 & ...
6
votes
0answers
376 views

Using Padé approximants for the quaternion exponential

On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for ...
5
votes
3answers
2k views

Euler angles and gimbal lock

Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
5
votes
2answers
91 views

Why is $(-1) \cdot j = j \cdot (-1)$ for quaternions?

I'm currently trying to understand the following part of a script (translated from German to English). It is the first part where quaternions get introduced, so I don't know anything about them except ...
5
votes
3answers
71 views

Show that for $\forall a\in\mathbb{H}, \ \exists b \in\mathbb{H}: ab =ba = 1$.

Show that $\forall a\in\mathbb{H}, \ \exists b \in\mathbb{H}: ab =ba = 1.$ I am pretty sure I can easily google the multiplicative inverse in $\mathbb{H}$, but can you give me a hint on how to ...
5
votes
1answer
334 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
5
votes
2answers
322 views

Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
5
votes
1answer
734 views

Using quaternions instead of 4x4 matrices for transformations

I'm interested in implementing a clean solution providing an alternative to 4x4 matrices for 3D transformation. Quaternions provide the equivalent of rotation, but no translation. Therefore, in ...
5
votes
1answer
205 views

Why do Quaternions and octonions exist?

Ok so I have known about imaginary numbers for quite some time now. I also understand why we want them to exist (to have a solution for $x^2=-1$). I also remember reading that the complex numbers are ...
5
votes
1answer
72 views

Is there much theory developed for analytic functions of quaternions or of octonions?

The quaternions are associative, so nonnegative integer powers of quaternions are well-defined, and one can consider analytic functions on $\mathbb{H}$ (functions that are given locally by power ...
5
votes
3answers
743 views

On multiplying quaternion matrices

Both matrix multiplication and quaternion multiplication are non-commutative; hence the use of terms like "premultiplication" and "postmultiplication". After encountering the concept of "quaternion ...
5
votes
1answer
1k views

Averaging quaternions

Given multiple quaternions representing orientations, and I want to average them. Each one has a different weight, and they all sum up to one. How can I get the average of them? Simple multiplication ...
5
votes
3answers
170 views

How to understand and create quaternions?

I have to multiply two quaternions to calculate a so called spherical linear interpolation between two $R^3$ coordinate systems within the interval $t = [0, 1]$. I understand how to do the ...
4
votes
4answers
77 views

Show that $Q_8$ can't be embedded in $M_{2 \times 2}(\mathbb{R})$ as a group.

So, suppose that we're working in a field $F$. Consider the ring $M_{n \times n} (F)$ which is the set all $n \times n$ matrices with entries in $F$. Is it possible to determine whether a matrix ...
4
votes
2answers
105 views

Does the square root of $i$ necessitate quaternions?

The square root of i is $\frac{\sqrt{2} + i \sqrt{2}}{2}$. But how is it valid to use a number in expressing the square root of that number?
4
votes
2answers
268 views

Square roots of $-1$ in quaternion ring

In this Wikipedia page it is said that the square roots of -1 in the quaternion ring are the elements of the imaginary sphere. I don't understand why this is so. I don't understand the system that's ...
4
votes
2answers
159 views

$x^2+1=0$ uncountable many solutions [duplicate]

Possible Duplicate: Why are the solutions of polynomial equations so unconstrained over the quaternions? Coudl someone explain me the following: Why should $x^2+1=0$ have uncountable ...
4
votes
1answer
3k views

Rotate Quaternion A by 180 degrees

Suppose you have an arbitrary quaternion - call it A - how do you rotate it by 180 degrees? Is there a way to do this without convert to angle-axis representation, i.e., keep it within the ...
4
votes
2answers
180 views

Trinonions, Quaternions, Quinonions, Sextonions, Septonions, Octonions

There are quaternions and octonions and even sextonions but what about trinonions, quinonions and septonions. Are there 3, 5, and 7 dimensional algebras which could be called trinonions, quinonions ...
4
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2answers
1k views

Combing rotation quaternions

If I combine 2 rotation quaternions by multiplying them, lets say one represents some rotation around x axis and other represents some rotation around some arbitrary axis. The the order of rotation ...
4
votes
3answers
258 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
4
votes
3answers
446 views

How to get a part of a quaternion? e.g. get half of the rotation of a quaternion?

if I have a quaternion which describes an arbitrary rotation, how can I get for example only the half rotation or something like 30% of this rotation? Thanks in advance!
4
votes
1answer
132 views

How to Perform Quaternion Multiplication

Everywhere that I've looked, it seems to be assumed that $i^{2} = j^{2} = k^{2} = - 1$, along with the other rules of quaternion multiplication. However - for my homework - I'm being asked to show ...
4
votes
1answer
287 views

Uniform distributions on the space of rotations in 3D

I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions. The Haar measure on $SO(3)$. The uniform ...
4
votes
1answer
49 views

Quarternionic Analysis

What is/are the current understanding/opinions about Quarternionic Analysis as a generalization of Complex Analysis with respect to a "Quarternionic Residue Calculus" (if such a thing exists)? i.e. ...