I have a little problem wrapping my head around quaternions, in particular I have problems about how to pair the usual "3D algebra" with the theoretic vision of a quaternion.

I know that informally a quaternion is a structure composed of 3 imaginary parts and 1 coefficient/real part .

Now back to elementary number theory, I know that I can visualize a pair composed of an imaginary number + a real part in $\mathbb{R^2}$, the so called complex numbers can be visulized this way with all their properties.

The problem comes when I have 3 imaginary parts and only 1 real, even more, I have to somehow group this into the same family with Euler angles and cartesian coordinates .

I know that mathematicians usually skip this part quite easily by telling you that $\mathbb{i}$ is just a "device" but I know a have a quaternion that is effectively using 4 dimensions, working in a 3 dimensional space while being comparable to other " native $\mathbb{R^3}$ solutions " such as Euler angles .

My question is why quaternions works so well in a 3D metric space and is a quaternion a 4D object ?

  • $\begingroup$ The quaternions are a number system that extend the set of complex numbers, namely $1,i,j,k$. An object could have quaternions for measurements (since they are essentially numbers), but there is no geometric object itself by the name 'quaternion'. There is a wikipedia page on it. $\endgroup$ – ghosts_in_the_code Jan 10 '15 at 15:47
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    $\begingroup$ @ghosts_in_the_code everyone can write an article on wikipedia, doesn't really sound like a reliable source, plus I posted this here, not as a new discussion on the wikipedia forums so I don't think that wikipedia is relevant . $\endgroup$ – user2485710 Jan 10 '15 at 15:56
  • $\begingroup$ I'm just answering your last question (Is a quaternion a 4D object?) by saying it is not a geometrical object, but a set of numbers. $\endgroup$ – ghosts_in_the_code Jan 10 '15 at 16:07
  • $\begingroup$ What is the right name for the space occupied by a real number?! "right name"="dimension"? $\endgroup$ – rschwieb Jan 13 '15 at 14:25

The quaternion algebra can be derived from a clifford algebra built upon 3d space. This construction proves the usefulness of quaternions for rotations as well.

Clifford algebra of 3d Euclidean spaces

The cliffod algebra of 3d space is built from a "geometric product" of vectors. Let $e_1, e_2, e_3$ be the basis of your 3d space. The geometric product obeys the following properties:

$$e_i e_j = \begin{cases} 1 & i =j \\ -e_j e_i & i \neq j\end{cases}$$

The product is also associative. This means that you get some products that can't be reduced to scalars: any individual vector can't be reduced, as well as products like $e_1 e_2$ or $e_1 e_2 e_3$. In general, these objects are called multivectors.

Clifford algebra and rotations and reflections

Given the geometric product, you can write rotations and reflections in a more compact manner. For instance, if $n$ is a unit vector normal to a plane, then any vector $a$ is reflected across that plane by $-nan$.

Any rotation can be performed by two reflections, so given two unit vectors $m$ and $n$, a rotation takes the form $mnanm$. The quantity $q = mn$ takes the following form:

$$q = mn = q_0 + q^{12} e_1 e_2 + q^{23} e_2 e_3 + q^{31} e_3 e_1$$

That looks like a quaternion, doesn't it? Indeed, see that $(e_1 e_2)^2 = -1$ and the same for the other basis "bivectors". In clifford algebra, objects like $q$ are called "versors" or "rotors", but they are in direct correspondence to quaternions, obey all the same multiplication rules, and behave in exactly the same manner in all the usual respects.

So, from clifford algebra built directly on top of 3d space, we can derive something identical to quaternions. We can do so in a way that makes the connection to rotations manifest. The geometrical interpretation of a "rotor" or "quaternion" itself may be a little more difficult to conceptualize, but we can see how this object stems from a composition of reflections at least.


"Is a quaternion a 4D object?"

The set of quaternions has four real dimensions, yes. The way you are asking the question makes it seem as if you think that dimension is an inherent property of an element of the vector space, but that is not really the idea. The usual thing to do is to model 3D geometry in the space of quaternions with zero real part. Would the quaternions in this set suddenly be less than 4D, if this were the case? So you can see, dimension is more about the whole set than single elements.

"Why do quaternions work so well in 3D metric space?"

Much like the complex numbers are able to capture Euclidean geometry on the plane, the properties of quaternions are enough to capture most 3D Euclidean geometry.


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