On this page, I’ll discuss a geometrical perspective of quaternions that I don’t completely understand, and haven’t found explained to my satisfaction.
To understand this, you need to know something about complex numbers, and also linear algebra, and something else that I don’t know myself.
Multiplication by a complex number can be viewed as a special linear transformation of the complex plane. To understand how it is special, let’s first look at general linear transformations of the plane.
A linear transformation of the plane onto itself is a geometrical transformation of the Euclidean plane that maps linear structures to linear structures: 0 is mapped to 0, lines are mapped to lines.
These transformations can be decomposed into combinations of simpler transformations of just a few categories. Depending on how you want to do it, there are just five categories: dilation (or scale), stretch, rotation, shear, flip. (Note: One can say a dilation is a special stretch, and a rotation is a combination of two flips. But this categorization suits our purposes.)
Here are some graphical illustrations of these transformations, applied to these objects in the plane:
A linear transformation effected by multiplication by a complex number consists precisely of a dilation together with a rotation.
These are precisely the transformations that map triangles into geometrically similar triangles. Under such a transformation, a triangle may be rotated and its size may be changed, but its angles (and the order of its angles) are unchanged.
This is the linear case of a conformal mapping.
In numbers of dimensions higher than 2, more things can happen, of course. However, a very nice result of linear algebra (e.g. the Jordan Canonical Form) is that, the above categorization of linear transformations is in a sense complete for any number of dimensions. Any weirder geometric behavior of linear transformations is due to the fact that one can combine the above categories in different ways in different directions.
Multiplication by a quaternion can be viewed as a linear transformation of four-dimensional Euclidean space onto itself.
However, the geometric behavior of such a transformation is qualitatively rather different than for the geometric transformations of the plane represented by multiplication by complex numbers.
Unlike the complex case, multiplication by a quaternion can effect a flip of four-space. (But there is more than one way to flip four-space; quaternions can effect some flips, but not others. More to come on this point, I hope.)
Like the complex case, dilation and rotations can be effected by multiplication by quaternions, but (non-trivial) stretches and shears cannot.
Note that there is much more room in four-space than in the plane. For example, in four-space, it is possible to rotate about two different axes independently (consider multiplying by the quaternion given by the pair (x, y), where x and y are complex numbers of unit length.). That can’t even happen in three-space!.
But, this is not the end of the story. There is left and right multiplication by quaternions. These can be shown to effect distinct linear transformations. (That is, there are quaternions such that, left multiplication by the quaternion corresponds to no right multiplication by any quaternion.)
We are left with the perplexing question: what is the geometrical distinction between the transformations effected by left and right quaternion multiplication?
The answer has to do with an action that can be performed in four-space, but not in lower-dimensional ones.
To get there, consider the geometric interpetation of complex multiplication. Multiplication by a complex number can be viewed as a linear transformation of the plane that preserves some geometric properties. Namely, the image of a triangle in the plane transformed by multiplication by a complex number has the same angles as the original, and the sense of the angles is the same. The triangle may be scaled and rotated, but not flipped, sheared, or stretched.
Something like this is true of multiplication by quaternions, but there is more than one way to transform four-space such that geometry is preserved.
If the plane is flipped on one axis, the sense of angles is reversed. If the plane is then flipped on another axis, the sense of angles returns. But two flips of the plane is equivalent to a rotation: that is, no flip at all.
In four-space, it is possible to do two flips that is not equivalent to a rotation. Namely, one can flip one two-dimensional subspace, and also flip a two-dimensional subspace perpendicular to the first. The resulting transformation is no rotation, and yet angles and sense are preserved.
Left multiplication by a quaternion effects this double-flip; right multiplication does not. (Depending on how quaternion muliplication is defined: there is an arbitrary choice involved.)
The space of linear transformations effected by left quaternion multiplication is a four-dimensional subspace of the space of linear transformations of four-space, and so also is the one effected by right multiplication. They intersect at the two-dimensional subspace of transformations corresponding to multiplication by complex numbers.
I first found it amazing that there is enough room for so many subspaces. But consider, the space of all linear transformations of four-space is 16-dimensional. There is plenty of room in this space for two four-dimensional subspaces.
The most general linear transformation effected by quaternion operations is that of multiplication on both left and right by fixed quaternions. Now, a complex multiple can be absorbed from either side due to commutativity, leaving (apparently) a six dimensional space of linear transformations. Note: Then correct number is between four and six inclusive, but I haven't pursued the question. I think it has to be more than four. And it can't be more than six. But wouldn't it be weird if it were five? What is a nice basis for these transformations?
One clue is that the determinant of the transformation of a quaternion multiplication on the left is −1, while that of a quaternion multiplication on the right is 1. (This is apparent from the complex matrix forms of the transformations, taking the determinant of the flip operator "~" to be −1).
Another thought concerns the analog of a tetrahedron in four-space, called a four-simplex. A four-simplex has
On the page concerning representing quaternions with matrices, it is shown that multiplication on the right by a quaternion can be viewed as a linear transformation of C×C, but a multiplication on the left involves a flip of one of the component complex subspaces.
Mark E. Shoulson’s beautiful Java applet Rotating Four-Dimensional Shapes
William Rowan Hamilton, On Quaternions
In this way, the algebra of this set of matrices models the algebra of the complex numbers. The correspondence between this matrix algebra and that of complex numbers is called a homomorphism; the two algebras are said to be homomorphic to one another.
So, the algebra of the complex numbers is a proper sub-algebra of the algebra of 2 × 2 matrices of real numbers.