This document can be viewed properly only in modern browsers with good support for CSS2 and the full HTML character set.

Geometry of Quaternions

Prerequisites

To understand this, you will need knowledge of basic linear algebra, such as is often taught in a high school second-year algebra course.

Geometry in different dimensions

Four-space has its own geometry, but in the context of quaternions, most of the geometrical literature is concerned with how multiplication by quaternions affect three-space as operations, in which case the fourth dimension can naturally be regarded as that of time.

An important theme in geometry (as in all of mathematics) is that of invariants. What transformations geometrical objects are unchanged under what transformations of space? For instance, geometric properties of 3-space generally are preserved under rotations of 3-space, but the property of being a cube is not preserved under a non-isometric scaling or a skew transformation of 3-space.

Change of orientation

The quaternions are often described as a “change of direction and rotation”, by which is meant, a general transformation of one 3-vector into another.

The physical notion is that of orientation, say, of an aircraft pilot looking ahead, who can change the direction the aircraft is flying, but also roll the aircraft about its directional axis. Together, these two actions amount to a general change of the pilot's orientation.

Is the dimensionality right? Consider what information is needed to transform one 3-vector into another.

• First, the unit normal to the plane containing the origin and both the vectors. That requires two degrees of freedom (a point on the unit sphere in 3-space).
• Then the angle between the two vectors
• Finally, the ratio of the lengths of the vectors.

Four degrees of freedom, altogether.

The quaternions are a four-dimensional space, so they should be able to model such a transformation.

Or a stretch and a turn

Another way to look at the transformation of one 3-vector into another:

In An Elementary Treatise on Quaternions, Chapter II, § 48, Tait writes:

Thus it appears that a quaternion, considered as the factor or agent which changes one definite vector into another, may itself be decomposed into two factors of which the order is immaterial.

The stretching factor, … is called the Tensor …

The turning factor, … is called the Versor…

Note that Hamilton’s term tensor differs from modern usage. Hamilton also coined the term vector in this context, with a meaning rather different from the modern one.

Geometry of non-commutativity

Above, it was shown how a quaternion corresponds to a transformation of one vector in 3-space into another. In particular, some quaternions correspond to rotations of 3-space.

Rotations of 3-space are not generally commutative. To see this, find a small rectangular box whose sides are marked, so they can be distinguished.

First:

1. Hold the box so that one side is facing you. Remember the markings on this side.
2. Rotate the box 180° forward (that is, halfway around, moving the top of the box toward you)
3. Rotate the box 90° to the left (that is, a quarter-way around moving the right side of the box toward from you).

Remember the markings that now face you. Then:

1. Return the box to the same position as before.
2. Rotate the box 90° to the left.
3. Rotate the box 180° forward.

The resulting markings are now different from the result of the previous steps. So, the act of rotating a box 180° forward then 90° to the left is different from that of rotating it 90° to the left then 180° forward.

Accordingly, the quaternions that correspond to these two rotations do not commute.

Regular polytopes

A regular polygon is a common idea of a planar shape, all of whose sides and angles are the same. There are infinitely many of these.

The three-space analogs are called regular polyhedra and are familiar too: tetrahedron, cube, octagon, dodecahedron, icosahedron. Exactly five; no more. Why five? The ancient Greeks knew this, and wondered about it.

(To be more precise, these are often called regular convex polyhedra: it is also possible to draw figures whose sides pass through one another, but are all of the same size.)

An interesting symmetry is apparent here. Notice that the cube and octahedron have the same number of edges, and the number of faces of the cube is the same as the number of edges of the octahedron, and vice-versa. The same relation holds between the dodecahedron and icosahedron. The tetrahedron—holds such a relation with itself. In fact, there is a transformation of solids that maps corners to faces, faces to corners, and edges to edges, called the projective dual. So in a sense, up to this duality, there are just three classes of regular polyhedra in three-space: the tetrahedron, the cube, and the dodecahedron.

Analogues to these polygons and polyhedra can be built up naturally, and are called generally polytopes. In n dimensions, the polytopes have boundaries composed of n−1 dimensional polytopes (generally called cells). So as polytopes in three-space are bounded by two-dimensional faces, polytopes in four-space are bounded by three-dimensional cells.

The analog of the tetrahedron (called a simplex), comes for free in any number of dimensions: a squashed simplex is formed by the origin and a unit vector along each of the coordinate axes. Likewise, the analog of a cube (called a hypercube) is formed from the coordinate axes in the same way as in two- and three-space. Also, the analog to the octagon comes for free in any number of dimensions as the dual to the analog of the cube.

In four-space, things get even more interesting. There are also polytopes somehow analogous to the icosahedron and dodecahedron, but these have far more corners. And then, there is one further polytope that has no analog at all in three dimensions.

Again, there is the same pattern of duality: but here corners are paired with cells, and edges with faces.

The first surprise is the sheer number of faces of the last two polytopes. Otherwise, these are analogous to the dodecahedron and icosahedron of three-space however, as the tesseract and 16-cell are analogous to the cube and octogon.

The extra polytope, the 24-cell, is even stranger. Like the simplex, opposite each cell is a vertex, and, it is its own dual. But it occupies a position unlike anything in three-space.

One might hope to shed more light on this by investigating further in five-space and so on. But another surprise awaits.

In five dimensions and higher, only the “for free” polytopes described above exist: the simplex, the hypercube and its dual. No others. (Again, why five?) In some sense, this loss of structure in higher dimensions throws the spotlight back on four-space.

See also