Maths - Rotations in 'n' Dimensions

Rotation is a transform which preserves distances and angles (for an introduction see this page). Here we explore the issues when we extend this concept to higher dimensional spaces.

Why study rotations in high dimensional spaces?

In most cases 3D rotations are all that is needed to model rotations in the world around us. However there might be some situations where we might want to go to higher dimensions:

Finite Rotations

Here we are looking at finite rotations, continuous rotations like angular velocity can be combined using vector addition in any dimensional space.

As we move to higher dimensional spaces then calculating and combining finite rotations gets more complicated. In two dimensions we can either add angles or multiply complex numbers or equivalent matrices. When we move upto three dimensions combining rotations is no longer commutative. When we move to 4 and above dimensions then combining (basic) rotations no longer necessarily gives another (basic) rotation, instead we need to work in terms of compound rotations which contain the complete Clifford even subalgebra rather than just scalars and bivectors. (see Penrose book below P209).

Effect of higher dimensional spaces:

  2D 3D 4D nD
combining rotations can be done in any order (transforms commute) yes no no no for n>2
combining rotations gives another rotation yes yes no no for n>3
degrees of freedom (basic rotation) 1 3 6 n*(n-1)/2
degrees of freedom (compound rotations) 1 3 7 2n/2 -1

The formula for the degrees of freedom for basic rotation comes from the number of combinations of k objects from n which is normally given by:

n!
----------
k!(n - k)!

So since rotations take place in a plane (i.e. two dimensions) we set k to 2 which gives:

n!
----------
2!(n - 2)!

which can be simplified to n*(n-1)/2

Types of Algebras:

There are two types of algebras that could be used:

  Matrices (tensors) Clifford Even Subalgebras
represent 2D rotation as: 2×2 matrix where:
[M][M]t=1 and det M =+1
complex numbers
represent 3D rotation as: 3×3 matrix where:
[M][M]t=1 and det M =+1
quaternions
represent 4D rotation as: ?

scalar + 6 bivector terms (+ 1 quadvector for compound)

(note: not octonons)

advantages: people sometimes find them easier to understand  
transform point p p' = [M]p

p' = r p r-1

where: r=scalar+bivector+(may have other even terms)

compound rotation multiply matrices multiply multivector

Interpretation of Geometry of Rotation:

We can analyse rotation in different ways, each has its pros and cons, especially when we extend these concepts to higher dimensions.

In terms of axis of rotation.

Here we define the rotation in terms of what does not change, that is the axis it rotates around, people often find this the most intuitive to understand. In two dimensions this axis is just a point around which other points rotate, in three dimensions it is a vector, in four dimensions it is an oriented plane and so on, this is for basic rotations, compound rotations in 4D and above get a bit more messy. Although this is the most intuitive I cant think of a way to construct the transform equations from this starting point.

In terms of plane of rotation.

Here we define the rotation in terms of what does change, that is the two dimensional plane within which points move as they rotate. We can factor the rotation into the following steps:

This method seems to work best when using matrices: see this page for full working . I don't think this method will work properly for compound rotations in 4 dimensions and above since we need to take into account the quadvector and other even blades generated.

In terms of 2 (or an even number) of reflections.

We can represent a rotation as an even number of reflections (as explained on this page).

I tend to have an intuitive problem with this method on the grounds that I think of finite rotations as related to continuous rotations (as discussed here) so that they are related by differentiation. In these terms it is quite difficult to think of a finite rotation as a number of 'jumps' and it is tempting to think of this as just a curious relationship between different types of transform.

However, I think its worth overcoming these objections, because this method seems to provide the most powerful way to analyse rotations. This method works in any dimensional space, whether the dimensions square to +ve or -ve (provided the dimensions anticommute) and it works with basic rotations (2 reflections) and compound rotations (even number of reflections).

Reflection swaps the left and right-handedness of the object being reflected:

So that is a reason that we need an even number of reflections.

This method works very well with Clifford algebras and explains why rotations are represented by even sub-algebras (more here).

The rotation angle produced by two mirrors depends on the angle between the two mirrors. There is a 2:1 ratio between the angle of rotation and the angle between the mirrors which applies, even when we move to higher dimensions, this allows us to relate this method to the spinor properties of rotations as described on this page.

In terms of exponential.

In two dimensions we have Euler's equation: r e = r (cos(θ) + i sin(θ)). This relates the complex number and angle representation of the rotation. Explained on this page.

However this does not work very well in higher dimensional spaces, because in 3D and higher rotations are not commutative there is no way that we are going to combine rotations using addition, in fact in 3D the equivalent of Euler's equation becomes a much more complicated series known as the Baker-Campbell-Hausdorff formula. As explained here.


We will now look at how these approaches can be applied in different dimensional spaces.

Rotations in 1 dimension

In one dimension a point can only move along the dimension, there are not enough dimensions for rotation.

A point with no dimensions can't rotate, or if it did, there is no way for us to know its rotating. In physics we talk about fundamental particles having spin. If this spin is a physical rotation then the particle must have non-zero dimensions or we need a new mathematical system to describe it?

Rotations in 2 dimensions

In 2 dimensions there is only one way that an object can rotate, so rotation has one degree of freedom, represented by an angle.

Of course we can rotate around different points in the plane. However rotation around a point is equivalent to rotation around the origin combined with a linear movement.

An orientation in 2D might therefore be denoted by θ

The speed of a continuous rotation is usually denoted by omega: ω
which is dθ/dt.

The position of a point, on an object being rotated, can be given by the linear relationship:

x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)

where:

If we use Clifford algebra then a 2D rotation is represented by the product of two 2D vectors (a e1 + b e2)(c e1 + d e2) which gives one scalar term and one bivector term (e12), this is equivalent (isomorphic to) complex numbers. Note the odd grades (vectors in this case) cancel out because the dimensions anticommute. More detail on this page).

Rotations in 3 dimensions

There is a lot of information on these pages about rotation in 3 dimensions as this is used for computer games.

There are 3 degrees of freedom for 3D rotations as discussed on this page. We can define a 3D rotation by an axis vector and an angle around that axis:

axis and rotation plane

However this way of describing rotations is specific to rotations in 3 dimensions.

A more general way to define rotations is by defining a rotation plane. Points in the rotation plane remain in the plane when rotated.

The reason that the axis vector defines rotations in 3D is that in 3D each plane is related to a given line, planes and lines are duals, but only in 3D.

Continuous Rotations in 3D

The rate of rotation can be measured in each plane, for instance the rotation in the x-y plane can be written: ωxy . For continuous rotations we can represent as a linear combination of the rotations in each plane, it is represented as a vector space:

yz, ωzx, ωxy)

This is a bivector (see 'directed area' box on right)

Because of the duality of planes and lines in 3D we could also represent each plane by the axis that it rotates round as follows:

x, ωy, ωz)

This is the usual notation for 3D.

Finite Rotations in 3D

Finite rotations, or orientations, we cant represent as a vector space, instead we need to use matrix of quaternion algebra.

There are a number of ways that these can be calculated, possibilities are:

In the second case, the matrix to map onto a plane (to get the parallel component to the plane) is:

basis1 basis2
basis1
basis2

we can rotate in a given plane as follows:

x'
y'
z'
=
b1x b2x
b1y b2y
b1z b2z
cos(θ) -sin(θ)
sin(θ) cos(θ)
b1x b1y b1z
b2x b2y b2z

where:

Example:

Rotation in x-z plane. In this case:

basis 1 = (1,0,0)

basis 2 = (0,0,1)

so:

x'
y'
z'
=
1 0
0 0
0 1
r cos(θ) - r sin(θ)
r sin(θ) r cos(θ)
1 0 0
0 0 1

multiplying matrices gives:

x'
y'
z'
=
r * cos(θ) 0 r * sin(θ)
0 1 0
-r * sin(θ) 0 r * cos(θ)

If we use Clifford algebra then a 3D rotation is represented by the product of two 3D vectors (a e1 + b e2 + c e3)(d e1 + e e2 + f e3) which gives one scalar term and three bivector terms (e12, e31 and e23), this is equivalent (isomorphic to) quaternions. Note the odd grades (vectors and trivectors) cancel out because the dimensions anticommute. More detail on this page).

Rotations in 4 dimensions

See Roger Penrose book (panel below) page 209.

If we use Clifford algebra (Which I find is the best way to investigate 4D and above rotations) then a 4D rotation is represented by the product of two 4D vectors (a e1 + b e2 + c e3 + d e4)(e e1 + f e2 + g e3 + h e4) which gives one scalar term and six bivector terms. However if we then go on to combine that with another rotation, that will be equivalent to 4 reflections, this means we may get an e1234 (quadvector term).

In general in 'n' dimensions we need to use the complete even subalgebra to represent any combination of rotations.

More detail on this page).

Note that this is not the same as octonons, as we explained here the algebras generated by the Cayley-Dickson doubling process diverge from the Clifford algebras in 4D and above.

Further Reading


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see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Roger Penrose - The Road to Reality: Partly a 'popular science' book as it tries to minimise the number of equations (Not that I'm complaining much his book 'Spinors and space-time' went over my head in the first few pages) it still has lots of interesting results that its difficult to find elsewhere.

 

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