# Rotations in Three Dimensions

In Rotating and Reflecting Vectors Using Matrices we learned how to express rotations in $2$-dimensional space using certain special $2\times 2$ matrices which form a group (see Groups) we call the special orthogonal group in dimension $2$, or $\text{SO}(2)$ (together with other matrices which express reflections, they form a bigger group that we call the orthogonal group in $2$ dimensions, or $\text{O}(2)$).

In this post, we will discuss rotations in $3$-dimensional space. As we will soon see, notations in $3$-dimensional space have certain interesting features not present in the $2$-dimensional case, and despite being seemingly simple and mundane, play very important roles in some of the deepest aspects of fundamental physics.

We will first discuss rotations in $3$-dimensional space as represented by the special orthogonal group in dimension $3$, written as $\text{SO}(3)$.

We recall some relevant terminology from Rotating and Reflecting Vectors Using Matrices. A matrix is called orthogonal if it preserves the magnitude of (real) vectors. The magnitude of the vector $v$ must be equal to the magnitude of the vector $Av$, for a matrix $A$, to be orthogonal. Alternatively, we may require, for the matrix $A$ to be orthogonal, that it satisfy the condition

$\displaystyle AA^{T}=A^{T}A=I$

where $A^{T}$ is the transpose of $A$ and $I$ is the identity matrix. The word “special” denotes that our matrices must have determinant equal to $1$. Therefore, the group $\text{SO}(3)$ consists of the $3\times3$ orthogonal matrices whose determinant is equal to $1$.

The idea of using the group $\text{SO}(3)$ to express rotations in $3$-dimensional space may be made more concrete using several different formalisms.

One popular formalism is given by the so-called Euler angles. In this formalism, we break down any arbitrary rotation in $3$-dimensional space into three separate rotations. The first, which we write here by $\varphi$, is expressed as a counterclockwise rotation about the $z$-axis. The second, $\theta$, is a counterclockwise rotation about an $x$-axis that rotates along with the object. Finally, the third, $\psi$, is expressed as a counterclockwise rotation about a $z$-axis that, once again, has rotated along with the object. For readers who may be confused, animations of these steps can be found among the references listed at the end of this post.

The matrix which expresses the rotation which is the product of these three rotations can then be written as

$\displaystyle g(\varphi,\theta,\psi) = \left(\begin{array}{ccc} \text{cos}(\varphi)\text{cos}(\psi)-\text{cos}(\theta)\text{sin}(\varphi)\text{sin}(\psi) & -\text{cos}(\varphi)\text{sin}(\psi)-\text{cos}(\theta)\text{sin}(\varphi)\text{cos}(\psi) & \text{sin}(\varphi)\text{sin}(\theta) \\ \text{sin}(\varphi)\text{cos}(\psi)+\text{cos}(\theta)\text{cos}(\varphi)\text{sin}(\psi) & -\text{sin}(\varphi)\text{sin}(\psi)+\text{cos}(\theta)\text{cos}(\varphi)\text{cos}(\psi) & -\text{cos}(\varphi)\text{sin}(\theta) \\ \text{sin}(\psi)\text{sin}(\theta) & \text{cos}(\psi)\text{sin}(\theta) & \text{cos}(\theta) \end{array}\right)$.

The reader may check that, in the case that the rotation is strictly in the $x$$y$ plane, i.e. $\theta$ and $\psi$ are zero, we will obtain

$\displaystyle g(\varphi,\theta,\psi) = \left(\begin{array}{ccc} \text{cos}(\varphi) & -\text{sin}(\varphi) & 0 \\ \text{sin}(\varphi) & \text{cos}(\varphi) & 0 \\ 0 & 0 & 1 \end{array}\right)$.

Note how the upper left part is an element of $\text{SO}(2)$, expressing a counterclockwise rotation by an angle $\varphi$, as we might expect.

Contrary to the case of $\text{SO}(2)$, which is an abelian group, the group $\text{SO}(3)$ is not an abelian group. This means that for two elements $a$ and $b$ of $\text{SO}(3)$, the product $ab$ may not always be equal to the product $ba$. One can check this explicitly, or simply consider rotating an object along different axes; for example, rotating an object first counterclockwise by 90 degrees along the $z$-axis, and then counterclockwise again by 90 degrees along the $x$-axis, will not end with the same result as performing the same operations in the opposite order.

We now know how to express rotations in $3$-dimensional space using $3\times 3$ orthogonal matrices. Now we discuss another way of expressing the same concept, but using “unitary”, instead of orthogonal, matrices. However, first we must revisit rotations in $2$ dimensions.

The group $\text{SO}(2)$ is not the only way we have of expressing rotations in $2$-dimensions. For example, we can also make use of the unitary (we will explain the meaning of this word shortly) group in $1$-dimension, also written $\text{U}(1)$. It is the group formed by the complex numbers with magnitude equal to $1$. The elements of this group can always be written in the form $e^{i\theta}$, where $\theta$ is the angle of our rotation. As we have seen in Connection and Curvature in Riemannian Geometry, this group is related to quantum electrodynamics, as it expresses the gauge symmetry of the theory.

The groups $\text{SO}(2)$ and $\text{U}(1)$ are actually isomorphic. There is a one-to-one correspondence between the elements of $\text{SO}(2)$ and the elements of $\text{U}(1)$ which respects the group operation. In other words, there is a bijective function $f:\text{SO}(2)\rightarrow\text{U}(1)$, which satisfies $ab=f(a)f(b)$ for $a$, $b$ elements of $\text{SO}(2)$. When two groups are isomorphic, we may consider them as being essentially the same group. For this reason, both $\text{SO}(2)$ and $U(1)$ are often referred to as the circle group.

We can now go back to rotations in $3$ dimensions and discuss the group $\text{SU}(2)$, the special unitary group in dimension $2$. The word “unitary” is in some way analogous to “orthogonal”, but applies to vectors with complex number entries.

Consider an arbitrary vector

$\displaystyle v=\left(\begin{array}{c}v_{1}\\v_{2}\\v_{3}\end{array}\right)$.

An orthogonal matrix, as we have discussed above, preserves the quantity (which is the square of what we have referred to earlier as the “magnitude” for vectors with real number entries)

$\displaystyle v_{1}^{2}+v_{2}^{2}+v_{3}^{2}$

while a unitary matrix preserves

$\displaystyle v_{1}^{*}v_{1}+v_{2}^{*}v_{2}+v_{3}^{*}v_{3}$

where $v_{i}^{*}$ denotes the complex conjugate of the complex number $v_{i}$. This is the square of the analogous notion of “magnitude” for vectors with complex number entries.

Just as orthogonal matrices must satisfy the condition

$\displaystyle AA^{T}=A^{T}A=I$,

unitary matrices are required to satisfy the condition

$\displaystyle AA^{\dagger}=A^{\dagger}A=I$

where $A^{\dagger}$ is the Hermitian conjugate of $A$, a matrix whose entries are the complex conjugates of the entries of the transpose $A^{T}$ of $A$.

An element of the group $\text{SU}(2)$ is therefore a $2\times 2$ unitary matrix whose determinant is equal to $1$. Like the group $\text{SO}(3)$, the group $\text{SU}(2)$ is also a group which is not abelian.

Unlike the analogous case in $2$ dimensions, the groups $\text{SO}(3)$ and $\text{SU}(2)$ are not isomorphic. There is no one-to-one correspondence between them. However, there is a homomorphism from $\text{SU}(2)$ to $\text{SO}(3)$ that is “two-to-one”, i.e. there are always two elements of $\text{SU}(2)$ that get mapped to the same element of $\text{SO}(3)$ under this homomorphism. Hence, $\text{SU}(2)$ is often referred to as a “double cover” of $\text{SO}(3)$.

In physics, this concept underlies the weird behavior of quantum-mechanical objects called spinors (such as electrons), which require a rotation of 720, not 360, degrees to return to its original state!

The groups we have so far discussed are not “merely” groups. They also possesses another kind of mathematical structure. They describe certain shapes which happen to have no sharp corners or edges. Technically, such a shape is called a manifold, and it is the object of study of the branch of mathematics called differential geometry, which we have discussed certain basic aspects of in Geometry on Curved Spaces and Connection and Curvature in Riemannian Geometry.

For the circle group, the manifold that it describes is itself a circle. The elements of the circle group correspond to the points of the circle. The group $\text{SU}(2)$ is the surface of the $4$– dimensional sphere, or what we call a $3$-sphere (for those who might be confused by the terminology, recall that we are only considering the surface of the sphere, not the entire volume, and this surface is a $3$-dimensional, not a $4$-dimensional, object). The group $\text{SO}(3)$ is $3$-dimensional real projective space, written $\mathbb{RP}^{3}$. It is a manifold which can be described using the concepts of projective geometry (see Projective Geometry).

A group that is also a manifold is called a Lie group (pronounced like “lee”) in honor of the mathematician Marius Sophus Lie who pioneered much of their study. Lie groups are very interesting objects of study in mathematics because they bring together the techniques of group theory and differential geometry, which teaches us about Lie groups on one hand, and on the other hand also teaches us more about both group theory and differential geometry themselves.

References:

Orthogonal Group on Wikipedia

Rotation Group SO(3) on Wikipedia

Euler Angles on Wikipedia

Unitary Group on Wikipedia

Spinor on Wikipedia

Lie Group on Wikipedia

Real Projective Space on Wikipedia

Algebra by Michael Artin

# Groups

Groups are some of the most basic concepts in mathematics. They are even more basic than the things we discussed in Rings, Fields, and Ideals. In fact, all these things require the concept of groups before they can even be defined rigorously. But apart from being a basic stepping stone toward other concepts, groups are also extremely useful on their own. They can be used to represent the permutations of a set. They can also be used to describe the symmetries of an object. Since symmetries are so important in physics, groups also play an important part in describing physical phenomena. The standard model of particle physics, for example, which describes the fundamental building blocks of our physical world such as quarks, electrons, and photons, is expressed as a “gauge theory” with symmetry group $U(1)\times SU(2)\times SU(3)$.

We will not discuss something of this magnitude for now, although perhaps in the future we will (at least electromagnetism, which is a gauge theory with symmetry group $U(1)$). Our intention in this post will be to define rigorously the abstract concept of groups, and to give a few simple examples. Whatever application we have in mind when we have the concept of groups, it will have the same rigorous definition, and perhaps express the same idea at its very core.

First we will define what a law of composition means. We have been using this concept implicitly in previous posts, in concepts such as addition, subtraction, and multiplication. The law of composition makes these concepts more formal. We quote from the book Algebra by Michael Artin:

A law of composition is a function of two variables, or a map

$\displaystyle S\times S\rightarrow S$

Here $S\times S\rightarrow S$ denotes, as always, the product set, whose elements are pairs $a, b$ of elements of $S$.

There are many ways to express a law of composition. The familiar ones include

$\displaystyle a+b=c$

$\displaystyle a\circ b=c$

$\displaystyle a\times b=c$

$\displaystyle ab=c$

From the same book we now quote the definition of a group:

A group is a set $G$ together with a law of composition that has the following properties:

• The law of composition is associative: $(ab)c=a(bc)$ for all $a$$b$, and $c$.
• $G$ contains an identity element $1$, such that $1a=a$ and $a1=a$ for all $a$ in $G$.
• Every element $a$ of $G$ has an inverse, an element $b$ such that $ab=1$ and $ba=1$.

Note that the definition has used one particular notation for the law of composition, but we can use different symbols for the sake of convenience or clarity. This is merely notation and the definition of a group does not change depending on the notation that we use.

All this is rather abstract. Perhaps things will be made clearer by considering a few examples. For our first example, we will consider the set of permutations of the set with three elements which we label $1$, $2$, and $3$. The first permutation is what we shall refer to as the identity permutation. This sends the element $1$ to $1$, the element $2$ to $2$, and the element $3$ to $3$.

Another permutation sends the element $1$ to $2$, the element $2$ to $1$, and the element $3$ to $3$. In other words, it exchanges the elements $1$ and $2$ while keeping the element $3$ fixed. There are two other permutations which are similar in a way, one which exchanges $2$ and $3$ while keeping $1$ fixed, and another permutation which exchanges $1$ and $3$ while keeping $2$ fixed. To more easily keep track of these three permutations, we shall refer to them as “reflections”.

We have now enumerated four permutations. There are two more. One permutation sends $1$ to $2$$2$ to $3$, and $3$ to $1$. The last permutation sends $1$ to $3$$2$ to $1$, and $3$ to $2$. Just as we have referred to the earlier three permutations as “reflections”, we shall now refer to these last two permutations as “rotations”.

We now have a total of six permutations, which agrees with the result one can find from combinatorics. Our claim is that these six permutations form a group, with the law of composition given by performing first one permutation followed by the other. Therefore the reflection that exchanges $2$ and $3$, followed by the reflection that exchanges $1$ and $3$, is the same as the rotation that sends $1$ to $3$$2$ to $1$, and $3$ to $2$, as one may check.

We can easily verify two of the properties required for a set to form a group. There exists an identity element in our set of permutations, namely the identity permutation. Permuting the three elements $1$, $2$, and $3$ via the identity permutation (i.e. doing nothing) followed by a rotation or reflection is the same as just applying the rotation or reflection alone. Similarly, applying a rotation or reflection, and then applying the identity permutation is the same as applying just the rotation or reflection alone.

Next we show that every element has an inverse. The rotation that sends $1$ to $2$$2$ to $3$, and $3$ to $1$ followed by the rotation that sends $1$ to $3$$2$ to $1$, and $3$ to $2$ results in the identity permutation. Also the rotation that sends $1$ to $3$$2$ to $1$, and $3$ to $2$ followed by the rotation that sends $1$ to $2$$2$ to $3$, and $3$ to $1$ results in the identity permutation once again. Therefore we see that the two rotations are inverses of each other. As for the reflections, we can see that doing the same reflection twice results in the identity permutation. Every reflection has itself as its inverse, and of course the same thing holds for the identity permutation.

The associative property holds for the set of permutations of three elements, but we will not prove this statement explicitly in this post, as it is perhaps best done by figuring out the law of composition for all the permutations, i.e. by figuring out which permutations result from performing two permutations successively. This will result in something that is analogous to a “multiplication table”. With all three properties shown to hold, the set of permutations of three elements forms a group, called the symmetric group $S_{3}$.

Although the definition of a group requires the law of composition to be associative, it does not require it to be commutative; for our example, two successive permutations might not give the same result when performed in the reverse order. When the law of composition of a group is commutative, the group is called an abelian group.

An example of an abelian group is provided by the integers, with the law of composition given by addition. Appropriately, we use the symbol $+$ to denote this law of composition. The identity element is provided by $0$, and the inverse of an integer $n$ is provided by the integer $-n$. We already know from basic arithmetic that addition is both associative and commutative, so this guarantees that under addition the integers form a group and moreover form an abelian group (sometimes called the additive group of integers).

That’s it for now, but the reader is encouraged to explore more about groups since the concept can be found essentially everywhere in mathematics. For example, the positive real numbers form a group under multiplication. The reader might want to check if they really do satisfy the three properties required for a set to form a group. Another thing to think about is the group of permutations of the set with three elements, and how they relate to the symmetries of an equilateral triangle. Once again the book of Artin provides a very reliable technical discussion of groups, but one more accessible book that stands out in its discussion of groups is Love and Math: The Heart of Hidden Reality by Edward Frenkel, which is part exposition and part autobiography. The connections between groups, symmetries, and physics are extensively explored in that book, as the author’s research explores the connection between quantum mechanics and the Langlands program, an active field of mathematical research where groups once again play a very important role. More on groups are planned for future posts on this blog.

References:

Groups on Wikipedia

Symmetric Group on Wikipedia

Dihedral Group on Wikipedia

Abelian Group on Wikipedia

Algebra by Michael Artin

Love and Math: The Heart of Hidden Reality by Edward Frenkel