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# Finite Groups of Isometries in the Euclidean Space

 ✅ Paper Type: Free Essay ✅ Subject: Physics ✅ Wordcount: 26126 words ✅ Published: 8th Feb 2020

Finite Groups of Isometries

Abstract

The aim of this report is to find the finite groups of isometries in the Euclidean space ${\mathbb{R}}^{n}$

. We shall specifically be considering finite groups of the special orthogonal group $\mathit{SO}\left(3\right)$

, which is a natural subgroup of the isometries in the Euclidean space. We shall assemble a set of definitions and theorems related to group theory, Euclidean geometry and spherical geometry to help gain an understanding of the finite groups of $\mathit{SO}\left(3\right)$

.

Introduction

This paper will explore the finite groups of isometries, specifically the special orthogonal group $\mathit{SO}\left(3,\mathbb{R}\mathbb{\right)}$

which is a natural subgroup of $\mathit{Isom}\left({\mathbb{R}}^{n}\right)$

. We will understand that all these finite groups are isomorphic to either a cyclic group, a dihedral group, or one of the groups of a Platonic solid.

As well as the aid of definitions and theorems leading up to the finite groups of $\mathit{SO}\left(3\right)$

, we shall study topics, such as polygons in the Euclidean plane and spherical triangles, which will be beneficial in gaining a larger insight to the subject. The main concept that we shall be investigating is the rotation groups of these finite subgroups.

1 Euclidean Geometry

1.1 Euclidean Space

$\mathbb{R}$

1 refers to the real line which is all real numbers from least to greatest. $\mathbb{R}$

2 is the plane, where points are represented as ordered pairs: $\left({x}_{1},{x}_{2}\right)$

. $\mathbb{R}$

n, which is the n-dimensional Euclidean space, is the space of n-tuples of real numbers: $\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$

.

In this report, the Euclidean space $\mathbb{R}$

n shall be used, equipped with the standard Euclidean inner-product

The inner-product is defined by

$\left(\mathbit{x},\mathbit{y}$

) = $\sum _{i=1}^{n}{x}_{i}{y}_{i}$

.

The Euclidean norm on $\mathbb{R}$

n is defined by

and the distance function $\mathcal{d}$

is defined by

.

Definition 1.1 A metric space is a set $X$

equipped with a metric $d$

, namely

, satisfying the following conditions:

• with

if and only if $P=Q$

.

for any points

.

The Euclidean distance $\mathcal{d}$

function is an example of a metric as it also satisfies the above conditions.

Lemma 1.2 A map

of metric spaces is continuous if and only if, under $f$

, the inverse image of every open subset of $Y$

is open in $X$

.

A homeomorphism between given metric spaces $\left(X,{d}_{X}\right)$

and $\left(Y,{d}_{Y}\right)$

is a continuous map with a continuous inverse.

A topological equivalence between the spaces is when the open sets in two spaces correspond under the bijection; the two spaces are then considered homeomorphic.

1.2 Isometries

Definition 1.3 Let $X$

and $Y$

be metric spaces with metrics ${d}_{X}$

and ${d}_{Y}$

. An isometry

is a distance-preserving transformation between metric spaces and is assumed to be bijective.

i.e.

Isometries are homeomorphisms since the second condition implies that an isometry and its inverse are continuous. A symmetry of the space is an isometry of a metric space to itself. $\mathit{Isom}\left(X\right)$

denotes the isometry group or the symmetry group, which are the isometries of a metric space $X$

to itself that form a group under composition of maps.

An isometry is a transformation in which the original figure is congruent to its image. Reflections, rotations and translations are isometries.

Definition 1.4 A group $G$

is a set of elements with a binary operation

$x,y\in G⟼x,y\in G$

called multiplication, satisfying three axioms:

1. $\left(\mathit{xy}\right)z=x\left(\mathit{yz}\right)\forall x,y,z\in G$

,

2. ,

3. There exists an inverse ${x}^{–1}\in G$

such that

.

Definition 1.5 A group $G$

is isomorphic to a group ${G}^{‘}$

if there is a bijection $\varphi$

from $G$

to ${G}^{‘}$

such that $\varphi \left(\mathit{xy}\right)=\varphi \left(x\right)\varphi \left(y\right)$

.

Definition 1.6 A group $G$

acts on a set $X$

if there is a map

; $\left(g,x\right)↦g\cdot x$

, such that

• $e\cdot x=x$

for the identity $e$

of $G$

and any point $x\in X$

• for

and any point $x\in X$

.

If for all

, there exists

with $g\left(x\right)=y$

then the action of $G$

is transitive.

For the case of the Euclidean space ${\mathbb{R}}^{n}$

, with its standard inner-product

and distance function $d$

, the isometry group $\mathit{Isom}\left({\mathbb{R}}^{n}\right)$

acts transitively on ${\mathbb{R}}^{n}$

since any translation of ${\mathbb{R}}^{n}$

is an isometry. A rigid motion is sometimes used to refer to an isometry of ${\mathbb{R}}^{n}$

.

Theorem 1.7 An isometry

is of the form $f\left(x\right)=\mathit{Ax}+b$

, for some orthogonal matrix $A$

and vector

.

Lemma 1.8 Given points

, there exists a hyperplane $H$

, consisting of the points of ${\mathbb{R}}^{n}$

which are equidistant from P and Q, for which the reflection ${R}_{H}$

swaps the points P and Q.

Theorem 1.9 Any isometry of ${\mathbb{R}}^{n}$

can be written as the composite of at most $\left(n+1\right)$

reflections.

1.3 The group $\mathbit{O}\mathbit{\left(}\mathbit{3}\mathbit{,}\mathbb{R}\mathbit{\right)}$

The orthogonal group, denoted $O\left(n\right)=O\left(n,\mathbb{R}\right)$

, is a natural subgroup $\mathit{of Isom}\left({\mathbb{R}}^{n}\right)$

which consists of those isometries that are fixed at the origin. These can therefore be written as a composite of at most n reflections. It is the group of

orthogonal matrices.

$O\left(n\right)≔\left\{A\in {M}_{n×n}\left(\mathbb{R}\right):{A}^{T}A=A{A}^{T}=I\right\}$

is a group with respect to matrix multiplication $X,Y⟼\mathit{XY}$

.

If

, then

and so $\mathit{detA}=1$

or $\mathit{detA}=–1$

.

The special orthogonal group, denoted $\mathit{SO}\left(n\right)$

,is the subgroup of $O\left(n\right)$

which consists of elements with $\mathit{detA}=1.$

Direct isometries of ${\mathbb{R}}^{n}$

are the isometries of ${\mathbb{R}}^{n}$

of the form $f\left(x\right)=A\mathbit{x}+\mathbit{b}$

, for some $A\in \mathit{SO}\left(n\right)$

and $b\in {\mathbb{R}}^{n}$

. They can be expressed as a product of an even number of reflections.

Suppose that $A\in O\left(3\right)$

. First consider the case where $A\in \mathit{SO}\left(3\right)$

, so $\mathit{detA}=1$

. Then

$\mathrm{det}\left(A–I\right)=\mathit{det}{\left(A}^{t}–I\right)=\mathit{det}{A\left(A}^{t}–I\right)=\mathrm{det}\left(I–A\right)$

i.e. $+1$

is an eigenvalue.

Therefore, there exists an eigenvector ${v}_{1}$

such that $A{v}_{1}={v}_{1}$

. $W={〈{v}_{1}〉}^{\perp }$

is set to be the orthogonal complement to the space spanned by ${v}_{1}$

. Then                                          $\left(\mathit{Aw},{v}_{1}\right)=\left(\mathit{Aw},A{v}_{1}\right)=\left(w,{v}_{1}\right)=0$

if $w\in W$

. Thus $A\left(W\right)\subset W$

and ${A|}_{W}$

is a rotation of the two-dimensional space $W$

, since it is an isometry of $W$

fixing the origin and has determinant $1$

. If $\left\{{v}_{1},{v}_{2}\right\}$

is an orthonormal basis for $W$

, the matrix

$\left(\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\theta & –\mathrm{sin}\theta \\ 0& \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right)$

represents the action of $A$

on ${\mathbb{R}}^{3}$

with respect to the orthonormal basis $\left\{{v}_{1},{v}_{2},{v}_{3}\right\}$

.

This is just rotation about the axis spanned by ${v}_{1}$

through an angle $\theta$

. It may be expressed as a product of two reflections.

Now suppose $\mathit{detA}=–1$

.

Using the previous result, there exists an orthonormal basis with respect to which $–A$

is a rotation of the above form, and so $A$

takes the form

$\left(\begin{array}{ccc}–1& 0& 0\\ 0& \mathrm{cos}\varphi & –\mathrm{sin}\varphi \\ 0& \mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right)$

With $\varphi =\theta +\pi$

. Such a matrix $A$

represents a rotated reflection, rotating through an angle $\varphi$

about a given axis and then reflecting in the plane orthogonal to the axis. In the special case $\varphi =0$

, $A$

is a pure reflection. The general rotated reflection may be expressed as a product of three reflections.

1.4 Curves and their lengths

Definition 1.10 A curve (or path) $\gamma$

in a metric space

is a continuous function

for some real closed interval $\left[a,b\right]$

.

If a continuous path can join any two points of $X$

, a metric is called path connected. Both connectedness and path connectedness are topological properties, in that they do not change under homeomorphisms. If X is path connected, then it is connected.

Definition 1.11 We consider dissections

of $\left[a,b\right]$

, with $N$

arbitrary, for a curve

on a metric space $\left(X,d\right)$

.

We set

and

The length $Ɩ$

of $\gamma$

is defined by

if this is finite.

For curves in ${\mathbb{R}}^{n}$

, this is illustrated below:

A straight-line segment is any curve linking the two endpoints which achieves this minimum length in the Euclidean space.

There are curves

which fail to have finite length but for sufficiently nice curves, this does not apply. A curve of finite length may connect any two points if $X$

denotes a path connected open subset or ${\mathbb{R}}^{n}$

.

A metric space $\left(X,d\right)$

is called a length space if

inf $\left\{$

length

:      $\gamma$

a curve joining $P$

to $Q$

},

for any two points $P,Q$

of $X$

.

The metric is sometimes called intrinsic metric.

We can identify a metric $d$

on $X$

, defining $d\left(P,Q\right)$

to be the infimum of lengths of curves joining the two points, if we start from a metric space $\left(X,{d}_{0}\right)$

that satisfies the property that any two points may be joined by a curve of finite length. This is a metric, and $\left(X,d\right)$

is then a length space.

Proposition 1.12 If

is continuously differentiable, then

,

where the integrand is the Euclidean norm of the vector

1.5 Completeness and compactness

Completeness and compactness are another two recognised conditions on metric spaces.

Definition 1.13 A sequence

of points in a metric space $\left(X,d\right)$

is called a Cauchy sequence if, for any

there exists an integer $N$

such that if

then

A metric space (X, d) in which every Cauchy sequence $\left({x}_{n}\right)$

converges to an element of X is called complete. This means that a point

such that $d\left({x}_{n},x\right)\to 0$

as $n\to \infty$

. These limits are unique.

The real line is complete since real Cauchy sequences converge. The Euclidean space ${\mathbb{R}}^{n}$

is also complete when this is applied to the coordinates of points in ${\mathbb{R}}^{n}$

. A subset $X$

of ${\mathbb{R}}^{n}$

will be complete if and only if it is closed.

Definition 1.14 Let $X$

be a metric space with metric d. If every open cover of X contains a finite subcover, $X$

is compact.

An open cover of $X$

is a collection

of open sets if every $x\in X$

belongs to at least one of the ${U}_{i}$

, with $i\in I$

. If the index $I$

is finite, then an open cover is finite.

Compactness is a property that establishes the notion of a subset of Euclidean space being closed and bounded. A subset being closed means to contain all its limit points. A subset being bounded means to have all its points lie within some fixed distance of each other. If every sequence in a X has a convergent subsequence, then a metric space $\left(X,d\right)$

is called sequentially compact.

Lemma 1.15 A continuous function

on a compact metric space $\left(X,d\right)$

is uniformly continuous.

i.e. given

there exists

such that if

, then

.

Lemma 1.16 If $Y$

is a closed subset of a compact metric space $X$

, then $Y$

is compact.

Since $X$

is a closed subset of some closed box ${\mathbb{R}}^{n}$

, we infer that any closed and bounded subset $X$

of ${\mathbb{R}}^{n}$

is compact.

Lemma 1.17 If

is a continuous surjective map of metric spaces, with $X$

compact, then so is $Y$

.

1.6 Polygons in the Euclidean Plane

Euclidean polygons in ${\mathbb{R}}^{2}$

will be considered as the ‘inside’ of a simple closed polygon curve.

Definition 1.18 For a metric space, a curve

is called closed if $\gamma \left(a\right)=\gamma \left(b\right)$

. It is called simple if, for

, we have $\gamma \left(t1\right)\ne \gamma \left(t2\right)$

, except for ${t}_{1}=a$

and ${t}_{2}=b$

, when the curve is closed.

Proposition 1.19 Let

be a simple closed polygonal curve, with $C\subset {\mathbb{R}}^{2}$

denoting the image $\gamma \left(\left[a,b\right]\right).$

Then ${\mathbb{R}}^{2}C$

has at most two path connected components.

Given a set $A\subset {\mathbb{C}}^{*}\mathbb{=}\mathbb{C}\left\{0\right\},$

a continuous function

such that $h\left(z\right)$

is an argument of $z$

for all $z\in A$

, is a continuous branch of the argument on $A$

.

A continuous branch of the argument exists on $A$

if and only if a continuous branch of the logarithm exists.

i.e. a continuous function

such that exp $g\left(z\right)=z$

for all $z\in A$

.

For a curve

; a continuous branch of the argument for $\gamma$

is a continuous function

such that $\theta \left(t\right)$

is an argument for $\gamma \left(t\right)$

for all $t\in \left[a,b\right].$

Continuous branches of the argument of curves in $\mathbb{C}$

* always exist, unlike continuous branches of the argument for subsets. The use of continuity of the curve can show that they exist locally on $\left[a,b\right]$

. Then, a continuous function overall of $\left[a,b\right]$

can be achieved using the compactness of $\left[a,b\right]$

.

For a closed curve

, the winding number of $\gamma$

about the origin, is any continuous branch of the argument $\theta$

for $\gamma$

. This is denoted $n\left(\gamma ,0\right)$

and is defined

.

Given a point $w$

not on a closed curve

,  the integer $n\left(\gamma ,w\right):=n\left(\gamma –w,0\right)$

defines the winding number of $\gamma$

about $w$

, where $\gamma –w$

is the curve whose value at $t\in \left[a,b\right]$

is $\gamma \left(t\right)–w$

. The integer $n\left(\gamma ,w\right)$

describes how many times the curve $\gamma$

‘winds around $w‘$

.

Elementary properties of the winding number of a closed curve $\gamma$

:

• The winding number does not change when reparametrising $\gamma$

or changing the starting point on the curve. However, if $–\gamma$

denotes the curve $\gamma$

travelled in the opposite direction i.e.

then for any $w$

not on the curve,

$n\left(\left(–\gamma \right),w\right)=–n\left(\gamma ,w\right).$

We have $n\left(\gamma ,w\right)=0$

for the constant curve $\gamma$

.

• $n\left(\gamma ,w\right)=0$

if a subset $A\subset {\mathbb{C}}^{*}$

contains the curve $\gamma –w$

on which a continuous branch of the argument can be defined. Therefore, if a closed ball $\stackrel{̅}{B}$

contains $\gamma$

, then $n\left(\gamma ,w\right)=0$

for all $w\notin \stackrel{̅}{B}$

.

• The winding number $n\left(\gamma ,w\right)$

is a constant on each path connected component of the complement of

, as a function of $w$

.

• If

are two closed curves with ${\mathrm{\gamma }}_{1}\left(0\right)={\mathrm{\gamma }}_{1}\left(1\right)={\mathrm{\gamma }}_{2}\left(0\right)={\mathrm{\gamma }}_{2}\left(1\right)$

, we can form the link

, defined by

Then, for $w$

not in the image of ${\mathrm{\gamma }}_{1}*{\mathrm{\gamma }}_{2}$

, we have

Definition 1.20 $C$

is compact for a simple closed polygonal curve with image $C\subset {\mathbb{R}}^{2}$

, and hence bounded. Therefore, some closed ball $\stackrel{̅}{B}$

contains $C$

. One of the two components of ${\mathbb{R}}^{2}/C$

contains the complement of $\stackrel{̅}{B}$

since any two points in the complement of $\stackrel{̅}{B}$

may be joined by a path and hence is unbounded, whilst the other component of ${\mathbb{R}}^{2}/C$

is contained in $\stackrel{̅}{B}$

, and hence is bounded. The closure of the bounded component will be a closed polygon in ${\mathbb{R}}^{2}$

or a Euclidean polygon. This consists of the bounded component together with $C$

. Since a Euclidean polygon is closed and bounded in ${\mathbb{R}}^{2}$

, it is also compact.

1.21 Exercise The rotation group for a cube centred at the origin in ${\mathbb{R}}^{3}$

is isomorphic to ${S}_{4}$

, considering the permutation group of the four diagonals.

Proof A cube has 4 diagonals and any rotation induces a permutation of these diagonals. However, we cannot assume different rotations correspond to different rotations.

We need to show all 24 permutations of the diagonals come from rotations.

Two perpendicular axes where ${{90}}^{{°}}$

rotations give the permutations

and

can be seen by numbering the diagonals as 1,2,3 and 4. These make an 8-element subgroup ${\left\{}{\epsilon }{,}{\alpha }{,}{{\alpha }}^{{2}}{,}{{\alpha }}^{{3}}{,}{{\beta }}^{{2}}{,}{{\beta }}^{{2}}{\alpha }{,}{{\beta }}^{{2}}{{\alpha }}^{{2}}{,}{{\beta }}^{{2}}{{\alpha }}^{{3}}{\right\}}$

and the 3-element subgroup ${\left\{}{\epsilon }{,}{\mathit{\alpha \beta }}{,}{\left({\mathit{\alpha \beta }}\right)}^{{2}}{\right\}}$

.

Thus, the rotations make all 24 permutations since ${\mathit{lcm}}\left({8}{,}{3}\right){=}{24}{=}{|}{{S}}_{{4}}{|}$

.

2 Spherical Geometry

2.1 Introduction

Let $S={S}^{2}$

denote a unit sphere in ${\mathbb{R}}^{3}$

with centre $O=\mathbit{0}$

.

The intersection of $S$

with a plane through the origin is a great circle on $S$

. This is the spherical lines on $S$

.

$S$

Definition 2.1 The distance $d\left(P,Q\right)$

between $P$

and $Q$

on $S$

is defined to be the length of the shorter of the two segments $\mathit{PQ}$

along the great circle. This is $\pi$

if $P$

and $Q$

are on opposite sides.

$d\left(P,Q\right)$

is the angle between $\mathbit{P}=\stackrel{⃗}{\mathit{OP}}$

and $\mathbit{Q}=\stackrel{⃗}{\mathit{OQ}}$

, and hence is just ${\mathit{cos}}^{–1}\left(\mathbit{P}\mathbit{,}\mathbit{Q}\right)$

, where $\mathbit{\left(}\mathbit{P}\mathbit{,}\mathbit{Q}\right)=\mathbit{P}\mathbit{·}\mathbit{Q}$

is the Euclidean inner-product on ${\mathbb{R}}^{3}$

.

2.2 Spherical Triangles

Definition 2.2 A spherical triangle $\mathit{ABC}$

on $S$

is defined by its vertices $A,B,C\in S$

, and sides

and

, where these are spherical line segments on $S$

of length

.

${S}^{2}$

The triangle $\mathit{ABC}$

is the region of the sphere with area

enclosed by these sides.

Setting

and

, $c={\mathit{cos}}^{–1}\left(\mathbit{A}\mathbit{·}\mathbit{B}\right)$

gives the length of the side $\mathit{AB}$

. For the lengths $a,b$

of the sides $\mathit{BC}$

and $\mathit{CA}$

, similar formulae are used.

The unit normals to the planes

are set by denoting the cross-product of vectors in ${\mathbb{R}}^{2}$

by $×$

;

.

Given a spherical triangle $∆\mathit{ABC}$

, the polar triangle $∆{A}^{‘}{B}^{‘}{C}^{‘}$

is the triangle with $A$

a pole of ${B}^{‘}{C}^{‘}$

on the same side as ${A}^{‘}$

, $B$

a pole of ${A}^{‘}{C}^{‘}$

on the same side as ${B}^{‘}$

, and $C$

a pole of ${A}^{‘}{B}^{‘}$

on the same side as ${C}^{‘}$

.

Theorem 2.3 If $∆{A}^{‘}{B}^{‘}{C}^{‘}$

is the polar triangle to $∆\mathit{ABC}$

, then $∆\mathit{ABC}$

is the polar triangle to $∆{A}^{‘}{B}^{‘}{C}^{‘}$

.

Theorem 2.4 If $∆{A}^{‘}{B}^{‘}{C}^{‘}$

is the polar triangle to $∆\mathit{ABC}$

, then

Theorem 2.5 (Spherical cosine formula)

Corollary 2.6 (Spherical Pythagoras theorem)

When $\gamma =\frac{\pi }{2}$

,

Theorem 2.7 (Spherical sine formula)

$\frac{\mathrm{sin}\left(a\right)}{\mathrm{sin}\left(\alpha \right)}=\frac{\mathrm{sin}\left(b\right)}{\mathrm{sin}\left(\beta \right)}=\frac{\mathrm{sin}\left(c\right)}{\mathrm{sin}\left(\gamma \right)}.$

Corollary 2.8 (Triangle inequality)

For

,

with equality if and only if $Q$

is on the line segment $\mathit{PR}$

.

Proposition 2.9 (Second cosine formula)

2.3 Curves on the sphere

The restriction to $S$

of the Euclidean metric on ${\mathbb{R}}^{3}$

and the spherical distance metric are two natural metrics defined on the sphere.

Proposition 2.10 These two concepts of length coincide, given a curve $\gamma$

on $S$

joining points

on

.

Proposition 2.11 Given a curve

on

joining points

and

, we have $Ɩ=$

length

is the spherical line segment

on

if $Ɩ=d\left(P,Q\right)$

.

A spherical line segment is a curve  $\gamma$

of minimum length joining

and

. So

length

for all $t$

. Therefore, the parameterisation is monotonic since

is strictly increasing as a function of $t$

.

2.4 Finite Groups of Isometries

Definition 2.12 Let $X=\left\{1,2,\dots ,n\right\}$

be a finite set. The symmetric group ${S}_{n}$

is the set of all permutations of $X$

. The order of ${S}_{n}$

is $\left|{S}_{n}\right|=n!=1\bullet 2\bullet \dots \bullet n$

.

Definition 2.13 The alternating group ${A}_{n}$

is the set of all even permutations in ${S}_{n}$

. The order of group ${A}_{n}$

is $\left|{A}_{n}\right|=\frac{|{S}_{n}|}{2}=\frac{n!}{2}$

.

Definition 2.14 The dihedral group ${D}_{n}$

is the symmetry group of a regular polygon with $n$

sides.

Definition 2.15 The cyclic group ${C}_{n}$

, with $n$

elements, is a group that is generated by combining a single element of the group multiple times.

A matrix in $O\left(3,\mathbb{R}\mathbb{\right)}$

determines an isometry of ${\mathbb{R}}^{3}$

which fixes the origin. Such a matrix preserves both the lengths of vectors and angles between vectors since it preserves the standard inner-product.

Any isometry

may be extended to a map

fixing the origin, which for non-zero $\mathbit{x}$

is defined by

With the standard inner-product

on ${\mathbb{R}}^{3}$

, $\left(g\left(\mathbit{x}\right),g\left(\mathbit{y}\right)\right)=\left(\mathbit{x}\mathbit{,}\mathbit{y}\right)$

for any $\mathbit{x}\mathbit{,}\mathbit{y}\in {\mathbb{R}}^{3}$

. For $\mathbit{x}\mathbit{,}\mathbit{y}$

non-zero, this follows since

From this we infer that $g$

is an isometry of ${\mathbb{R}}^{3}$

which fixes the origin and is given by a matrix in $O\left(3\right).$

Therefore, $\mathit{Isom}\left({S}^{2}\right)$

is naturally acknowledged with the group

.

The restriction to ${S}^{2}$

of the isometry ${R}_{H}$

of ${\mathbb{R}}^{3}$

, the reflection of ${\mathbb{R}}^{3}$

in the hyperplane $H$

is defined as the reflection of ${S}^{2}$

in a spherical line $Ɩ$

. Therefore, three such reflections are the most any element of $\mathit{Isom}\left({S}^{2}\right)$

can be composite of. Isometries that are just rotations of ${S}^{2}$

and are the composite of two reflections are an index two subgroup of $\mathit{Isom}\left({S}^{2}\right)$

corresponding to the subgroup $\mathit{SO}\left(3\right)\subset O\left(3\right)$

. The group $O\left(3\right)$

is isomorphic to $\mathit{SO}\left(3\right)×{C}_{2}$

, since any element of $O\left(3\right)$

is of the form $±A$

, with $A\in \mathit{SO}\left(3\right)$

.

Any finite subgroup $G$

of $\mathit{Isom}\left({\mathbb{R}}^{3}\right)$

has a fixed point in ${\mathbb{R}}^{3}$

,

and corresponds to a finite subgroup of $\mathit{Isom}\left({S}^{2}\right)$

. Since any finite subgroup of $\mathit{Isom}\left({\mathbb{R}}^{2}\right)$

has a fixed point, it is either a cyclic or dihedral group.

We consider the group of rotations $\mathit{SO}\left(3\right)$

. All finite subgroups of $\mathit{SO}\left(3\right)$

are isomorphic to either the cyclic group, the dihedral group, or one of the groups of a Platonic solid. There are five platonic solids: the icosahedron, the dodecahedron, the tetrahedron, the octahedron and the cube.

Copies of a cyclic group ${C}_{n}$

are contained in

by considering rotations of ${S}^{2}$

about the $z$

-axis through angles which are multiples of

. We generate a new subgroup of $\mathit{SO}\left(3\right)$

by also including the rotation of ${S}^{2}$

-axis through an angle $\pi$

which is isomorphic to the group of symmetries ${D}_{2n}$

of the regular $n–$

gon for

. We have the special case

when $n=2$

.

However, corresponding to the rotation groups of the regular solids, there are further finite subgroups of $\mathit{SO}\left(3\right)$

. The tetrahedron has rotation group ${A}_{4}$

, the cube has rotation group ${S}_{4}$

and the octahedron is dual to the cube. Dual solids are solids that can be constructed from other solids; their faces and vertices can be interchanged. The dodecahedron and the icosahedron are also dual solids and have rotation group ${A}_{5}$

.

Proposition 2.16 The finite subgroups of $\mathit{SO}\left(3\right)$

are of isomorphism types ${C}_{n}$

for

, ${D}_{2n}$

for

,

, the last three being the rotation groups arising from the regular solids.

Since

is a subgroup of $O\left(3\right)$

of twice the order if $G$

is a finite subgroup of $\mathit{SO}\left(3\right)$

, with elements $±A$

for $A\in G$

.

The reason why extra finite groups do not occur for either the Euclidean or hyperbolic cases but does occur for the sphere is because we can consider the subgroup of isometries $G$

generated by the reflections in the sides of the triangle, if we have a spherical triangle $\Delta$

with angles

and

with $r\ge q\ge p\ge 2$

.

The tessellation of ${S}^{2}$

is by the images of Δ under the elements of $G$

by the theory of reflection groups. This means that the spherical triangles $g\left($

Δ $\right)$

for $g\in G$

covers ${S}^{2}$

and that any two such images have disjoint interiors. A special type of geodesic triangulation for which all triangles are congruent is developed by such a tessellated ${S}^{2}$

. Therefore, the reflection group $G$

is finite.

From Gauss-Bonnet Theorem, the area of Δ is

, and hence

.

The only solutions are:

• with

. The area of Δ is $\pi /n$

.

• . The area of Δ is $\pi /6$

.

• . The area of Δ is $\pi /12$

.

• . The area of Δ is $\pi /30$

.

$G$

has order 4n, 24, 48 and 120 in these cases. This is implied from the tessellation of ${S}^{2}$

by the images of Δ under $G$

. It is then clear that $G$

is

in the first case, and it is the full symmetry group of the tetrahedron, cube and dodecahedron in the remaining cases.

2.5 Gauss-Bonnet and Spherical Polygons

The statement that angles of a Euclidean triangle add up to $\pi$

is the Euclidean version of Gauss-Bonnet.

Proposition 2.17 If Δ is a spherical triangle with angles $\alpha ,\beta ,\gamma$

, its area is

.

For a spherical triangle,

. We obtain the Euclidean case;

in the limit as area

.

We can subdivide the triangle, whose sides have length less than

, into smaller ones if one of the sides of the spherical triangle has length

. The area of the original triangle is still

when applying Gauss-Bonnet to the two smaller triangles and adding.

The Gauss-Bonnet can be extended to spherical polygons on ${S}^{2}$

. Consider a simple closed polygonal curve $C$

on ${S}^{2}$

, where spherical line segments are the segments of $C$

. Suppose that the north pole does not lie on $C$

. We consider a simple closed curve in $\mathbb{C}$

the image

of $C$

under stereographic projection. Stereographic projection is a mapping that projects a sphere onto a plane.

Arcs of certain circles or segments of certain lines are the segments of $┌$

. A bounded and an unbounded component are contained by the complement of $┌$

in $\mathbb{C}$

. Therefore, two path connected components are also contained in the complement of $C$

in ${S}^{2}$

. Each component corresponds to the bounded component in the image of a stereographic projection. A spherical polygon is determined by the information of the polygonal curve $C$

and a choice of a connected component of its complement in ${S}^{2}$

.

A subset $A$

of ${S}^{2}$

is called convex if there is a unique spherical line segment of minimum length joining

to

, for any points $P,Q\in A$

and this line segment is contained in $A$

.

Theorem 2.18 If $\prod \subset {S}^{2}$

is a spherical $n$

-gon, contained in some open hemisphere, with interior angles ${\alpha }_{1},...,{\alpha }_{n}$

, its area is

2.6 Möbius Geometry

Möbius transformations on the extended complex plane

is closely related to spherical geometry, with a coordinate $ϛ$

. The stereographic projection map

,

defined geometrically by the diagram below provides this connection.

The point of intersection of the line through $N$

and $P$

with $\mathbb{C}$

is $\pi \left(P\right)$

, where the plane $z=0$

identifies $\mathbb{C}$

, and where we define $\pi \left(N\right):=\infty$

; $\pi$

is a bijection.

Using the geometry of similar triangles, an explicit formula for $\pi$

can be formed;

since in the diagram below $\frac{r}{R}=\frac{1–z}{1}$

and so $R=\frac{r}{1–z}$

.

Lemma 2.19 If

denotes the stereographic projection from the south pole, then

for any $P\in {S}^{2}.$

The map

is just inversion in the unit circle, $ϛ↦1/\stackrel{̅}{ϛ}$

.

If $P=\left(x,y,z\right)\in {S}^{2}$

, then

.

The antipodal point

has $\pi \left(–P\right)=–\frac{x+\mathit{iy}}{1+z}$

and so

Therefore

The group $G$

, of Möbius transformations, is acting on ${\mathbb{C}}_{\infty }$

. $A$

defines a Möbius transformation on ${\mathbb{C}}_{\infty }$

by

$\varsigma ↦\frac{\mathit{a\varsigma }+b}{\mathit{c\varsigma }+d}$

if $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathit{GL}\left(2,\mathbb{C}\mathbb{\right)}$

.

$\mathit{\lambda A}$

defines the same Möbius transformation for any $\lambda \in {\mathbb{C}}^{*}\mathbb{=}\mathbb{C}\mathbb{}\mathbb{\left\{}0\right\}$

.

Conversely, if ${A}_{1},{A}_{2}$

define the same Möbius transformation, then the identity transformation is identified by ${{A}_{2}}^{–1}{A}_{1}$

. This simplifies that

for some $\lambda \in {\mathbb{C}}^{*},$

and hence that ${A}_{1}=\lambda {A}_{2}$

. Therefore

,

identifying elements of $\mathit{GL}\left(2,\mathbb{C}\right)$

attains the group on the right, which are non-zero multiples of each other.

If

and ${A}_{1}=\lambda {A}_{2}$

, then ${\lambda }^{2}=1$

, and so $\lambda =±1$

. Therefore

,

where identifying elements of $\mathit{SL}\left(2,\mathbb{C}\mathbb{\right)}$

which differ only by a sign attains the group on the right. The quotient map

is a surjective group homomorphism which is 2-1. $\mathit{SL}\left(2,\mathbb{C}\mathbb{\right)}$

is a double cover of $G$

.

1. The group $G$

of Möbius transformations is generated by elements of the form

1. Any circle/straight line in $\mathbb{C}$

is of the form

for $a,c\mathbb{\in }\mathbb{R}$

,

such that

, and therefore is determined by an indefinite hermitian

matrix

$\left(\begin{array}{cc}a& w\\ \stackrel{̅}{w}& c\end{array}\right).$

1. Möbius transformations send circles/straight lines to circles/straight lines.
2. There exists a unique Möbius transformation $T$

such that

,

$T\left(z\right)=\frac{z–{z}_{1}}{z–{z}_{3}}\frac{{z}_{2}–{z}_{3}}{{z}_{2}–{z}_{1}},$

given distinct points ${z}_{1},{z}_{2},{z}_{3}\in {\mathbb{C}}_{\infty }$

.

1. The image of ${z}_{4}$

under the unique map $T$

defined above in iv. is defined by the cross-ratio $\left[{z}_{1},{z}_{2},{z}_{3},{z}_{4}\right]$

of distinct points of ${\mathbb{C}}_{\infty }$

.

There exists a unique Möbius transformation $T$

sending $R\left({z}_{1}\right),R\left({z}_{2}\right)$

and $R\left({z}_{3}\right)$

to

and

, given distinct points ${z}_{1},{z}_{2},{z}_{3},{z}_{4}$

and a Möbius transformation $\mathbb{R}$

. The composite $\mathit{TR}$

is therefore the unique Möbius transformation sending ${z}_{1},{z}_{2}$

and ${z}_{3}$

to

and

. Our definition of cross-ratio then implies that

$\left[R{z}_{1},R{z}_{2},R{z}_{3},{\mathit{Rz}}_{4}\right]$ $\left[{z}_{1},{z}_{2},{z}_{3},{z}_{4}\right].$

2.7 The double cover of $\mathbit{SO}\mathbit{\left(}\mathbit{3}\mathbit{\right)}$

We have an index two subgroup of the full isometry group $O\left(3\right)$

, the rotations $\mathit{SO}\left(3\right)$

on ${S}^{2}$

. The section aims to show that the group $\mathit{SO}\left(3\right)$

is established isomorphically with the group $\mathit{PSU}\left(2\right)$

by the stereographic projection map $\pi$

. There is a surjective homomorphism of groups $\mathit{SU}\left(2\right)\to \mathit{SO}\left(3\right)$

, which is $2–1$

map.

Theorem 2.20 Every rotation of ${S}^{2}$

corresponds to a Möbius transformation of ${\mathbb{C}}_{\infty }$

in $\mathit{PSU}\left(2\right)$

via the map π.

Theorem 2.21 The group of rotations

acting on ${S}^{2}$

corresponds isomorphically with the subgroup $\mathit{PSU}\left(2\right)=\mathit{SU}\left(2\right)/\left\{±1\right\}$

of Möbius transformations acting on ${\mathbb{C}}_{\infty }$

­.

Corollary 2.22 The isometries of ${S}^{2}$

which are not rotations correspond under stereographic projection precisely to the transformations of ${\mathbb{C}}_{\infty }$

of the form

$z↦\frac{a\stackrel{̅}{z}–b}{\stackrel{̅}{b}\stackrel{̅}{z}+\stackrel{̅}{a}}$

with ${|a|}^{2}+{|b|}^{2}=1.$

There exists a 2-1 map

$\mathit{SU}\left(2\right)\to \mathit{PSU}\left(2\right)\cong \mathit{SO}\left(3\right).$

This map is usually produced using quaternions.

This is the reason why a non-closed path of transformations in

going from

to $–I$

exists, corresponding to a closed path in $\mathit{SO}\left(3\right)$

starting and ending at $\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

.

Since $\mathit{SU}\left(2\right)$

consist of matrices of the form $\left(\begin{array}{cc}a& –b\\ \stackrel{̅}{b}& \stackrel{̅}{a}\end{array}\right)$

with

, geometrically it is ${S}^{3}\subset {\mathbb{R}}^{4}.$

There are finite subgroups of $\mathit{SU}\left(2\right)$

of double the order corresponding to finite subgroups of $\mathit{SO}\left(3\right)$

, specifically cyclic, dihedral and the rotation groups of the tetrahedron, cube and dodecahedron.

2.8 Circles on ${\mathbit{S}}^{\mathbit{2}}$

We consider the locus of points on ${S}^{2}$

, whose spherical distance from $P$

is $\rho$

, given an arbitrary point $P$

on ${S}^{2}$

and

. In spherical geometry, this is what is meant by a circle.

To ensure the point $P$

is always at the north pole, we may rotate the sphere, as shown below:

Therefore, the circle is also a Euclidean circle of radius $\mathit{sin}\left(\rho \right)$

and that it is the intersection of a plane with ${S}^{2}$

. Conversely, a plane cuts out a circle if its intersection with ${S}^{2}$

consists of more than one point. Great circles correspond to the planes passing through the origin. The area of such a circle is calculated by

which, from the Euclidean case, is always less than the area $\pi {\rho }^{2}.$

For small $\rho$

this may be expanded as

$\pi {\rho }^{2}\left(1–\frac{1}{12}{\rho }^{2}+O\left({\rho }^{4}\right)\right).$

2.23 Exercise Two spherical triangles ${∆}_{1},{∆}_{2}$

on a sphere ${S}^{2}$

are said to be congruent if there is an isometry of ${S}^{2}$

that takes ${∆}_{1}$

to ${∆}_{2}$

. ${∆}_{1},{∆}_{2}$

are congruent if and only if they have equal angles.

Proof Let $∆\mathit{ABC}$

and $∆\mathit{DEF}$

have $\angle A=\angle D$

etc and let $∆{A}^{‘}{B}^{‘}{C}^{‘}$

and $∆{D}^{‘}{E}^{‘}{F}^{‘}$

be the polar triangles. By theorem 2.18,

and so on. So, by the three sides, $∆{A}^{‘}{B}^{‘}{C}^{‘}$

is congruent to $∆{D}^{‘}{E}^{‘}{F}^{‘}$

which means that they have the same angles. Now theorem 2.17 implies that $∆\mathit{ABC}$

and $∆\mathit{DEF}$

are the polar triangles of $∆{A}^{‘}{B}^{‘}{C}^{‘}$

and $∆{D}^{‘}{E}^{‘}{F}^{‘}$

. Thus, with roles reversed, theorem 2.18 can be applied to get

and so on. Therefore, the original triangles are congruent.

Conclusion

In conclusion, in this report we have discussed isometries and the group $O\left(3,\mathbb{R}\mathbb{\right)}$

, including the special orthogonal group $\mathit{SO}\left(3\right)$

. As well as exploring related concepts within Euclidean geometry and spherical geometry, we have analysed the finite groups of $\mathit{SO}\left(3\right)$

and classified their symmetry groups by considering their rotational symmetry.

We also checked two examples: one which aided to understand the rotational symmetry of a cube, which is one of the finite subgroups of $\mathit{SO}\left(3\right)$

and one which helped us understand the congruence of spherical triangles under certain circumstances.

References

• Wilson, P. M. H. (2007). Curved spaces: from classical geometries to elementary differential geometry. Cambridge University Press.
• Armstrong, M. A. (2013). Groups and symmetry. Springer Science & Business Media.

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