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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

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Finite Groups of Isometries


The aim of this report is to find the finite groups of isometries in the Euclidean space Rn

. We shall specifically be considering finite groups of the special orthogonal group SO(3)

, 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 SO(3)



This paper will explore the finite groups of isometries, specifically the special orthogonal group SO(3,R)

which is a natural subgroup of Isom(Rn)

. 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 SO(3)

, 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


1 refers to the real line which is all real numbers from least to greatest. R

2 is the plane, where points are represented as ordered pairs: (x1,x2)

. R

n, which is the n-dimensional Euclidean space, is the space of n-tuples of real numbers: (x1,x2,,xn)


In this report, the Euclidean space R

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

The inner-product is defined by


) = i=1nxiyi


The Euclidean norm on R

n is defined by

x =(x,x)

and the distance function d

  is defined by

dx,y= xy


Definition 1.1 A metric space is a set X

equipped with a metric d

, namely d: X × X  R

, satisfying the following conditions:

  • dP,Q0 

    with dP,Q=0 

    if and only if P=Q


  • dP,Q= d(Q,P)
  • dP,Q+ dQ,R= d(P,R)

for any points P, Q, R


The Euclidean distance d

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

Lemma 1.2 A map f: X  Y

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 (X,dX)

and Y,dY

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 dX

and dY

. An isometry     f:(X, dX)  (Y, dY)

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


dYfx1,fx2= dXx1,x2    x1, x2 X.


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. Isom(X)

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


called multiplication, satisfying three axioms:

  1. xyz=xyzx,y,zG


  2. xe=ex=x  x,eG


  3. There exists an inverse x1G

    such that xx1=x1x=e  xG


Definition 1.5 A group G

is isomorphic to a group G

if there is a bijection ϕ

from G

to G

such that ϕxy=ϕ(x)ϕ(y)



Definition 1.6 A group G

acts on a set X

if there is a map × X  X

; g,xgx

, such that

  • ex=x

    for the identity e

    of G

    and any point xX

  • ghx=(gh)

    for g,G

    and any point xX


If for all x,X

, there exists G

with gx=y

then the action of G

is transitive.

For the case of the Euclidean space Rn

, with its standard inner-product ( , )

and distance function d

, the isometry group Isom(Rn)

 acts transitively on Rn

since any translation of Rn

is an isometry. A rigid motion is sometimes used to refer to an isometry of Rn


Theorem 1.7 An isometry f:Rn  Rn

is of the form f(x)=Ax+b

, for some orthogonal matrix A

and vector  bRn


Lemma 1.8 Given points  Q in Rn

, there exists a hyperplane H

, consisting of the points of Rn

which are equidistant from P and Q, for which the reflection RH

swaps the points P and Q.


Theorem 1.9 Any isometry of Rn

can be written as the composite of at most (n+1)




1.3 The group O(3,R)

The orthogonal group, denoted O(n)=O(n,R)

, is a natural subgroup of Isom(Rn)

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 × n

orthogonal matrices.


is a group with respect to matrix multiplication X,YXY


If  O(n)

, then

detA detAt= det(A)2 = 1,

and so detA=1

or detA=1


The special orthogonal group, denoted SO(n)

,is the subgroup of O(n)

which consists of elements with detA=1.

Direct isometries of Rn

are the isometries of Rn

of the form f(x)=Ax+b

, for some ASO(n)

and bRn

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

Suppose that AO(3)

. First consider the case where ASO(3)

, so detA=1

. Then



i.e. +1

is an eigenvalue.

Therefore, there exists an eigenvector v1

such that Av1=v1

. W=v1

is set to be the orthogonal complement to the space spanned by v1

. Then                                          Aw,v1=Aw,Av1=w,v1=0

if wW

. Thus A(W)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 {v1,v2}

is an orthonormal basis for W

, the matrix


represents the action of A

on R3

with respect to the orthonormal basis {v1,v2,v3}


This is just rotation about the axis spanned by v1

through an angle θ

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

Now suppose 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


With ϕ=θ+π

. Such a matrix A

represents a rotated reflection, rotating through an angle ϕ

about a given axis and then reflecting in the plane orthogonal to the axis. In the special case ϕ=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) γ

in a metric space (X,d) 

is a continuous function                    γ :[a,b]X,

for some real closed interval [a,b]


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

D: a = t0 < t1 < ... < tN = b

of [a,b]

, with N

arbitrary, for a curve  γ : [a,b]  X

on a metric space (X,d)


We set Pi= γ (ti)

and SD:=  d(Pi,Pi+1).

The length Ɩ

of γ

is defined by

Ɩ = sup SD

if this is finite.

For curves in Rn

, 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  γ  : [a,b]  R2

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 Rn


A metric space (X,d)

is called a length space if

d(P,Q) = 

inf {

length ( γ )

:      γ

  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(P,Q)

to be the infimum of lengths of curves joining the two points, if we start from a metric space (X,d0)

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

is then a length space.

Proposition 1.12 If  γ  : [a,b]  R3

is continuously differentiable, then

length  γ  =    γ(t)  dt


where the integrand is the Euclidean norm of the vector  γ tR3.


1.5 Completeness and compactness

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

Definition 1.13 A sequence x1,x2, 

of points in a metric space (X,d)

is called a Cauchy sequence if, for any ε > 0

there exists an integer N

such that if m, N

then d(xm,xn) < ε.

A metric space (X, d) in which every Cauchy sequence (xn)

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

  such that d(xn,x)0

as n

. These limits are unique.

The real line is complete since real Cauchy sequences converge. The Euclidean space Rn

is also complete when this is applied to the coordinates of points in Rn

. A subset X

of Rn

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 {Ui}I

of open sets if every xX

belongs to at least one of the Ui

, with iI

. 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 (X,d)

is called sequentially compact.

Lemma 1.15 A continuous function f: X  R

on a compact metric space (X,d)

is uniformly continuous.

i.e. given ε > 0

there exists δ > 0

such that if d(x,y) < δ

, then |f(x)f(y)| < ε



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 Rn

, we infer that any closed and bounded subset X

of Rn

is compact.


Lemma 1.17 If f: X  Y

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 R2

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

Definition 1.18 For a metric space, a curve γ: [a,b]X

is called closed if γ(a)=γ(b)

. It is called simple if, for t1 <t2

, we have γ(t1)γ(t2)

, except for t1=a

and t2=b

, when the curve is closed.

Proposition 1.19 Let γ: [a,b]  R2

be a simple closed polygonal curve, with CR2

denoting the image γ([a,b]).

Then R2C

has at most two path connected components.

Given a set AC*=C{0},

a continuous function h: AR

such that h(z)

is an argument of z

for all zA

, 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 g: AR

such that exp g(z)=z

for all zA


For a curve γ: a,bC*

; a continuous branch of the argument for γ

is a continuous function θ: [a,b]R

such that θ(t)

is an argument for γ(t)

for all t[a,b].

Continuous branches of the argument of curves in 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 [a,b]

. Then, a continuous function overall of [a,b]

can be achieved using the compactness of [a,b]


For a closed curve γ: [a,b]C*

, the winding number of γ

about the origin, is any continuous branch of the argument θ

for γ

. This is denoted n(γ,0)

and is defined

n(γ,0) =  θ(b)  θ(a)2π


Given a point w

not on a closed curve γ: [a,b]C=R2

,  the integer n(γ,w):=n(γw,0)

defines the winding number of γ

about w

, where γw

is the curve whose value at t[a,b]

is γ(t)w

. The integer n(γ,w)

describes how many times the curve γ

‘winds around w


Elementary properties of the winding number of a closed curve γ


  • The winding number does not change when reparametrising γ

    or changing the starting point on the curve. However, if γ

    denotes the curve γ

    travelled in the opposite direction i.e. (γ)(t)= γ(b(ba)t),

    then for any w

    not on the curve,


We have n(γ,w)=0

for the constant curve γ


  • n(γ,w)=0

    if a subset AC*

    contains the curve γw

    on which a continuous branch of the argument can be defined. Therefore, if a closed ball B̅

    contains γ

    , then n(γ,w)=0

    for all wB̅


  • The winding number n(γ,w)

    is a constant on each path connected component of the complement of C:= γa,b

    , as a function of w


  • If γ1,γ2: [0,1]  C

    are two closed curves with γ10=γ11=γ20=γ2(1)

    , we can form the link γ=γ1*γ2: [0,1]C

    , defined by

γt=       γ1t   for 0  t  1

                               γ2(t1)   for 1  t  2.

Then, for w

not in the image of γ1*γ2

, we have

n(γ1*γ2,w) = n(γ1,w) + n(γ2,w).

Definition 1.20 C

is compact for a simple closed polygonal curve with image CR2

, and hence bounded. Therefore, some closed ball B̅

contains C

. One of the two components of R2/C

contains the complement of B̅

since any two points in the complement of B̅

may be joined by a path and hence is unbounded, whilst the other component of R2/C

is contained in B̅

, and hence is bounded. The closure of the bounded component will be a closed polygon in R2

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

. Since a Euclidean polygon is closed and bounded in R2

, it is also compact.


1.21 Exercise The rotation group for a cube centred at the origin in R3

is isomorphic to S4

, 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 α=(1 2 3 4)

and β=(1 4 3 2)

can be seen by numbering the diagonals as 1,2,3 and 4. These make an 8-element subgroup {ε,α,α2,α3,β2,β2α,β2α2,β2α3}

and the 3-element subgroup {ε,αβ,αβ2}


Thus, the rotations make all 24 permutations since lcm8,3=24=|S4|


2 Spherical Geometry

2.1 Introduction

Let S=S2

denote a unit sphere in R3

with centre O=0


The intersection of S

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

. This is the spherical lines on S






Definition 2.1 The distance d(P,Q)

between P

and Q

on S

is defined to be the length of the shorter of the two segments PQ

along the great circle. This is π

if P

and Q

are on opposite sides.


is the angle between P=OP

and Q=OQ

, and hence is just cos1(P,Q)

, where (P,Q)=P·Q

is the Euclidean inner-product on R3








2.2 Spherical Triangles

Definition 2.2 A spherical triangle ABC

on S

is defined by its vertices A,B,CS

, and sides AB, BC 

and  AC

, where these are spherical line segments on S

of length < π



The triangle ABC

is the region of the sphere with area < 2π

enclosed by these sides.

Setting A=OA, B=OB 

and  C=OC

, c=cos1(A·B)

gives the length of the side AB

. For the lengths a,b

of the sides BC

and CA

, similar formulae are used.

The unit normals to the planes OBC, OAC, OBA

  are set by denoting the cross-product of vectors in R2

by ×


n1 = C × B / sin a

n2= A × C / sin b

n3 = B × A / sin c


Given a spherical triangle ABC

, the polar triangle ABC

is the triangle with A

a pole of BC

on the same side as A

, B

a pole of AC

on the same side as B

, and C

a pole of AB

on the same side as C


Theorem 2.3 If ABC

is the polar triangle to ABC

, then ABC

is the polar triangle to ABC


Theorem 2.4 If ABC

is the polar triangle to ABC

, then + BC = π.


Theorem 2.5 (Spherical cosine formula)

sin(a) sin(b) cos(γ) = cos(c)  cos(a) cos(b).


Corollary 2.6 (Spherical Pythagoras theorem) 

When γ=π2


cos(c)=cos(a) cos(b).

Theorem 2.7 (Spherical sine formula)


Corollary 2.8 (Triangle inequality)

For P,Q,S2


d(P,Q) + d(Q,R)  d(P,R)

with equality if and only if Q

is on the line segment PR


Proposition 2.9 (Second cosine formula)

sin(α) sin(β) cos(c) =cos(γ)  cos(α) cos(β).


2.3 Curves on the sphere

The restriction to S

of the Euclidean metric on R3

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 γ

on S

joining points P,

on  S


Proposition 2.11 Given a curve  γ  

on  S

joining points

and  Q

, we have Ɩ=

length  γ   d(P,Q).

In addition, the image of  γ 

is the spherical line segment PQ 

on  S

if Ɩ=d(P,Q)


A spherical line segment is a curve  γ

   of minimum length joining  P 

and  Q

. So

length  γ  |[0,1] = d(P, γ (t)),

for all t

. Therefore, the parameterisation is monotonic since d(P, γ (t))

is strictly increasing as a function of t



2.4 Finite Groups of Isometries

Definition 2.12 Let X={1,2,,n}

be a finite set. The symmetric group Sn

is the set of all permutations of X

. The order of Sn

is Sn=n!=12n


Definition 2.13 The alternating group An

is the set of all even permutations in Sn

. The order of group An

is An=|Sn|2=n!2


Definition 2.14 The dihedral group Dn

is the symmetry group of a regular polygon with n


Definition 2.15 The cyclic group Cn

, with n

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

A matrix in O(3,R)

determines an isometry of R3

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 f: S2S2

may be extended to a map g: R3R3

fixing the origin, which for non-zero x

is defined by

g(x):= x f(x/x).

With the standard inner-product ( , )

on R3

, (g(x),g(y))=(x,y)

for any x,yR3

. For x,y

non-zero, this follows since

(g(x),g(y)) = x y (f(x/x),f(y/y))

= x y (x/x,y/y) = (x,y).

From this we infer that g

is an isometry of R3

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

  Therefore, Isom(S2)

is naturally acknowledged with the group O(3, R)


The restriction to S2

of the isometry RH

of R3

, the reflection of R3

in the hyperplane H

is defined as the reflection of S2

in a spherical line Ɩ

. Therefore, three such reflections are the most any element of Isom(S2)

can be composite of. Isometries that are just rotations of S2

and are the composite of two reflections are an index two subgroup of Isom(S2)

corresponding to the subgroup SO(3)O(3)

. The group O(3)

is isomorphic to SO(3)×C2

, since any element of O(3)

is of the form ±A

, with ASO(3)


Any finite subgroup G

of Isom(R3)

has a fixed point in R3


1|G| g(0)R3,

and corresponds to a finite subgroup of Isom(S2)

. Since any finite subgroup of Isom(R2)

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

We consider the group of rotations SO(3)

. All finite subgroups of SO(3)

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 Cn

are contained in SO(3) 

by considering rotations of S2

about the z

-axis through angles which are multiples of  2π/n

. We generate a new subgroup of SO(3)

by also including the rotation of S2

about the  x

-axis through an angle π

which is isomorphic to the group of symmetries D2n

of the regular n

gon for  n > 2

. We have the special case D4=C2×C2 

when n=2


However, corresponding to the rotation groups of the regular solids, there are further finite subgroups of SO3

. The tetrahedron has rotation group A4

, the cube has rotation group S4

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 A5


Proposition 2.16 The finite subgroups of SO(3)

are of isomorphism types Cn

for  1

, D2n

for  2

, A4, S4, A5

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

Since IO(3)  SO(3), H=C2× G

is a subgroup of O(3)

of twice the order if G

is a finite subgroup of SO(3)

, with elements ±A

for AG


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 Δ

with angles π/p, π/

and  π/

with rqp2


The tessellation of S2

is by the images of Δ under the elements of G

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

Δ )

for gG

covers S2

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 S2

. Therefore, the reflection group G

is finite.

From Gauss-Bonnet Theorem, the area of Δ is π(1/+ 1/+ 1/r1)

, and hence 1/+ 1/+ 1/> 1


The only solutions are:

  • (p,q,r) = (2,2,n)

    with  2

    . The area of Δ is π/n


  • (p,q,r) = (2,3,3)

    . The area of Δ is π/6


  • (p,q,r) = (2,3,4)

    . The area of Δ is π/12


  • (p,q,r) = (2,3,5)

    . The area of Δ is π/30



has order 4n, 24, 48 and 120 in these cases. This is implied from the tessellation of S2

by the images of Δ under G

. It is then clear that G

is C2× D2n

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 π

is the Euclidean version of Gauss-Bonnet.

Proposition 2.17 If Δ is a spherical triangle with angles α,β,γ

, its area is (α+β+γ) π


For a spherical triangle, α+β+γ > π

. We obtain the Euclidean case; α+β+γ = π

in the limit as area Δ  0


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

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

. The area of the original triangle is still

α+β+γ+π2π = α+β+γπ

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

The Gauss-Bonnet can be extended to spherical polygons on S2

. Consider a simple closed polygonal curve C

on S2

, 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 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 C

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

in S2

. 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 S2


A subset A

of S2

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

to  Q

, for any points P,QA

and this line segment is contained in A



Theorem 2.18 If S2

is a spherical n

-gon, contained in some open hemisphere, with interior angles α1,...,αn

, its area is

α1+...+αn (n2) π.


2.6 Möbius Geometry

Möbius transformations on the extended complex plane C={}

is closely related to spherical geometry, with a coordinate ϛ

. The stereographic projection map

π: S2C


defined geometrically by the diagram below provides this connection.

The point of intersection of the line through N

and P

with C

is π(P)

, where the plane z=0

identifies C

, and where we define πN:=

; π

is a bijection.

Using the geometry of similar triangles, an explicit formula for π

can be formed;

π(x,y,z) =x+iy1z 

since in the diagram below rR=1z1

and so R=r1z





Lemma 2.19 If π: S2 C

denotes the stereographic projection from the south pole, then

π(P) = 1 / π(P)̅

for any PS2.

The map ππ1 :CC

is just inversion in the unit circle, ϛ1/ϛ̅


If P=(x,y,z)S2

, then πP=ϛ== x + iy1z


The antipodal point = (x,yz)

has πP=x+iy1+z

and so



π(P) =  1 / π(P).̅

The group G

, of Möbius transformations, is acting on C

. A

defines a Möbius transformation on C



if A=abcdGL(2,C)



defines the same Möbius transformation for any λC*=C{0}


Conversely, if A1,A2

define the same Möbius transformation, then the identity transformation is identified by A21A1

. This simplifies that A21A1= λI

for some λC*,

and hence that A1=λA2

. Therefore

= PGL(2,C) := GL(2,C) / C*


identifying elements of GL(2,C)

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

If det A1=1=det A2

and A1=λA2

, then λ2=1

, and so λ=±1

. Therefore

= PSL(2,C) := SL(2,C) / {±1}


where identifying elements of SL(2,C)

which differ only by a sign attains the group on the right. The quotient map  SL(2,C)  G

is a surjective group homomorphism which is 2-1. SL(2,C)

is a double cover of G


Elementary facts about Möbius transformation

  1. The group G

    of Möbius transformations is generated by elements of the form

  • z+ a   for aC
  • zaz        for aC* = {0}
  • 1/z. 
  1. Any circle/straight line in C

    is of the form

azz̅  w̅ wz̅ + c = 0,

for a,cR

, w C

such that |w|2> ac

, and therefore is determined by an indefinite hermitian 2 × 2



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

    such that  

T(z1)=0, T(z2)=1, T(z3)=



given distinct points z1,z2,z3C


  1. The image of z4

    under the unique map T

    defined above in iv. is defined by the cross-ratio [z1,z2,z3,z4]

    of distinct points of C


There exists a unique Möbius transformation T

sending R(z1),R(z2)

and R(z3)

to 0,1 


, given distinct points z1,z2,z3,z4

and a Möbius transformation R

. The composite TR

is therefore the unique Möbius transformation sending z1,z2

and z3

to 0,1 


. Our definition of cross-ratio then implies that

[Rz1,Rz2,Rz3,Rz4] = T(Rz4) = (TR) z4= z1,z2,z3,z4.


2.7 The double cover of SO(3)

We have an index two subgroup of the full isometry group O(3)

, the rotations SO(3)

on S2

. The section aims to show that the group SO(3)

is established isomorphically with the group PSU(2)

by the stereographic projection map π

. There is a surjective homomorphism of groups SU(2)SO(3)

, which is 21



Theorem 2.20 Every rotation of S2

corresponds to a Möbius transformation of C

in PSU2

via the map π.

Theorem 2.21 The group of rotations SO(3) 

acting on S2

corresponds isomorphically with the subgroup PSU(2)=SU(2)/{±1}

of Möbius transformations acting on C


Corollary 2.22 The isometries of S2

which are not rotations correspond under stereographic projection precisely to the transformations of C

of the form


with |a|2+|b|2=1.

There exists a 2-1 map


This map is usually produced using quaternions.

This is the reason why a non-closed path of transformations in SU(2) 

going from

to I

exists, corresponding to a closed path in SO(3)

starting and ending at 100010001


Since SU(2)

consist of matrices of the form abb̅a̅


with |a|2+|b| 2=1

, geometrically it is S3R4.

There are finite subgroups of SU(2)

of double the order corresponding to finite subgroups of SO(3)

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

2.8 Circles on S2

We consider the locus of points on S2

, whose spherical distance from P

is ρ

, given an arbitrary point P

on S2

and 0  p  π

. 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 sin(ρ)

and that it is the intersection of a plane with S2

. Conversely, a plane cuts out a circle if its intersection with S2

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

2π1cosρ=4π sin2ρ2,

which, from the Euclidean case, is always less than the area πρ2.

For small ρ

this may be expanded as


2.23 Exercise Two spherical triangles 1,2

on a sphere S2

are said to be congruent if there is an isometry of S2

that takes 1

to 2

. 1,2

are congruent if and only if they have equal angles.

Proof Let ABC

and DEF

have A=D

etc and let ABC

and DEF

be the polar triangles. By theorem 2.18,

B C = π  = π  =EF

and so on. So, by the three sides, ABC

is congruent to DEF

which means that they have the same angles. Now theorem 2.17 implies that ABC

and DEF

are the polar triangles of ABC

and DEF

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

BC = π  0 = π  0 = EF

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


In conclusion, in this report we have discussed isometries and the group O(3,R)

, including the special orthogonal group SO(3)

. As well as exploring related concepts within Euclidean geometry and spherical geometry, we have analysed the finite groups of SO(3)

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 SO(3)

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


  • 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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