Finite Groups of Isometries

Abstract

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)
.

Introduction

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

R
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

(x,y
) =
?i=1nxiyi
.

The Euclidean norm on
R
n is defined by

?x? =(x,x)

and the distance function
d
is defined by

dx,y= ?x–y?
.

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,Q?0
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.

i.e.

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

x,y?G?x,y?G

called multiplication, satisfying three axioms:

xyz=xyz?x,y,z?G
,

xe=ex=x ?x,e?G
,

There exists an inverse
x–1?G
such that
xx–1=x–1x=e ?x?G
.

Definition 1.5 A group
G
is isomorphic to a group
G‘
if there is a bijection
?
from
G
to
G‘
such that
?xy=?(x)?(y)
.

Definition 1.6 A group
G
acts on a set
X
if there is a map
G × X ? X
;
g,x?g?x
, such that

e?x=x
for the identity
e
of
G
and any point
x?X

g?h?x=(gh)?x
for
g,h ?G
and any point
x?X
.

If for all
x,y ?X
, there exists
g ?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
b?Rn
.

Lemma 1.8 Given points
P ? 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)
reflections.

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 × n
orthogonal matrices.

On?{A?Mn×nR:ATA=AAT=I}

is a group with respect to matrix multiplication
X,Y?XY
.

If
A ? 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
A?SO(n)
and
b?Rn
. They can be expressed as a product of an even number of reflections.

Suppose that
A?O(3)
. First consider the case where
A?SO(3)
, so
detA=1
. Then

det?A–I=det(At–I)=detA(At–I)=det?(I–A)

? det?A–I=0,

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
w?W
. 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

1000cos??–sin??0sin??cos??

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
?
. 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

–1000cos??–sin??0sin??cos??

With
?=?+?
. Such a matrix
A
represents a rotated reflection, rotating through an angle
?
about a given axis and then reflecting in the plane orthogonal to the axis. In the special case
?=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)
?
in a metric space
(X,d)
is a continuous function
? :[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 ? : [a,b] ? X on a metric space (X,d) . We set Pi= ? (ti) and SD:= ? d(Pi,Pi+1). The length ? of ? is defined by ? = 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 ? : [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 ( ? ) : ? 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 ? : [a,b] ? R3 is continuously differentiable, then length ? = ? ? ?‘(t) ? dt , where the integrand is the Euclidean norm of the vector ?‘ t?R3. 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 ? > 0
there exists an integer
N
such that if
m,n ? N
then
d(xm,xn) < ?. 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 ? 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 ?I of open sets if every x?X belongs to at least one of the Ui , with i?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 (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 ? > 0
there exists
? > 0
such that if
d(x,y) < ? , then |f(x)–f(y)| < ? . 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 ?: [a,b]?X is called closed if ?(a)=?(b) . It is called simple if, for t1 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
n ? 1
,
D2n
for
n ? 2
,
A4, S4, A5
, the last three being the rotation groups arising from the regular solids.

Since
–I?O(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
A?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
?
with angles
?/p, ?/q
and
?/r
with
r?q?p?2
.

The tessellation of
S2
is by the images of ? under the elements of
G
by the theory of reflection groups. This means that the spherical triangles
g(
?
)
for
g?G
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 ? is
?(1/p + 1/q + 1/r–1)
, and hence
1/p + 1/q + 1/r > 1
.

The only solutions are:

(p,q,r) = (2,2,n)
with
n ? 2
. The area of ? is
?/n
.

(p,q,r) = (2,3,3)
. The area of ? is
?/6
.

(p,q,r) = (2,3,4)
. The area of ? is
?/12
.

(p,q,r) = (2,3,5)
. The area of ? is
?/30
.

G
has order 4n, 24, 48 and 120 in these cases. This is implied from the tessellation of
S2
by the images of ? 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
?
is the Euclidean version of Gauss-Bonnet.

Proposition 2.17 If ? is a spherical triangle with angles
?,?,?
, its area is
(?+?+?)– ?
.

For a spherical triangle,
?+?+? > ?
. We obtain the Euclidean case;
?+?+? = ?
in the limit as area
? ? 0
.

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

?+?+?+?–2? = ?+?+?–?

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
P
to
Q
, for any points
P,Q?A
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
?1,…,?n
, its area is

?1+…+?n– (n–2) ?.

2.6 Möbius Geometry

Möbius transformations on the extended complex plane
C?=C U {?}
is closely related to spherical geometry, with a coordinate
?
. The stereographic projection map

?: S2?C?
,

defined geometrically by the diagram below provides this connection.

The point of intersection of the line through
N
and
P
with
C
is
?(P)
, where the plane
z=0
identifies
C
, and where we define
?N:=?
;
?
is a bijection.

Using the geometry of similar triangles, an explicit formula for
?
can be formed;

?(x,y,z) =x+iy1–z

since in the diagram below
rR=1–z1
and so
R=r1–z
.

Lemma 2.19 If
?‘: S2? C?
denotes the stereographic projection from the south pole, then

?‘(P) = 1 / ?(P)?

for any
P?S2.

The map
?‘??–1 :C??C?
is just inversion in the unit circle,
??1/??
.

If
P=(x,y,z)?S2
, then
?P=?== x + iy1–z
.

The antipodal point
–P = (–x,–y–z)
has
?–P=–x+iy1+z
and so

?P?–P?=–x2+y21–z2=–1.

Therefore

?(–P) = – 1 / ?(P).?

The group
G
, of Möbius transformations, is acting on
C?
.
A
defines a Möbius transformation on
C?
by

??a?+bc?+d

if
A=abcd?GL(2,C)
.

?A
defines the same Möbius transformation for any
??C*=C{0}
.

Conversely, if
A1,A2
define the same Möbius transformation, then the identity transformation is identified by
A2–1A1
. This simplifies that
A2–1A1= ?I
for some
??C*,
and hence that
A1=?A2
. Therefore

G = 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=?A2
, then
?2=1
, and so
?=±1
. Therefore

G = 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

The group
G
of Möbius transformations is generated by elements of the form

z?z + a for a?C

z?az for a?C* =C {0}

z ?1/z.

Any circle/straight line in
C
is of the form

azz? – w?z – wz? + c = 0,

for
a,c?R
,
w ?C
such that
|w|2> ac
, and therefore is determined by an indefinite hermitian
2 × 2
matrix

aww?c.

Möbius transformations send circles/straight lines to circles/straight lines.

There exists a unique Möbius transformation
T
such that

T(z1)=0, T(z2)=1, T(z3)=?
,

Tz=z–z1z–z3z2–z3z2–z1,

given distinct points
z1,z2,z3?C?
.

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
and
?
, 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
and
?
. 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
?
. There is a surjective homomorphism of groups
SU(2)?SO(3)
, which is
2–1
map.

Theorem 2.20 Every rotation of
S2
corresponds to a Möbius transformation of
C?
in
PSU2
via the map ?.

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

z?az?–bb?z?+a?

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

There exists a 2-1 map

SU(2)?PSU(2)?SO(3).

This map is usually produced using quaternions.

This is the reason why a non-closed path of transformations in
SU(2)
going from
I
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
a–bb?a?

with
|a|2+|b| 2=1
, geometrically it is
S3?R4.

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
?
, given an arbitrary point
P
on
S2
and
0 ? p ? ?
. 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(?)
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

2?1–cos??=4? sin2?2,

which, from the Euclidean case, is always less than the area
??2.
For small
?
this may be expanded as

??21–112?2+O?4.

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
?A‘B‘C‘
and
?D‘E‘F‘
be the polar triangles. By theorem 2.18,

B‘ C‘ = ? – ?A = ? – ?D =E‘F‘

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
?ABC
and
?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

BC = ? – ?A 0 = ? – ?D 0 = EF

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

Conclusion

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.

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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