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
. We shall specifically be considering finite groups of the special orthogonal group
, 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
.
Introduction
This paper will explore the finite groups of isometries, specifically the special orthogonal group
which is a natural subgroup of
. 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
, 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.
2 is the plane, where points are represented as ordered pairs:
.
n, which is the n-dimensional Euclidean space, is the space of n-tuples of real numbers:
.
In this report, the Euclidean space
n shall be used, equipped with the standard Euclidean inner-product
The inner-product is defined by
) =
.
The Euclidean norm on
n is defined by
and the distance function
is defined by
.
Definition 1.1 A metric space is a set
equipped with a metric
, namely
, satisfying the following conditions:
-
with
if and only if
.
for any points
.
The Euclidean distance
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
, the inverse image of every open subset of
is open in
.
A homeomorphism between given metric spaces
and
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
and
be metric spaces with metrics
and
. 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.
denotes the isometry group or the symmetry group, which are the isometries of a metric space
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
is a set of elements with a binary operation
called multiplication, satisfying three axioms:
-
,
-
,
-
There exists an inverse
such that
.
Definition 1.5 A group
is isomorphic to a group
if there is a bijection
from
to
such that
.
Definition 1.6 A group
acts on a set
if there is a map
;
, such that
-
for the identity
of
and any point
-
for
and any point
.
If for all
, there exists
with
then the action of
is transitive.
For the case of the Euclidean space
, with its standard inner-product
and distance function
, the isometry group
acts transitively on
since any translation of
is an isometry. A rigid motion is sometimes used to refer to an isometry of
.
Theorem 1.7 An isometry
is of the form
, for some orthogonal matrix
and vector
.
Lemma 1.8 Given points
, there exists a hyperplane
, consisting of the points of
which are equidistant from P and Q, for which the reflection
swaps the points P and Q.
Theorem 1.9 Any isometry of
can be written as the composite of at most
reflections.
1.3 The group
The orthogonal group, denoted
, is a natural subgroup
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.
is a group with respect to matrix multiplication
.
If
, then
and so
or
.
The special orthogonal group, denoted
,is the subgroup of
which consists of elements with
Direct isometries of
are the isometries of
of the form
, for some
and
. They can be expressed as a product of an even number of reflections.
Suppose that
. First consider the case where
, so
. Then
i.e.
is an eigenvalue.
Therefore, there exists an eigenvector
such that
.
is set to be the orthogonal complement to the space spanned by
. Then
if
. Thus
and
is a rotation of the two-dimensional space
, since it is an isometry of
fixing the origin and has determinant
. If
is an orthonormal basis for
, the matrix
represents the action of
on
with respect to the orthonormal basis
.
This is just rotation about the axis spanned by
through an angle
. It may be expressed as a product of two reflections.
Now suppose
.
Using the previous result, there exists an orthonormal basis with respect to which
is a rotation of the above form, and so
takes the form
With
. Such a matrix
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
,
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
is a continuous function
for some real closed interval
.
If a continuous path can join any two points of
, 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
, with
arbitrary, for a curve
on a metric space
.
We set
and
The length
of
is defined by
if this is finite.
For curves in
, 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
denotes a path connected open subset or
.
A metric space
is called a length space if
inf
length
:
a curve joining
to
},
for any two points
of
.
The metric is sometimes called intrinsic metric.
We can identify a metric
on
, defining
to be the infimum of lengths of curves joining the two points, if we start from a metric space
that satisfies the property that any two points may be joined by a curve of finite length. This is a metric, and
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
is called a Cauchy sequence if, for any
there exists an integer
such that if
then
A metric space (X, d) in which every Cauchy sequence
converges to an element of X is called complete. This means that a point
such that
as
. These limits are unique.
The real line is complete since real Cauchy sequences converge. The Euclidean space
is also complete when this is applied to the coordinates of points in
. A subset
of
will be complete if and only if it is closed.
Definition 1.14 Let
be a metric space with metric d. If every open cover of X contains a finite subcover,
is compact.
An open cover of
is a collection
of open sets if every
belongs to at least one of the
, with
. If the index
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
is called sequentially compact.
Lemma 1.15 A continuous function
on a compact metric space
is uniformly continuous.
i.e. given
there exists
such that if
, then
.
Lemma 1.16 If
is a closed subset of a compact metric space
, then
is compact.
Since
is a closed subset of some closed box
, we infer that any closed and bounded subset
of
is compact.
Lemma 1.17 If
is a continuous surjective map of metric spaces, with
compact, then so is
.
1.6 Polygons in the Euclidean Plane
Euclidean polygons in
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
. It is called simple if, for
, we have
, except for
and
, when the curve is closed.
Proposition 1.19 Let
be a simple closed polygonal curve, with
denoting the image
Then
has at most two path connected components.
Given a set
a continuous function
such that
is an argument of
for all
, is a continuous branch of the argument on
.
A continuous branch of the argument exists on
if and only if a continuous branch of the logarithm exists.
i.e. a continuous function
such that exp
for all
.
For a curve
; a continuous branch of the argument for
is a continuous function
such that
is an argument for
for all
Continuous branches of the argument of curves in
* always exist, unlike continuous branches of the argument for subsets. The use of continuity of the curve can show that they exist locally on
. Then, a continuous function overall of
can be achieved using the compactness of
.
For a closed curve
, the winding number of
about the origin, is any continuous branch of the argument
for
. This is denoted
and is defined
.
Given a point
not on a closed curve
, the integer
defines the winding number of
about
, where
is the curve whose value at
is
. The integer
describes how many times the curve
‘winds around
.
Elementary properties of the winding number of a closed curve
:
-
The winding number does not change when reparametrising
or changing the starting point on the curve. However, if
denotes the curve
travelled in the opposite direction i.e.
then for any
not on the curve,
We have
for the constant curve
.
-
if a subset
contains the curve
on which a continuous branch of the argument can be defined. Therefore, if a closed ball
contains
, then
for all
.
-
The winding number
is a constant on each path connected component of the complement of
, as a function of
.
-
If
are two closed curves with
, we can form the link
, defined by
Then, for
not in the image of
, we have
Definition 1.20
is compact for a simple closed polygonal curve with image
, and hence bounded. Therefore, some closed ball
contains
. One of the two components of
contains the complement of
since any two points in the complement of
may be joined by a path and hence is unbounded, whilst the other component of
is contained in
, and hence is bounded. The closure of the bounded component will be a closed polygon in
or a Euclidean polygon. This consists of the bounded component together with
. Since a Euclidean polygon is closed and bounded in
, it is also compact.
1.21 Exercise The rotation group for a cube centred at the origin in
is isomorphic to
, 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
rotations give the permutations
and
can be seen by numbering the diagonals as 1,2,3 and 4. These make an 8-element subgroup
and the 3-element subgroup
.
Thus, the rotations make all 24 permutations since
.
2 Spherical Geometry
2.1 Introduction
Let
denote a unit sphere in
with centre
.
The intersection of
with a plane through the origin is a great circle on
. This is the spherical lines on
.
Definition 2.1 The distance
between
and
on
is defined to be the length of the shorter of the two segments
along the great circle. This is
if
and
are on opposite sides.
is the angle between
and
, and hence is just
, where
is the Euclidean inner-product on
.
2.2 Spherical Triangles
Definition 2.2 A spherical triangle
on
is defined by its vertices
, and sides
and
, where these are spherical line segments on
of length
.
The triangle
is the region of the sphere with area
enclosed by these sides.
Setting
and
,
gives the length of the side
. For the lengths
of the sides
and
, similar formulae are used.
The unit normals to the planes
are set by denoting the cross-product of vectors in
by
;
.
Given a spherical triangle
, the polar triangle
is the triangle with
a pole of
on the same side as
,
a pole of
on the same side as
, and
a pole of
on the same side as
.
Theorem 2.3 If
is the polar triangle to
, then
is the polar triangle to
.
Theorem 2.4 If
is the polar triangle to
, then
Theorem 2.5 (Spherical cosine formula)
Corollary 2.6 (Spherical Pythagoras theorem)
When
,
Theorem 2.7 (Spherical sine formula)
Corollary 2.8 (Triangle inequality)
For
,
with equality if and only if
is on the line segment
.
Proposition 2.9 (Second cosine formula)
2.3 Curves on the sphere
The restriction to
of the Euclidean metric on
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
on
joining points
on
.
Proposition 2.11 Given a curve
on
joining points
and
, we have
length
In addition, the image of
is the spherical line segment
on
if
.
A spherical line segment is a curve
of minimum length joining
and
. So
length
for all
. Therefore, the parameterisation is monotonic since
is strictly increasing as a function of
.
2.4 Finite Groups of Isometries
Definition 2.12 Let
be a finite set. The symmetric group
is the set of all permutations of
. The order of
is
.
Definition 2.13 The alternating group
is the set of all even permutations in
. The order of group
is
.
Definition 2.14 The dihedral group
is the symmetry group of a regular polygon with
sides.
Definition 2.15 The cyclic group
, with
elements, is a group that is generated by combining a single element of the group multiple times.
A matrix in
determines an isometry of
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
is defined by
With the standard inner-product
on
,
for any
. For
non-zero, this follows since
From this we infer that
is an isometry of
which fixes the origin and is given by a matrix in
Therefore,
is naturally acknowledged with the group
.
The restriction to
of the isometry
of
, the reflection of
in the hyperplane
is defined as the reflection of
in a spherical line
. Therefore, three such reflections are the most any element of
can be composite of. Isometries that are just rotations of
and are the composite of two reflections are an index two subgroup of
corresponding to the subgroup
. The group
is isomorphic to
, since any element of
is of the form
, with
.
Any finite subgroup
of
has a fixed point in
,
and corresponds to a finite subgroup of
. Since any finite subgroup of
has a fixed point, it is either a cyclic or dihedral group.
We consider the group of rotations
. All finite subgroups of
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
are contained in
by considering rotations of
about the
-axis through angles which are multiples of
. We generate a new subgroup of
by also including the rotation of
about the
-axis through an angle
which is isomorphic to the group of symmetries
of the regular
gon for
. We have the special case
when
.
However, corresponding to the rotation groups of the regular solids, there are further finite subgroups of
. The tetrahedron has rotation group
, the cube has rotation group
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
.
Proposition 2.16 The finite subgroups of
are of isomorphism types
for
,
for
,
, the last three being the rotation groups arising from the regular solids.
Since
is a subgroup of
of twice the order if
is a finite subgroup of
, with elements
for
.
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
generated by the reflections in the sides of the triangle, if we have a spherical triangle
with angles
and
with
.
The tessellation of
is by the images of Δ under the elements of
by the theory of reflection groups. This means that the spherical triangles
Δ
for
covers
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
. Therefore, the reflection group
is finite.
From Gauss-Bonnet Theorem, the area of Δ is
, and hence
.
The only solutions are:
-
with
. The area of Δ is
.
-
. The area of Δ is
.
-
. The area of Δ is
.
-
. The area of Δ is
.
has order 4n, 24, 48 and 120 in these cases. This is implied from the tessellation of
by the images of Δ under
. It is then clear that
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
is the Euclidean version of Gauss-Bonnet.
Proposition 2.17 If Δ is a spherical triangle with angles
, 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
. Consider a simple closed polygonal curve
on
, where spherical line segments are the segments of
. Suppose that the north pole does not lie on
. We consider a simple closed curve in
the image
of
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
. Therefore, two path connected components are also contained in the complement of
in
. 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
and a choice of a connected component of its complement in
.
A subset
of
is called convex if there is a unique spherical line segment of minimum length joining
to
, for any points
and this line segment is contained in
.
Theorem 2.18 If
is a spherical
-gon, contained in some open hemisphere, with interior angles
, its area is
2.6 Möbius Geometry
Möbius transformations on the extended complex plane
is closely related to spherical geometry, with a coordinate
. The stereographic projection map
,
defined geometrically by the diagram below provides this connection.
The point of intersection of the line through
and
with
is
, where the plane
identifies
, and where we define
;
is a bijection.
Using the geometry of similar triangles, an explicit formula for
can be formed;
since in the diagram below
and so
.
Lemma 2.19 If
denotes the stereographic projection from the south pole, then
for any
The map
is just inversion in the unit circle,
.
If
, then
.
The antipodal point
has
and so
Therefore
The group
, of Möbius transformations, is acting on
.
defines a Möbius transformation on
by
if
.
defines the same Möbius transformation for any
.
Conversely, if
define the same Möbius transformation, then the identity transformation is identified by
. This simplifies that
for some
and hence that
. Therefore
,
identifying elements of
attains the group on the right, which are non-zero multiples of each other.
If
and
, then
, and so
. Therefore
,
where identifying elements of
which differ only by a sign attains the group on the right. The quotient map
is a surjective group homomorphism which is 2-1.
is a double cover of
.
Elementary facts about Möbius transformation
-
The group
of Möbius transformations is generated by elements of the form
-
Any circle/straight line in
is of the form
for
,
such that
, and therefore is determined by an indefinite hermitian
matrix
- Möbius transformations send circles/straight lines to circles/straight lines.
-
There exists a unique Möbius transformation
such that
,
given distinct points
.
-
The image of
under the unique map
defined above in iv. is defined by the cross-ratio
of distinct points of
.
There exists a unique Möbius transformation
sending
and
to
and
, given distinct points
and a Möbius transformation
. The composite
is therefore the unique Möbius transformation sending
and
to
and
. Our definition of cross-ratio then implies that
2.7 The double cover of
We have an index two subgroup of the full isometry group
, the rotations
on
. The section aims to show that the group
is established isomorphically with the group
by the stereographic projection map
. There is a surjective homomorphism of groups
, which is
map.
Theorem 2.20 Every rotation of
corresponds to a Möbius transformation of
in
via the map π.
Theorem 2.21 The group of rotations
acting on
corresponds isomorphically with the subgroup
of Möbius transformations acting on
.
Corollary 2.22 The isometries of
which are not rotations correspond under stereographic projection precisely to the transformations of
of the form
with
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
going from
to
exists, corresponding to a closed path in
starting and ending at
.
Since
consist of matrices of the form
with
, geometrically it is
There are finite subgroups of
of double the order corresponding to finite subgroups of
, specifically cyclic, dihedral and the rotation groups of the tetrahedron, cube and dodecahedron.
2.8 Circles on
We consider the locus of points on
, whose spherical distance from
is
, given an arbitrary point
on
and
. In spherical geometry, this is what is meant by a circle.
To ensure the point
is always at the north pole, we may rotate the sphere, as shown below:
Therefore, the circle is also a Euclidean circle of radius
and that it is the intersection of a plane with
. Conversely, a plane cuts out a circle if its intersection with
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
For small
this may be expanded as
2.23 Exercise Two spherical triangles
on a sphere
are said to be congruent if there is an isometry of
that takes
to
.
are congruent if and only if they have equal angles.
Proof Let
and
have
etc and let
and
be the polar triangles. By theorem 2.18,
and so on. So, by the three sides,
is congruent to
which means that they have the same angles. Now theorem 2.17 implies that
and
are the polar triangles of
and
. 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
, including the special orthogonal group
. As well as exploring related concepts within Euclidean geometry and spherical geometry, we have analysed the finite groups of
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
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.
Cite This Work
To export a reference to this article please select a referencing stye below:
Related Services
View allDMCA / Removal Request
If you are the original writer of this essay and no longer wish to have your work published on UKEssays.com then please: