Notation

• 0 \in \N and \N^+ :=\ \N \setminus \{0\}.
• Empty set is denoted with \varnothing.
• d_1 \>|\>d_2\>|\>\cdots\>|\>d_r means d_1 divides d_2 divides d_3...

Status

These notes are still under work therefore not complete, also

• there are not many exercises,
• proofs might be absent,
• typos are quite possible,
• ordering is generally good but should be improved, and
• more visuals and intuition should be provided.

Def. Isometry

Let (X, d_X) and (Y, d_Y) be metric spaces. A map f: X \to Y is called an isometry or distance-preserving map if for all a, b \in X,

Two metric spaces (X, d_X) and (Y, d_Y) are called isometric if there is a bijective isometry between them.

Thm. Basic Properties of Isometries

Let f: X \to Y be an isometry on the metric spaces (X, d_X) and (Y, d_Y), then

1. f is always injective.
2. f is not necessarily surjective. For example x \mapsto (x,0) is an isometry but not surjective on \R \to \R^2.
3. Composition of isometries are an isometry.

Exercise

Thm. Inverse of an Isometry

Let f: X \to Y be an isometry. If f is bijective then f^{-1} is also an isometry. Consider the case f is not bijective, then we can construct an bijective isometry

so that since f is injective (and thus f') and f' is surjective we have, by construction, f' bijective.

Thm. Isometries are Continuous

An isometry f: X \to Y is always continuous.

Def. \text{Isom}(X)

Let (X, d) be a metric space, then set of all bijective isometries on X denoted \text{Isom}(X) forms a group under composition called the (full) isometry group of X.

Exercise.

Thm. \text{Isom}(\Z)

We have

where \bold{D}_\infty is the infinite dihedral group.

Exercises

Def. Symmetry

Let (X,d) be a metric space and S \subseteq X. A symmetry of S is an isometry f: X \to X such that

Def. \text{Sym}(S)

Let (X,d) be a metric space and S \subseteq X, then the set of all symmetries of S forms a group, and thus is a subgroup of \text{Isom}(X).

Exercise

Example. \text{Sym}(\Z)

The group of symmetries of \Z is generated by two elements, particularly by a translation and a reflection so that

where t(n) = n+1 and r(n) = -n.

Def. Euclidian Isometry

From now on we will call an isometry f: \R^n \to \R^n under the Euclidian distance an Euclidian isometry.

Notice that \tilde{f}(x) := f(x) - f(0), which we will call reduced Euclidian isometry of f, is an origin-fixing Euclidian isometry so that \tilde{f}(0) = 0.

Thm. Euclidian Isometries are Bijective

Let f: \R^n \to \R^n be any Euclidian isometry, then f is bijective.

Thm. \text{Sym}(\R^n) = \text{Isom}(\R^n)

Obvious as each Euclidian isometry is bijective.

Thm. Norm Preservation

Let \tilde{f} be an origin-fixing Euclidian isometry, then \tilde{f} preserves the norm so that, for all x \in \R^n

Exercise

Thm. Inner Product Preservation

Let \tilde{f} be an origin-fixing Euclidian isometry, then \tilde{f} preserves the inner product so that, for all x,y \in \R^n

Thm. Additivity

Let \tilde{f} be an origin-fixing Euclidian isometry, then \tilde{f} is additive so that, for all x,y \in \R^n

TODO: Add not so simple proof which make use of norm expension and preservation

Thm. Euclidian Isometry Linearity

An Euclidian isometry f: \R^n \to \R^n is linear if and only if it fixes the origin.

Thm. General Euclidian Isometry

Let f: \R^n \to \R^n be any Euclidian isometry, then there exists an orthogonal matrix \bold{A} \in \R^{n \times n} and a vector b \in \R^n such that

for all x \in \R^n where b = f(0) and \bold{A} is uniquely determined.

Thm. \text{Isom}(\R)

Let f \in \text{Isom}(\R), then f is of the form

with \epsilon = \pm 1 and b \in \R. Moreover, if

• \epsilon = 1, then it is a translation, and
• \epsilon = -1, then it is a reflection

Direct result of the theorem General Euclidian Isometry proven above.

Thm. Euclidian Isometries Preserve the Midpoint

Let f: \R^n \to \R^n be an Euclidian isometry, then it preserves the midpoints so that

Exercise.

Note that this is not necessarily the case for other isometries nor other other norms on \R^n.

Thm. \text{Isom}(\R^n)

Let \text{Isom}(\R^n) denote the set of all isometries \R^n \to \R^n, then it forms a group under composition.

Def. Euclidian Group

Def. Polygon, n-gon

There is no established definition for a polygon. Currently think of it as an geometric object with a closed surface connected by closed polygonal chain.

An n-gon is a polygon with n edges.

Def. Regular n-gon

For n \geq 3, a regular n-gon denoted by \bold{P}_n is a (closed) polygon with n sides of equal length joined together at equal angles.

Thm. Regular n-gon Symmetries

A regular n-gon has 2n different symmetries which are n rotational symmetries and n reflection symmetries.

Thm. Dihedral Group

The symmetries of the regular n-gon \bold{P}_n forms a group under composition called the dihedral group denoted by \bold{D}_{2n}.

Some authors prefer the notation \bold{D}_n or \bold{Dih}_n, however notice that in our case 2n denotes the number of symmetries and its order.

\bold{D}_{2n} is the semidirect product of C_2 acting on C_n via the automorphism \varphi_s(r) = r^{-1}, therefore:

The group representation of \bold{D}_{2n} is as follows:

where r denotes a rotation and s denotes a reflection.

Notice that, by srs=r^{-1} condition, we have by induction: sr^k = r^{-k}s.

Def. Convex

A set is said to be convex if any two points A and B are contained in the set, the entire line segment [AB] also lies within the set.

Def. Polyhedron

A polyhedron is an union of finitely many convex polygons (called faces), arranged in space so that the intersection of any two polygons is a shared vertex or edge or the empty set.

This is the definition by O'Rourke (1993). Definitions of polyhedron (although differs) are equivalent and agreed upon.

Def. Platonic Solid

A platonic solid is a convex regular polyhedron in three-dimensional space. Therefore, its faces are congruent regular n-gons and at each vertex the same number of polygons (called faces) meet.

Thm. Euler's Polyhedron Formula

Any three-dimensional convex polyhedrons surface has an Euler characteristic of \chi = 2 so that

For the proof, please check out various online resources that are both geometric and algebraic.

Thm. Platonic Solids

There are only 5 platonic solids called (regular) tetrahedron, cube, (regular) octahedron, (regular) dodacahedron and (regular) icosahedron.

Thm. Cube Symmetries

Let G be the symmetry group of the cube including the rotations and the reflections, then

1. A cube has 24 rotations in total that consists of
   1. 9 rotations through 3 face axes
      1. 90\degree
      2. 180\degree
      3. 270\degree
   2. 8 rotations through 4 vertex axes
      1. 120\degree
      2. 240\degree
   3. 6 rotations through 6 edges
   4. the identity
2. G \cong \bold{S}_4 \times \Z_2 where \bold{S}_4 consists of rotations and \Z_2 consists of \Set{1,d} where d is the central inversion.
3. Every rotation of the cube maps vertices to vertices, so each diagonal is mapped to another diagonal.

Thm. Rotations on Cube

1. A non-trivial rotation fixes exactly one axis.
2. Two non-trivial rotations commute if and only if
   • They are on the same axis, or
   • Axes are perpendicular and both rotations are 180\degree.
3. Two rotations commute if and only if they preserve each others axis, that is, sends the others axis to itself.

Def. Inner Product Space

An inner product space is a vector space \mathcal{V} over the field \mathbb{F} together with an inner product

that satisfies, for all vectors x,y,z \in \mathcal{V} and scalars a,b \in \mathbb{F}

1. \Braket{x,y} = \overline{\Braket{y, x}} called symmetry axiom
2. \Braket{x + y, z} = \Braket{x,z} + \Braket{y,z} called additivity axiom
3. \Braket{ax, y} = a\Braket{x,y} called homogenity axiom
4. \Braket{x,x} \geq 0 and \Braket{x,x} = 0 \iff x = 0 called the positivity axiom

Thm. Cauchy-Schwarz Inequality

Let x,y be two vectors in the inner product space \mathcal{V}, then

where \Vert \cdot \Vert is the norm induced by the inner product.

Def. Kronecker Delta Function

Def. Orthonormality

Let \mathcal{V} be an inner product space. A set of vectors \Set{v_1, v_2, ...} \subseteq \mathcal{V} is called orthonormal if

Each vector is orthogonal to each other.

Def. Orthogonal Matrix

An orthogonal (or orthonormal) matrix is a real square matrix whose columns and rows are orthonormal vectors. Equivalently, a matrix Q is orthogonal if

An orthogonal matrix is called special if its determinant is 1.

Thm. Basic Properties of Orthonal Matrices

Let \bold{O} be an orthogonal matrix, then

1. \bold{O} is invertible.
2. \bold{O}^T \bold{O} = \bold{I}
3. The determinant of \bold{O} is either 1 or -1.
4. If \bold{O} is special orthogonal matrix, then — as a linear transformation — it acts as a rotation around a fixed point if n \geq 2 in \R^n.

Def. Semidirect Product

Let G be a group with the identity e, H \leq G and N \trianglelefteq G such that N \cap H = \Set{e}. We say G is the semidirect product of N and H if G=NH denoted by G = N \rtimes H or G = H \ltimes N.

Notice how \rtimes points toward the normal group similar to \trianglelefteq.

This is not a good definition of the semidirect product but a simple one. Curious reader might want to check out other resources.

Thm.

Let G be a group with a (normal) subgroup N of index 2. If there exists g \in G \setminus N of order 2, then

Thm. Orbit-Stabilizer Theorem

Let G be a group and x \in G, then

In order to denote the orbit of x, the notations

• \bar{x},
• G(x)
• Gx

are also used. Similarly, for the stabilizer of x, the notations

• G_x

could be used as well.

Thm. Burnside's Lemma

where

• |X/G| denotes the number of orbits, and
• |X^g| denotes the elements in X fixed by g

Notice X^g is not the stabilizer of x which are the elements in G fixed by g but rather in X.

Thm. Center of \bold{D}_{2n}