MTH3003 Lecture 6

Following on from dihedral groups and regular $n$-gons, we can expand our understanding of symmetries to more general shapes in ${\mathbb{R}}^n$. These symmetry groups consist of isometries - maps that preserve distance.

Symmetry Groups of More General Shapes

Given any shape $\Pi$ in ${\mathbb{R}}^n$, its symmetries (distance-preserving maps from ${\mathbb{R}}^n$ to ${\mathbb{R}}^n$ that send $\Pi$ to $\Pi$) form a group called the symmetry group of $\Pi$, denoted as $\text{Isom}(\Pi )$.

If $\Pi$ doesn’t have corners, we can define $X$ as all the points on its surface; the symmetry group of $\Pi$ is then a subgroup of $\text{Sym}(X)$. Because these maps preserve distance, we often refer to this as the symmetry group of isometries of $\Pi$.

Isometry Group of a Cube

Let $C$ be a cube. Any symmetry of $C$ is a permutation of its corners. If we pick a face $F$, the symmetry group of $F$ is $D_8$ (as it is a square). There are 6 faces that $F$ can be sent to via 3D rotations (let’s call them $R_1,\dots ,R_6$).

Applying a symmetry from $D_8$ to $F$ and then one of these 6 rotations gives every symmetry of the cube. Thus, the total symmetries equal $6\times |D_8|=48$, giving the group $\text{Isom}(C)=D_8{R}_1\cup \cdots \cup D_8{R}_6$.

Rotations of a Cube

There is a natural subgroup of $\text{Isom}(C)$ consisting only of rotations, denoted $\text{Rot}(C)$. Because our face $F$ has 4 rotations ($e,\rho ,{\rho }^2,{\rho }^3$ from $D_8$) and there are 6 rotations moving $F$ to each face, we combine them to find exactly $24$ rotations. So, $|\text{Rot}(C)|=24$.

These 24 rotations break down into 5 distinct types based on what they fix:

1. $1$ identity rotation: Does nothing (fixes all faces, corners, edges).
2. $6$ $90^{\circ }$ face rotations: One per face (fixes precisely two faces, no corners, no edges).
3. $3$ $180^{\circ }$ face rotations: One per pair of opposite faces (fixes precisely two faces, no corners, no edges).
4. $8$ $120^{\circ }$ corner rotations: One per corner (fixes precisely two corners).
5. $6$ $180^{\circ }$ edge rotations: Rotating an edge about its midpoint (fixes precisely two edges).

Infinite Isometry Groups

Isometry groups are not restricted to finite shapes. Consider an infinite ladder-like shape $S$ extending forever horizontally with nodes $a_i$ on top and $b_i$ on the bottom. Its symmetry group $G=\text{Isom}(S)$ contains:

• Translations (e.g., shifting everything one unit left/right): $\rho =(\dots a_{-1}{a}_0{a}_1\dots )(\dots b_{-1}{b}_0{b}_1\dots )$.
• Horizontal reflection (flipping top to bottom): $\sigma =\dots (a_0{b}_0)(a_1{b}_1)\dots$.
• Vertical reflections through an edge: ${\theta }_1=(a_{-1}{a}_1)(b_{-1}{b}_1)\dots$.
• Vertical reflections not through an edge: ${\theta }_2=\dots (a_{-1}{a}_0)(b_{-1}{b}_0)\dots$.

All other translations and reflections are powers or combinations of these, meaning the entire infinite group can be generated by just these four elements: $G=⟨\rho ,\sigma ,{\theta }_1,{\theta }_2⟩$.

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Pre-Lecture Notes from University Notes

• General shapes in ${\mathbb{R}}^n$ and defining $\text{Isom}(\Pi )$ for distance-preserving isometries.
• Worked example of a cube: 48 total symmetries combining $D_8$ and 6 face rotations.
• Breaking down the 24 specific rotations of a cube ($\text{Rot}(C)$) by their fixed elements.
• Looking at infinite isometry groups using a repeating grid/ladder shape, generated by translations and reflections.