An *isometry* is a transformation of the plane that takes each line segment to a congruent line segment and each angle to a congruent angle. Basic types of isometries include reflections, rotations, and translations.

In the last last post, we saw that a symmetry of a geometrical figure is an isometry that preserves the figure. A *group* is a collection of symmetries satisfying: (1) if a symmetry T is in the group, then its inverse is also in the group; and (2) if T and U are in the group, then the composition U ∘ T is also in the group.

We discussed the dihedral groups D_{3} of the equilateral triangle and D_{4} of the square. Now consider the set of symmetries that preserves the regular pentagon.

In this case we have five reflections: R_{1}, R_{2}, R_{3}, R_{4}, and R_{5}. In addition, we have clockwise rotation S through one fifth of a complete revolution, or 72°, and its compositions with itself: S^{2}, S^{3}, S^{4}. So the dihedral group D_{5} consists of ten symmetries: I, S, S^{2}, S^{3}, S^{4}, R_{1}, R_{2}, R_{3}, R_{4}, and R_{5}. Once again, we can write D_{5} = 〈R_{1},R_{2}〉, where R_{1} and R_{2} are reflections across adjacent lines making an angle of 36° with one another. For instance, R_{1} ∘ R_{2} is a 72°-rotation.

Now imagine extending the sides of the pentagon as shown.

This “breaks” the symmetry. No longer can we reflect. However, the rotations still preserve the figure. So the symmetry group now consists of five symmetries: I, S, S^{2}, S^{3}, and S^{4}. This is referred to as the *cyclic group* C_{5}. It’s generated by S, so we can write C_{5} = 〈S〉. The cyclic group is a subgroup of the dihedral group.

In general, the cyclic group C_{n} is generated by a rotation through 360°/n, or one n^{th} of a revolution, while the dihedral group D_{n} is generated by two reflections whose lines make an angle of 360°/2n. Cyclic and dihedral groups abound in ornamental symmetry; Owen Jones’ 1910 *Grammar of Ornament* contains a wealth of examples.

The group D_{1} contains a single reflection; it may be thought of as the set of symmetries of the letter **V**. This is bilateral symmetry, as exhibited in the following Egyptian ornament.

The group D_{2} contains two reflections across perpendicular lines; it may be thought of as the set of symmetries of the letter **H**. We see examples of it in the following Greek and Pompeian ornaments.

Notice that if we ignore the color in the Pompeian ornament, then we have a D_{4} symmetry instead. A true example of D_{4} symmetry is given in the following Persian ornament.

Larger dihedral groups are rarer but still occur. The following ornaments are, respectively, Assyrian, Chinese, and Byzantine; they have symmetry groups D_{5}, D_{8}, and D_{22}.

Ornaments that exhibit only a cyclic symmetry generally have some kind of overlapping knot pattern or swirl element which prevents reflection. The cyclic group C_{1} is the trivial group consisting of just the identity.

The cyclic group C_{2} contains a single rotation through 180°. This may be thought of as the set of symmetries of the letter **N**. The following Celtic ornament has C_{2} symmetry; the overlapping of the knot prevents its having D_{2} symmetry.

The cyclic group C_{4} is generated by a 90°-rotation. Examples include the following medieval European and Moorish ornaments.

The group C_{6} is generated by a 60°-rotation, while C_{8} is generated by a 45°-rotation; these are exhibited in the following medieval European and Arabian ornaments.

In the latter case, we have to ignore the box design around the circular ornament. If we don’t, then the symmetry group is C_{4} rather than C_{8}. In general, C_{n} contains C_{m} as a subgroup if and only if m is a divisor of n. So C_{8} contains C_{1}, C_{2}, and C_{4}. Similarly, C_{6} contains C_{1}, C_{2}, and C_{3}. We’ll return to this in the next post on the topic.