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 D3 of the equilateral triangle and D4 of the square. Now consider the set of symmetries that preserves the regular pentagon.
In this case we have five reflections: R1, R2, R3, R4, and R5. In addition, we have clockwise rotation S through one fifth of a complete revolution, or 72°, and its compositions with itself: S2, S3, S4. So the dihedral group D5 consists of ten symmetries: I, S, S2, S3, S4, R1, R2, R3, R4, and R5. Once again, we can write D5 = 〈R1,R2〉, where R1 and R2 are reflections across adjacent lines making an angle of 36° with one another. For instance, R1 ∘ R2 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, S2, S3, and S4. This is referred to as the cyclic group C5. It’s generated by S, so we can write C5 = 〈S〉. The cyclic group is a subgroup of the dihedral group.
In general, the cyclic group Cn is generated by a rotation through 360°/n, or one nth of a revolution, while the dihedral group Dn 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 D1 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 D2 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 D4 symmetry instead. A true example of D4 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 D5, D8, and D22.
Ornaments that exhibit only a cyclic symmetry generally have some kind of overlapping knot pattern or swirl element which prevents reflection. The cyclic group C1 is the trivial group consisting of just the identity.
The cyclic group C2 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 C2 symmetry; the overlapping of the knot prevents its having D2 symmetry.
The cyclic group C4 is generated by a 90°-rotation. Examples include the following medieval European and Moorish ornaments.
The group C6 is generated by a 60°-rotation, while C8 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 C4 rather than C8. In general, Cn contains Cm as a subgroup if and only if m is a divisor of n. So C8 contains C1, C2, and C4. Similarly, C6 contains C1, C2, and C3. We’ll return to this in the next post on the topic.