Many people know symmetry when they see it, even though they may not be able to describe what makes a figure or object symmetrical. In general, a symmetrical figure, design or pattern has some sort of regular repetition. That repetition is defined by a transformation.

A TRANSFORMATION is a rule that establishes a one-to-one correspondence between each point and its image. Your reflection in a ‘funny’ mirror at a carnival or a picture taken through a ‘fisheye’ lens are examples of transformations whose image is a distortion of the original, because it alters the shape of the original.

A DILATION is a special type of transformation that preserves shape. The image of a pattern under a dilation will always be similar to the original, and may be smaller, the same size, or larger than the original.

An ISOMETRY is a special type of dilation that preserves shape and size. Isometries are sometimes referred to as rigid transformations because they keep the size and shape rigidly fixed. The image under an isometry will always be congruent to the original. There are four types of isometries, and they are described in detail below.

Speaking mathematically, a symmetry of a figure or pattern is an isometry that leaves the whole pattern and its position apparently unchanged even after it has been applied. That is, the image coincides with the original. A pattern that has one or more symmetries is described as symmetrical. A figure with no symmetry or repetition is described as asymmetrical.

We could think about symmetry in terms of the points or elements of a pattern instead of the pattern itself. From this point of view, a figure, design or pattern is symmetrical if there is a correspondence between each point and at least one other point* in the design so that they fit into the pattern in exactly the same way. All such points are described as the set of corresponding or equivalent points.

The connection between corresponding points of a symmetrical pattern is defined by a SYMMETRY OPERATION, which is the action that is applied to a point to locate an image. We say that the symmetry operation is a mapping, or that it maps each point onto its corresponding image point(s).

There are four basic symmetry operations: rotation, reflection, translation, and glide reflection. Each of these operations has an associated symmetry tool that is used to create or define the action. Using any of the basic operations, or combinations of them, causes some sort of well-behaved and predictable repetition. The repeating part of a pattern is often described as a pattern element or motif. In this investigation you will explore two of the basic symmetry operations.

A ROTATION is the symmetry operation that rotates or "spins" a point or pattern element about some rotocenter for a specific angle of rotation. A rotocenter and angle, is the symmetry tool for a rotation. As a pattern is rotated a full 360° around the rotocenter, the number of times N that it coincides with its original position is described as its N-fold rotational symmetry, or its cyclic degree.

A REFLECTION is the symmetry operation that reflects or "flips" a point or design element across a mirror (or line or axis of symmetry) to locate its image. In a figure with line symmetry (or mirror symmetry) the mirror divides the design into two congruent halves. These halves are mirror images of each other; they would coincide if the figure were folded along the mirror line. Note that the image of a point is directly across and the same distance from the mirror as the original point.

A design with exactly one line of symmetry has exactly two matching parts, and is said to have simple bilateral symmetry.

Note that we can now identify two essential types of centered patterns -- those with a center point for rotation, and those with a single mirror. Rotocenters and points on a single mirror are special points that are mapped onto themselves by an isometry, so there would be no other point that fits into the pattern in exactly the same way. Thus, these points are referred to as unique points. (*Unique points are the the exceptions to the idea of a figure being symmetrical only if every point has at least one other point that fits into the pattern in the same way.)

A design can have more than one mirror. These will be most interesting when there are multiple mirrors that are either parallel or all intersect at a point.

We will be exploring and describing the symmetries of figures and patterns. The set of isometries found in a pattern will be described as that pattern's symmetry group. While the word group has a technical definition in mathematics, it is used informally here to mean only a set of symmetries.

Patterns that have only rotational symmetry will be described as having or exhibiting a CYCLIC group. We will use the symbol C_N to describe a pattern that has only N-fold rotational symmetry.

Note that it is understood that N>1. In fact, C₁ is the designation for an asymmetrical design, since it would repeat itself exactly one time in a full rotation.

Two intersecting mirrors multiply the number of images each point creates, since the image from one mirror reflects in the other mirror. The total number of images created by each point is a function of the angle between the mirrors, described as a dihedral angle. The result is a pattern with the intersection point of the mirrors as the center of rotation. Thus, there are DIHEDRAL groups. We will use the symbol D_N to describe a pattern that has N-fold rotational symmetry with N apparent mirrors.

As with cyclic patterns, it is understood that N>1 for dihedral patterns. In fact, D₁ is the designation for a design that has only one mirror, and is described as having simple bilateral (two sided) symmetry. Such a design may or may not have rotational symmetry.

Action

1. Working with a partner or two, use two blocks to create a simple asymmetrical shape. Using two copies of this shape to act as an original and its image, illustrate a few examples of reflections and rotations. Identify the mirrors for reflections, and the centers and angles of rotations. Illustrate how moving a mirror or center changes the location of the image.

2. Investigate the rotational symmetry of the classic block shapes. Use the template or a block to trace the outline of the shape onto a piece paper and onto a piece of transparent film. Mark the approximate position of the rotocenter on the film. Place the film on top of the paper so the outlines coincide, press down lightly on the rotocenter with a pencil or pin, and rotate the film. How accurately did you locate the rotocenter? Describe the symmetry group for each of the block shapes. That is, describe the set of symmetries each block exhibits. For example, the triangle exhibits 3-fold rotational symmetry centered about its centroid, with a rotation of (360/3 = 120°). Since the triangle also exhibits mirror symmetry, it would exhibit D₃ symmetry.

3. Use at least two different block types to make a pattern block design that you think exhibits C₄ (4-fold) rotational symmetry. Approximate the position of the rotocenter.

Similarly, create a D₄ design and locate its rotocenter.

4. Which cyclic groups can you illustrate with classic pattern block designs while honoring the basic rules? Record an attractive example of a pattern block design for each possible cyclic group. Confirm that each rotocenter is a centroid or "natural center";, vertex or midpoint. Also confirm that no other point corresponds to the rotocenter.

5. What kind of symmetry does D₁ symbolize? What does "center"; mean for this type of symmetry? Create an attractive D₁ pattern. Pick some point M on the mirror and another point X not on the mirror; show the images as M' and X'.

6. Which dihedral groups can you illustrate with classic pattern block designs? Record an attractive example of a pattern block design for each dihedral group possible. Again, confirm that each rotocenter is a centroid, vertex or midpoint.

7. Use a real mirror or draw a line down the middle of a transparency film to represent a mirror. Trace the outline of each of the block shapes, and sketch all symmetry mirrors. How many distinct mirrors does each shape have? For each shape, describe how the mirrors are related to each other. Is there any mathematical relationship between the number of mirrors and the N-fold rotational symmetry of each shape? Is this relationship true for any shape?

8. On the transparency, mark two or three convenient points and their images. Connect each point and its image with a segment. What is the relationship between the mirror and the segments?

9. Begin your own design that is centered around a hexagon and exhibits cyclic or dihedral symmetry. Like a mathematical atom, each layer or "shell" adds complexity to the structure. After adding two or three complete shells, begin a next shell and challenge a partner to complete what you've started, while you do the same for their design.

10. Plan the next outer shell for your atom, and sketch or write what you plan to add. Ask your partner what they would add next to see if they were thinking about the same next shell.

What do these last two explorations suggest about continuing a centered pattern?

11. Trace your design. Locate the rotocenter in black. Isolate a convenient point in the first shell. Highlight the point and all equivalent points in the same color. Connect these point in clockwise order. Describe what figure is made. How does the rotocenter relate to this figure?

If you do this same procedure with some set of equivalent points in the outer shell, do you always get a similar figure?

12. Create a figure with tan rhoms surrounding a square in some predictable arrangement. Does every rhom have at least one other rhom that fits into the pattern in exactly the same way? ... more than one?

13. Can you form predictable patterns from tan rhoms surrounding any of the other pattern block shapes? Which ones do and do not work this way?

14. Surround a hexagon as the center of a predictable pattern of blue rhoms. Which other block shapes can be the center of a predictable pattern of blue rhoms?

Compare how the tan and blue rhoms 'work' to form patterns like these.

15. Squares are also rhombuses. Will squares form similar predictable patterns around any other block shapes?