Slide and glide: translation, glide reflection, friezes
Patterns that repeat along a strip
Teaching this? Open projector mode
So far each move kept one point fixed (the mirror line or the centre). Now we add two moves that travel.
- A translation slides every point the same distance in the same direction. It is described by a vector (a length and a direction). A translation by a non-zero vector has no fixed point: nothing stays where it was. Sliding a shape never turns it or flips it.
- A glide reflection is a reflection followed by a slide along the mirror line. The footprints of a person walking along a beach make the standard picture: the left footprint and the right footprint are mirror images of each other, and each is also shifted forward. Reflect the left print in the centre line and slide it forward by one step and it lands on the next right print.
A glide reflection is not the same thing as a translation (the shape ends up flipped), and not the same as a plain reflection (the shape has moved along the line). Like a translation, it has no fixed point.
Together, the four moves are called the isometries of the plane: reflection, rotation, translation and glide reflection. “Isometry” means “same measure”: distances and angles are kept. There are no other isometries of the plane. Every move that keeps all distances is one of these four.
Friezes. A frieze pattern is a pattern that repeats forever in one direction along a strip, like a border on a wall, a band on a pot or a row of fence posts. Mathematicians have proved that, up to the type of symmetry, there are exactly 7 frieze groups, no more and no fewer. Each one is a different combination of these ingredients: a translation along the strip, a horizontal mirror line (along the strip), vertical mirror lines, half-turns, and glide reflections. A handy way to tell them apart is a row of the same motif:
| Frieze type | Typical example | Symmetries besides translation |
|---|---|---|
| Hop | a row of the letter F | none |
| Step | alternating left and right footprints | glide reflection |
| Jump | a row of the letter D | horizontal mirror line |
| Sidle | a row of the letter A | vertical mirror lines |
| Spinning hop | a row of the letter N | half-turns |
| Spinning sidle | a zigzag line | vertical mirrors, half-turns, glides |
| Spinning jump | a row of the letter H | horizontal and vertical mirrors, half-turns |
The names “hop”, “step”, “jump”, “sidle”, “spinning hop”, “spinning sidle” and “spinning jump” come from the mathematician John Conway’s playful way of describing the steps a walker or dancer would take to copy the pattern. You do not have to memorise them. What matters is that the list is complete: take any border pattern, find which of the symmetries it has, and it falls into exactly one of the 7 boxes.
Which symmetry does the pattern of left and right footprints along a straight path have?
How many different frieze groups are there?
Does a non-zero translation have a fixed point (a point that stays where it was)?