Seven kinds of border
Tonight's idea
A border is a pattern that repeats along a strip: the edge of a rug, a row of tiles, a line of footprints. Slide it along by one repeat and it lands on top of itself. Every border can do that. The question is what else it can do. Four other moves might leave it looking unchanged.
- Left-right flip: reflect it in a mirror standing upright across the strip.
- Up-down flip: reflect it in a mirror lying along the middle of the strip.
- Half turn: spin the strip 180° about a point on its middle line.
- Glide: slide it half a repeat, then flip it up-down. Footprints do this.
Four moves, each allowed or not, sounds like sixteen kinds. But the moves are tied together. A border with two of them is often forced to have a third, and some pairs cannot live together. Only seven kinds survive, and every border ever drawn is one of them.
Watch one: the border p q p q p q
Imagine the letters going on for ever both ways. Test each move.
Left-right flip: a mirror standing between a p and a q swaps them, and the row looks the same. Yes.
Up-down flip: every p becomes a b. No. A glide ends with an up-down flip, so it makes b's as well. No.
Half turn: p becomes d. No.
Only the left-right flip works, so this is kind 3.
Try slides, flips and turns on a shape: mathdetours.com/demos/rigid.html.
Your turn
- Treat each row of plain capital letters as a border that goes on for ever. Which kind, 1 to 7, is it? (a) F F F F F (b) A A A A A (c) D D D D D (d) N N N N N (e) H H H H H
- Now three rows of small letters. Say which moves each one allows, and its kind. (a) b d b d b d (b) b p b p b p (c) b q b q b q
- Footprints in sand make a border. Which kind is left by someone (a) walking: left, right, left, right; (b) jumping along with both feet side by side; (c) hopping on the right foot only?
- Start with the letter b. Flip it up-down, then give the result a half turn. (a) Which letter do you have now? (b) Which single move takes b straight to that letter? (c) Use this to explain why no border can allow an up-down flip and a half turn without allowing a left-right flip too.
Stretch: two mirrors
In border 3 the upright mirror lines are half a repeat apart. Flip the border in one mirror, then flip the result in the next mirror along. The two flips together are the same as one move you already know. Which move, and how far?
Answers
- (a) F has no flip or turn of its own: kind 1. (b) A has a left-right flip: kind 3. (c) D has an up-down flip: kind 6. (d) N looks the same after a half turn but not after either flip: kind 4. (e) H allows both flips and the half turn: kind 7.
- (a) b and d are left-right mirror images, so a left-right flip works and nothing else does: kind 3. (b) An up-down flip turns b into p. Slide one letter along and then flip, and the row matches itself: a glide and nothing else, kind 2. (c) A half turn takes b to q, so a half turn about the point midway between a b and a q works. Nothing else does: kind 4.
- (a) Each print is the one before it, slid along and flipped across the line of travel: a glide, kind 2. (b) The left and right prints are mirror images across the middle line: an up-down flip, kind 6. (c) Slide only, kind 1. (A foot has a heel and a toe, so no left-right flip works.)
- (a) b flips to p, and p turns to d. (b) A left-right flip takes b to d. (c) If the border looks unchanged after each move, it looks unchanged after one followed by the other. But an up-down flip followed by a half turn is a left-right flip. So the border allows that too. This is why kind 7 has all three and no kind has just those two.
Stretch. A slide. The first flip reverses the border and the second reverses it back, so nothing is left facing the wrong way. A flag sitting on the first mirror ends up twice as far along as the gap between the mirrors. The gap is half a repeat, so the two flips make a slide of exactly one repeat.