How can we understand the figure? There seems to be a repeating pattern. (1) Can you precisely describe a procedure to draw the pattern so that someone can draw the tiling on their own, based only on your description? (2) Can you justify why your procedure works?
- (1) Can you precisely describe a procedure to draw the pattern so that someone can draw the tiling on their own, based only on your description?
- (2) Can you justify why your procedure works?
Step-by-step solution
Idea: Turning twice through 180° about two different points has the same effect as one slide. That is why the half-turn pattern repeats by slides along the diagonals of SOME.
(1) Can you precisely describe a procedure to draw the pattern so that someone can draw the tiling on their own, based only on your description?
- Step 1: Draw SOME. Mark the midpoint of side OM. Draw the copy of SOME turned through 180° about that midpoint; it lies on the other side of OM. Call SOME and this copy a “pair”.1 mark
- Step 2: Draw copies of the pair slid (without turning) by the vector from S to M, any number of times forwards or backwards; then slide all of these by the vector from O to E, any number of times forwards or backwards. The copies of SOME and of its turned partner together fill the plane.1 mark
(2) Can you justify why your procedure works?
- A half-turn about the midpoint of OM swaps O and M, so the turned copy shares the whole edge OM. In the same way each copy across any edge is a half-turn about that edge’s midpoint.½ mark
- Two half-turns about different points equal one slide. For example, the half-turn about the midpoint of OM followed by the half-turn about the midpoint of the partner’s next edge moves SOME by the vector SM. So the copies produced by repeated half-turns are exactly the slid copies of the pair in step 2: the rule is consistent.1 mark
- At every corner point, four copies meet bringing angles 1, 2, 3, 4 once each, total 360°, and neighbouring copies share whole edges. So there are no gaps or overlaps, and the slides spread the pattern over the whole plane.½ mark
Check: Slide check with coordinates: S(0, 3), O(4, 3.6), M(5, 0.6), E(−0.6, 0). A half-turn about point c sends X to 2c − X. Half-turn about the midpoint of OM, then about the midpoint of the partner’s edge from M towards O + M − S, moves every point by M − S = (5, −2.4), the vector SM ✓.
Answer to write in the exam
(1)
Draw SOME; draw its copy turned 180° about the midpoint of OM (the pair)
Slide the pair by vector SM repeatedly, both ways
Slide all of these by vector OE repeatedly, both ways
(2)
Half-turn about midpoint of an edge maps the edge onto itself ⇒ neighbours share whole edges
Two half-turns about different points = one slide (by the diagonals SM, OE) ⇒ pattern repeats
At each vertex angles 1 + 2 + 3 + 4 = 360° ⇒ no gap, no overlap
Common mistakes that cost marks
- Describing only “rotate the tile” without saying about which point and by how much (180° about the midpoint of a side).
- Sliding the single tile instead of the pair: SOME alone slid by SM leaves gaps that the turned copies fill.
- Justifying by “it looks right in the picture”. The reasons are the shared edges and the 360° at every corner.
How this can come in the exam
A point X is turned through 180° about point A(1, 2) and then through 180° about point B(4, 3). The result is X moved by
- (3, 1)
- (6, 2)
- (5, 5)
- (−6, −2)
Show answer
(B) (6, 2)
Two half-turns: X → 2A − X → 2B − (2A − X) = X + 2(B − A) = X + 2(3, 1) = X + (6, 2).
Try one yourself
A tile is turned through 180° about the midpoint of one edge, and the result is turned through 180° about the midpoint of the same edge again. Where does the tile end up?
Show answer
Back where it started: two half-turns about the same point make a full turn of 360°, i.e. a slide by zero.
More questions like this
- Note two interesting things about the second step: (1) each new copy can be obtained by rotating any one of its neighbours, and both ways give the same result. (2) The new copies fit perfectly. Can you explain these facts by reasoning? This is needed to prove that the method works!
- Do you see which of the three possibilities shown in the figure occurs in the tiling?
- A grid of Varignon parallelograms of SOME is given in the figure. To begin with, focus only on the 9 green coloured copies of SOME in the figure. We will first place only these 9 copies in the figure. The corresponding parallelograms are shaded in the figure. Place 9 copies of SOME so as to match the way the green coloured copies of SOME are placed in the figure. If done carefully, you will observe that each 4-gon you placed meets other placed copies exactly at vertices. Now see how 4 copies of SOME are made automatically in the gaps! Carefully place four more copies of SOME in these gaps. Continuing this process with more copies of SOME will give us the desired tiling.
- The following shape— the first of its kind!— that leads to aperiodic tilings was discovered only recently in 2023 by a team of four mathematicians (Smith, Myers, Kaplan and Goodman-Strauss). Cut out 15 identical copies of the following shape (known as the ‘hat’) and start to tile with it. Do you see any pattern?
- Justify why the plane cannot be tiled with a regular pentagon. (Hint: Read the first 3 sentences of ‘Think and Reflect’ in the section on tiling.) (There are many ways to tile the plane using a suitable irregular pentagon. The most recent method was found in 2015.)
All Quadrilaterals and parallelograms questions · All maths questions