Do you see which of the three possibilities shown in the figure occurs in the tiling?
Step-by-step solution
Idea: In the tiling, neighbouring copies are half-turns of each other about the midpoint of their shared edge. Follow the angles round one corner, say M, using that rule.
- At corner M of SOME, the copy across edge OM is SOME turned about the midpoint of OM; it puts its angle 2 (from O) at M, next to angle 3. The copy across edge ME puts its angle 4 (from E) at M, on the other side of angle 3.1 mark
- So round M the angles go 2, 3, 4 and the last copy fills the gap with angle 1: the order is 1, 2, 3, 4 round the point (angle 1 opposite angle 3). This is the third arrangement.1 mark
Answer to write in the exam
Copy across OM brings angle 2 to M; copy across ME brings angle 4 to M
Round M: 2, 3, 4, then 1 ⇒ order 1, 2, 3, 4 (1 opposite 3, 2 opposite 4)
∴ The third possibility occurs in the tiling.
Common mistakes that cost marks
- Choosing an arrangement because it “looks like” the picture without following the angles round a corner.
- Thinking the order is different at different corners. In this tiling every corner has the order 1, 2, 3, 4.
How this can come in the exam
In a tiling by half-turns about edge midpoints, which angle of the tile lies opposite angle 2 at every corner point?
- Angle 1
- Angle 2
- Angle 3
- Angle 4
Show answer
(D) Angle 4
Round each corner the angles go 1, 2, 3, 4, so angle 2 is opposite angle 4.
Try one yourself
At a corner of the tiling, the angles of copies are 1 and 2 next to each other. Which two angles complete the corner, and in what order?
Show answer
Angles 3 and 4, continuing the order: 1, 2, 3, 4 round the point.
More questions like this
- 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.)
- Draw a non-convex 4-gon DART. Show how we can tile the plane with copies of DART. Both methods that we discussed earlier will work. Which do you prefer?
- Using a fact about parallelograms, show how to tile the plane using any given triangle. (Hint: Can you use the parallelogram tiling in the introduction?)
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