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?
Step-by-step solution
Idea: A periodic tiling repeats a fixed block by sliding it in two directions, like the quadrilateral tilings. The hat can cover the whole plane, but never in such a repeating way.
- Copies of the hat can be fitted together edge to edge with no gaps, though as you add more you find that some copies have to be flipped over (mirror images) to keep going.1 mark
- Looking for a repeating block, you never find one: patches look similar in places, but the arrangement as a whole never repeats by sliding. This is why the tiling is called aperiodic, unlike the parallelogram or quadrilateral tilings, which repeat.1 mark
Answer to write in the exam
Hats fit together with no gaps or overlaps (some copies flipped over)
No block repeats by sliding in two directions
∴ The hat tiles the plane, but only aperiodically (no repeating pattern).
Common mistakes that cost marks
- Concluding that the hat cannot tile because 15 copies did not form a neat repeating block. It does tile, just not periodically.
- Thinking “aperiodic” means “random”. The arrangement follows strict rules; it simply never repeats by sliding.
- Refusing to flip copies over. The hat tiling uses some mirror-image copies.
How this can come in the exam
Assertion (A): Copies of any quadrilateral can tile the plane in a pattern that repeats.
Reason (R): In the half-turn tiling, the pattern is repeated by sliding along the two diagonals of the quadrilateral.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
Both are true, and the repetition by slides along the diagonals is exactly what makes the quadrilateral tiling periodic.
Try one yourself
Is a tiling of the floor by identical square tiles periodic or aperiodic? Explain.
Show answer
Periodic: sliding the whole pattern by one tile width (sideways or up) carries it onto itself.
More questions like this
- 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?)
- Mark the midpoint of the line drawn on the paper (see the figure), given that the horizontal lines are equally spaced. Justify your answer.
- You know that the sum of angles of a quadrilateral is 360°, even for a non-convex quadrilateral. (Recall the proof.) Now consider a self-intersecting quadrilateral ABCD, where AB and CD intersect at point E. Show that ∠A + ∠B + ∠C + ∠D < 360°. Can you construct ABCD such that ∠A + ∠B + ∠C + ∠D = 2°?
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