There are multiple ways of doing this, as shown in the figure. Can you use any of these ways to continue fitting further copies of SOME to tile the plane? Try it with the 15 copies you made!
Step-by-step solution
Idea: To keep going we need copies that meet edge to edge, so that every new corner point again gets angles 1, 2, 3, 4. Half-turns about midpoints of edges give exactly that.
- In the arrangement with angles in the order 1, 2, 3, 4 round the point, each pair of neighbouring copies shares a whole edge: one copy is the other turned through 180° about the midpoint of that edge.1 mark
- Keep doing the same at every free edge: turn the copy through 180° about the midpoint of the edge. At every new corner the four angles 1, 2, 3, 4 meet again (total 360°), so the copies close up with no gaps or overlaps. With 15 copies you can already cover a patch like the picture, and the process never gets stuck. (The other two arrangements do not meet edge to edge, so they are not continued this way.)1 mark
Answer to write in the exam
Choose the arrangement with angles 1, 2, 3, 4 in order round the point
Each copy = neighbour turned 180° about the midpoint of the shared edge ⇒ whole edges match
Every new vertex again has angles 1, 2, 3, 4 = 360°
∴ Yes, the copies can be continued to tile the plane.
Common mistakes that cost marks
- Sliding every copy without turning it. For an irregular quadrilateral, the edges then do not match.
- Trying to extend an arrangement where copies meet along edges of different lengths; gaps appear at the ends of those edges.
- Stopping after one ring of copies; the point is that the same rule works at every edge forever.
How this can come in the exam
A tile is turned through 180° about the midpoint of one of its edges. Explain why the turned copy fits along that whole edge.
Show answer
A half-turn about the midpoint of an edge sends each end of the edge to the other end (1 mark). So the edge is carried onto itself, and the turned copy lies on the other side of the same edge, sharing it completely (1 mark).Try one yourself
Copies of a quadrilateral tile the plane by half-turns about edge midpoints. How many copies meet at each corner point, and what angles do they bring?
Show answer
Four copies meet at each corner, bringing the four different angles 1, 2, 3, 4 of the quadrilateral, which add up to 360°.
More questions like this
- 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?
- 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?
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