Tiling the plane with quadrilaterals · 3 marks

Suppose we have a tiling of the plane. Consider any vertex. As we go around this vertex and consider the angles made by consecutive lines, the total of these angles must be 360°. This suggests an idea. What if we take 4 copies of SOME and fit them together around a common point so that each angle is used once as we go around?

Answer: It works: angles 1 + 2 + 3 + 4 = 360° (the angle sum of a quadrilateral), so four copies with one of each angle at the point fill the full turn exactly, with no gap and no overlap around that point. There are three essentially different orders in which the four angles can go round.

Step-by-step solution

Idea: Around any point of a tiling the angles must total 360°. A quadrilateral’s four angles total exactly 360°, so one copy of each angle is the perfect fit.

Trace SOME, number its angles 1, 2, 3, 4 and cut out copies. The four angles add up to 360°, because any quadrilateral splits into two triangles by a diagonal (180° + 180°).1 mark
Put four copies round one point so that angle 1, angle 2, angle 3 and angle 4 each touch the point once. Their total is 360°, a full turn, so the copies close up round the point: no gap and no overlap there.1 mark
The order of the angles round the point can be chosen in three different ways (which angle sits opposite angle 1: angle 2, 3 or 4). The arrangement in the picture, where the angles go round in the order 1, 2, 3, 4, is the one in which neighbouring copies share whole edges, and it can be continued to tile the plane.1 mark

Four copies fit exactly round the point, because the four angles of SOME add up to 360°; this is the starting idea for tiling with any quadrilateral.

Check: For the tile in the picture the angles are about 109.8°, 99.9°, 77.7° and 72.6°; 109.8 + 99.9 + 77.7 + 72.6 = 360.0 ✓.

Answer to write in the exam

∠1 + ∠2 + ∠3 + ∠4 = 360° (angle sum of quadrilateral SOME)

Four copies round a point, each angle used once ⇒ angles at the point total 360°

∴ The copies fit exactly round the point (no gap, no overlap); three different orders are possible.

Common mistakes that cost marks

Using the same angle twice at a point (for example two copies of angle 1). Then the total is usually not 360°.
Thinking the copies must all point the same way. Some copies have to be turned round.
Forgetting that a non-convex quadrilateral’s reflex angle is still one of the four angles that add to 360°.

How this can come in the exam

MCQ (1 mark)

Three angles of a quadrilateral tile are 80°, 95° and 115°. For four copies to fit round a point using each angle once, the fourth angle must be

60°
70°
80°
90°

Show answer

(B) 70°
360° − (80° + 95° + 115°) = 360° − 290° = 70°, which is just the fourth angle of the quadrilateral.

Try one yourself

Could four copies of a triangle with angles 50°, 60°, 70° fit round a point using each angle at most once? How many triangle corners are needed round a point?