Parallelogram 32 Level 1 11 Apr 2024Tiles and tribulations

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. It all fits together nicely

Tiles are all around us, on pavements and walls, but they're usually made in the classic (sometimes boring) rectangle, which leads to a repetitive pattern.

However, mathematicians prefer something more interesting.

Below is an amazing kind of tiling which is made from two different tiles.

Incredibly, these two tiles can form a pattern that never repeats!

Perhaps a little tricky for the bathroom floor, but very fun for the mathematical mind!

Watch this short video and find out more about the structure that underpins this non-repeating tiling pattern.

(If you have problems watching the video, right click to open it in a new window)

1 mark

1.1 Which of these shapes will not tesselate?

  • Rectangle
  • Trapezium
  • Regular octagon
  • Triangle
  • (Not answered)
1 mark

1.2 Look closely at the Penrose tiling itself, what shape are the tiles?

  • Square
  • Rhombus
  • Diamond
  • Cube
  • (Not answered)
2 marks

1.3 Penrose tiling is called “quasi-periodic”. What does “quasi” mean?

  • Not
  • Resembling something, without actually being that thing
  • Very
  • (Not answered)

In March 2023 a brand new tiling with this same “non-repeating” property was found:

2 marks

1.4 What shape are the tiles?

  • Nonagon (9 sides)
  • Decagon (10 sides)
  • Dodecagon (12 sides)
  • Tridecagon (13 sides)
  • (Not answered)

2. Some puzzling fun

2 marks

2.1 Craig the spider weaves a circular web of circumference 40cm.

The web is made up of sections, where the angle at the centre is exactly 45 degrees.

If he travels around the edge of one of the sections of web, how far will he have travelled?

  • 5cm
  • 8cm
  • 10cm
  • 15cm
  • 20cm
  • (Not answered)
Show Hint (–1 mark)
–1 mark

90 degrees would be one quarter of the web.

Think about what fraction of the full 360 degrees he passes through.

2 marks

2.2 Which of these solid shapes will not tesselate in three dimensions?

  • A
  • B
  • C
  • D
  • E
  • (Not answered)

Shape A cannot tesselate, due to the curved edge.

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and collect your reward points.
  • You might earn a new badge... if not, then maybe next week.
  • Make sure you go through the solution sheet – it is massively important.
  • A score of less than 50% is ok – it means you can learn lots from your mistakes.
  • The next Parallelogram is next week, at 3pm on Thursday.
  • Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon and Ayliean.