Falling apart

Introduction

Mathematical dissections make fun jigsaw puzzles. Here are a few of the more popular.

The Pythagorean Theorem

The Pythagorean Theorem

This Theorem, Euclid 1.47, is perhaps the most famous mathematical result. A geometrical statement is that for a planar right-angled triangle

the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
This geometrical statement suggests dissecting and comparing physical areas.

 
Dudeney's dissection

Dudeney's dissection

Dudeney's dissection is a dissection of an equilateral triangle into pieces which can be rearranged into a square. The solution to this problem is often attributed to Henry Ernest Dudeney (1857-1930), one of the greatest nineteenth century puzzlers.

It is particularly satisfying that the pieces may be hinged. This makes it possible to produce a table, teapot stand or other objects which are useful for two, three, four or six people! What is even more surprising about this particular dissection is that the grain of the wood lines up correctly in both configurations!

 
Click here for a video clip.

Duijvestijn's dissection

Duijvestijn's dissection

There are many dissections of squares into other squares, no two of which are the same. The smallest dissection of a large square into squares which do not use a regular sequence of smaller squares was first discovered by A. J. W. Duijvestijn, and published in Scientific American in 1978. This consists of 21 squares, with sides of length 2, 4, 6, 7, 8, 9, 11, 15, 16, 17, 18, 19, 24, 25, 27, 29, 33, 35, 37, 42 and 50. This fits into a square with sides of length 112 units per side.

 
Click here for a video clip.
Problems

Here are some problems for you to solve.

  1. Cut a (unit) square piece of toast into two equal right angled triangular slices. You want to cut each slice into two pieces of the same area. What is the length of the shortest straight cut and how is this done?
  2. How do you cut a solid cube into three identical pieces?
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