Paper is one of the most commonly used materials in daily life. It is hard to imagine a world without it. The word paper is derived from “papyrus” - a reed that grows on the banks of the river Nile. The early Egyptians flattened the stalks of this reed and wrote on it - the Dead Sea Scrolls are written on it. But this “paper” was very brittle and could not be folded. Paper as we know it was invented in China in 105 AD. T’sa Lin, a minister in the Imperial Palace, discovered how to make it by boiling rags and old fishing nets and beating them into pulp. Papermaking was probably the first industry to use recycled materials, and it still does.

In many countries paper has played a vital role in cultural expression and as an artistic medium for local traditions. The art of paper folding is a favourite pastime in Japan. In recent times there has been a revival of the creative use of paper, especially in packaging and pop-up structures. However, paper folding is more than a creative pastime. It can be an entertaining and imaginative way to learn mathematics, particularly its geometry.

A-sized paper

Metric (A-sized) paper, of which A4 is the most common, was first used in 1922 in Germany, where it was called “Din A”, but its use did not spread until after 1945. Now it is to be found in almost every country in the world, with the notable exception of the United States, where they still use paper that is shorter and wider. Previously, paper sizes depended very much on the individual paper mills and their facilities. There was little relationship between one size and another.

The seemingly magical properties of A-sizing rely on a system that is rational, systematic and mathematical. Each A-size rectangular sheet is made by folding in half the size numerically below it. For example, folding A3 in half creates A4 (Figs 1a and 1b). Two similar rectangles are produced and, if the process were continued, it would make a never-ending family of similar shapes (Fig 2). This is because the sides of any A-size paper are in the ratio 1: C 2 (1.4142...).

Of course, the process can be reversed, with two A4 sheets combined to create an A3 and two A3 to create an A2 and so on until A0 is reached; A0 has an area of one square metre (Fig 3), giving the name “metric paper”.

The sizes of metric paper, in millimetres, are: A0 841 x 1,189

A1 594 x 841

A2 420 x 594

A3 297 x 420

A4 210 x 297

A5 148 x 210.

The ratio 1: C 2 can be illustrated visually by taking three same-sized sheets of A-sized paper and folding two of them in half to get two sheets of the size numerically below. Arrange the three pieces so that their lengths (or breadths) form sides of an isosceles right-angled triangle (Fig 4).

Using Pythagoras: If CD = AB = BC = 1 unit Then AC2 = 12 + 12 = 2 units Therefore AC = C 2 units.

Algebraically, the ratio 1:C2 can be proved by taking any two consecutive sheets of A-size paper. Let the width of he smaller rectangle be 1 unit and its length be x units. Then, the bigger rectangle has width x units and length 2 units. The two rectangles are similar (Fig 5), so: x2 = 2

x = C 2

William Gibbs is a retired maths lecturer and Liz Meenan is 4Learning education officer

RECTANGLE FAMILIES

A-size paper has the magical property of being easily folded (or cut) to create families of similar rectangles. So if we use A-size paper to fold other polygons we can create other families of similar shapes. And by using coloured A-size paper we can easily create very interesting and colourful patterns.

POLYGONS

Metric paper can be folded, usually with only a few folds, to form a variety of polygons, for example, squares, equilateral triangles, isosceles triangles, kites, rhombi, pentagons, hexagons, octagons. No compasses, rulers, protractors or previous experience are needed to obtain large, elegant and easily handled polygons (for example an equilateral triangle of side approximately 24.5cm can be folded from A4 paper). These can be used for display, for making striking coloured designs and for exploring the properties of polygons. Simple illustrated diagrams can explain the folding process so that pupils can produce a variety of different sized polygons. Remember to start with A-sized paper. After folding a few, pupils can start to be creative. They can explore the numerous large patterns they can make, see how easy it is, and learn some maths as they go along.

* Nesting patterns

Part of the magic of shapes made from A-sized paper is that shapes folded in the same way from different sizes will be similar. Some of these similar shapes will nest beautifully inside each other to create large attractive patterns.

* Nesting Squares Make squares from A4, A5, A6, and A7 paper.

A4 Square

A7 Square

If the corners are placed to touch the middle of the sides, these squares nest exactly inside each other.

If you make the squares from two contrasting colours then you can build up a simple but beautiful nesting pattern.

* Nesting triangles

Triangles do not nest in the same way. Instead they form spiral patterns - like ‘Curves of Pursuit’.

Regular octagon using nesting isosceles triangles Nesting equilateral triangles

* Stacking Patterns Families of similar polygons in contrasting colours can be used to form stacking patterns.

* Stacking in the middle Kites can be stacked in the middle to form this pattern.

RESOURCES

Further information and suggestions on paper folding (and other such activities including tiling, printing, weaving and reflecting) are included in the Shape, Space and Measures (maths from design) video (pound;14.99), teacher’s guide (pound;3.95) and activity book (pound;6.95). These resources are available from 4Learning, PO Box 100, Warwick, CV34 6TZ. Tel: 01926 436444. The Shape, Space and Measures programmes on the video are aimed at 7 to 11-year-olds and are to be broadcast each Tuesday from May 15 to June 19 at 10.00-10.15am on Channel 4.