Maths stories from CSIRO's Double Helix Magazine

Stories on this page:

Spaced-out planets
How many times can you fold a piece of paper in half?

Spaced-out planets

by Malcolm Miller

Humans see patterns in many of the things around them. Some say this ability helped our ancestors spot predators hiding in the bushes. Our brains seem to be ‘hard-wired’ towards putting things into patterns, groups, orders, and sequences to help us understand them.

An example is the prime number series - numbers that are divisible only by themselves and one (for example, 3, 11, 29, and 101). For thousands of years, people have attempted to find a rule that shows how prime numbers are distributed, and have failed.

The distances of the planets make a kind of series: they’re close together near the Sun, and further apart at great distances from it. People interested in numbers looked at these distances and tried to find a simple mathematical rule or formula that explained them. In 1766, a person by the name of Johann Titus discovered a rule that fit the pattern, which he published. It became known as the Titius-Bode Law.

Cassini Mission portrait of Jupiter

The law is based on the sequence: 0, 3, 6, 12, 24... Every number in the sequence (except the first two numbers) is double the previous number. The numbers then have four added to them, and are then divided by 10. The resulting series, 0.4, 0.7, 1.0, 1.6, 2.8, 5.2... is close to the distances of the planets from the Sun when they are measured in astronomical units (AU). An astronomical unit is the average distance from the Sun to the Earth – 150 million kilometres. The Titius-Bode Law doesn't work well for Neptune, and we needn't be surprised that it also doesn't work for Pluto, since it isn't really a planet.

The ‘law’ also suggested that a planet existed between Mars and Jupiter. In 1801, Giuseppe Piazzi found not a planet but the first asteroid, one of millions of pieces of rock ranging in size from pebbles to hundreds of kilometres in diameter.

The spacing of the planets makes it hard to draw the Solar System to scale, since the inner planets are too close, and the outer ones too far apart to fit neatly on a page.

There is a more mathematical way to draw the Solar System. Instead of making concentric circles scaled from the planet's distance from the Sun, try making them proportional to the logarithm of the true distances. You'll find that they fit closer together and are almost equally spaced.

Voyager image of Neptune

The logarithmic scale diagram has a practical application. If you had a spaceship that could constantly accelerate, its travel time to each planet would be proportional to the logarithm of its distance. One day we may have rockets that run continuously, instead of for a few minutes, and the log scale might be a useful map for navigating.

An interesting exercise to demonstrate the distance of the planets from the Sun uses a roll of toilet paper. By multiplying the distance in astronomical units by 10, you can mark out the distance of each planet using sheets, or squares, of toilet paper. Is this a scientific diagram of where we live?

Images courtesy of NASA

What else can you see in the night sky? Visit Australia Sky.

How many times can you fold a piece of paper in half?

Did you know that it is physically impossible to fold it in half more than eight times? The reason is exponential growth.

Starting with a piece of paper and folding it in half, you’ll notice that its thickness doubles to become two sheets thick. If you fold it again it becomes four sheets thick. This is double what you had before. In fact, each fold doubles the thickness of the stack of paper.

This relationship can be expressed mathematically as 2 x 2 x 2 x . . ., and so on, or 2n where n is equal to the number of folds.

When the paper is folded once, n equals one, therefore the thickness of the pile is 2¹ or two. When you fold it again, the thickness is 2² or 2 x 2 = 4. After the seventh fold, the thickness of the stack is 2⁷ = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128.

What prevents the paper from folding any further is its inability to wrap around the thickness of the stack.

If you measure a stack of 10 sheets of paper, you’ll find that they have a thickness of approximately one millimetre. Therefore, each sheet averages 0.1 millimetres thick.

If this sheet were folded in half seven times, then its thickness would be 12.8 mm.

What if it was possible to continue folding the paper? Amazingly, the stack of paper would be over 1.6 metres thick (as tall as you are), after the 14th fold!

In fact, by the 28th fold it would be higher than Mount Everest and only five more folds would see it tower past the Moon (over 400 000 kilometres).

Many organisms, such as bacteria, multiply according to this simple mathematical principle. It’s amazing how a simple thing like doubling can cause such a rapid increase.