The Hypercube

Heaven comes down to Earth

The Universe is very big, so if we try to draw a map of it we would have to use a very small scale. The trouble is, such a map doesn't show any of the beautiful stars, galaxies or planets up there: it's just billions of points of light. To represent these objects, astronomers need to use several different scales. We know more about things closer to us, so the scale gets smaller with increasing distance: as we look further away, we can see a larger area, but in less detail. A scale which varies like this is called a logarithmic scale.

Armagh Astropark's "Hyper-Cube" illustrates the concept of logarithmic scale using a series of nested cubes, each with sides ten times longer than the cube inside:
 

Small cube Medium cube Large cube
Cube 1: 8 cm. Cube 2: 80 cm. Cube 3: 8 m.

Centred at the same point in Armagh, number 4 would be the height of a ten-storey building (40m.), number 5 would dominate any skyline (400m.), and so on. . .

6 (4 km.) Covers all of Armagh City and the surrounding area, as far out as Navan. Going up, it reaches the typical cruising altitude of a short-haul jet airliner.

7 (40 km.) Encloses an area including Newry, Monaghan and Lough Neigh. Going down, the cube's lower face is buried deep in the Earth's Mantle of molten rock.

8 (400 km.) Reaches Cork to the South, Edinburgh to the North, and the height of many satellites and spacecraft above.

9 (4,000 km.) Takes a large chunk out of the Earth, including Western Europe, New York and the North Pole.

11 (400,000 km.) Swallows up the Moon's orbit around Earth.

14 (2.7 AU) Includes the inner Solar System: the Sun, Mercury, Venus, Earth, Mars and much of the asteroid belt.

15 (27 AU) This cube ends between Uranus and Neptune, the outermost gas giant planets.

19 (4.2 light-years) Beyond Proxima Centauri, the nearest star.

23 (42,000 Light-years) Large enough to include our entire galaxy, which contains hundreds of billions of stars.

28 (420 million light-years) This is about the limit of present-day telescopes, and so of the observable Universe.


Why "Logarithmic"?

The "logarithm" (or just "log") is a mathematical function, which tells us how many times we have to multiply by one number to get another. For example, if the log of 10 is 1, then the log of 100 is 2, the log of 1000 is 3, and so on.

A logarithmic scale means that we count in intervals of numbers' logs, not in the numbers themselves: instead of writing out 10, 100, 1000, astronomers just write 1,2,3:

10x is 1 followed by x "0"s.

They would write 20,000,000,000 as 2 x 1010.

Multiplying or dividing numbers is the same as adding or subtracting their logarithms, which makes them useful in complicated calculations. A slide rule is a device made from two logarithmic rulers, which slide back and forth to display multiplication tables. The rule below shows multiples of 3:

Slide Rule

Slide rules seem primitive compared to modern electronic calculators, but they could be accurate and were used until very recently. Compact computers were not available in the early days of the space programme, for example, so the Apollo astronauts carried slide rules to plan their landings on the moon.


The Chessboard Parable

Logarithmic scales allow a large range of numbers to be compressed into a much smaller one. An old story involves a foolish king who promises to give one of his subjects a grain of rice, then double it for every square on a chessboard:

wpe4.gif (10304 bytes)
 
 

By the final square (no. 64), the king owes more than 9 x 1018 (or 9 million trillion) grains, too much to produce even if the entire Earth were used for rice fields until the Sun burns out.

This kind of doubling is called exponential growth. It occurs often in nature: a single cell becomes a baby in only nine months by dividing into 2, then 4, then 8, then 16. . .

Space itself is also growing exponentially, which means that distant galaxies seem to be moving away from us.


Logarithmic scales aren't just used to measure space and time. They're everywhere, and we use them all the time --- although we don't always notice it.
 

The "Learning Curve"

People often find that learning a new skill starts off very difficult, but gets easier. Mathematicians express this on a logarithmic scale, which when plotted as a graph is called a "leaning curve". The effort required to learn more of the skill can be compared to cycling up the curve.

Cycling up the Learning Curve
Sound

Have you ever wondered why the higher settings on a volume control all sound so similar? Imagine one which is marked into ten divisions, such as the dial below. Most people notice a massive increase in volume between "1" and "2", but can barely tell "9" and "10" apart:

Volume Control

The reason is that we hear logarithmically: our ears are more sensitive to quiet than loud noises. Moving from "9" to "10" makes the speakers pump out just as much extra energy as changing from "1" to "2", but we don't notice it. People's perception of sound is measured in decibels, a logarithmic scale which does match what we hear.
 
 

Earthquakes

The Richter Scale is logarithmic: a one point increase in an Earthquake's "magnitude" means ten times more shaking of the ground. So, an Earthquake described as "7" is not just a bit worse than one measuring "6"; it's ten times as bad. And one of "8" is a hundred times more damaging.


Back In Time

Just like space, time is often best represented on a logarithmic scale: we know a lot about different decades in the present century, but (to us) one decade in the Jurassic era would be much the same as any other. We measure:

As we go further back, the appropriate time-scale becomes larger. An exception to this rule is the Big Bang itself: the event that created the Universe. Cosmology is an entire science devoted mainly to studying the first second of time!
 
 

The Size of Space

Earthly units such as the kilometre are too small to represent distances in space conveniently, so astronomers have developed their own units:

The Astronomical Unit is the average distance from Earth to the Sun, used within the solar system.1 AU = 150,000,000 km.

The light-year is the distance that light travels in a year, used outside of our solar system. 1 ly = 63,000 AU.

For historical reasons, astronomers sometimes use an even larger unit called the parsec instead. This is based on the way that distances to stars were first measured. 1 pc = 3.26 ly.
 

Last Revised: 2010 January 29th