# Size matters. . . and so does scale

How big, how far, how small, how much? These are the questions about our universe, and everything in it, that have fascinated us humans for millennia. And science helps us answer them, writes MARIE BORAN

EVER SINCE mankind gazed up at the night sky, it has inspired questions that form the basis of modern science: how big, how far, how small, how much? If twinkling stars appear to be nothing more than tiny shimmering lights in the sky, how can we tell how big they are or what distance they are from Earth?

It’s all a matter of working out scale, as Ted patiently explains to a perplexed Dougal in Father Ted. Holding up a plastic toy cow while simultaneously pointing to a herd of cattle outside the window, Ted says: “These are small . . . but the ones out there are far away.” This is obviously a silly example. We know how big something is when we hold it on our hands.

Simply put, it is either roughly bigger or smaller than our hand. In fact, this is how the ancient Egyptians and Babylonians first measured objects in the third millennium BC. A cubit was the distance from the forearm to the tip of the middle finger, and a half-cubit was the span of the hand.

As time went on this evolved and changed into the standard inch, foot and yard that we use today.

Thousands of years later, in the third century BC, the Greek astronomer and mathematician Aristarchus took one giant leap and attempted to work out the distance to the moon. He did this by observing the size of the Earth’s shadow in relation to the moon during a lunar eclipse.

Aristarchus also estimated that the moon was a quarter the size of the Earth, and was about 60 times the radius of Earth, which is fairly close to modern calculations. He wasn’t so successful with his guess on how far we are from the sun but he was ahead of his time – and 17 centuries ahead of Copernicus. He suggested that the Earth was not the centre of the universe but instead moved around the sun.

Unlike physically measuring something in cubits, these kinds of calculations showed that it was possible to infer the size or distance of an object indirectly by using information about related objects. Unfortunately there is only so much you can observe and calculate with the naked eye.

The next great leap in understanding the scale of the world around us came with the invention of the telescope and microscope. These instruments appeared around the same time in the 17th century, when Dutch lens-grinder Jan Lippershey is documented as having made the first telescopic device. Galileo, however, was the Steve Jobs of his time: while he didn’t invent the telescope he was the one who made it popular, and improved upon its design.

Galileo was not only busying himself with bringing us beyond the solar system by unveiling the Milky Way, he also turned the lens down upon more earthly things with the microscope. Illustrations of the legs, wings and other body parts of bees from 1625 are the first documented use of this scientific instrument.

Besides asking how big the universe is we were now asking how small things were. The first measurement for things smaller than the human eye was the micrometer, which was scientifically quantified in the mid 18th century.

The micrometer fits into a scale of universal measurements alongside the millimetre, the centimetre and so on. It can be difficult to conceptualise micrometres and even smaller subatomic measurements but you can use order-of-magnitude reference objects to visualise very small or very large scales.

Imagine you are 1mm tall, even tinier than the Lilliputians in Gulliver’s Travels. The eye of a needle now becomes a doorway and a single hair is as thick as a tree trunk. If you were to pick up red blood cells they would be the size of MMs.

Another quick way to visualise scale is to substitute microscopic objects with stuff you have lying around the house. Take those MMs again and think of them as red blood cells, roughly 10 um (micrometers) in size. If you placed a sugar grain beside the MM that’s what a bacteria would look like in comparison.

Additionally, a hair from your head would be the size of a poster tube and an actual sugar grain would be the size of an A4 cardboard box.

The reason we can “see” so much of the universe is because of powerful telescopes such as the Hubble Space Telescope. This telescope was launched into space on April 24th, 1990 aboard the space shuttle Discoveryand is almost the size of a large school bus.

It is so powerful that it can lock onto a target within the accuracy of 7/1000th of an arcsecond, or the width of a human hair seen from one mile away.

Hubble not only sees many magnitudes farther than the human eye but also sees wavelengths not visible to us: near ultra-violet to near infrared wavelength. It has given us staggeringly beautiful images of galaxies billions of light years away and allowed scientists to observe the birth of stars and the prevalence of black holes.

The telescope, and the astronomer it’s named after, Dr Edwin Hubble, helped us learn more about the size of our universe, that it is expanding, and introduced us to the Big Bang theory.

The Big Bang theory tells us that between 12 to 14 billion years ago the universe was only a few millimetres across but expanded in a hot dense state into all matter in the known universe today.

Studying this matter helps us understand how the universe came about and this means going small. The job of particle physicists is to examine a world we cannot see: the subatomic world of quarks, leptons and bosons.

If you remember protons from class then imagine that this is comprised of smaller particles: two up quarks and one down quark to be precise.

On this scale, it becomes tricky to quantify matter. Scientists don’t know exactly how small quarks are but because 99.999999999999 per cent of an atom’s volume is empty space, if you were to scale an atom’s diameter to the length of 30 American football fields, electrons and quarks would be less than the diameter of a human hair.

Calculating anything on the back of an envelope

Physicist Enrico Fermi (1901-1954) was famous for his back-of-the-envelope calculations. He delighted in thinking up tricky mathematical problems and working out approximate answers with a pencil and paper. These are now known as Fermi questions.

A Fermi question uses limited data and asks further questions in order to work out a fairly accurate answer that doesn’t require any specialist scientific knowledge. For example, what is the circumference of the Earth?

The most famous Fermi question, and one the physicist posed to his students, is the piano tuner problem. He challenged his class to work out the number of piano tuners in Chicago given a single piece of information: the city’s overall population.

Back in those days you couldn’t do an internet search for such an obscure question, which was just as well because Fermi was teaching his class to become scientists and mathematicians by using the most powerful computer in their possession: the human brain.

Fermi knew from census figures that Chicago had a population of about 3 million. He then assumed that the average family has about four members, which makes 750,000 families in Chicago. If we assume that one in five families owns a piano there are 150,000 pianos in the city.

The average piano tuner would: 1. Service four pianos each day; 2. Work a typical five-day week; 3. Take two weeks of holidays a year. We can now begin calculating. In one year the average piano tuner would service four (pianos) x five (days) x 52 (weeks) minus the 10 days off (40 pianos-worth). That’s 1,000 pianos serviced in a year. If there are 150,000 pianos in the city then 150,000/1,000 gives us 150 piano tuners to meet demand.

Using this method you can work out the circumference of the Earth in a few minutes. First, we take something we already know. Using an atlas we can see that the distance between New York and Los Angeles is roughly 3,000 miles. In those 3,000 miles we pass through three different time zones. From this we work out there are roughly 1,000 miles per time zone. But how many time zones are there in the world? There are 24 because there are 24 hours in the day.

This means that it takes 24,000 (24 x 1,000) miles to travel around the Earth. From class, you will remember that the formula for the circumference of a circle is 2 Pi r where r is the radius and Pi is 3 (we’re using rough numbers here because we’re doing it with pencil and paper). So we have: 24,000/6 = r = 4,000. The diameter is twice the radius, so the diameter of the Earth is 8,000 miles. The scientifically accurate diameter of the Earth is 7,901 miles so for the back-of-an-envelope calculation you can be pretty accurate.