ObserveTheSky › Explanations › When a picture squashes the numbers

When a picture squashes the numbers

Sometimes a drawing has to fit a very small number and a very big one side by side.

Two rulers one normal, one squashed

Look at these two lines. On both of them, every gap is the same width on your screen.

Same gaps on the screen. Wildly different gaps in real life.

On the top ruler, going from 0 to 1 is one step. One step on the screen is one step in real life.

On the bottom ruler, going from 1 to 10 looks like exactly the same little hop. But it is not one step. It is nine more steps added to the one you already took. So you are ten steps from the start, which is exactly what the number says. The next hop, from 10 to 100, is ninety more. The one after that is nine hundred.

Count the zeroes on the red ruler. Now look at the green one above it. Coincidence?

It is not. 1 has no zeroes, and the green ruler above it says 0. 10 has one zero, and the green ruler says 1. A thousand has three, and the green ruler says three.

The green number is simply how many zeroes the red one is carrying. Which is the same as saying how many hops you have made.

Why anybody would do this

Say you want one picture showing something 1 metre long and something 100,000 metres long.

On a normal ruler, if the big one fits across your screen, the small one is a twentieth of a hair’s width. You cannot see it. It is not there.

On the squashed ruler both fit, and you can see both. You pay for it by giving up the widths. The gaps on the picture are no longer the gaps in real life.

Where this site does it on purpose

One drawing here uses a squashed ruler, and it has to: the one about where the solar system ends. Its three answers are 30.07, 121.6 and about 100,000 times the Earth’s distance from the Sun. Draw those on a normal ruler and the first two sit on top of each other in the corner. One control does the same: the zoom slider in Space.

Where does the solar system end? →

And where it does not which is nearly everywhere

A squashed picture is rare here, and it would be easy to assume everything is one.

In the Space view, the distances are real. If Neptune is thirty times further from the Sun than Earth is, it is thirty times further across your screen. That holds at every zoom, all the way out to the galaxies. The size-and-distance charts on the object pages are the same. One fixed scale, true ratios.

So when the space between the planets looks enormous and empty, that is not a drawing trick. That is the measurement.

How to tell, in one question

Look at the numbers along the edge of a picture.

If they go 1, 2, 3, 4, 5, adding the same amount each time, the picture’s widths are true.

If they go 1, 10, 100, 1000, multiplying each time, it is squashed. The gaps are lying to you, in a way the numbers own up to.

And what about the zoom slider? →

The name for it

A ruler where each step multiplies instead of adds is called a logarithmic scale, and the number of hops is the logarithm. There is nothing more to it than the bottom ruler above.

It is used all over the place once you start looking. Earthquakes, loudness, how sour something is. All counted in hops. That is why a magnitude 7 earthquake is not a bit worse than a magnitude 6. It is ten times as much shaking.