Of course you know the answer to this: the Sun. (Though often when I ask people they tend to look sheepish and say nothing, for fear of being wrong!)

Actually, it’s not at all obvious which one is bigger: after all, they look the same size. You know which one is bigger because you were told it when you were very young. Which rather spoils the pleasure of trying to work it out. But *how* do we know the Sun is bigger?

It’s fairly easy to establish that the Sun is a least a bit bigger than the Moon. They look the same size in the sky, but during a solar eclipse the Moon completely covers the face of the Sun. So the Sun must be further away than the Moon and therefore it must be bigger. But *how* much bigger?

To establish this we need another rare sighting in the sky: have you ever seen the Moon and the Sun in the sky at the same time? It does happen — as the picture above shows. (Indeed, Lewis Carroll wrote about it in the first two verses of ** The Walrus and the Carpenter**.)

But imagine that you can see both the Sun and the Moon in the sky together and that the Moon is a *half *Moon. What must the geometry of the solar system be at that moment?

The Sun-Moon-Earth forms a giant right-angled triangle, with the Moon at the right-angle. If we can measure the Sun-Earth-Moon angle, then we can use trigonometry to find the relative distances of the Sun and the Moon from Earth. Since the Sun and the Moon look the same size in the sky, these relative distances must also be the relative sizes of the Sun and Moon themselves.

The hard part is measuring the angle. You need it to be *exactly* a half Moon. You need to measure the angle between the Sun (**without** looking at it directly), you and the Moon.

But it gets worse, if you’re even *slightly* wrong with your measurement, the relative distances and sizes change by a surprisingly large amount. This is because the angle is very nearly 90° — and that is because the Sun is really *very* much farther away than the Moon and *very* much bigger. In fact the angle is about 89 5/6°, which indicates that the Sun is about 340 times the size of the Moon. But if you measure it as 89 4/6°, you’ll get the Sun being only 170 times the size of the Moon, in other words *half* the size (and therefore one-eighth of the volume, and one-eighth of the mass). So your measurement needs to be incredibly precise!

**Notes**

The diagram above is merely intended to indicate the relative positions of the Sun, Moon and Earth. The sizes are not to the same scale (compared with the Earth, the Sun should be vastly bigger than shown and the Moon slightly smaller) and the distances are not in proportion (the Earth-Sun distance is substantially bigger than the Earth-Moon distance – indeed, the triangle should look almost like two parallel lines).

The Sun is, in fact, almost exactly 400 times the size of the Moon, by which I mean its radius is about 400 times the radius of the Moon. This means that its volume is 400×400×400 times the volume of the Moon, which makes it nearly three quarters of a million times as large. (The ratio of the masses of the Sun and the Moon is different, however, because they do not have the same average density.)

The figures of 340 and 170 come from tan(89 5/6°) and tan(89 4/6°), which actually represent the Moon:Sun and Moon:Earth distance ratios. It’s an incredible coincidence that, although the angle differs by one-sixth of a degree, one tangent is (almost exactly) double the size of the other. (In fact, it’s double to one part in a hundred thousand!)

]]>Why do we use radians? After all, there’s nothing wrong with degrees. Everyone understands degrees. Almost any fraction of a circle is a whole number of degrees. In radians, it seems like every angle is an irrational number. So what’s the point?

The truth is that no-one uses radians to measure angles. We use radians because it makes the graph of sin(*x*) look nice. If you draw the **graph of sin( x) in degrees**, it is a virtually flat, featureless graph: the

This is not a trivial point. Look at the **graph of sin( x) in radians near to the origin**. It looks like a straight line. Specificially, it looks like the line

So what is sin(18°)? Well, 18° is a tenth of 180°, so it’s a tenth of π, i.e. about 0.31. So the sine of 18° is equal to the sine of 0.31 radian. But the sine of angle in radians is approximately equal to the angle itself. Thus sin(18°) ≈ sin(0.31) ≈ 0.31. Easy!

(If you check this on a calculator, you’ll see it’s correct!)

But how do we find sin(54°) – for angles that large, the sine graph doesn’t look at all like a straight line. So what do we do then? What does the calculator do? (Hint: the **graph of sin( x) looks a little bit like a cubic** equation between –180° and +180°.)

I was idly browsing **Quora**, when I came upon the following sequence:

1, ∞, 5, 6, 3, 3, 3, ...

It is highly unusual to have infinity as a term in the middle of a sequence – whatever could it be?

Well, the *n*th term is the number of **convex regular polytopes in n dimensions**. Ah! That explains it?

It’s easiest to start with the second term of the sequence. It’s the number of **regular polygons** that you can make. A regular polygon is a plane figure with straight sides all of equal length. The most obvious is perhaps a square. But we additionally require the polygons to be *convex:* this means that the sides cannot turn in on themselves. The figure below on the left is a convex regular pentagon; the figure on the right is also a regular pentagon, but it is not convex.

It should be fairly obvious that you can construct regular polygons with any number of sides:

So the number of convex regular polygons is infinite. And that’s the second term of the sequence.

What about the others? Let’s look at the third term: 5. This is the number of **regular polyhedra**. That is, the number of three dimensional shapes, where each face is a regular polygon. (Again, we require them to be convex.) The five regular polyhedra are called the **Platonic solids**, and they are illustrated in the photograph at the top of this blog. From left to right: icosahedron, dodecahedron, cube, octahedron and tetrahedron. It is not possible to construct any other regular polyhedra: the angles won’t fit together to form a closed solid.

In informal language, then, the sequence is the number of regular shapes in one dimension, two dimensions, three dimensions, and so on. The word “polytope” is the generic term that covers polygons (two dimensions), polyhedra (three dimensions), and all the others in higher dimensions. Perhaps surprisingly, it is the two-dimensional world that offers the greatest variety.

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