Anthony the Ant
Pictures by Maria Borovik
Part I: Anthony the Ant Discovers Angular Deficit
Anthony the Ant lived with his fellow ants on a desk in one of the offices in the Mathematics Department. He ate bread crumbs from the occasional sandwich and was quite happy.
Of course, ants are creatures of low horizons, but clever enough to navigate the most difficult terrain. The most tough, of course, were sheets of plain paper - no landmarks! But Anthony knew some geometry and could measure distances and angles. In particular, he knew a useful fact: if on his way he did several turns and returned to the point from which he had started, the turns summed up to the full turn, 360 degrees.
Here are some other examples:
Of course, Anthony added the turns to the left and subtracted the turns to the right. No matter how tricky was his path, the result was always 360 degrees.
But one fateful day, DISASTER STRUCK!
Huge steel scissors cut from the sheet of paper - where Anthony happened to be - a net for making a cube.
And Anthony found himself on the cube which was hanging on a thin thread from the office ceiling. He did not know that, of course. As I said, Anthony had low horizons and could see only a small patch of the surface of the cube around him. Since he could walk upside down, there was no difference for him between horizontal and vertical faces: he even did not feel edges when he crawled over them. What made a difference for him was that his new world had eight sticky points - vertices - where glue came out from the seams. This glue was nasty and sticky and so Anthony was afraid to approach these points. He could only walk around them, and to his surprise he discovered that unlike any other places in his world, these points were enchanted! The sum of turns in a path around a sticky point was 270 degrees! 90 degrees had been lost! And for all 8 sticky points Anthony found 720 degrees - two full turns - were missing!
And here Anthony received a call from his friend Betsy: Betsy told him that something horrible happened with her sheet of paper: now it had four nasty points with glue and half a turn, 180 degrees, was lost around every point. Altogether - 720 degrees, two full turns!
And then called Cindy - with the same news, two full turns were missing!
In the case of Darren, though, the story was different - his world lost 90 degrees in walks around eight sticky points, but gained 90 degrees in walks around some other eight points, and the balance was zero!
And Edward even gained two full turns!
· Check that the ants were not mistaken in the assessment of their gains and losses.
· You have probably understood that gain or loss in turns around sticky points has something to do with the number of `holes' in the ants' worlds (Darren's world has one hole and Edward's two, while Anthony's, Betsy's and Cindy's none). Can you conjecture a formula and check it for Fred's world, which has three holes?
· And, if you are REALLY ambitious: can you PROVE the formula?
Part II: People as Ants
Well, when you look at people on the Earth; we are no more than ants, and our horizons are pretty low. Unlike our clever ant Anthony and his friends, we do not always notice that the Earth is curved and that its geometry is quite different from that of a sheet of paper. We got used to measure the position of a point on the Earth by its latitude and longitude. One of the paradoxes of the Earths' geometry is that if two cities are on the same latitude, the shortest way from one to another is not along the parallel; the shortest paths on the sphere are arcs of big circles (that is, circles cut on the surface of the sphere by planes passing through its center). For example, Ankara and New York are at almost the same latitude, but the shortest route for a plane from Ankara to New York goes well north of Iceland!
Therefore when a traveller T travels around the globe staying at the same latitude, he deviates all the time from the straight path; he slightly turns, with every step, to one side. When he returns to the starting point, what is the total turn?
To see that the traveller actually turns, let us do the same trick as in the story about ants and glue to the Earth a huge paper cone which touches the Earth tangently at out traveller's parallel:
Now our traveller walks, simultaneously, on the sphere and on the cone - and, being a person of low horizons, does not see much difference between the two. But, unlike the sphere, the cone can be cut along the "meridian" line and unrolled to a flat sheet of paper:
Now it is obvious that, indeed, the small turns accumulate up to a quite considerable total turn f equal to the angle at the vertex of the sector. Of course, this happened because the Earth is curved; what we did when we glued the cone to it is that we collected all the curvature of the Earth north of the parallel and concentrated it in one point, the vertex of the cone, making it the "sticky point", like in the story about the ants; at the same time, the rest of the cone is essentially flat: it is just bent sheet of paper, and an ant, sitting on it, will not notice the difference between the plane and the cone.
Moreover, the cone gives us the chance to calculate the angle of the turn accumulated in the round-the-world travel at the given latitude, and do so using only elementary geometry. It is simple indeed: in the radian measure of angles, the length of arc TT' spanned by the sector equals f |PT| ; but this is also is the length of our parallel on the Globe!
To compute its length in terms of the latitude, we denote the latitude by a, then, in the picture below, the radius |O'T| of the circle in the base of the cone is |OR| cos a .
Hence the length of the parallel is 2p |OR| cos a. On the other hand, one can easily see from the similar triangles that the angle O'PT is a , hence
|PT| = |O'T|/ sin a = |OR| cos a / sin a
and the length of the parallel is
f |PT| = f|OR| cos a / sin a .
Comparing two expressions for the length of the parallel, we see that
2p |OR| cos a = f|OR| cos a / sin a
f = 2p sin a .
In particular, when you travel along the equator, your do not deviate from the shortest path because the equator is a big circle. Correspondingly, in our formula a = 0 and f = 0; out cone,of course, in this extreme case becomes a cylinder embracing the Earth. When a is very close to 90 degrees, we walk around the North Pole on a sheet of ice, which, because of its small size, is undistinguishable from the flat plane, and, in our formula, the total turn amounts to 2p, as we should expect from the plane.
But when a = 30 degrees (the latitude of Cairo), the total turn is p, or 180 degrees. Hence Cairo has a remarkable property: if you head from Cairo straight to the East, and continue to walk along the parallel at the same latitude, but look all the time in the same direction as you looked when you started your way, you will eventually come to Cairo from the West, walking backwards, looking to the West!
If you trust computers more than you trust yourself and mathematics, then you can find on Internet, at
a remarkable applet which allows you to drag over the sphere an arrow which looks into the original direction; try it, and you will see that, when you circumnavigate the globe at the latitude of about 30 degrees, the arrow returns home turned into the opposite direction.
Of course, the discerning reader rightly suspects that these two simple tales contain much more mathematics than was just told; so, we shall return to Anthony and his friend on another occasion.
To be continued. In the next episode: Anthony discovers Euler characteristic.
© Alexandre Borovik