Tag Archives: Geometry

Which is Bigger, the Sun or the Earth?  Check it Yourself!

Once you’ve convinced yourself the Earth’s a spinning sphere of diameter about 8000 miles (13000 km), and you’ve estimated the Moon’s size and distance (diameter about 1/4 Earth’s, and distance about 30 times Earth’s diameter), it’s easy to convince yourself the Sun’s bigger than the Earth, and much further than the Moon.  It just takes a couple of triangles, and a bit of Moon-gazing.

Since that’s all there is to it, you can guess that the ancient Greek astronomers, masters of geometry, already knew the Sun’s the larger of the two.  That said, they never did quite figure out how big and far the Sun actually is; we need modern methods for that.

It’s Just a Phase

The Moon goes through a monthly cycle of phases, lasting about 291/2 Earth days, in which the part that glows brightly with reflected sunlight grows and shrinks, from crescent to full and back again.  The phases arise because there are two simple ways of dividing the Moon in half:

  • At any moment, the half of the Moon that faces Earth — let’s call it the near half of the Moon — is the only half that we can potentially see. (We’d only be able to see the far half, facing away from Earth, if the Moon were transparent, or a big mirror was sitting beyond the Moon.)
  • At any moment, the half of the Moon that faces the Sun is brightly lit — let’s call it the lit half.  The other half is dark, and its presence can only be detected by the fact that it can block stars that it moves in front of, and through a very dim glow in which it reflects sunlight that first reflected from the Earth (called “Earthshine.”)  

The phases arise because the lit half and the near half aren’t the same, and the relationship between them changes from night to night.   See the diagram below. When the Moon is more or less between the Sun and the Earth (it rarely passes exactly between, because its orbit is tilted by a few degrees out of the plane of the drawing below) then the Moon’s lit half is its far half, and the near half is unlit. We call this dark view of the Moon the “New Moon” because it is traditionally viewed as the start of the Moon’s monthly cycle. 

Figure 1: The Moon’s phases, assuming the Sun’s much further than the Moon. When the Moon is roughly between the Earth and Sun, its near half coincides with the unlit half, making it invisible (New Moon). As the cycle proceeds, more of the near half intersects with the lit half; after 1/4 or the cycle, the Moon’s near half is half lit and half unlit, giving us a “half Moon.” At the cycle’s midpoint, the near side coincides with the lit half and the Moon appears full. The cycle then reverses, with the other half Moon occurring after 3/4 of the cycle.

When the Moon is on the opposite side of the Earth from the Sun (but again, rarely eclipsed by Earth’s shadow because of its tilted orbit), then its near side is its lit side, and that creates the “Full Moon”, a complete white disk in the sky. 

At any other time, the near side of the Moon is partly lit and partly unlit. When the line between the Moon and Earth is perpendicular to the Earth-Sun line, then the lit side and unlit side slice the near side in half, and the Moon appears as a half-disk cut down the middle.

When I was a child, I wondered why half this half-lit phase of the Moon, midway between New Moon (invisible) and Full Moon (the bright full disk), was called “First Quarter”, when in fact the Moon at that time is half lit.  Why not “First Half?”  Two weeks later, the other half of the near-side of the Moon is lit, and why is that called “Third Quarter” and not, say, “Other Half”?

This turns out to have been an excellent question. The fact that a Half Moon is also a First Quarter Moon tells us that the Sun is large and far away!

Continue reading

How to Figure Out the Distance to the Moon Yourself

Last time I described an easy way for you to determine the size of the Moon — easier than the famous techniques used by the classical Greeks. (We don’t need to know the Earth’s circumference, as they did, if we’re ok with a moderately precise estimate.) Once you’ve done that, there’s an simple method, well known since classical times, for figuring out how far away the Earth’s companion is. That’s what I’ll describe in this short post.

(What’s not so easy is to determine the distance and size of the Sun. The classical Greeks failed in their efforts. We’ll need a more modern approach… but that’s for next week.)

Size Versus Distance

Even the early classical Greeks knew something about the Sun, just from the fact that the Moon and Sun appear roughly the same size to our eyes — that is, they occupy about the same amount of sky. If the Sun is twice as far away as the Moon, its diameter must be twice as big, in order that it appear the same size. That’s illustrated in the figure below. If it is ten times as far away, its diameter must be ten times as big. If it’s four hundred times as far away, its diameter is four hundred times as big. (Spoiler: that last one’s the truth; but we’ll get to it later.)

If the Moon is a distance L away from you, and another object twice as far away appears to be the same size in the sky, then that object’s diameter must be twice the Moon’s diameter D. This logic applies more generally to objects further and nearer than the Moon.

You can run this logic in the other direction; if something perfectly blocks the Moon, then if it’s ten times closer than the Moon its diameter must be ten times smaller. If it’s a billion times closer than the Moon, it must be a billion times smaller.

Continue reading

How to Figure Out the Size of the Moon Yourself


Having confirmed we live on a spherical, spinning Earth whose circumference, diameter and radius are roughly 25000, 8000, and 4000 miles (40000, 13000, and 6500 km) respectively, it’s time to ask about the properties of the objects that are most obvious in the sky: the Sun and Moon. How big are they, and how far away?

If the Moon were close to Earth, then at any one time it would only be visible over a small part of the Earth, as indicated in light blue. But in fact (except at new moon) about half the Earth can see it at a time.

Historically, many peoples thought they were quite close. With our global society, it’s clear that neither can be, because they can be seen everywhere around the world. Even the highest clouds, up to 10 miles high, can only be seen by those within a couple of hundred miles or so. If the Moon were close, only a small fraction of us could see it at any one time, as shown in the figure at right. But in fact, almost everyone in the nighttime half of the Earth can see the full Moon at the same time, so it must be much further away than a couple of Earth diameters. And since the Moon eclipses the Sun periodically by blocking its light, the Sun must be further than the Moon.

The classical Greeks were expert geometers, and used eclipses, both lunar and solar, to figure out how big the Moon is and how far away. (To do this they needed to know the size of the Earth too, which Eratosthenes figured out to within a few percent.) They achieved this and much more by working carefully with the geometry of right-angle triangles and circles, and using trigonometry (or its precursors.)

The method we’ll use here is similar, but much easier, requiring no trigonometry and barely any geometry. We’ll use eclipses in which the Moon goes in front of a distant star or planet, which are also called “occultations”. I’m not aware of evidence that the Greeks used this method, though I don’t know why they wouldn’t have done so. Perhaps a reader has some insight? It may be that the empires they were a part of weren’t quite extensive enough for a good measurement.

Continue reading

The Earth’s Shape and Size? You Can Measure it Yourself — Part 2

In my last post, I showed, using only simple arithmetic, that the observed atmospheric effects from the January 15th volcanic explosion in the Kingdom on Tonga are consistent with a round Earth. From the timing of the observed spikes in pressure, seen around the world, one can work out how long the pressure wave took to do a round-trip of our planet. It’s clear that the pressure wave from the eruption moved outward and circled the Earth, moving in all directions over the same amount of time (35-36 hours, to within 5% or so). This uniformity is what we would expect if the Earth’s approximately a sphere and the pressure wave had a roughly constant speed.

But at the end of the post I pointed out that this isn’t yet a proof that the Earth’s spherical; there are loopholes, involving possibilities such as an ellipsoidal Earth with the Tonga eruption at one end. And there’s even a flat version of Earth that we can’t rule out with this data!

So in this post, we’ll look at why most shapes for the Earth are ruled out, see why there’s a loophole — why a small number of non-spherical shapes are still consistent with the data — and look at how we might close that loophole.

A Square Earth

Flat Earth’s aren’t plausible, but they are easy to draw and visualize, so I’m going to start by showing why most (but not quite all) flat Earth’s are inconsistent with the data I used in the last post. Once you see the origins of the inconsistencies, the same principles will apply to other shapes that would be much harder to visualize if you didn’t already know what to look for.

Let’s start with a square Earth (yes, square, not a cube — though a cube would have similar problems). This Earth has edges, and we have to figure out what happens to the pressure wave when it gets there. Leaving aside the obvious difficulty that we have no idea how a square planet would hold on to its atmosphere at the edges, there are three easy options for what happens to the pressure wave at the edge:

  • It disappears.
  • It bounces back (i.e., it reflects).
  • It somehow goes round to the back side, crosses it, and reappears.

Disappearance is ruled out immediately, because then the pressure wave would pass each point on Earth once, whereas the data shows it appears multiple times. So let’s focus on the second possibility, the reflecting square. The problems we’ll find here will also affect the third possibility.

There’s another question we have to answer: where is the volcano inside this square? Well, let’s start with the simplest case, where the volcano is dead center. After we see what’s wrong with that, it will be easy to see that an off-center volcano is even worse.

On a square with reflections, the pressure wave expands and then bounces back from the walls, rather than going all the way around as on a sphere. In other words, a round trip from the volcano to a chosen location and back to the volcano involves some reflections instead of a continuous trip. That’s okay in principle, but what’s not okay can be seen in the Figure below. Trips north-south and east-west have the same length, but trips northeast-southwest and northwest-southeast are longer by a square root of 2, about 40% longer. We would certainly have seen this in the pressure spike data; if north-south trips took 36 hours, then northeast-southwest trips would have taken almost 51 hours.

(Left) On a square Earth with reflective edges and a central volcano, round trip times are different for diagonal paths than for horizontal or vertical paths. (Right) In fact round trips can be extremely long at certain angles.

And actually it’s worse than this, because the reflections would make a total mess of the pressure wave. You can get a little intuition for this by tracing the path of the bit of wave that moves west-southwest. It bounces around several times before returning to the volcano!

More generally, what is happening is that the wave is becoming very complex as it reflects multiple times. In the animation below I’ve shown what would happen to a pressure wave on a square. There’s no way we would have seen a simple pattern of spikes in the data around the world had it been square.

On a square Earth with a reflective edge, a circular blast from a centrally located volcano would lead to a very complicated pattern of pressure spikes that do not correspond to what is observed.

Is there any way out of this argument? So far I’ve assumed that the wave travels at a constant speed as it moves away from the volcano. What if it didn’t? What if, instead of forming a circle, it formed a square, which could move out uniformly and bounce back uniformly from the edges, so that all round trips were of the same duration? This would require that the wave’s speed heading toward the corners of the square is 40% faster than it’s speed heading north, south, east and west. That’s a clever idea, and so far, what I’ve told you doesn’t exclude it. But in a later post we’ll use pressure spike data to measure the wave’s speed in various directions, and we won’t see such large variation; so we will rule this out soon enough.

The spike patterns would be at least as complicated, and generally worse, if

  1. the volcano were not dead center on the reflecting square (making the pattern of reflections even more complex — see the figure below);
  2. the pressure wave went round the back of the square Earth;
  3. the square was instead a rectangle with sides of different length; or
  4. the square was instead a triangle, hexagon, parallelogram, a five-pointed star, a crescent, or some irregular shape;

In short, a flat Earth is completely excluded — ruled out by the data — except for one very special shape.

On a square Earth with a reflective edge, a circular blast from a non-centrally located volcano would lead to an even more complicated pattern of pressure spikes.

The Flat Disk Earth

Imagine the Earth’s a flat disk, and put the volcano at the exact center. Then, you can get exactly the same pressure spike data as we actually observe. Let’s see why.

If a pressure wave moves off at a constant speed from an explosion at the center of a disk, it will form a ring that moves outward, reflects off the walls, and comes right back to the volcano. And it will do this over and over again. In all directions from the volcano, the out-and-back trips all take the same amount of time; and at each location on Earth, the pressure wave will pass twice during this out-and-back trip. You can go further and check that the equations I used to determine the round-trip time on a spherical Earth will work for a disk Earth too, where T is now the out-and-back time. The spike pattern from a volcano centered on a disk looks identical to that of a volcano on a sphere.

On a disk-shaped Earth with a reflective edge, a blast from a central volcano would reproduce the pressure spikes observed following this month’s explosion.

This is only if the volcano is dead center, however. For example, in the figure below, the trip to the right is longer than the trip to the left; and yet again, because the volcano’s not in the center, the reflections off the edges will quickly make the wave extremely complex and lead to a highly irregular pattern of spikes around the world. So an off-center volcano is ruled out. (The situation is no better if the waves, rather than reflecting off the edges, somehow go round the back.)

A disk Earth with a volcanic blast away from the center will not have equal round-trip times; the full pattern is very complex, but just the two paths shown are enough to give different round-trip times, not seen in the actual pressure spike data.

So the only way to interpret our data, if the Earth is flat, is to conclude that Tonga sits in the very middle of a flat disk. But this is quite a loophole! How can we prove the Earth is not flat?


The Flat-Earthers’ Flat Earth

By the way, what I’ve just told you means that the pressure spike data rules out the flat-disk Earth most popular with flat Earthers. That silly model of Earth puts the north pole at the center and stretches the south pole out into a circle tens of thousands of miles around, with the idea that no one ever actually flies over the south pole to check it out.

The flat earth with north pole at center, popular with a certain set, with a red dot showing the far-off-center location of the Tonga volcano. Pressure waves from that location could not have created the observed pressure spikes.

Well, let’s leave aside the fact that many scientists, including personal friends of mine, have experiments (Ice Cube, BICEP, South Pole Telescope, and many more…) running within a mile or so of the south pole, and they (and the pilots who fly them there) can confirm it is a point, not an arc tens of thousands of miles wide. But we now have an argument that’s not hearsay: given where the Tonga volcano is located on this flat-disk Earth, an explosion there would never have been able to generate the observed regular and simple pattern of pressure spikes. A 12-year-old can prove the flat-earthers’ model of Earth is definitively ruled out.

And these considerations also show us why a flat Earth that puts Tonga dead center is ruled out too, though not from the pressure spike data. Just as the flat-earther’s model of Earth, with the north pole at the center, spreads the south pole into an arc tens of thousands of miles long, one with Tonga at the center would spread southern Algeria, the region exactly opposite, into an arc tens of thousands of miles long. But even though that’s in the desert, people live there. There are a few roads and a few towns. Residents there would certainly know if driving to the nearest town took many weeks instead of a few hours.

A flat Earth centered on region of Tonga would spread southern Algeria across tens of thousands of miles; residents of that country would beg to differ.

So that one remaining flat Earth is dead too. Good-bye, and good riddance.

But I went through this argument carefully for a reason. Once we understand why a Tonga-centered flat disk Earth is consistent with the pressure spike data, we can understand all the other loopholes, such as ellipsoidal Earths — and we’ll also see how to rule them out too.

A Symmetry

Why was it that every flat Earth gave the wrong pattern for the spike timing except for the flat disk with the volcano at dead center? What was special about that case?

The study in my last post showed that any bit of the pressure wave, as it started at and headed out from the volcano, took the same amount of time to travel outward and back to its starting point. In other words, as far as the pressure wave was concerned, all directions leading away from Tonga are equivalent to one another. East, north, northwest, south-southwest — it doesn’t matter, the length of the round-trip path was always the same.

A fancier way to say that is that there is a symmetry, a rotational symmetry in particular. If you were to put a spike straight through the Earth starting right at the volcano and going through the Earth’s center, and then you rotated the Earth around the spike, the Earth’s shape would stay the same as you did so. If that weren’t true, then not all directions would look the same, and not all round-trip times would be equal.

So what my analysis of the data actually shows is not that the Earth’s a sphere, but only that it is symmetrical around the Tonga volcano — all directions are equivalent. That’s true of a sphere. But it’s also true of a flat disk with the volcano at its center — or of a bowl. And it’s also true of an ellipsoid with the volcano at one end, or of a gourd shape, or of half a sphere.

So how are we going to check that the right shape for the Earth is truly a sphere?

A pressure wave starting from any point on a sphere, or from a point at either end of an ellipsoid, or from the center of a disk or hemisphere (with reflecting edges), will give a similar pattern to the one observed after the Tonga volcanic explosion. Note this is not true at any other point on the non-spherical surfaces. (Image made with Mathematica 12.)

Special Points vs. Typical Points

The symmetry that I just described requires that either

  • the Tonga volcano is at a very special point on a non-spherical shape, or
  • the Tonga volcano is at a typical point on a sphere.

We already saw this for the flat disk; we could only reproduce the data if the volcano were at the center, and not if it were off-center (as in the flat-earthers’ flat Earth.) And while it’s true for an ellipsoid with a circular cross-section if the volcano is exactly at one end, it wouldn’t be true if the volcano were anywhere else.

That makes all the non-spherical shapes somewhat implausible, because they require that the Tonga volcano be located at a unique, special place — one of at most two on Earth. And what are the chances that the first big volcanic blast of the internet era would occur at such a special location? There are so many other volcanoes — Vesuvius, Mount Rainier, Mount Erebus, Cotopaxi, Taal, Merapi, and hundreds more — any volcano that isn’t on exactly the opposite side of the Earth from the Tonga volcano would have given asymmetric data, with round trip times that vary widely. Only on a sphere is the Tonga volcano at a typical point, with nothing unusual about it.

So a sphere seems much more plausible. But, hey, that’s just a plausibility argument, and coincidences do happen sometimes. If you want to prove the Earth’s a sphere, this argument is not enough.

Fortunately, it’s now clear where proof would come from. We just need to wait for another similarly-sized eruption, from some other volcano, to create another pressure wave that goes round the Earth. Even if the Tonga volcano were somehow located at a special point on Earth, the next big volcanic blast will almost certainly originate from a typical point. It’s very unlikely that it will lie exactly on the opposite side of the Earth from Tonga. If, after this second blast, we do the same measurement of round-trip times using its pressure spikes, and we again find they all show equal round-trip distances in all directions, then we’ll know the Earth is symmetric around that volcano too. And that’s enough, because only a sphere can be rotationally symmetric around two points (unless those two points are exactly at the opposite ends of an ellipsoid or similar shape.)

The only thing that’s too bad (although it’s also quite fortunate) is that explosions this size don’t happen often. We may not be able to close this loophole for quite a few decades to come…

…unless, rather than looking to the future, we look to the past…?

There is, after all, Krakatoa.

(to be continued)

Geometry From Within: Evidence for a Round Earth

It’s a lot easier to map the Earth than it used to be.  Before satellites, you had to do many careful measurements of distances and directions, at many different locations around the world, and combine them all to build a picture of a world you couldn’t see.  That’s part of why maps and globes made in past centuries had so many inaccuracies and distortions; it was a tough business.  

How that changed in the 1960s!   The first full photograph of the Earth that I’m aware of was made in 1967 by the ATS-3 satellite (were there earlier ones?)  So much simpler… the whole planet laid out in front of you.  You just need a few photographs like this, and the era of measuring from one point on the ground to another is mostly over.

But the challenge of trying to measure things when you’re stuck within them, and can’t step outside them, hasn’t gone away.  Just as we could see in telescopes that the Moon and Mars are ball-shaped, long before we could observe the Earth itself, today we can see other galaxies in great detail, but we still struggle to build a complete picture of our own, the Milky Way. The Gaia satellite is trying hard.

To determine the Earth’s two-dimensional surface is really round took some clever thinking. Aristotle, in ancient Greek times, noted that the Earth’s shadow on the Moon during a lunar eclipse is always curved in the same way — it doesn’t matter what time of day or year the eclipse occurs, or whether the shadow is on the north, east, west or south side of the Moon.  This feature is to be expected if the Earth’s a ball, like the Moon and Sun, and very difficult to explain otherwise.  [Try to figure out what you might see if it were cylinder-shaped!]

But there are other tricks you can use if you have a hunch that the place you live on, or in, is of finite size.

One Dimension: the Possibly Circular Canal

Suppose you live on the banks of a canal, a long thin channel extending off to the horizon, like a river without any flow.  And suppose you suspect that this canal forms a loop, surrounding a large island.  How could you check?   Well, if you had a boat, you could row yourself down the canal; or you could walk along the shore. If the canal is really in the shape of a loop, you’ll eventually come back to your starting point.  But maybe you’re worried such a journey would be too long, difficult, risky, expensive. Do you have other options?  

Here’s one: suppose you could make a big wave moving in the clockwise direction around the canal.  The wave, unlike you, wouldn’t need any food and drink or fuel for the journey — so time and money would not be a problem. The wave would move down the canal at a definite speed [I’m assuming here that it maintains its height], and if the canal were really a loop, then after some time T you’d see the wave return, still moving clockwise, and pass by you.  If you waited the same amount of time T again, you’d see the same wave a second time, again clockwise.  After the same amount of time T, you’d see it a third time. 

If instead the canal were a finite strip, then the wave would reflect off the end, and so the wave would return from the opposite direction. If it were infinite in length, it would never return. And if it had a complicated shape — perhaps a P or an R or a B instead of an O — you would get multiple waves in a complex pattern. But the simple pattern in which the waves return again and again, from the same direction, after a time T, is consistent with the canal being a simple loop.

You could try sending a wave counterclockwise too, and you’d expect the same pattern if the canal’s a loop.

As the wave passes you, you can also estimate its speed v. Having also measured T, you can now determine the length L of the canal. It’s the wave speed times the time T for the wave to go round once:

  • L = v T
Figure 1: You live on the shore of a canal, which you suspect is circular. You could find out how big it is by sending a large wave in either direction, and measuring the time T that it takes to return.

Perhaps making such a wave is too difficult for you, but if you’re lucky, someone or something down the canal may make a giant splash. Then you’ll see the ripples from the splash come by in a similar pattern. Now waves will travel both counterclockwise and clockwise around the canal, and they probably won’t arrive at the same time. That doesn’t matter, though. You’ll see the clockwise waves repeat after a time T, and you’ll see the same for the counterclockwise waves. Seeing both of them repeat after the same time T will give you confidence that the canal’s really a simple loop

To be specific, let’s call t1 the time you measure the first wave, t2 the second wave, t3 the third, t4 the fourth, and so on; if the first wave is counterclockwise, then the second is clockwise (see Figure 2), the third counterclockwise, and so on. (This won’t be true if instead of a loop the canal is in the form of a line segment! A reflection off the end could make the first two waves come from the same direction.) As the clockwise waves will repeat after a time T, and the same for the counterclockwise waves, it will be the case, if the canal’s a loop, that

  • t3 – t1 = t4 – t2 = T
  • L = v T

There’s more; if you know the time ts when the splash happened and you know the wave speed, then you can learn how far away the splash was from you:

  • D = v ( t1 – ts )

But even if you don’t know what time the splash happened, you can figure it out; see Figure 2. The distance traveled by the counterclockwise wave to get to you, plus the distance traveled by the clockwise wave to do the same, equals the full distance round the circle (Figure 2), so the time that the counterclockwise wave required to reach you ( t1 – ts ) plus the corresponding time for the clockwise wave ( t2 – ts ) must be equal to T.

  • T = ( t1 – ts ) + ( t2 – ts ) = t1 + t2 – 2ts , which implies ts = 1/2 (t1 + t2 – T)

If you look closely at these four bold-faced equations, they tell you that you can determine T, L, D and ts , properties of the loop and the splash, if you know t1, t2 and t3 and v, which are all things that you can measure without going anywhere. From this point of view t4 is a bonus, a nice check that things are working as expected.

Even better, if you have a friend down the canal who makes the same measurements, that friend won’t get the same answers for t1, t2, t3 and t4 ; the waves arrive at different times for your friend than for you. But when you obtain T and L and ts from the waves you see, and your friend does the same, you’d better get the same answer — because these are properties of the loop and splash, and don’t care where either you or your friend is located.

Figure 2: A large splash occurs at time ts, and waves travel both counterclockwise (green), in which case they reach you at time t1, and clockwise (red), reaching you at time t2.

By themselves, these equations do not prove the canal is round, though they are consistent with it. They only tell you that it’s a loop of length L, with no kinks which could cause extra reflections. Still, it’s a lot of information for a very low price, without taking a boat around the loop, walking all around it, or sending up a drone to take a photograph. The waves have done all the work for you.

Figure 3: After the counterclockwise wave passes you at time t1, it continues round the canal, and passes you again at time t3 = t1 + T.

Two Dimensions: the Possibly Round Surface of the Earth

What would be different if you lived on a sphere?  (A subtlety of language: by “sphere,” I do not mean “ball”, which is three-dimensional; I mean the surface of the ball, which is two-dimensional.  In this terminology, the Earth is a ball, while its surface is a sphere, approximately.)  Again, of course, you always have the option of traveling round the sphere yourself and exploring it, checking that no matter what direction you go in, if you walk in a perfectly straight line, you will always come back to your starting point after you travel the circumference of the sphere.  But that’s expensive and time-consuming and not very practical.  What other options do you have?

You could wait for a big splash in the atmosphere — a natural one like a volcanic eruption, or an artificial one of similar size (fortunately now forbidden by nuclear testing treaties).   This opportunity, if you want to call it that, came this past week, unfortunately near an inhabited area and at the ocean’s surface within the Kingdom of Tonga, with ensuing loss of life, as well as the destruction of crops and homes; the resulting tsunami even took lives far across the Pacific ocean.  It’s not an experiment we would happily have chosen. But nature has carried it out without asking us; we may as well learn what we can from it.

When water hits hot magma and turns to steam, there’s an immense release of energy, especially if the magma is itself packed with compressed gasses. This is partly why some of the largest explosions in the last two hundred years have occurred when volcanic islands self-destructed; Krakatoa is the most famous.  The latest estimate as of the time of writing is that the one in Tonga last week was overall perhaps only 1/20 times as powerful as Krakatoa, but its plume was enormous, and its shock waves were strong enough to be detected multiple times, in many places, as they traveled round and round the Earth.

The shock wave emanated from the explosion in all directions, moving outward as an ever expanding circle, as you can guess by pure reasoning but also as confirmed by satellite.  After traveling 1/4 of the way around the Earth, the wave front reached a maximum extent — the same size and shape as the equator, though with a different orientation — and then shrank again, converging to a point in Africa exactly halfway around the Earth from the explosion’s location. (A nice visualization of this, and of what I’ll say next, can be found here.) Then the shockwave continued onward, again expanding to the Earth’s full extent, and then shrinking and converging on the very spot where it was created in the first place.   And this process repeated, until the shock wave, gradually losing its energy, faded beyond the point of detectability.

This pattern of outward expansion, convergence to the opposite point, return-ward expansion, and convergence to the original point, means that the waves from the explosion passed every point on Earth multiple times, and did so first moving away from the explosion, then returning, then again moving away, and again returning, until finally they were too small to observe.  That this pattern was seen everywhere, in countries widely spread around the globe, by both professional and civilian weather stations, gives some qualitative evidence that the Earth’s a smooth object with a rounded surface of some type.  For example, here is the pattern of multiple waves crossing, returning, re-crossing and re-returning as measured by weather stations in China; we can see three wave passages clearly (the fourth is too dim to measure well).  And here is a similar pattern in the Netherlands; though it’s only at one location, and only the main shock wave is detected, the shock is seen six times. 

What’s nice is that for a sphere — and only for a sphere [see caveat below] — the equations I wrote earlier for a circular canal still hold, and importantly, they hold everywhere, and have to give the same circumnavigation time T and the same splash time ts. That’s because if you are on a sphere, motion away from the volcano (or indeed any point), in any direction, will take you on a circular path of length equal to the sphere’s circumference. On any other shape, this won’t be true.

[To be fair, I am making a couple of assumptions: for instance, that the volcano was located on a random, not special, point on the Earth. (For example, if the Earth’s surface was oblong instead of circular, then the two points at either end of the oblong are special.) To make a long story short, there are still loopholes to the argument I’m giving here, but they are only relevant if there are very special and unlikely coincidences. Additional volcanoes, would quickly close the loopholes.]

In particular, the equations I introduced earlier should hold in China, about 1/4 around the Earth from Tonga. And they should also hold in the Netherlands, much further from Tonga, in a quite different direction. If the Earth had an uneven shape, then the time to go round the Earth in the direction from Tonga to Beijing would be different from the time to go round it from Tonga to the Netherlands; you wouldn’t get the same T. And if the Earth had edges (as in the absurd flat-earth map), you would see reflection waves; you wouldn’t get the same T or the same ts, and the second big wave across China wouldn’t look like the original one retracing its steps (a fact which already gives qualitative evidence for a round Earth.)

Using publicly available data from anywhere in the world, including what I’ve shown you from China and from the Netherlands, we can check ourselves that the Earth’s a ball and measure its circumference. Let’s do it.

So as not to spoil the fun, I’m going to wait until after the weekend to post the results. You are all encouraged to gather your children together and to try to measure:

  • T, the time it took for the waves to travel around the Earth; do this both with the data from the Netherlands and that from China; do you get similar answers?
  • ts, the time when the eruption occurred; use both the data from the Netherlands and the data from China (make sure you’re using UTC time, so you don’t get confused by time zones). Do you get similar and roughly accurate answers? Is it close to the time reported in this article?
  • v, the speed of the waves, which you can determine by watching how long it takes them to cross a part of China and comparing that time with the distance of that path; caution, make sure you trace a path perpendicular to the wave front.
  • C = T v , the circumference of the Earth, equal to the time it took for the waves to circumnavigate the Earth times their speed. Can you get fairly close?

Caution: You’re not going to get exactly the precise scientifically-known answers, nor will your answers be perfectly consistent, because the data I’ve linked to was neither taken nor presented with scientific levels of precision. But you should be able to get within 10-20%, enough to convince you the Earth’s surface pretty darn close to a sphere. If you want more precision, I’m sure precision data is available (anybody have a good link?) [Also note that there are some extra waves seen in the China map, some of them reverberations from the original explosion, and some due to later, smaller explosions; they travel in the same directions as the original ones, showing they come from the same place. For our purpose here, just keep your focus on the biggest waves.]

The point is that we can learn the Earth is ball-shaped without ever stepping off the Earth, and in fact without even traveling; and we can even learn, from the timing, how big the Earth is.  All it takes is a natural explosion, measurements from a few places, some logic, and simple algebra.   The data is now publicly available, and every science teacher in the world ought to encourage their teenage students to do this exercise!  Not only does it confirm we live on a sphere, it shows that one needs neither a photograph taken from outer space, nor a flight around the world, nor specialized map-making skills, to obtain that proof.

Three Dimensions: The Universe

Now what about the universe as whole?  The Earth and Sun are carrying us along as they travel within a three-dimensional surface.  What is its shape?  How can we know?  [There is also the question of the four-dimensional surface that makes up the space and time of the universe.  I’m not addressing that here, that’s even more complex.]

A circle is a one-dimensional sphere; the surface of the Earth (not its interior) is a two-dimensional sphere. Could the universe be a three-dimensional sphere?   We can’t stand outside it to find out.  In fact it’s far from clear there is meaning to “outside” since, after all, it’s the universe, and might be everything there is. Nevertheless, we can imagine, at least, trying to do a similar experiment.  If there were a huge supernova explosion, or a tremendous flare from a distant black hole as it ripped apart a star, maybe we would see the light arrive from one side of us, and then later see it arrive from the other side, and yet again from the first direction, and so on.

Back before we knew the huge scale of the universe and the tiny speed of light, that might have seemed plausible.  We can’t hope to do anything like this, unfortunately.   But it’s not because the question makes no sense.  The natural Tonga volcano experiment worked thanks to the fact that it’s a small world (after all) and the speed of sound is relatively fast, so it all took less than a day or two.  In the universe, it’s the reverse; it’s a big place and the speed of light is relatively slow.  Our own galaxy, the Milky Way, is itself 100,000 light-years across [i.e. it’s so big that it takes 100,000 years for light, traveling at the fastest speed our universe allows, to cross it], so even if our galaxy were the entire cosmos, as was thought until the 1920s, it would take at least 100,000 years to do this experiment.  And of course we now know the universe is immensely larger than our own galaxy; indeed the most recent map of galaxies extends out, for the brightest galaxies, as far as 10,000,000,000 light years.  Hopeless.

Nevertheless, the possibility that the universe has an interesting shape, and though huge might be small enough that we could see some evidence of its shape, remains a topic of research.  The light from events in the distant past might give us clues.  While a blast wave isn’t something we’d be able to see from multiple perspectives, a long-lasting bright spot on the sky could potentially be seen reaching us from different paths around a complex universe.  The fact that the universe has been expanding over the billions of years since the Hot Big Bang began complicates the thinking, but also provides opportunities.

To give insight into how this could be done is beyond the scope of this blog post, but if you’re curious about it, you might try this long-form article from Quanta Magazine (a highly recommended source for interesting articles.)  

The Lesson for Humankind

The big lesson here: geometry can be learned from the inside.  You don’t need to be outside an object to map it and learn its shape and size. That this is possible explains how mapmakers knew the shapes of continents long before satellites, and how one can determine that the universe is expanding while remaining within it (though the story of how scientists did this, without using the methods described in this post, is for another day.) And if the object is finite, so that no wave can travel forever without eventually returning to you, then it’s possible to infer its shape just by learning how waves travel and bounce around the object. That’s how the depth of the ocean’s deepest point was recently measured, as I described in my last post; and that’s how children (of all ages) should prove for themselves, using publicly available data from last weekend and simple algebra, that the Earth is indeed round.