Professor Stewart's Hoard of Mathematical Treasures (23 page)

Loyd’s four-piece attempt to dissect a mitre into a square.

Speaking of quibble-cooks: I’m going to set exactly the same puzzle as one I set in Cabinet (page 199), where the answer was ‘impossible’. But I’m looking for a different answer, because this time I’ll allow any suitably clever quibble-cook.

Three houses have to be connected to three utility companies - water, gas, electricity. Each house must be connected to all three utilities. Can you do this without the connections crossing? (Work in the plane, no third direction to pass pipes over or under cables. And you are not allowed to pass the cables through a house or a utility company building.)

Actually, I should have said: ‘You are not allowed to pass the cables or pipes through a house or a utility company building.’ I
think that was clear from the context, but just in case you disagree, assume that too.

Connect houses to utilities obeying all conditions.

 

In the earliest days of the telescope, Galileo Galilei discovered that the planet Jupiter had four moons, now named Io, Europa, Ganymede and Callisto. Astronomers now know of at least 63 moons of Jupiter, but the rest are much smaller than these four ‘Galilean’ satellites, and some are very small indeed. The times the Galilean satellites take to go once round Jupiter, in days, are respectively 1.769, 3.551, 7.155 and 16.689. What is remarkable about these numbers is that each is roughly twice the previous one. In fact,
The first two ratios are very close to 2; the third one is less impressive.

The simple numerical relationships between the first three periods are not coincidental: they result from a dynamic resonance, in which configurations of moons or planets tend to repeat themselves at regular periods. Europa and Io are in 2:1 resonance, and so are Ganymede and Europa. The ratio is that of
the orbital periods of the two moons concerned; the numbers of orbits they make in the same time are in the opposite ratio, 1 : 2.

Resonances arise because the corresponding orbits are especially stable, so they are not disrupted by any other bodies in the vicinity, such as the other moons of Jupiter. However, to make things more difficult, some types of resonance are especially unstable, depending on the ratio concerned and the physical system involved. We don’t fully understand the reasons for this. But this type of 2:1 resonance is very stable, and this is why we find it in Jupiter’s larger moons.

The other main orbital resonances within the Solar System are:
where all bodies listed except Pluto and Neptune are moons of Saturn.
²⁴

When thinking about resonances, it is important to realise that any ratio can be approximated by exact fractions, and there can be ‘accidental resonances’ that are unrelated to dynamic influences between the two orbits concerned. All the above are genuine resonances, showing features such as ‘precession of perihelion’ - movement of the orbital position nearest to the Sun - that lock the orbits stably together. Among the accidental resonances that can be found by searching tables of astronomical data are:

Some important genuine resonances occur for asteroids - mainly small bodies, most of which orbit between Mars and Jupiter. Resonances with Jupiter cause asteroids to ‘clump’ at some distances from the Sun, and to avoid other distances.
²⁵
More asteroids than average have orbits that are in 2:3, 3:4 and 1:1 resonance with Jupiter (the Hilda family, Thule, and the Trojans) because these resonances stabilise the orbits. In contrast, the resonances 1 : 3, 2:5, 3:7 and 1:2 destabilise the orbits: rings and belts are different from individual bodies. As a result, there are very few asteroids at the corresponding distances from the Sun, called Kirkwood gaps.

Kirkwood gaps and Hilda clumps (1 AU is the Earth-Sun distance).

Similar effects occur in Saturn’s rings. For instance, the Cassini Division - a prominent gap in the rings - is caused by a
2:1 resonance with Mimas, which this time is unstable. The ‘A ring’ does not slowly fuzz out, because a 6:7 resonance with Janus sweeps material away from the outer edge.

One of the weirdest resonances occurs in the rings of Neptune, a ratio of 43 : 42. Despite the big numbers, this one seems to be a genuine dynamic effect. Neptune’s Adams ring is a complete, though narrow, ring, and it is much denser in some places than others, so the dense regions create a series of short arcs. The problem is to explain how these arcs are spaced along that orbit, and a 43:42 resonance with the moon Galatea, which lies just inside the Adams ring, is thought to be responsible. The arcs should then be placed at some of the 84 equilibrium points associated with this resonance, which form the vertices of a regular 84-sided polygon, and pictures from Voyager 2 support this.

A section of the Adams ring: grey, resonance islands; black, ring material.

Resonances are not confined to the orbital periods of moons and planets. Our own Moon always turns the same face towards our planet, so that the ‘far side’ remains hidden. The Moon wobbles a bit, but 82 per cent of the far side is never visible from Earth. This is a 1:1 resonance between the Moon’s period of rotation around its axis and its period of revolution around the Earth. This type of effect is called spin-orbital resonance, and again there are plenty of examples. It used to be thought that the planet Mercury did the same as our Moon, so that one side —
facing the Sun - was tremendously hot, and the other one tremendously cold. This turned out to be a mistake, caused by the difficulty of observing the planet when it was close to the Sun and the absence of any surface markings visible in the telescopes then available. In fact, the periods of revolution and rotation of Mercury are 87.97 days and 58.65 days, with a ratio of 1.4999 - a very precise 3:2 resonance.

Astronomers now know that many stars also have planets; indeed, a total of 344 ‘extrasolar’ planets has been found
²⁶
since the first one was detected in 1989. For example, two planets of the star Gliese 876, known as Gliese 876b and Gliese 876c, are in 2:1 resonance. Extrasolar planets are normally detected either by their (tiny) gravitational effects on their parent star, or by changes in the star’s light as and if the planets pass across its face as seen from Earth. But in 2007 the first telescopic image of such a planet was obtained, around a star rejoicing in the name HR8799.
²⁷
The main difficulty here is that the light from the star swamps that of the planet, so various mathematical techniques are used to ‘subtract’ the star’s light. Early in 2009, it was discovered that one of these stars can be extracted, by similar image-processing methods, from a photo of the star taken by the Hubble telescope in 1998, but that’s an aside. The point is that the dynamics of this three-planet system is unstable, so that we would be unlikely to observe it unless the planets are in 4:2:1 resonance. An important consequence of this line of thinking is that such resonances improve the chances of other stable planetary systems existing. Which, perhaps, also improves the prospect of alien life existing somewhere.

A good site for this topic is:

en.wikipedia.org/wiki/Orbital_resonance
which has a long list of ‘accidental’ resonances, more explanation of the dynamics involved, and an animation of Jupiter’s moons’ 1:2:4 resonance. There is also an animation showing how planets cause a star’s position to ‘wobble’ at:
www.gavinrymill.com/dinosaurs/extra-solar-planets.html

What’s special about the number 0588235294117647? (That leading zero does matter here.) Try multiplying it by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, and you’ll see. You do need a calculator or software that works with 16-digit numbers. I find that the human brain, a piece of paper and a pen does that job pretty well.

What happens when you multiply it by 17?

 

Which is bigger: e
^π
or π
^e
?

They are surprisingly close together. Recall that e ≈ 2.71828 and π ≈ 3.14159.

Answer on page 306

They sound like a childhood nightmare, but sums where you never get to the end are among the most important mathematical inventions. Of course, you don’t work them out by doing an infinitely long calculation, but, conceptually, they open up very powerful practical ways to calculate things that mathematicians and scientists want to know.

Back in the 18th century, mathematicians were coming to grips with - or often not coming to grips with - the paradoxical behaviour of infinite sums (or series). They were happy to use sums like
(where the . . . means that the series never stops) and they were also happy that this particular sum is exactly equal to 2. Indeed, if the sum is s, then
so s = 2.