Professor Stewart's Cabinet of Mathematical Curiosities
Triangle of Cards
Turnip for the Books
The Four-Colour Theorem
Shaggy Dog Story
Shaggy Cat Story
Rabbits in the Hat
River Crossing 1 – Farm Produce
More Curious Calculations
Extracting the Cherry
Make Me a Pentagon
What is π?
Legislating the Value of π
If They Had Passed It …
Much Undo About Knotting
To Find Fake Coin
Mathematical Jokes 1
An Age-Old Old-Age Problem
Why Does Minus Times Minus Make Plus?
How to Unmake a Greek Cross
How to Remember a Round Number
The Bridges of Königsberg
How to do Lots of Mathematics
Euler's Pentagonal Holiday
Who Was Pythagoras?
Proofs of Pythagoras
A Constant Bore
Fermat's Last Theorem
A Little-Known Pythagorean Curiosity
Squaring the Square
Squares of Squares
Ring a-Ring a-Ringroad
Pure v. Applied
How Old Was Diophantus?
If You Thought Mathematicians Were Good at Arithmetic …
The Sphinx is a Reptile
Six Degrees of Separation
Duplicating the Cube
Curves of Constant Width
The Stolen Car
The Square Wheel
Why Can't I Divide by Zero?
River Crossing 2 – Marital Mistrust
Wherefore Art Thou Borromeo?
Kinds of People
The Sausage Conjecture
Tom Fool's Knot
What is the Golden Number?
What are the Fibonacci Numbers?
The Plastic Number
Don't Let Go!
The Most Likely Digit
Why Call It a Witch?
Möbius and His Band
Three More Quickies
Miles of Tiles
Why No Nobel for Maths?
Is There a Perfect Cuboid?
When Will My MP3 Player Repeat?
The Poincaré Conjecture
Pig on a Rope
The Surprise Examination
Mathematical Jokes 2
Why Gauss Became a Mathematician
What Shape is a Crescent Moon?
What is a Mersenne Prime?
The Goldbach Conjecture
Turtles All the Way Down
A Puzzling Dissection
A Really Puzzling Dissection
Nothing Up My Sleeve …
Nothing Down My Leg …
Can You Hear the Shape of a Drum?
What is e, and Why?
May Husband and Ay …
Many Knees, Many Seats
What Day is It?
Logical or Not?
A Question of Breeding
The Sixth Deadly Sin
How Deep is the Well?
What is the Square Root of Minus One?
The Most Beautiful Formula
Why is Euler's Beautiful Formula True?
Your Call May be Monitored for Training Purposes
Archimedes, You Old Fraud!
Fractals – The Geometry of Nature
The Missing Symbol
Where There's a Wall, There's a Way
Constants to 50 Places
Are Hard Problems Easy? or How to Win a Million Dollars by Proving the Obvious
Don't Get the Goat
All Triangles are Isosceles
If π isn't a Fraction, How Can You Calculate It?
Let Fate Decide
What Shape is a Rainbow?
The Riemann Hypothesis
Disproof of the Riemann Hypothesis
Murder in the Park
The Cube of Cheese
The Game of Life
Drawing an Ellipse – and More?
Mathematical Jokes 3
The Kepler Problem
The Milk Crate Problem
Spherical Sliced Bread
Professor Stewart's Cunning Crib Sheet
The Kepler Problem // 231
That ought to give some interesting new kinds of curves. So why don't the mathematics books mention them?
Answer on page 304
Mathematical Jokes 3 Two mathematicians in a cocktail bar are arguing about how much maths the ordinary person knows. One thinks they're hopelessly ignorant; the other says that quite a few people actually know a lot about the subject.
`Bet you twenty pounds I'm right,' says the first, as he heads for the gents. While he is gone, his colleague calls the waitress over.
`Listen, there's ten pounds in it for you if you come over when my friend gets back and answer a question. The answer is ``one-third x cubed.'' Got that?'
`Ten pounds for saying ``One thirdex cue?'' '
`No, one-third x cubed.'
`One thir dex cubed?'
`Yeah, that'll do.'
The other mathematician comes back, and the waitress comes over.
`Hey what's the integral of x squared?'
`One third x cubed,' says the waitress. As she walks away, she
........................................... adds, over her shoulder, `Plus a constant.'
The Kepler Problem Mathematicians have learned that apparently simple questions are often hard to answer, and apparently obvious facts may be false, or may be true but extremely hard to prove. The Kepler problem is a case in point: it took nearly three hundred years to 232 // The Kepler Problem
solve it, even though everyone knew the correct answer from the start.
It all began in 1611 when Johannes Kepler, a mathematician and astrologer (yes, he cast horoscopes; lots of mathematicians did at that time it was a quick way to make money) wanted to give his sponsor a New Year's gift. The sponsor rejoiced in the
¨ name Johannes Mathaus Wacker of Wackenfels, and Kepler wanted to say `thanks for all the cash' without actually spending any of it. So he wrote a book, and presented it to his sponsor. Its title (in Latin) was The Six-Cornered Snowflake. Kepler started with the curious shapes of snowflakes, which often form beautiful sixfold symmetric crystals, and asked why this happened.
A typical `dendritic'
It is often said that `no two snowflakes are alike'. The logician in me objects `How can you tell?' but a back-of-the-envelope calculation suggests that there are so many features in a `dendritic' snowflake, of the kind illustrated, that the chance of two being identical is pretty much zero.
No matter. What matters here is that Kepler's analysis of the snowflake led him to the idea that its sixfold symmetry arises because that's the most efficient way to pack circles in a plane.
Take a lot of coins, of the same denomination pennies, say. If you lay them on a table and push them together tightly, you The Kepler Problem // 233
quickly discover that they fit perfectly into a honeycomb pattern, or `hexagonal lattice':
(Left) The closest way to
pack circles, and (right)
a less efficient lattice
And this is the closest packing the one that fills space most efficiently, in the ideal case of infinitely many circles arranged on a plane. Alternatives, such as the square lattice on the right, are less efficient.
Mind you, this innocent assertion wasn't proved until 1940,
´ ´ ´ when Laszlo Fejes Toth managed it. (Axel Thue sketched out a proof in 1892, and gave more details in 1910, but he left some
´ gaps.) Toth's proof was quite hard. Why the difficulty? We don't know, to begin with, that the most efficient packing forms a regular lattice. Maybe something more random could work better. (For finite packings, say inside a square, this can actually happen see the next puzzle, about a milk crate.)
Along the way, Kepler came very close indeed to the idea that all matter is made from tiny indivisible components, which we now call `atoms'. This is impressive, given that he did no experiments in the course of writing his book. Atomic theory, introduced by the Greek Democritus, was not established experimentally until about 1900.
Kepler had his eye on something a bit more complicated, though: the closest way to pack identical spheres in space. He was aware of three regular `lattice' packings, which we now call the hexagonal, cubic and face-centred cubic lattices. The first of these is formed by stacking lots of honeycomb layers of spheres on top of one another, with the centres of corresponding spheres forming a vertical line. The second is made from square-lattice layers, also stacked vertically. For the third, we stack hexagonal layers, but fit the spheres in any given layer into the hollows in the one below. 234 // The Kepler Problem
You can get the same result, though tilted, by similarly stacking square-lattice layers so that the spheres in any given layer fit into the hollows in the one below this isn't entirely obvious, and like the milk crate puzzle it shows that intuition may not be a good guide in this area. The picture shows how this happens: the horizontal layers are square, but the slanting layers are hexagonal.
Part of a face-centred
Now, every greengrocer knows that the way to stack oranges is to use the face-centred cubic lattice.* By thinking about pomegranate seeds, Kepler was led to the casual remark that with the face-centred cubic lattice, `the packing will be the tightest possible'.
That was in 1611. The proof that Kepler was right had to wait until 1998, when Thomas Hales announced that he had achieved this with massive computer assistance. Basically, Hales consid- ered all possible ways to surround a sphere with other spheres, and showed that if the arrangement wasn't the one found in the face-centred cubic lattice, then the spheres could be shoved
´ closer together. Toth's proof in the plane used the same ideas, but he only had to check about forty cases.
Hales had to check thousands, so he rephrased the problem in terms that could be verified by a computer. This led to a huge computation but each step in it is essentially trivial. Almost all of the proof has been checked independently, but a very tiny level of doubt still remains. So Hales has started a new computer-
* They don't say it that way, but they stack it that way.
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