Professor Stewart's Hoard of Mathematical Treasures Read Online Free PDF Page B
Author: Ian Stewart
Tags: General, Mathematics
lengths may change as the polyhedron flexes. Somehow the formula has to be tinkered with to get rid of these unwanted edges.
A heroic calculation led to the amazing conclusion that such a formula does exist for an octahedron - a solid with eight triangular faces. It involves the 16th power of the volume, not the square. By 1996, Sabitov had found a way to do the same for any polyhedron, but it was very complicated, which may have been why the great mathematicians of earlier times had missed it. In 1997, however, Connelly, Sabitov and Walz found a far simpler approach, and the bellows conjecture became a theorem.
Same edges, different volumes.
I’d better point out that the existence of this formula does not imply that the volume of a polyhedron is uniquely determined by the lengths of its edges. A house with a roof has a smaller volume if you turn the roof upside down. These are two
different solutions of the same polynomial equation, and that causes no problems in the proof of the bellows conjecture - you can’t flex the roof into the downward position without bending something.
Digital Cubes
The number 153 is equal to the sum of the cubes of its digits:
1 3 + 5 3 + 3 3 = 1 + 125 + 27 = 153
There are three other 3-digit numbers with the same property, excluding numbers like 001 with a leading zero. Can you find them?
Answer on page 283
Nothing Which Appeals Much to a Mathematician
In his celebrated book A Mathematician’s Apology of 1940, the English mathematician Godfrey Harold Hardy had this to say about the digital cubes puzzle:
‘This is an odd fact, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in it which appeals much to a mathematician ... One reason ... is the extreme speciality of both the enunciation and the proof, which is not capable of any significant generalisation.’
In his 1962 book Profiles of the Future, Arthur C. Clarke stated three laws about prediction. The first is:
• When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.
This is called Clarke’s first law, or often just Clarke’s law, and there are good reasons to claim that it applies to Hardy’s statement. To be fair, the point Hardy was trying to make is a good one, but you can pretty much guarantee that, whenever anyone cites a specific example to drive such an argument home, it will turn out to be a bad choice. In 2007, a trio of mathematicians - Alf van der Poorten, Kurth Thomsen and Mark Weibe - took an imaginative look at Hardy’s assertion. Here’s what they found.
It was all triggered by a ‘cute observation’ made by the number-theorist Hendrik Lenstra:
12 2 + 33 2 = 1, 233
This is about squares, not cubes, but it hints that maybe there is more to this sort of question than first meets the eye. Suppose that a and b are 2-digit numbers, and that
a 2 + b 2 = 100a + b
which is what you get by stringing the digits of a and b together. Then some algebra shows that
(100 - 2a) 2 + (2b - 1) 2 = 10,001
So we can find a and b by splitting 10,001 into a sum of two squares. There is an easy way:
10,001 = 100 2 + 1 2
But 100 has three digits, not two. However, there is also a less obvious way:
10,001 = 76 2 + 65 2
So 100 - 2a = 76 and 2b - 1 = 65. Therefore a = 12 and b = 33, which leads to Lenstra’s observation.
A second solution is hidden here, because we could take 2a - 100 = 76 instead. Now a = 88, and we discover that
88 2 + 33 2 = 8,833
Similar examples can be found by splitting numbers like 1,000,001 or 100,000,001 into a sum of squares. Number theorists know a general technique for this, based on the prime factors of those numbers. After a lot of detail that I won’t go into here, this leads to things like
588 2 + 2,353 2 = 5,882,353
This is all very well, but what about cubes? Most mathematicians would probably guess that 153 is a
A heroic calculation led to the amazing conclusion that such a formula does exist for an octahedron - a solid with eight triangular faces. It involves the 16th power of the volume, not the square. By 1996, Sabitov had found a way to do the same for any polyhedron, but it was very complicated, which may have been why the great mathematicians of earlier times had missed it. In 1997, however, Connelly, Sabitov and Walz found a far simpler approach, and the bellows conjecture became a theorem.
Same edges, different volumes.
I’d better point out that the existence of this formula does not imply that the volume of a polyhedron is uniquely determined by the lengths of its edges. A house with a roof has a smaller volume if you turn the roof upside down. These are two
different solutions of the same polynomial equation, and that causes no problems in the proof of the bellows conjecture - you can’t flex the roof into the downward position without bending something.
Digital Cubes
The number 153 is equal to the sum of the cubes of its digits:
1 3 + 5 3 + 3 3 = 1 + 125 + 27 = 153
There are three other 3-digit numbers with the same property, excluding numbers like 001 with a leading zero. Can you find them?
Answer on page 283
Nothing Which Appeals Much to a Mathematician
In his celebrated book A Mathematician’s Apology of 1940, the English mathematician Godfrey Harold Hardy had this to say about the digital cubes puzzle:
‘This is an odd fact, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in it which appeals much to a mathematician ... One reason ... is the extreme speciality of both the enunciation and the proof, which is not capable of any significant generalisation.’
In his 1962 book Profiles of the Future, Arthur C. Clarke stated three laws about prediction. The first is:
• When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.
This is called Clarke’s first law, or often just Clarke’s law, and there are good reasons to claim that it applies to Hardy’s statement. To be fair, the point Hardy was trying to make is a good one, but you can pretty much guarantee that, whenever anyone cites a specific example to drive such an argument home, it will turn out to be a bad choice. In 2007, a trio of mathematicians - Alf van der Poorten, Kurth Thomsen and Mark Weibe - took an imaginative look at Hardy’s assertion. Here’s what they found.
It was all triggered by a ‘cute observation’ made by the number-theorist Hendrik Lenstra:
12 2 + 33 2 = 1, 233
This is about squares, not cubes, but it hints that maybe there is more to this sort of question than first meets the eye. Suppose that a and b are 2-digit numbers, and that
a 2 + b 2 = 100a + b
which is what you get by stringing the digits of a and b together. Then some algebra shows that
(100 - 2a) 2 + (2b - 1) 2 = 10,001
So we can find a and b by splitting 10,001 into a sum of two squares. There is an easy way:
10,001 = 100 2 + 1 2
But 100 has three digits, not two. However, there is also a less obvious way:
10,001 = 76 2 + 65 2
So 100 - 2a = 76 and 2b - 1 = 65. Therefore a = 12 and b = 33, which leads to Lenstra’s observation.
A second solution is hidden here, because we could take 2a - 100 = 76 instead. Now a = 88, and we discover that
88 2 + 33 2 = 8,833
Similar examples can be found by splitting numbers like 1,000,001 or 100,000,001 into a sum of squares. Number theorists know a general technique for this, based on the prime factors of those numbers. After a lot of detail that I won’t go into here, this leads to things like
588 2 + 2,353 2 = 5,882,353
This is all very well, but what about cubes? Most mathematicians would probably guess that 153 is a
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