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
146 // Why Gauss Became a Mathematician
hose, and so on, he uses the hose to put out the fire with the minimum expenditure of energy.
Later, the mathematician wakes up and smells smoke. He goes into the hallway and sees a (third) fire. He notices the fire hose on the wall, and thinks for a moment . . . Then he says, `OK,
........................................... a solution exists!' and goes back to bed.
Why Gauss Became a Mathematician
Carl Friedrich Gauss.
Carl Friedrich Gauss was born in Brunswick in 1777 and died
¨ in Gottingen in 1855. His parents were uneducated manual workers, but he became one of the greatest mathematicians ever; many consider him the best. He was precocious he is said to have pointed out a mistake in his father's financial calculations when he was three. At the age of nineteen he had to decide whether to study mathematics or languages, and the decision was made for him when he discovered how to construct a regular 17-sided polygon using the traditional Euclidean tools of an unmarked ruler and a compass.
This may not sound like much, but it was totally unprece- dented, and the discovery led to a new branch of number theory. Euclid's Elements contains constructions for regular polygons (all Why Gauss Became a Mathematician // 147
sides equal length, all angles equal) with 3, 4, 5, 6 and 15 sides, and the ancient Greeks knew that the number of sides could be doubled as often as you wish. Up to 100, the number of sides in a constructible (regular) polygon as far as the Greeks knew must be
2; 3; 4; 5; 6; 8; 10; 12; 15; 16; 20; 24; 30; 32; 40;
48; 60; 64; 80; 96
For more than two thousand years, everyone assumed that no other polygons were constructible. In particular, Euclid does not tell us how to construct 7-gons or 9-gons, and the reason is that he had no idea how this might be done. Gauss's discovery was a bombshell, adding 17, 34 and 68 to the list. Even more amazingly, his methods prove that other numbers, such as 7, 9, 11 and 13, are impossible. (The polygons do exist, but you can't construct them by Euclidean methods.)
Gauss's construction depends on two simple facts about the number 17: it is prime, and it is one greater than a power of 2. The whole problem pretty much reduces to finding which prime numbers correspond to constructible polygons, and powers of 2 come into the story because every Euclidean construction boils down to taking a series of square roots which in particular implies that the lengths of any lines that feature in the construction must satisfy algebraic equations whose degree is a power of two. The key equation for the 17-gon is
x16 þ x15 þ x14 þ x13 þ x12 þ x11 þ x10 þ x9 þ x8 þ x7 þ x6
þ x5 þ x4 þ x3 þ x2 þ x þ 1 ¼ 0
where x is a complex number. The 16 solutions, together with the number 1, form the vertices of a regular 17-gon in the complex plane. Since 16 is a power of 2, Gauss realised that he was in with a chance. He did some clever calculations, and proved that the 148 // Why Gauss Became a Mathematician
17-gon can be constructed provided you can construct a line whose length is
1 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi
À 1 þ 17 þ 34 À 2 17 þ
pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi ffi
68 þ 12 17 À 16 34 þ 2 17 À 2ð1 À 17Þð 34 À 2 17Þ
Since you can always construct square roots, this effectively solves the problem, and Gauss didn't bother to describe the precise steps needed the formula itself does that. Later, other mathematicians wrote down explicit constructions. Ulrich von Huguenin published the first in 1803, and H.W. Richmond found a simpler one in 1893.
Richmond's method for constructing a regular 17-gon. Take two
perpendicular radii, AOP0 and BOC, of a circle. Make 4OB ¼ 1
and angle 4OJP0 ¼ 1. Find F Such that angle EJF is 458. Draw a
circle with FP0 as diameter, meeting OB at K. Draw the circle
with centre E through K, cutting AP0 in G and H. Draw HP3 and
GP5 perpendicular to AP0. Then P0, P3 and P5 are respectively
the 0th, 3rd and 5th vertices of a regular 17-gon, and the other
vertices are now easily constructed.
Gauss's method proves that a regular n-gon can be con- structed whenever n is a prime of the form 2k þ 1. Primes like this are called Fermat primes, because Fermat investigated them. In Why Gauss Became a Mathematician // 149
particular he noticed that k must itself be a power of 2 if 2k þ 1 is going to be prime. The values k = 1, 2, 4, 8 and 16 yield the Fermat primes 3, 5, 17, 257 and 65,537. However, 232 þ 1 ¼ 4;294;967;297 ¼ 64166;700;417 is not prime. Gauss was aware that the regular n-gon is constructible if and only if n is a power of 2, or a power of 2 multiplied by distinct Fermat primes. But he didn't give a complete proof probably because to him it was obvious.
His results prove that it is impossible to construct regular 7-, 11- or 13-gons by Euclidean methods, because these are prime but not of Fermat type. The analogous equation for the 7-gon, for instance, is x6 þ x5 þ x4 þ x3 þ x2 þ x þ 1 ¼ 0, and that has degree 6, which is not a power of 2. The 9-gon is not constructible because 9 is not a product of distinct Fermat primes it is 363, and 3 is a Fermat prime, but the same prime occurs twice here.
The Fermat primes just listed are the only known ones. If there is another, it must be absolutely gigantic: in the current state of knowledge the first candidate is 233;554;432 þ 1, where 33;554;432 ¼ 225 . Although we're still not sure exactly which regular polygons are constructible, the only obstacle is the possible existence of very large Fermat primes. A useful website for Fermat primes is mathworld.wolfram.com/FermatNumber.html
In 1832 Friedrich Julius Richelot published a construction for the regular 257-gon. Johann Gustav Hermes of Lingen University devoted ten years to the 65,537-gon, and his unpublished work
¨ can be found at the University of Gottingen, but it probably contains errors.
With more general construction techniques, other numbers are possible. If you use a gadget for trisecting angles, then the 9-gon is easy. The 7-gon turns out to be possible too, but that's
........................................... nowhere near as obvious.
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