Influential Mathematicians: Gauss (1)

I decided to start a new series on influential mathematicians, starting with Gauss, one of my personal favourites. Carl Friedrich Gauss (1777-1855) is considered to be one of the greatest mathematicians in the 19th century, and is sometimes referred to as the “Prince of Mathematics”.

His discoveries influenced and left a lasting mark in a variety of different areas, including number theory, astronomy, geodesy, and physics, particularly the study of electromagnetism.

Born in Brunswick, Germany to poor, working class parents, he was discouraged from attending school from his father, a gardner and brick-layer, who expected Gauss would follow one of the family trades. However, Gauss’ mother and uncle recognised Gauss’ early genius and knew he must develop this gift with a proper education.

In arithmetic class, at the age of 10, Gauss showed his skills as a maths prodigy. A well known anecdote about Gauss and his early school education is about when the strict schoolmaster gave the following assignment:

“Write down all the whole numbers from 1 to 100 and add up their sum.”

They expected this assignment to take a while to complete but after a few seconds, to the teacher’s surprise, Carl placed his slate on the desk in front of the teacher, showing he was done with the question. His other classmates took a much longer time to complete the assignment. At the end of class time, although most other students answers were wrong, Gauss’ was correct: 5050. Carl then explained to the teacher that he found the result as he could see that 1+100 = 101, 2+99=101, etc. So he could find 50 pairs of numbers that each add up to 101, and so 50*101 = 5050. I don’t know about you but I definitely could not come up with this sort of argument at the age of 10…

Although his family was poor, Gauss’ intellectual abilities drew the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum at the age of 15, and then to the University of Göttingen – a very prestigious university – where he stayed from 1795 to 1798. During this period, Gauss discovered many important theorems.

Prime Numbers

No pattern had previously been found in the occurrence of prime numbers until Gauss. Although the occurrence of the primes seems to be completely random, by approaching the problem from a different angle and graphing the incidence of primes as the numbers increased, he noticed a rough trend: as numbers increased by 10, the probability of the numbers reduced by a factor of around 2. However, as his method only gave him an approximation, and as he could not definitively prove his findings, he kept them a secret until much later in his life.

Graphs of the density of prime numbers

1796

1796 is known as Gauss’ “annus mirabilis” (means “wonderful year” and is used to refer to several years during which events of major importance are remembered).  In 1796:

  • Gauss constructed a regular 17-sided heptadecagon, which had previously been unknown, using only a ruler and a compass. This was a major advance in geometry since the time of the Greeks.
  • Gauss formulated this prime number theorem on the distribution of prime numbers among the integers, which states that \displaystyle \lim_{n\rightarrow\infty}\left[ \frac{\pi(n)}{n/\log n} \right] = 1 \,.  Here {\pi(n)} is the number of primes less than or equal to n. We can also write {\pi(n) \sim n/\log n}.
  •  Gauss proved that every positive integer can be represented as the sum of at most 3 triangular number

More about Gauss in the next post!

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MATHS BITE: Brouwer Fixed Point Theorem

The Brouwer Fixed Point Theorem is a result in topology that has proven to be extremely useful in mathematics.

2D

Suppose you take two sheets of paper with one lying directly above the other. If you crumple the top sheet and place it on top of the other sheet, then Brouwer’s theorem states that there is at least one point that has not moved, i.e. there is at least one point on the top sheet that is directly above the corresponding point on the bottom sheet.

3D

Take a cup of coffee, and mix it around. The theorem states that after mixing there must be some point in the coffee, which is in the exact spot that it was before the mixing. Furthermore, if you try to move that point out of its original position then you will inevitably move another point into its original position.

Formally

A continuous function from an n-ball into an n-ball must have a fixed point. Note that continuity of the function is essential.

Click here for the proof.

 

Rings: The Basics

Last year, on first learning about groups I created two blog posts introducing the group axioms and subgroups. This year, I was introduced to rings, so I thought I’d create a blog post on these!

What is a ring?

ring is a quintuple (R, +, *, 0R, 1R) where 0R, 1R ∈ R and +, *: R x R –> R are binary operations such that:

  1. (R, +,0R) is an abelian group
  2. The operation *: RxR –> R satisfies:
    • associativity –  a*(b*c) = (a*b)*c;
    • identity – 1R*r = r*1R = r.
  3. Multiplication distributes over addition
    • r1 * (r2 + r3) = (r1 * r2) + (r1 * r3)
    • (r1 + r2) * r3  = (r1 * r3) + (r2 * r3)

We say that a ring is commutative if a*b = b*a for all a,b ∈ R. These types of rings are much easier to study.

Subring

Let (R, +, *, 0R, 1R) be a ring and S ⊆ R is a subset. We say that S is a subring of R if 0R, 1R ∈ S and the operations +,* make S into a ring in its own right. We normally denote this by S ≤ R.

Examples

  • The familiar number systems are all rings: Z ≤ Q ≤ R ≤ C, under the usual 0, 1, +, *.
  • Q[√2] = {a + b√2 ∈ R: a,b ∈ Q} ≤ R.
  • The set Z[i] = {a + bi: a, b ∈ Z} ≤ C is the Gaussian integers, which is a ring.
gaussian-and-eisenstein
Gaussian Integer Lattice | Source: cp4space.wordpress.com

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Maths in Buildings?

Maths and architecture are very closely linked. In today’s blog post I thought I would talk about some famous buildings and describe how maths is related to them.

The Great Pyramid of Giza – Cairo, Egypt

800px-Kheops-Pyramid.jpg

The Great Pyramid of Giza, one of the Seven Wonders of the Ancient World, is the largest and the oldest of the three pyramids and was the tallest man-made structure in the world for 3,800 years.

In cubits (the first recorded unit of length) the pyramid’s perimeter is 365.24, which is the number of days in the year! Also, if you divide the pyramid’s perimeter by twice its height, you will get pi! The last amazing fact is that the measurements in the King’s chamber are based on the famous Pythagorean triangle 3, 4, 5.

Taj Mahal – Agra, India

The main mathematical feature of the Taj Mahal is its impeccable symmetry. It is symmetric about a vertical line down the middle, as well as a horizontal line along the waterline, as one can see its reflection in the water.

The Eden Project – Cornwall, UK

800px-Eden_project.jpg

The Eden project opened in 2001 and is composed of two biomes containing plants from many different climates and environments.

The greenhouses are geodesic domes made up of hexagonal and pentagonal cells.

Furthermore, in 2005 an education centre was built. This building draws inspiration from plants, using Fibonacci numbers to reflect the nature onsite. Also, the structure of is derived from phyllotaxis which is the mathematical basis for most plant growth – a pattern of opposing spirals.

Parthenon – Athens, Greece

800px-The_Parthenon_in_Athens.jpg

The Parthenon was constructed in 430/440 BC on the Ancient Greek ideals of harmony, which is shown in the perfect proportions of the building, for example the width to height ratio of 9:4, the spacing of the columns, etc.

In the Ancient Greek’s quest for beauty, they wanted to make their columns appear completely straight. However, to achieve this they made their columns slightly thicker in the middle due to the fact that if they had not, an optical illusion would make them seem thinner in the middle.

In addition, it’s been suggested that the Parthenon’s proportions are based on the Golden Ratio.

Chichen Itza – Mexico

800px-Chichen_Itza_3.jpg

The Mayan civilisation  were incredible mathematicians. The structure shown above, called El Castillo, within Chichen Itza is based on the astrological system:

  • The 52 panels on each side of the pyramid represent the number of years in the Mayan cycle;
  • The stairways dividing the 18 tiers correspond to the Mayan calendar of 18 months;
  • The steps within El Castillo mirror the solar year, with a total of 365 steps i.e. 1 step for each day of the year.

Sagrada Familia – Barcelona, Spain

800px-Sagrada_Familia_01.jpg

This structure was designed by Antoni Gaudi and is one of Spain’s top tourist destinations. Gaudi used hyperbolic paraboloid structures, which can be seen within particular façades.

This quadratic and doubly ruled surface given by the Cartesian equation:

 z=(y^2)/(b^2)-(x^2)/(a^2)

What does this look like? Pringles resemble a hyperbolic paraboloid!

Additionally, the Sagrada Familia features a magic square within the Passion façade. This is an arrangement where the number in all columns, rows and diagonals add up to the same sum, which in this case is 33. The magic constant M is the constant sum in every row, column and diagonal and can be represented by the following formula:

M = n(n2+1)/2

(All pictures from Wikipedia)

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F.T.A. via Complex Analysis

Although this requires a bit of knowledge on Complex Anlaysis, I recently discovered this new way to prove the Fundamental Theorem of Algebra and I couldn’t help but share it.

First of all, what is the Fundamental Theorem of Algebra (FTA)? This very important (hence the name!) result states that:

Every non-constant polynomial with complex coefficients has a complex root. 

In order to prove this, we must first be aware of Liouville’s Theorem:

Every bounded, entire function is constant. 


Definitions

Bounded: a function on a set X is said to be bounded if there exists a real number M such that

|f(x)|\le M

for all x in X.

Entire: An entire function is a holomorphic function on the entire complex plane.


Liouville’s theorem is proved using the Cauchy integral formula for a disc, one of the most important results in Complex Analysis. Although I will not describe how to prove it or what it states in this blog post, I encourage you to read about here it as it is truly a remarkable result.

Now armed with Liouville’s Theorem we can prove the FTA.

Proof

Let P(z) = zn + cn-1zn-1 + … + c1z + c0 be a polynomial of degree n > 0. Then |P(z)| –> ∞ as |z| –> ∞, so there exists R such that |P(z)| > 1 for all z with |z| > R.

Consider f(z) = 1/P(z). If P has no complex zeros then f is entire. So, as f is continuous, f is bounded on {|z| ≤ R}.

As |f(z)| < 1 when |z| > R, f is a bounded entire function, so by Liouville’s Theorem f is constant, which is a contradiction.

The only thing we assumed was that P had no complex zeros, and so we contradicted this fact. Hence, P must have at least one complex zero. Amazing right!

 

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Catalan’s Conjecture

Although popularly known as Catalan’s conjecture, this is in fact a theorem in number theory, proven by Preda Mihăilescu in 2002, 158 years after it was conjecture in 1844 by French and Belgian mathematician Eugène Charles Catalan.

The theorem states that the only solution in the natural numbers of

{\displaystyle x^{a}-y^{b}=1}

for a,b>1; x,y>0 is x = 3, a=2, y=2, b=3.

Catalan ‘Near Misses’ (i.e. xa – yb <= 10; 2 <= x,y,a,b <= 100; a,b prime)

  • 33 – 52 = 2
  • 27 – 53 = 3
  • 23 – 22 = 62 – 25 = 53 – 112 = 4
  • 25 – 33 = 5
  • 25 – 52 = 42 – 32 = 27 – 112 = 7
  • 42 – 23 = 8
  • 52 – 42 = 62 – 33 = 152 – 63 = 9
  • 133 – 37 = 10

Proving the Conjecture

Similarly to Fermat’s last theorem, the solution to this conjecture was assembled over a long period of time. In 1850, Victor Lebesgue established that b cannot equal 2. But then it was only after around 100 years, in 1960, that Chao Ko constructed a proof for the other quadratic case: a cannot equal 2 unless x = 3.

Hence, this left a,b odd primes. Expressing the equation as (x-1)(xa-1)/(x-1) = yb one can show that the greatest common divisor of the two left hand factors is either 1 or a. The case where the gcd = 1 was eliminated by J.W.S Cassels in 1960. Hence, only the case where gcd = a remained.

It was this “last, formidable, hurdle” that Mihailescu surmounted. In 2000, he showed that a and b would have to be a ‘Wieferich pair’, i.e. they would have to satisfy ab-1 ≡ 1 (mod b2) and ba-1 ≡ 1 (mod a2). In 2002,  he showed that such solutions were impossible!

Video

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Algorithms: PART 2

Read part 1 here!

QR Algorithm

From 1959-1961, John G.F Francis worked on finding a stable method to compute eigenvalues and thus created the QR algorithm. Eigenvalues are one of the most essential numbers associated with matrices, however they can be quite difficult to compute.

This algorithm is based on the fact that is relatively easy to transform a square matrix into a matrix that is almost upper triangular (one extra set of no-zero entries just below the main diagonal). Based on QR decomposition, which writes a matrix A as a product of an orthogonal matrix Q and an upper triangular matrix R, the QR algorithm iteratively changes Ai = QiRi to Ai+1RiQi

“The algorithm is numerically stable because it proceeds by orthogonal similarity transforms.”

-Wikipedia

Under certain conditions, the matrices Ai converge to a triangular matrix and the eigenvalues of a triangular matrix are listed on the diagonal so the eigenvalue problem is solved.

By the mid-1960s, the QR algorithm had turned once-formidable eigenvalue problems into routine calculations.

Quicksort

The Quicksort algorithm was developed by Tony Hoare in 1962. It puts N things in numerical or alphabetical order by using a recursive strategy:

  1. Pick one element as a pivot
  2. Separate the rest into piles of big and small elements, as compared with the pivot
  3. Repeat this procedure on each pile

Although you can get stuck doing all N(N-1)/2 comparisons, on average Quicksort runs on average with O(N logN) efficiency, which is very fast. Its simplicity has made Quicksort a “poster child of computational complexity“.

Fast Fourier Transform

The fast Fourier Transform was developed by James Cooley and John Tukey in 1965. The FFT revolutionised signal processing. Although the idea can be traced back to Gauss, it was a paper by Cooley and Tukey that made it clear how easily Fourier transforms can be computed. It relies on a ‘divide-and-conquer‘ strategy and hence reduces it from an O(N^2) to an O(N logN) computation.

Integer Relation Detection Algorithm

Given real numbers x1, …, xn, are there integers a1, …, an (not all 0) for which a1x1 + … + anxn = 0? Helaman Ferguson and Rodney Forcade found an algorithm – the Integer Relation Detection Algorithm – to answer this question. This is easily solved in the case where n = 2 by Euclid’s algorithm, computing terms in the continued-fraction expansion of x1/x2 – if x1/x2 is rational, then the expansion terminates.

The detection algorithm has, for example, been used to find the precise coefficients of the polynomials satisfied by the third and fourth bifurcation points of the logistic map.

Logistic Map | Source: geoffboeing.com

It has also proved useful in simplifying calculations with Feynman diagrams in quantum field theory.

Fast Multipole Algorithm

This algorithm overcomes one of the biggest problems of N-body simulations, which is the fact that accurate calculations of the motions of N particles interaction via gravitational or electrostatic forces seemingly requires O(N^2) calculations, i.e. one for each pair of particles. This algorithm, developed in 1987 by Leslie Greengard and Vladimir Rokhlin, does it with O(N) computations.

How does it do this? It uses multipole expansions to approximate the effects of a distant group of particles on a local group. Then we define ever-larger groups as distances increase. One of the big advantages of the fast multipole algorithm is that it comes with rigorous error estimates, a feature that a lot of other methods lack.

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