gauss

MATHS BITE: Heptadecagon

A heptadecagon (or a 17-gon) is a seventeen sided polygon.

File:Regular polygon 17 annotated.svg

Regular Heptadecagon | Wikipedia

Constructing the Heptadecagon

In 1796, Gauss proved, at the age of 19 (let that sink in…) that the heptadecagon is constructible with a compass and a straightedge, such as a ruler. His proof of the constructibility of an n-gon relies on two things:

  • the fact that “constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions“;
  • the odd prime factors of n are distinct Fermat primes.

Constructing the regular heptadecagon involves finding the expression for the cosine of  2\pi /17 in terms of square roots, which Gauss gave in his book Disquistiones Arithmeticae:

{\displaystyle {\begin{aligned}16\,\cos {\frac {2\pi }{17}}=&-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+\\&2{\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}\\=&-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+\\&2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}.\end{aligned}}}

Source: Wikipdia

An explicit construction was given by Herbert Willian Richmond in 1893.

Regular Heptadecagon Using Carlyle Circle.gif

Source: Wikipedia

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MATHS BITE: Shoelace Theorem

The Shoelace theorem is a useful formula for finding the area of a polygon when we know the coordinates of its vertices. The formula was described by Meister in 1769, and then by Gauss in 1795.

Formula

Let’s suppose that a polygon P has vertices (a1, b1), (a2, b2), …, (an, bn), in clockwise order. Then the area of P is given by

\[\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)|\]

The name of this theorem comes from the fact that if you were to list the coordinates in a column and mark the pairs to be multiplied, then the image looks like laced-up shoes.

Screen Shot 2017-08-04 at 11.59.29 AM.png

Proof

(Note: this proof is taken from artofproblemsolving.)

Let $\Omega$ be the set of points that belong to the polygon. Then

\[A=\int_{\Omega}\alpha,\]

where $\alpha=dx\wedge dy$.

Note that the volume form $\alpha$ is an exact form since $d\omega=\alpha$, where

\[\omega=\frac{x\,dy}{2}-\frac{y\,dx}{2}.\label{omega}\]

Substitute this in to give us

\[\int_{\Omega}\alpha=\int_{\Omega}d\omega.\]

and then use Stokes’ theorem (a key theorem in vector calculus) to obtain

\[\int_{\Omega}d\omega=\int_{\partial\Omega}\omega.\]

where

$\partial \Omega=\bigcup A(i)$

and $A(i)$ is the line segment from $(x_i,y_i)$ to $(x_{i+1},y_{i+1})$, i.e. Screen Shot 2017-08-04 at 12.05.20 PM.png is the boundary of the polygon.

Next we substitute for $\omega$:

\[\sum_{i=1}^n\int_{A(i)}\omega=\frac{1}{2}\sum_{i=1}^n\int_{A(i)}{x\,dy}-{y\,dx}.\]

Parameterising this expression gives us

\[\frac{1}{2}\sum_{i=1}^n\int_0^1{(x_i+(x_{i+1}-x_i)t)(y_{i+1}-y_i)}-{(y_i+(y_{i+1}-y_i)t)(x_{i+1}-x_i)\,dt}.\]

Then, by integrating this we obtain

\[\frac{1}{2}\sum_{i=1}^n\frac{1}{2}[(x_i+x_{i+1})(y_{i+1}-y_i)- (y_{i}+y_{i+1})(x_{i+1}-x_i)].\]

This then yields, after further manipulation, the shoelace formula:

\[\frac{1}{2}\sum_{i=1}^n(x_iy_{i+1}-x_{i+1}y_i).\]

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Prime Number Theorem

Today I thought I’d quickly discuss a extremely important theorem in one of my favourite areas in mathematics: Number Theory (as you can probably tell by the number of posts that I’ve published about primes!).

Perhaps the first property of π(x) – the number of primes less than or equal to x – is that π(x) tends to infinity as x tends to infinity. In other words, the prime numbers are infinite, which was proved by Euclid in “Elements”. A more precise result, established by Euler in 1737, was that the series of reciprocals of the prime numbers:

Screen Shot 2016-10-19 at 6.09.35 PM.png

is a divergent series. In doing so, Euler found an alternative way to prove that there was an infinite number of primes, as if there wasn’t then the series would have a finite value.

The Prime Number Theorem states that if π(x) is the number of primes less than or equal to x, then

img39.gif

Although the notation ~ may be unfamiliar, it simply means that π(x) is asymptotically equal to x/lnx, i.e.

 img41.gif

Note that the prime number theorem is equivalent to saying that the nth prime number pn satisfies the following relationship:

img44.gif

The PNT was proposed by Gauss in 1792 when he was only 15 years old! (Makes you wonder what you’ve been doing with your life so far…) He later refined this estimate to

\begin{displaymath}\pi(x) \sim \int_2^x \frac{d u}{\ln{u}}.\end{displaymath}

HAPPY HALLOWEEN! M x

Beautiful Equations II

‘Beautiful Equations I’ was one of my favourite posts to write, so I decided to continue with this series and write a part 2.

Euler’s Identity

Leonhard Euler was one of the most influential and prolific mathematicians in history, laying the foundations of an array of areas in mathematics for his successors to build upon. His output was immense; he published more than 500 books and papers during his lifetime and a further 400 appeared posthumously.

065028gett62angangag6g

Euler’s Identity is often considered the most beautiful equation in mathematics as it combines five of the most fundamental mathematical constants:

  • e: the base of natural logarithms
  • i: the imaginary unit of complex numbers, equivalent to the square root of -1
  • π: the ratio of a circle’s circumference to its diameter
  • 1: the multiplicative identity
  • 0: the additive identity

Feynman described it as “the most remarkable formula in mathematics”, and I must say that I completely agree.

So where does this identity come from?

If you’ve studied complex numbers you will know Euler’s relation:

2000px-Euler's_formula.svg

Simply substitute the angle as π!

euler21

For more information on where Euler’s relation comes from, click here.

Boltzmann’s Entropy Formula

As a chemistry student, one of my favourite topics (apart from my beloved organic chemistry) is entropy. Put simply, entropy is the degree of disorder in the system. For a reaction to occur, entropy must always increase.

boltzmann-equation

Boltzmann’s formula relates entropy (S) of an ideal gas and the number of ways that the atoms or molecules can be arranged (k log W). The more ways the particles can be arranged, the greater the disorder and therefore entropy of the system. K is Boltzmann’s constant and W is the number of microscopic elements of a system in a macroscopic system in a state of balance.

Schrödinger Equation

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Edwin Schrödinger’s famous partial differential equation illustrates how subatomic particles change with time when under the influence of a force. Any particular atom or molecule is described by its wave function (represented by the Greek letter psi), which predicts the probability of where and when the particles appear.

However, physicists are still unsure on how to interpret this equation. Some believe that it’s just a useful calculation tool, but does not actually correspond to anything real, whilst others argue that it demonstrates the limit to the amount that we can learn about the universe, as we can only learn about a particle once it’s measured.

Schrödinger believed that the wavefunction represented a real, physical object and rejected the interpretation that a particle only collapses when it’s measured. In fact, his famous cat experiment actually intended to demonstrate the weakness of this interpretation.

The Gaussian Integral

gaussian integral

The function in the Gaussian integral is a very hard function to integrate. However, when analyzed over the whole real line – from minus infinity to infinity – the answer is surprisingly neat. This formula is of extreme use and has a range of applications. For example, it is used to calculate the normalising constant of the normal distribution.

The Analytic Continuation of the Factorial

The factorial function is commonly defined as

factorial-formula

However, this only works for positive integers. Therefore, by using this integral:

analytic continuation of the factorial

mathematicians are able to compute factorials for fractions, decimals, negative numbers and even complex numbers. The gamma function is an extension of this, using n – 1 instead of n.

gamma function

It’s used in various probability-distribution functions, and so highly applicable to probability, statistics and combinatorics.

The Explicit Formula for the Fibonacci Sequence

The Explicit Formula for the Fibonacci Sequence

This formula, derived by Binet in 1843 (although the result was known to Euler, Daniel Bernoulli and de Moivre more than a century earlier) can be used to calculate the nth Fibonacci number in the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, etc., where each number is the sum of the previous two numbers). Although this sequence a classic and known by the vast majority of people, this formula is known to few. Remarkably, despite the formula having square roots and divisions, the answer is always an exact positive integer.

The phi in the formula represents the golden ratio, where

gr value

Two quantities are in the golden ratio when their ratio is the same as the ratio of their sum to the larger of the two quantities. When a > b > 0, this can be expressed as

gr

I personally find the golden ratio a fascinating part of mathematics. Would you like me to do a blog post on this? x