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How to Solve It

In 1945, Hungarian mathematician George Pólya wrote an extremely successful book called How to Solve It, which sold over one million copies and has been translated into 17 languages. In this book, Pólya identifies the four basic principles of problem solving.

1. Understand the Problem

Firstly, you must understand the problem you are tackling.  Although this may seem like common sense, it is often overlooked; the amount of times I’ve been staring at a problem for hours because I haven’t fully understood what it is asking! Consider asking the following questions:

  • What is the unknown? What are the data? What is the condition?
  • What are you asked to find or show?
  • Can you restate the problem in your own words?
  • Do you understand all the words used in stating the problem?

For some areas in maths, such as mechanics, drawing a labelled figure can often be extremely helpful to visualise what is going on.

Resultado de imagem para free body diagram

Example of Diagram


2. Devise a Plan

There are many different suitable ways to solve a problem, as mentioned by Pólya. However, it is important to choose an appropriate strategy, which is a skill learnt by solving many problems. Some strategies include:

  • Guess and check
  • Use symmetry
  • Consider special cases
  • Solve an equation or use a formula
  • Look for a pattern
  • Solve a simpler problem
  • Work backwards
  • Eliminate possibilities

When choosing a strategy, it may help to consider whether you have solved a related problem already or whether you can think of a familiar problem that has the same or similar unknown.


3. Carry out the Plan

Once you have devised the plan, this step is simple if you already have the necessary skills, often only required care and patience -try to avoid silly mistakes!

Persist with the plan that you have chosen, only discarding it after multiple failed attempts. Then, return to step 2.

4. Look back

Pólya highlights how a lot can be gained from looking back and reflecting on the work you have done, asking yourself what worked and what didn’t. This way you can implement strategies that were successful for future problems that are similar.


Read more here.

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Where have I been?

So I have a confession to make… I have completely neglected my blog for the past month. Although it really upset me that I couldn’t upload regular (or in fact any) content, I have just been so busy with work that it was impossible. In these last few months I have been in complete exam mode, and although it’s been absolutely exhausting, it has also been extremely rewarding.

But now, I am finally finished and can concentrate on uploading more regularly. Thank you all so much for your patience, new blog posts coming soon!

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12th Polymath Project

The Polymath Project is a collaboration among mathematicians to solve important problem in mathematics by providing a platform for mathematicians to communicate with each other on how to find the best route to the solution.

It began in January 2009 when Tim Gowers posted a problem on his blog and asked readers to reply with partial ideas or answers. This experiment resulted in a new answer to a difficult problem, proving the benefits of collaboration.

Previous Polymath projects that have successfully led to proofs incude the density version of the Hales-Jewett theorem and the Erdös discrepancy problem, as well as famously reducing the bound on the smallest gaps between primes.

Recently the 12th Polymath Project has started; Timothy Chow of MIT has proposed a new Polymath Project – resolve Rota’s basis conjecture.

What is the Rota’s Basis Conjecture?

The Rota’s Basis Conjecture states:

“If B1, B2,…., Bn are n bases of an n-dimensional vector space V (not necessarily distinct or disjoint), then there exists an n x n grid of vectors (vij) such that:

  1. the n vectors in row i are the members of the ith basis Bi (in some order), and
  2. in each column of the matrix, the n vectors in that column form a basis of V.”

Although easy to state, this conjecture has revealed itself hard to prove (like Fermat’s Last Theorem)!

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Intrinsic Geometry

Today I wanted to discuss the geometry of curves and surfaces.

Curves, Curvature and Normals

First let us consider a curve r(s) which is parameterised by s, the arc length.

Now, t(s) = Screen Shot 2017-02-14 at 8.05.47 AM.png is a unit tangent vector and so t2 = 1, thus t.t = 1. If we differentiate this, we get that t.t‘ = 0, which specifies a direction normal to the curve, provided t‘ is not equal to zero. This is because if the dot product of two vectors is zero, then those two vectors are perpendicular to each other.

Let us define t’ = Kwhere the unit vector n(s) is called the principal normal and K(s) is called the curvature. Note that we can always make K positive by choosing an appropriate direction for n.

Another interesting quantity is the radius of curvature, a, which is given by

a = 1/curvature

Now that we have n and t we can define a new vector x n, which is orthonormal to both t and n. This is called the binormal. Using this, we can then examine the torsion of the curve, which is given by

T(s) = –b’.n

Image result for binormal tangent

Intrinsic Geometry

As the plane is rotated about n we can find a range

Screen Shot 2017-02-14 at 8.16.01 AM.pngwhereScreen Shot 2017-02-14 at 8.16.08 AM.png and Screen Shot 2017-02-14 at 8.16.14 AM.png are the principal curvatures. Then

Screen Shot 2017-02-14 at 8.17.34 AM.png

is called the Gaussian curvature.

Gauss’ Theorema Egregium (which literally translates to ‘Remarkable Theorem’!) says that K is intrinsic to the surface. This means that it can be expressed in terms of lengths, angles, etc. which are measured entirely on the surface!

For example, consider a geodesic triangle on a surface S.

Screen Shot 2017-02-14 at 8.20.37 AM.png

Let θ1, θ2, θ3 be the interior angles. Then the Gauss-Bonnet theorem tells us that

Screen Shot 2017-02-14 at 8.22.10 AM.png

which generalises the angle sum of a triangle to curved space.

Let us check this when S is a sphere of radius a, for which the geodesics are great circles. We can see that Screen Shot 2017-02-14 at 8.16.08 AM.png=Screen Shot 2017-02-14 at 8.16.14 AM.png= 1/a, and so K = 1/a2, a constant. As shown below, we have a family of geodesic triangles D with θ1 = α, θ2 = θ3 = π/2.

screen-shot-2017-02-14-at-8-26-15-am

Since K is constant over S,

Screen Shot 2017-02-14 at 8.27.09 AM.png

Then θ1 + θ2 + θ3 = π + α, agreeing with the prediction of the theorem.

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VIDEO: Banach–Tarski Paradox

The Banach-Tarski Paradox is a theorem in geometry which states that:

“It is possible to decompose a ball into five pieces which can be reassembled by rigid motions to form two balls of the same size as the original.”

It was first stated in 1924, and is called a paradox as it contradicts basic geometric intuition.

An alternate version of this theorem tells us that:

“It is possible to take a solid ball the size of a pea, and by cutting it into a finite number of pieces, reassemble it to form a solid ball the size of the sun.”

Below is an awesome video explaining how this paradox works:

 

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Mandelbrot and Julia Sets

In today’s post I wanted to quickly highlight a cool relationship between Mandelbrot and Julia sets.

Consider the function, which depends of complex parameter z:

f(z) = x2 z 

Fixing this z, f(z) defines a map from the complex plane to itself. We can start from any value of x and apply this function over and over, which would give us a sequence of numbers. This sequence can either go off to infinity, or not. The boundary of the set of values of x where it doesn’t is the Julia set for this particular z, which we fixed initially.

Conversely, starting with x = 0, we can draw the set of numbers for which the resulting sequence does not go off to infinity. This is called the Mandelbrot set. (Note the subtle difference between the two).

Okay, so the cool relationship is that, near the number z, the Mandelbrot looks like the Julia set for the number z, or as Wikipedia describes:

“There is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set.”

To illustrate this, consider the following Julia set:

relationship_between_mandelbrot_sets_and_julia_sets-1

Zooming into the Mandelbrot set at the same value of z gives us this image:

Relationship_between_Mandelbrot_sets_and_Julia_sets.PNG

They are extremely similar! So, essentially, the Mandelbrot sets looks like a lot of Julia sets! (Click here to explore this in more detail).

This amazing result is used in lots of results on the Mandelbrot set, for example, it was exploited by Shishikura to prove that “for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff dimension two, and then transfers this information to the parameter plane“.

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