Hiatus

Hi everybody!

This semester for me is the dreaded exam term… Because of this I’ll be going on a hiatus until June.

See you all in 2 months!

M x

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MATHS BITE: Pinwheel Tiling

Pinwheel tiling is a type non-periodic tiling that was defined by American mathematician Charles Radin based on a construction by John Conway. They are the first known non-periodic tilings where the tiles appear in infinitely many orientations.

Conway Tessellation

Given a right angle triangle T with side lengths 1, 2 and {\sqrt {5}}, Conway noticed that we can divide it into five copies of its image by the dilation of factor 1/{\sqrt  {5}}

Source: Wikipedia
By rescaling, translating and rotating, we can iterate this to obtain and infinite increasing sequence of growing triangles. If we take the union of these triangles, we obtain T. It is this increasing sequence of triangles that defines the Conway tiling.

We observe that the triangles appear in infinitely many orientations. (This is because arctan(1/2) and arctan(2), two angles in the triangles, are both not proportionate to \pi ). Extraordinarily, despite this all the vertices have rational coordinates!

Tiling

Based on this tessellation, Radin defined a tiling:

Source: Wikipedia

Architecture

Federation Square, a building complex in Melbourne, Australia has this pinwheel tiling.

Federation Square - Melbourne, Australia

File:Federation-square-sandstone-facade.jpg

M x

Top Moments in Math 2017

As 2018 arrived, I couldn’t help but reflect on the past year in mathematics, some tragic moments along with some glorious feats. I thought I would share what I believe to be the 3 biggest moments of mathematics in the media in 2017.

Hidden Figures

Mathematics hit the mainstream when it featured in a big blockbuster released in December 2016 (although personally I saw it in 2017). Hidden Figures is a biographical film about the black female mathematicians who worked at NASA during the Space Race, including Katherine Johnson – a mathematician who calculated flight trajectories for Project Mercury and other missions. During the promotion of the movie, the films stars appeared on TV programs to talk about the extraordinary women mathematicians featured in the movie. This movie inspired book clubs, schools to bring students to the film, and even a set of LEGOs of some of the featured women. In 2017, NASA honoured Katherine Johnson by “dedicating a building in her name at the space agency’s Langley Research Center in Hampton, Virginia“.

Maryam Mirzakhani: May 3rd 1977 – July 14th 2017

 

This year saw a tragic death in mathematics. An inspiration to many female mathematics, Maryam Mirzakhani – the only woman to win a Fields Medal – died at the age of 40. See my posts about her here and here.

Weapons of Math Destruction

Weapons of Math Destruction is a high-selling book of 2017 written by Cathy O’Neil. When talking in interviews and radio shows, O’Neil expresses how people don’t “understand the mathematical models, algorithms, and scoring systems that impact us in so many ways”, such as college admissions, elections, social networks, financial systems, education, etc. In an interview with EdSurge O’Neil said:

“Algorithms, at the end of the day, are typically scoring systems. As soon as you have a good scoring system, then you can game the scoring system. If you game it enough, it’ll stop making sense. That’s essentially what happened.”

As well as this, O’Neil gave a TED Talk, which I highly recommend, inspired by the issues discussed in her book. Watch the TED talk here.

Happy New Year!

M x

 

 

 

 

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.

M x

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!

M x

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)!

M x