Magical, mind-bending Mathematics

Mathematics expert Dr Katie Steckles gave QE boys plenty to puzzle over and enjoy during an entertaining, interactive talk.

An accomplished communicator who works with the organisation, Think Maths, she was invited in to School to speak to the whole of Year 10 – the seventh time a Thinks Maths speaker has visited QE, but the first such visit from Dr Steckles.

Since finishing her PhD in 2011, she has talked about Mathematics in schools, at science and music festivals, on BBC radio, in theatre shows and on the internet. In 2016, Dr Steckles was awarded the Joshua Phillips Award for Innovation in Science Engagement, a national award established to recognise and support rising talent in science communication, which resulted in her becoming science communicator-in-residence at the Manchester Science Festival 2016.

Entitled Shapes and Smoke Rings, her QE lecture began with a puzzle: given these shapes (pictured), could the boys draw a line to connect the matching shapes without the lines crossing over each other or going outside the rectangle?  The top three shapes are touching the top of the rectangle and the lines do not have to be straight, Dr Steckles said.

She then set the boys the challenge of moving their arm from a position where the palm faced downwards to one where it faced upwards without moving their elbow. Assistant Head of Mathematics Wendy Fung said: “After a little experimentation, the boys were given a clue of ‘use your elbow’!”

This led on to Dr Steckles’ third-favourite shape, the sphere. Spheres, she said, are ubiquitous – seen in objects ranging from basketballs to the dengue fever and ebola viruses – yet they are strange: on a sphere, it is possible to create a triangle whose angles add up to more than 180 degrees.

She chose pupil Aryan Jain to go up to the stage where he was instructed to draw some dots on a balloon and join up any combination of the dots using some lines. He had to count the number of dots, the number of lines and the number of regions, and use these values in a formula which she gave him. Next he had to convert the resulting number to a letter using A = 1, B = 2, C = 3, etc. and write down a fruit which begins with that letter. Aryan came up with ‘banana’ just as Dr Steckles was eating one on the other side of the screen! How was that possible? The key is knowing the Euler characteristic of a sphere (which is 2), she explained.

The talk moved onto the Mobius Band and all its curious properties. The boys were amazed that a single Mobius strip could be turned into interlocking rings and that two Mobius strips joined together perpendicularly could transform into interlocking hearts, Miss Fung said. Dr Steckles talked about her favourite shape – the torus – explaining how to make one by taking a cylinder and joining the two ends together. It also has an Euler characteristic, but in this case it is zero. She finished her lecture by creating tori using a smoke machine and showing that they are a stable vortex (unlike tornadoes).  “There were audible gasps of awe as the smoke rings drifted across the Main Hall,” said Miss Fung.

“The boys really enjoyed the lecture as it gave them a wider understanding of how Maths can be applied, as well as its purely theoretical side,” Miss Fung concluded.  “It introduced Year 10 to shapes that are not on the School syllabus and encouraged them to think about why formulae work and where they come from. Most importantly of all, it reminded them that Maths is magical.”

Afterwards, several of the boys reflected on what they had enjoyed in the lecture:

• Reza Sair: “I liked the Mobius Bands and how it was explained – very cool.”
• Vithusan Kuganathan: “The idea that topology can link everyday 3D shapes like spheres to weird shapes like the torus was mind-bending.”
• Shai Kuganesan: “I liked seeing the information in real life, rather than on videos – it made me appreciate it more.”
• Shayan Sadjady: “The tricks were magical.”
• Suvir Rathore: “I enjoyed the puzzles.”