Euler’s characteristic

From the obvious examples such as shopping, making sure everyone gets a slice of pizza and planning a trip, to the more obscure such as making phone apps or calculating the best This activity should fill a club angle for the perfect shot on goal, maths is an important timeslot as a one- part of every day life. off project. In this session students can explore some real-world applications of maths they learn at school.

Introduction

There are many maths equations that have changed the way we look at the world. One equation that has had an impact in many areas, from computing to the shape of a football, is Euler’s characteristic.

What you need

Pens or pencils

Paper

Scissors

Glue

Note for STEM Ambassadors: schools will have many of the items you need, so check with the teacher before your session. A risk assessment should be done before starting this activity.

in partnership with What to do

Euler’s characteristics Hexahedron Tetrahedron Octahedron V – E +F = χ V = number of vertices E = number of edges F = number of faces χ = Euler’s characteristic Dodecahedron Icosahedron Construct the nets and make the following platonic solids:

Tetrahedron Hexahedron Explanation Octahedron Euler’s characteristic Dodecahedron For convex polyhedra χ=2. However, for non-convex Icosahedron polyhedra or general surfaces other formulas can be used to calculate the Euler characteristic. What do you notice about χ? This is the foundation of topology and is used in a wide Is this true for other polyhedra? variety of different engineering fields such as example Nets for platonic solids can be found on the internet computer chip design, sensor networks, the study of the for students that need support. For example, nets universe, robotics design and medical imaging. For example, can be found here http://tinyurl.com/solidnets Euler's characteristic can be used to diagnose osteoporosis. Two objects with the same Euler characteristic are said to be 'homeomorphic' (from Greek homoios = identical Seven bridges of Königsberg and morphe = shape)

Can you plan a route across the bridges so that Euler’s circuit theorem each bridge is crossed once, but no bridge is crossed more than once? The Euler characteristic for connected planar graphs is also V – E +F, where F is the number of faces in the graph, Some topologists think including the exterior face. a coffee mug is the same as a donut, If a graph is connected and every vertex has an even are they correct? number of edges, then it has at least one Euler circuit, a path that starts and ends at the same vertex and uses every edge of the graph exactly once. If a graph has any odd vertices, then it does not have an Euler circuit. The seven bridge problem, the land masses represent vertices and the bridges represent edges. As each vertex is connected to an odd number of edges there is no Euler circuit, therefore it is not possible to plan a route across the bridges so that each bridge is crossed once, but no bridge is crossed more than once.

Osteoporosis: normal bone (left) and abnormal bone (right)

2 Euler’s characteristic Profile

Konstantinos Krestenitis CEO of Anumerics

My job involves the transformation of contact mechanics into a programming code that can run on large supercomputer clusters, which use the latest processor nanotechnology.

The job is challenging but very satisfying. It is challenging because in order to do mathematical computations on millions of processors in parallel, it requires splitting the large problem into easier- to-solve sub-problems; this is not trivial.

The work is very beneficial in various applications in industry; for example, creating new walking robots that can climb stairs, balance on a ball, or even predict how a space station can be assembled in space while in orbit safely.

How did you get where you are today? What do you like about your job?

I was interested in computer security at What I like about my job is the creativity GCSE level and that's how I learned my first involved in engineering work. programming language PASCAL. My current work in robotics using a Later on, I discovered that programming humanoid (Project Vega) involves enabled me to achieve many great tricks. me using the mathematics I studied I learned more programming languages to make the robot's walking more (C and Assembly) and knew by then that stable. Eventually it can be used for I wanted to be a computer scientist. So I tasks that humans can't or shouldn't did; I went on to study a computer science do, such as space exploration, fire- degree and learned everything about fighting or anti-terrorism. computers. What does engineering mean to you? The learning didn’t stop there as I enrolled to do a PhD in computational mechanics and Engineering is about appreciation of high performance computation. mathematical beauty; it is to create new technologies and mathematical Then I went on to create my own company models using our understanding that specialises in engineering and of nature so that we can benefit computational sciences with the aim of humans for the better. benefiting people using computers and mathematics.

I do a lot of rewarding work now that involves the use of supercomputing centres and the creation of an ultimate artificial intelligence.

Euler’s characteristic 3 This is engineering

The image of an engineer that students often have in their heads is one of a person (normally a man) working in a hard hat.

Engineering is so much more than that.

Engineering is the designing, testing, Next steps manufacturing and improving of products to do something useful. It is the application of science, The following websites might be of use maths and creative skills to achieve a goal. to you and your students: What engineering is and careers in engineering Curriculum links www.tomorrowsengineers.org.uk

Maths: Solve problems, Geometry and measure Resources

www.raeng.org.uk/education/schools/ teaching-and-learning-resources

This site includes more activities like this one as well as longer extended STEM projects.

There are also various teacher networks and ways of collaborating with STEM teachers. Contact the Royal Academy of Engineering to find out more.

Royal Academy of Engineering Generously supported by Prince Philip House, 3 Carlton House Terrace, London SW1Y 5DG Tel: +44 (0)20 7766 0600 www.raeng.org.uk Registered charity number 293074 www.baesystems.com/education