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
angle for the perfect shot on goal, maths is an important
part of every day life.
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
This activity
should fill a club
timeslot as a oneoff project.
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
Explanation
Hexahedron
Octahedron
Euler’s characteristic
Dodecahedron
For convex polyhedra χ=2. However, for non-convex
polyhedra or general surfaces other formulas can be used
to calculate the Euler characteristic.
Icosahedron
What do you notice about χ?
Is this true for other polyhedra?
Nets for platonic solids can be found on the internet
for students that need support. For example, nets
can be found here http://tinyurl.com/solidnets
Seven bridges of Königsberg
Can you plan a route across the bridges so that
each bridge is crossed once, but no bridge is
crossed more than once?
Some topologists think
a coffee mug is the
same as a donut,
are they correct?
This is the foundation of topology and is used in a wide
variety of different engineering fields such as example
computer chip design, sensor networks, the study of the
universe, robotics design and medical imaging. For example,
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
and morphe = shape)
Euler’s circuit theorem
The Euler characteristic for connected planar graphs is
also V – E +F, where F is the number of faces in the graph,
including the exterior face.
If a graph is connected and every vertex has an even
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 easierto-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?
I was interested in computer security at
GCSE level and that's how I learned my first
programming language PASCAL.
Later on, I discovered that programming
enabled me to achieve many great tricks.
I learned more programming languages
(C and Assembly) and knew by then that
I wanted to be a computer scientist. So I
did; I went on to study a computer science
degree and learned everything about
computers.
The learning didn’t stop there as I enrolled
to do a PhD in computational mechanics and
high performance computation.
Then I went on to create my own company
that specialises in engineering and
computational sciences with the aim of
benefiting people using computers and
mathematics.
What do you like about your job?
What I like about my job is the creativity
involved in engineering work.
My current work in robotics using a
humanoid (Project Vega) involves
me using the mathematics I studied
to make the robot's walking more
stable. Eventually it can be used for
tasks that humans can't or shouldn't
do, such as space exploration, firefighting or anti-terrorism.
What does engineering mean to you?
Engineering is about appreciation of
mathematical beauty; it is to create
new technologies and mathematical
models using our understanding
of nature so that we can benefit
humans for the better.
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,
manufacturing and improving of products to do
something useful. It is the application of science,
maths and creative skills to achieve a goal.
Curriculum links
Maths: Solve problems, Geometry
and measure
Next steps
The following websites might be of use
to you and your students:
What engineering is and careers in
engineering
www.tomorrowsengineers.org.uk
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
Prince Philip House, 3 Carlton House Terrace, London SW1Y 5DG
Tel: +44 (0)20 7766 0600
www.raeng.org.uk
Registered charity number 293074
Generously supported by
www.baesystems.com/education