Fullerene / Buckyball teacher pack
Mathematics Masterclasses www.rigb.org
3rd June 2009
The Royal Institution of Great Britain
Lesson organisation and overview
This document is part of the Ri teacher pack Fullerene/ Buckyball.
It provides guidance
on one possible outline of the activity, suggestions of lesson time, resources and
information on how to prepare the lessons. You will find further guidance in the worksheet
-- written in italics. The activity is inspired in the work developed by the Ri and
teachers. We have also obtained feedback from teachers. If you wish to contact these
teachers please email maths@ri.ac.uk.
Fullerene / Buckyball
Can mathematics aid chemistry? In this activity you travel in time to 1985 when radio
astronomers and chemists were duelling with the shape of the molecule of a new
compound of carbon found in outer space. Your work will recreate achievements of the
1996 Nobel Prize Laureates!
This activity intends to give learners the opportunity to
explore 3d shapes and place 3d skills in the context of the
demands of architecture, engineering and scientific
advancement. It explores purely mathematical concepts also
allowing for a natural and accurate positioning of mathematics
as a tool to the progress of other sciences (in this case
chemistry) and to solve problems in engineering (like folding
of airbags in cars, telescopes, solar panels, surgical
implants). The rich and real context allows pupils to explore
the topics further as out of the classroom investigations.
The activities proposed allow the teacher to work on a variety of
Figure 1: Buckyball
frameworks:

Series of whole-class sessions with the option of pulling out some activities for
after school clubs or to stretch the gifted and talented cohort

Whole day activity (e.g. STEM day, Activity week)

After school clubs in technology, mathematics, origami, etc.
Curricular links
M in STEM: chemistry, engineering, architecture,
medicine, nanotechnology, weather forecasting,
sculpture, paper folding technology
Fullerene / Buckyball teacher pack
Mathematics Masterclasses www.rigb.org
Lesson organisation and overview
4th June 2009
The Royal Institution of Great Britain
Mathematics: angle, tessellation, algebra, counting of faces, edges and vertices of 3d
shapes leading into Euler’s formula, Platonic solids, Archimedean solids.
Learning approaches: kinaesthetic activities, team work, spatial
Figure 2: boys at Woolwich
awareness, investigation.
Content
In this activity learners explore 3d shapes, investigating a relationship between the number
of vertices, edges and faces of shapes (Euler’s formula).
Pupils build:

dodecahedron and a football using paper folding

a ‘ball’ using pentagons and hexagons only
Pupils investigate:

uses of the Euler’s formula to define the shape of a ‘new’ carbon compound

ways of counting vertices, edges and faces of 3d
shapes systematically
Pupils are asked to research:

geodesic domes

Buckminster Fuller

Platonic and Archimedean solids

uses of paper folding in industry and science: airbag
folding, Eyeglass telescope, etc

mathematically interesting buildings in the neighbourhood

carbon nanotubes
Figure 3: Buckyball by Simon
Thomas
Outline of sessions
We outline two possible approaches:
A – including paper folding activity
B – using ready-made hexagons and pentagons
The indications below are based on trials run with
whole-classes in schools, teacher development
Figure 4: boys at Woolwich folding Phizz-units
sessions and masterclass series.
This activity has been inspired and informed by the work of the sculptor Simon Thomas. http://simonthomassculpture.com/.
Further ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
The masterclass was developed with the generous support of The Esmée Fairbairn Foundation.
Further questions? Contact Sara Santos on ssantos@ri.ac.uk.
Page 2 of 8
Fullerene / Buckyball teacher pack
Mathematics Masterclasses www.rigb.org
Lesson organisation and overview
4th June 2009
The Royal Institution of Great Britain
Approach A – paper folding
Equipment:

95 A4 sheets per group, cut into squares

your demo kit (illustrating )

sets of 3D shapes to be distributed amongst the groups, ideally 5
different solids per group (if you have a greater variety then you
may allow groups to swap shapes)

worksheets (as in the resource pack)

instructions for the Phizz-unit (from the ‘Project Origami’ sample)
Figure 5: step-bystep demo kit
Preparation:

have a quick read of the chapter on Phizz-units (I have never heard of this
name before I’ve run this activity, it is just the name of the strip of paper you will
fold) from ‘Project Origami’ – see the resource pack. Follow the You Tube video
on building the units and try it yourself: you need basic A4 paper (a tooth-pick
may help but it optional) and a flat surface. Read the chapter again.

Once you know how to fold the strips prepare big strips of paper (from A3) for
demo, one a each stage of the folding and number them – some pupils can refer
to this during the lesson instead of having you explaining.

For the session, have A4 sheets cut into a square (using the shortest edge),
each group will need 90 Phizz-units (add some extra for the spoiled ones). You
may wish to have a mixture of plain paper, scrap paper and colour paper. If you
use three different papers you may leave as a question to investigate if one can
arrange the edges so that touching edges have different colours (pupils can
investigate this a posteriori ). We have tried this with groups of 4 or 5: it is a
good number in terms of producing the units but requires the pupils to divide tasks
when it comes to assembling the units.

Work through the worksheets provided and consult the chapter from ‘Project
Origami’ to compare your solutions. Prepare a way of conveying the counting
arguments used to find equations relating vertices and edges or pentagons,
hexagons and vertices. Go back to the book as often as you need. Don’t get
disheartened if it seems hard at first! Once you master it, it is a fabulous idea to
use in the classroom!

Once you teach one group of pupils you can pick a few enthusiastic children to
become your assistants in one of your other classes or to help another teacher in
This activity has been inspired and informed by the work of the sculptor Simon Thomas. http://simonthomassculpture.com/.
Further ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
The masterclass was developed with the generous support of The Esmée Fairbairn Foundation.
Further questions? Contact Sara Santos on ssantos@ri.ac.uk.
Page 3 of 8
Fullerene / Buckyball teacher pack
Mathematics Masterclasses www.rigb.org
Lesson organisation and overview
4th June 2009
The Royal Institution of Great Britain
your school. We tried this with Woolwich Polytechnic School for boys and it paid
off.
A standard guillotine will be enough to cut the paper but nowadays some schools have a
bigger guillotine that can automatically cut large amounts of paper.
Part 1: Euler’s formula and the Buckyball
Time: approximately two periods of 50 minutes each.
5 mins
Starter: Vowelless (from http://www.transum.org/Starter, with kind permission
from the website owner).
20
mins
Phase1: identify given 3D shapes, register number of faces, edges and
vertices, find a relationship between these numbers (Euler’s formula).
30 mins
if in
groups of
Phase2: fold paper units in groups.
Phase3: assemble two and then three units.
4
20 mins
Phase 4: in groups assemble a Buckyball.
15 mins
Phase 5: count vertices and faces; use Euler’s formula to work out the
edges; count and compare results.
Homework: research on the buckyballs, its history, the origin of the name, the
link with architecture, the link with Carbon technology, Platonic solids, and
Archimedean solids, anything else related.
Part 2: a short tale in chemistry
Time: two periods of 50 minutes each.
10 mins
Starter: quiz on 3D shapes (based on previous lesson’s findings)
10 mins
Phase 1: pupils share their findings.
30 mins
Phase 2: back in the lab in 1984, how do we know this is the shape of
the fullerene? How many pentagons? How many hexagons?
30 mins
Phase 3: finding algebraic relationships between parts of the buckyball. Use
Euler’s formula and the above to find the number of pentagons and
This activity has been inspired and informed by the work of the sculptor Simon Thomas. http://simonthomassculpture.com/.
Further ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
The masterclass was developed with the generous support of The Esmée Fairbairn Foundation.
Further questions? Contact Sara Santos on ssantos@ri.ac.uk.
Page 4 of 8
Fullerene / Buckyball teacher pack
Mathematics Masterclasses www.rigb.org
Lesson organisation and overview
4th June 2009
The Royal Institution of Great Britain
hexagons.
10 mins
Phase 4: conclusion and highlighting links with science.
Start in class and finish as homework: produce a poster and a webpage on
the findings.
Wider reach: if you wish to engage pupils that are fonder of drama and literature than
science you may ask them to write a dramatised account of the events, stage it and film
it! Bounce the idea to the Drama, English, ICT and Science departments.
Approach B – using ready-made hexagons and pentagons
Equipment:

sets of 3D shapes already made or make
some using the material below

Frameworks, Polydron or any other
construction set (for example Zometool
Buckyball kit). You will need to allow for
each group to have at least 12 pentagons
and 20 hexagons, but you should provide
plenty more hexagons but only one spare
pentagon. If you involve the design and
Figure 6: shapes made of pentagons and
hexagons
technology department early on you could get the children to design the shapes
and cut them in plastic or wood

worksheets (as provided)
Preparation:

try the activity yourself and be prepared for the possibility of the children created
shapes you will not know

arrange the construction pieces in packs for each group to save lesson time

work through the worksheets provided and consult the chapter from ‘Project Origami’
to compare your solutions. Prepare a way of conveying the counting arguments
used to find equations relating vertices and edges or pentagons, hexagons and
vertices. Go back to the book as often as you need. Once you master it, it is a
fabulous idea to use in the classroom!
This activity has been inspired and informed by the work of the sculptor Simon Thomas. http://simonthomassculpture.com/.
Further ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
The masterclass was developed with the generous support of The Esmée Fairbairn Foundation.
Further questions? Contact Sara Santos on ssantos@ri.ac.uk.
Page 5 of 8
Fullerene / Buckyball teacher pack
Mathematics Masterclasses www.rigb.org
Lesson organisation and overview
4th June 2009
The Royal Institution of Great Britain
Suggestions: involve the science, art and technology departments early on so that you can
all think of planning it together. There are classroom resources on carbon technology
(carbon nanotubes) for science classes (see the link on the PowerPoint presentation).
With regards to the art and technology, we have developed a masterclass based on the
work of the sculptor Simon Thomas and another one on the architectural work of Foster +
Partners – contact us on the email below!
Euler’s formula and the Buckyball
Timings: see the guidance on Approach A above.
Starter: Vowelless (from http://www.transum.org/Starter, with kind permission from the
website owner)
Phase1: identify given 3D shapes, register number of faces, edges and vertices, find a
relationship between these numbers (Euler’s formula)
Phase 2: back in the lab in 1984, how do we know this is the shape of the fullerene?
How many pentagons? How many hexagons? Finding algebraic relationships between parts
of the buckyball.
Phase 3: use Euler’s formula and the above to find the number of pentagons and
hexagons.
Phase 4: conclusion and highlighting links with science.
Homework: research on the buckyballs, its history, the origin of the name, the link with
architecture, the link with Carbon technology, Platonic solids, and Archimedean solids,
anything else related.
Further Homework: produce a poster and a webpage for the school on the findings.
Wider reach: if you wish to engage pupils that are fonder of drama and literature than
science you may ask them to write a dramatised account of the events, stage it and film
it! Bounce the idea to the Drama, English, ICT and Science departments.
References
AcessNano
http://www.accessnano.org/
accessed 03/06/2009
Cromwell, P (2008), Polyhedra, Cambridge University Press
This activity has been inspired and informed by the work of the sculptor Simon Thomas. http://simonthomassculpture.com/.
Further ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
The masterclass was developed with the generous support of The Esmée Fairbairn Foundation.
Further questions? Contact Sara Santos on ssantos@ri.ac.uk.
Page 6 of 8
Fullerene / Buckyball teacher pack
Mathematics Masterclasses www.rigb.org
Lesson organisation and overview
4th June 2009
The Royal Institution of Great Britain
Freiberger, M. (2007) Perfect buildings: the maths of modern architecture, Plus+
magazine, 42, http://plus.maths.org/issue42/features/foster/
accessed 03/06/2009
Hull, T. (2006) Project Origami – Activities for exploring mathematics, Wellesley,
Massachusetts, A K Peters
Jenkins, G. and Wild, A. (2006) Make Shapes 1, Tarquin
Maths starter of the day, Vowelless,
http://www.transum.org/Software/SW/Starter_of_the_day/starter_October1.asp
accessed 03/06/2009
NRICH, Building with solid shapes,
http://nrich.maths.org/public/viewer.php?obj_id=239,
accessed 03/06/2009
NRICH, Paper folding – models of the Platonic solids,
http://nrich.maths.org/public/viewer.php?obj_id=5480,
accessed 04/06/2009
Paper folding in medicine and technology: http://wwwcivil.eng.ox.ac.uk/people/zy/research/origamistentgraft.html
Swan, M. (2005) Standards Unit - Improving learning in mathematics: challenges and
strategies, DfES, http://www.ncetm.org.uk/files/224/improving_learning_in_mathematicsi.pdf
accessed 03/06/2009
The Royal Institution Christmas Lecture 2006, Elusive shapes,
http://www.rigb.org/christmaslectures2006/20_10.html
accessed 03/06/2009
The Royal Institution, Engaging mathematics activities,
http://www.rigb.org/contentControl?action=displayContent&id=00000001862,
accessed 03/06/2009
The Royal Institution Mathematics Masterclasses for secondary pupils,
http://www.rigb.org/contentControl?action=displayContent&id=00000001857
accessed 03/06/2009
This activity has been inspired and informed by the work of the sculptor Simon Thomas. http://simonthomassculpture.com/.
Further ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
The masterclass was developed with the generous support of The Esmée Fairbairn Foundation.
Further questions? Contact Sara Santos on ssantos@ri.ac.uk.
Page 7 of 8
Fullerene / Buckyball teacher pack
Mathematics Masterclasses www.rigb.org
Lesson organisation and overview
4th June 2009
The Royal Institution of Great Britain
The Vega Science Trust Videos, C60, the Celestial Sphere that Fell to Earth - Science
Video Lecture, Sir Harry Kroto, The Royal Institution Friday Evening Discourse, 1995,
http://vega.org.uk/video/programme/65
accessed 03/06/2009
Wikipedia, Archimedean solid,
http://en.wikipedia.org/wiki/Archimedean_solid,
accessed 03/06/2009
This activity has been inspired and informed by the work of the sculptor Simon Thomas. http://simonthomassculpture.com/.
Further ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
The masterclass was developed with the generous support of The Esmée Fairbairn Foundation.
Further questions? Contact Sara Santos on ssantos@ri.ac.uk.
Page 8 of 8