Optimal geometry of soap bubbles
Submitting InstitutionAberystwyth University
Unit of AssessmentMathematical Sciences
Summary Impact TypeSocietal
Research Subject Area(s)
Mathematical Sciences: Pure Mathematics, Numerical and Computational Mathematics
Information and Computing Sciences: Computation Theory and Mathematics
Summary of the impact
Research on the optimal arrangements of soap bubbles and soap films has
been used as a vehicle for public engagement in mathematics. Presentations
and demonstrations have been given in both Welsh and English at various
events. These have had an impact on the awareness and interest of school
children in geometry and mathematics.
Soap films minimize surface energy, which is directly proportional to
surface area (or perimeter, in two dimensions), if the surface tension is
constant, as is usually the case. The usefulness of wire frames dipped in
soap solution to demonstrate fundamental concepts in minimization and
geometry, building on Plateau's work in the 19th century,
continues to this day, acting as a spur to both scientific endeavour and
public engagement. It is of particular relevance to the development of
mathematical models of the structure and dynamics of aqueous foams, which
consist of collections of soap films, of relevance to industries include
ore separation and oil recovery.
In a wire frame in the shape of a triangular prism, there is a transition
between different film structures as the length of the prism changes. The
transition is hysteretic, occurring at different lengths in extension and
compression, and Cox and co-authors were able to show that the transition
occurs "pre-emptively", that is, before energetic arguments suggest that
it should [3.1].
Mathematics, and in particular Plateau's rules, is also an important
feature of calculations that explore the energy landscape of
perimeter-minimizing clusters of bubbles. Similar to problems of shortest
distance (cf. Steiner networks, or the Travelling Salesman problem), soap
bubbles give rise to the following question: given N bubbles of
given areas, which arrangement has least perimeter, and hence energy. This
is a packing problem, the solution to which can indicate how best
(in the sense of least interface, and possibly least deformation) to fit
deformable objects within some given boundary. Ongoing effort in
Aberystwyth seeks to provide candidate solutions to this problem [3.2],
and to explore the conjectured solutions in the large N limit
The elementary geometrical rules can even be used as the basis of
predictions of the rheology of foams consisting of many bubbles. More
usually, the Aberystwyth group uses energy minimization to simulate the
structure, the static rheology (e.g. calculations of shear modulus [3.4]),
and the slow (quasi-static) flow of foams [3.5], for example in their
EPSRC/P&G-funded work [3.6].
References to the research
[3.1] S. Hutzler, D. Weaire, S.J. Cox, A. Van der Net, and E. Janiaud
(2007) Pre-empting Plateau: the nature of topological transitions in foam.
Europhys. Lett. 77: 28002.
[3.3] S.J. Cox, F. Morgan and F. Graner (2013) Are large
perimeter-minimizing two-dimensional clusters of equal-area bubbles
hexagonal or circular? Proc. Roy. Soc A 469: 20120392.
[3.4] S.J Cox, and E.L Whittick (2006) Shear modulus of two-dimensional
foams: The effect of area dispersity and disorder. Euro. Phys. J. E
[3.5] I.T. Davies and S.J. Cox (2010) Sedimentation of an elliptical
object in a two-dimensional foam. J. Non-Newt. Fl. Mech. 165:793-799.
[3.6] EPSRC Strategic Partnership (P&G), EP/F000049/1
Characterisation, Modification and Mathematical Modelling of Sudsing,
Details of the impact
Aberystwyth researchers Cox and Davies have developed a number of talks
and demonstrations, aimed mainly at children of secondary school age, that
describe soap bubble geometry and the mathematics of minimization. These
have been given under various names, including "Show me the (shortest) way
to go home", "Mathematics of Soap Bubbles" and "Bubble Magic". The
material is based upon
(i) demonstrations of Plateau's laws in wire frames, including the
hysteretic transition between different minima;
(ii) seeking least area arrangements of soap films, comparing the
conjectured solutions in  with children's intuition;
(iii) a discussion of soap-film solutions of Steiner-like problems in the
plane (for example the shortest road network joining 3 cities, 4 cities,
and 5 towns/cities in Wales) and local minima in complicated energy
landscapes. Here numerical solutions are used to indicate the relevance of
Interaction with participants is increased by offering the opportunity
for attendees to do experiments themselves, and by running a quiz on
Plateau's laws, with prizes of book tokens.
Exemplars of this activity include:
- Cox gave an invited evening talk for the Lancashire and North-west
Branch of the Institute of Mathematics and its Applications and the
Lancashire and Cumbria Branch of the Institute of Physics, hosted by the
University of Central Lancashire, in December 2010. The audience was a
mixture of academics and members of the public, including A-level pupils
- Cox brought Frank Morgan (Williams College, US) to the UK to give a
talk "Soap Bubbles and Mathematics" associated with an ICMS workshop
[5.2], and publicised on his Huffington post Blog [5.3]. The event was
held at Dynamic Earth in Edinburgh, attended by about sixty 14-15 year
olds [5.4]. He used Cox's research results [3.2] in the talk. The children
also tried out experiments for themselves (from the IMA's Big Box). A
report and photograph were used as an exemplar of ICMS outreach in the LMS
newsletter [5.5], and feedback from teachers and pupils included "the
speaker was excellent; enthusiastic, sense of humour, involved the
audience" and "Showing the applications of maths is incredibly useful"
- Davies talks and gives demonstrations at the annual Welsh
"Eisteddfod"s, both for young people (the Urdd Eisteddfod) and adults (the
National Eisteddfod), which attract over 10,000 people every year. In 2012
his work was publicised by amgylchedd.com [5.7] and, in May 2013, 55
children completed a questionnaire which indicated their increased grasp
of the material.
- School visits to Aberystwyth to hear soap film talks have covered much
of the country, including Llanfyllin (2009) and Bro Ddyfi, Machynlleth
(2013). The feedback from Bro Ddyfi included the following comment on
Twitter: "Thanks for having us, the presentations were great, very
interesting" [5.8] and an email from a teacher: "The pupils enjoyed the
experience, and it has certainly enriched their understanding of
Mathematics in higher education" [5.9].
Sources to corroborate the impact
[5.5] LMS Newsletter No. 415 June 2012, pg 7. http://newsletter.lms.ac.uk/415/415_issue.pdf
[5.6] Email from the Knowledge Transfer Officer at ICMS.
(no longer active). [Welsh]
[5.9] Email from the numeracy coordinator at Ysgol Bro Ddyfi.