Mathematics + billiard or billiard + mathematics: the SmartPool project connects two communities

Written by Evgenia Sendova, IMI-BAS

Mathematics billiard – past and present

Do mathematicians study billiards because they like playing it and improving their skills? Not necessarily. Idealized variations of the game are models for dynamical systems that occur in nature. In fact, the interest of mathematicians in the study of billiard comes originally from celestial mechanics. The first question considered was about the existence of periodic trajectories (corresponding to periodic orbits in mechanics). The detailed mathematical study of billiards trajectories gives rise to many interesting problems dealing with unusual table shapes such as circles, ellipses  and polygons. What is generally accepted in the idealized versions of the game of billiards though is the assumption that the incidence and the reflection angles are the same. Furthermore, the balls have no mass and move with no spin, which means they travel without friction at a constant speed in a straight line. Besides, there are no pockets that can swallow the ball. This means that the ball will bounce infinitely on the sides of the billiard table.

You may ask what real-life applications the mathematical billiard could offer (except potentially to the real game). Well, more than at least I expected. It turns out that billiards are models of many systems not only in mechanics, but also in optics, acoustics, thermodynamics . . .

Analysis of billiards path can involve sophisticated use of ergodic theory and dynamical systems. The billiard-ball computer, also known as a conservative logic circuit, is an idealized model of a reversible mechanical computer based on Newtonian dynamics, proposed in 1982 by Edward Fredkin and Tommaso Toffoli. Instead of using electronic signals like a conventional computer, it relies on the motion of spherical billiard balls in a friction-free environment made of buffers against which the balls bounce perfectly. This model played a significant role in the development of the quantum computer. A different application of billiard ball dynamics to semiconductor device physics, was the  Billiard model of a ballistic multiprobe conductor (1989). Some potential applications to the design of optical cavities for lasers have also been considered.

No wonder that the mathematical billiard is a popular object of study: a MathSciNet and Google search shows that about 2000 publications devoted to billiards have appeared in mathematical and physical literatures over the years.

Thus, it seems reasonable to identify part of the mathematical billiard that could be incorporated into the school math curriculum. So far, the so called arithmetic billiards have been discussed as mathematical puzzles by Hugo Steinhaus and Martin Gardner,  and are popular among mathematics teachers under the name Paper Pool.

But to design a whole package of resources (real and foldable pool table, a handbook integrating mathematics and pool assignments with natural connections between them, dynamic geometry applets for explorations with various configurations of billiard balls) to be integrated in the national curriculum of 5 European countries and led by a team of math teachers and pool coaches, was an initiative, first in its nature, launched as the SmartPool Project.

The SmartPool Project

SmartPool is originally a Dutch initiative, launched in June 2018. The teaching materials were developed by Erik Van Haren, a math teacher and a pool player, in response of an appeal from the Royal Dutch Billiard Association. Thanks to its success in the Netherlands, SmartPool spread in Europe within the Erasmus+ EU’s program as a project of the European Pocket Billiard Federation, in cooperation with the National billiard federations of Bulgaria, Finland, Germany, Cyprus, The Netherlands and mathematics organizations, across Europe. The expertise and networks of the billiard federations, the EU-MATHS-IN (European Service Network of Mathematics for Industry and Innovation) and PWN (Platform Wiskunde Nederland), combined with the research expertise of Mulier Institute are joining their efforts to assure the project’s implementation and dissemination. SmartPool combines pool billiards with mathematics/arithmetic for young students (9 to 15 years). A school curriculum is being developed in which the elements of pool have been translated into mathematical topics (for example, finding patterns in billiard ball configurations, hitting the right point or finding the correct angle by visualizing pool billiard situations on paper, by explorations in computer simulations and by demonstrations in real game setting).

SmartPool in Bulgaria

The management and scientific body of SmartPool in Bulgaria is represented by billiard specialists from the BoandBo billiard club in Petrich and mathematics/informatics educators from the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences (IMI-BAS), a member of ECMI (European Consortium for Mathematics in Industry). These are Georgi Bozhkov (Secretary General of the EEBC International Billiard Academy), Radoslav Milkov (billiards coach and player), Evgenia Sendova (associate member of IMI-BAS) and Prof. Toni Chehlarova (IMI-BAS). The invitation to the mathematics component of the Project management came from Prof. Wil Schilders (executive director of PWN) via Prof. Stefka Dimova (former member of the Council of ECMI).

The activities carried out by the Bulgarian SmartPool team so far cover the following tasks:

• Stakeholder analysis and needs assessment in a Bulgarian setting
• Translation of the Dutch version of the SmartPool Materials
• Country-specific feasibility check and conditions for implementation based on the translated version, reporting on adaptations needed
• Development of a country-specific adaptation, enriched with dynamic geometry applets for explorations of various billiard situations
• Participation in the Project meeting in Amsterdam
• Launching of a qualification course for teachers of Mathematics with billiards (16 hours of attendance, including synchronous distance learning in an electronic environment, and 16 hours of distance learning) by the Scientific Council of IMI-BAS
• Presenting the project at the National seminar on Inquiry Based Math Education
• Holding a working meeting of the Bulgarian Project team in Petrich

The meeting in Petrich (December 1-4, 2021)

According to the Project plan, the pilot experiment will be held with 5-7 graders (11-13 years old) in three schools in Petrich. The classes will be led by three teachers in mathematics and IT and three billiard coaches from the BoandBo billiard club.  

The meeting in Petrich was the first project event, in which the billiard and math specialists met and worked in person.  The whole atmosphere was stimulating for discussions and hands-on activities using the Bulgarian version of the Dutch handbook (translated by Radoslav Milkov and edited by Evgenia Sendova), the teaching resources of Georgi Bozhkov, the dynamic geometry resources in the Virtual Math Laboratory of IMI-BAS developed by Toni Chehlarova:

The support of the Municipalty of Petrich was guaranteed by the Ms Nina Popova, Deputy-mayor, in charge of humanitarian activities. She joined the discussions with specific ideas of the teachers’ involvement in the teacher education courses to be held in mid-January. Zaprjan Kandilarov, one of the pilot teachers, also expressed his views on how to implement the teaching resources.

Very exciting for us, the mathematicians, were the first steps in pool billiard under the coaches Georgi Bozhkov, Radoslav Milkov and Milcho Makenjiev. After trying hard to determine and keep the different body angles required, we were shown master hits even with a single hand…

The event attracted the attention of the local TV:

Here is what the math and billiard specialists in the project share at the end of the meeting about their motivation and the expected effect of the project:

Georgi Bozhkov: My main motivation to coordinate the SmartPool project in Bulgaria is the desire to work with children. On one hand you can move forward and develop the sport you love. On the other, you change your attitude to mathematics – from our Project meetings we understood that we know more about mathematics than we were aware of. When playing, I imagine triangles, intersecting lines, I think of interesting combinations. Now, I see the mathematical interpretations of numerous situations I have experienced on the green table. As for the effect I hope to see, this would be that SmartPool becomes a new subject/sport at school, aiming at an easier access to mathematics and at an even more attractive game supported by math and science.

 Radoslav Milkov: I feel motivated by the fact that I will have the opportunity to show young children my greatest passion in life. I used to work with children, I love children and I want to pass on my knowledge to them to demonstrate how wonderful a means of concentration, but also fun, this game is.

First impressions of our meeting in Petrich are that a group of people focused on a joint cause and integrating their passion and high professional standards can produce miracles.

Toni Chehlarova: Our role will be to adapt the teaching/learning materials to the national curriculum and enhance them with virtual resources. This will bring an intermediate stage between solving the assignments in the worksheets and the practical exercises on the billiard table – explorations with dynamic geometry software. After synchronizing the key terms and notions in billiard and mathematics, the students are expected to see some general common ideas, make multiple-step predictions and verify them in different settings.

As for me, one of the greatest achievements of the Project is its role in raising the public awareness of mathematics and more importantly, in bringing together different communities in the pursuit of a common goal – upbringing and educating the children to love  learning and to enjoy life to its fullest.