Three participants write about their experience.
BY MRINALINI BISWAS, SHARANYAA BISWAS, and SHLOKA SURAJ
The International Centre for Theoretical Sciences (ICTS), TIFR Bengaluru, hosted the event MathSpark 2024 on the 21st December 2024. Kaushik Basu, an instructor in Mathematics and Physics at the Graduate School of Education, University of California, Berkeley, facilitated the event. It was based on exploration of volumes and areas of interesting shapes and solids through Cavalieri’s Principle and Mamikon’s Sweeping Tangent Theorem.
We started off going through the exploratory problems, most of which looked complicated, leading us to feel they required complex proofs. The problems were divided into four sets with different topics and difficulty levels. Based on our comfort level, we were free to choose and work on any of these problems throughout the session.
As we all gathered in the foyer area of the Chandrasekhar auditorium, Kaushik introduced a unique experiment consisting of cycles and their paths. Some of our team members were required to ride cycles with tyres made wet with water and trace out their paths using chalk while the rest of us were not looking. Our puzzle was to figure out in which direction the cycle went by looking at the wheel tracks. As the volunteers who wanted to ride the cycles and trace their paths went to do so, Kaushik acquainted us with a fascinating problem. We were given two concentric circles. A chord (of length ‘a’) of the bigger circle is tangent to the smaller circle (Fig. 1). We were then asked to find the area between the two circles. At first glance, it looked like we had to calculate the area of the two circles to figure out the final area. However, with only the variable a, how were we supposed to find the answer? And this was our introduction to Mamikon’s Sweeping Tangent Theorem.
In the two concentric circles, we can draw several tangents (of length a/2) to the inner circle. According to Mamikon’s Sweeping Tangent Theorem, we can shrink the inner circle into a point. This will form a circle with a radius of a/2 (length of tangent). Then, we can calculate the area of this new circle. Since the area of the annular region between the two concentric circles (which we started with) is made up of the same tangents as the new circle, their areas would be equal. So, the area of the annular regions is a2/4π.
Following this, we had a tea break with scrumptious snacks. Then we all went to analyse the cycle tracks. The tracks had been shuffled so that no team worked on the track their group had made. Everyone was puzzled over the problem. How do we determine the direction a cycle was driven in, with only its tyre tracks, a string, and mathematics? We pondered over this engrossing question for a very long time — each team trying to measure distances in the most arbitrary ways one could think of!
The atmosphere was filled with frustration and confusion. How do we solve a problem that we do not know how to approach? Everyone was deep in thought.
The method of attempting to solve complex problems using physical tools is not a traditional method of problem solving in India and hence was an intriguing journey for all of us.
After an appetising lunch, we all gathered once more in the foyer area. This time, we were enlightened with the solution to the problem we had all been perplexed over. The solution was quite simple and beautiful and was not as complex as we had thought it would be.
A cycle has two tyres, one front and one back. As the front tyre is pivoted, its track is a wobbly one. As the back tyre cannot be pivoted, it leaves a steady track (Fig. 2). In this manner, we can find the front and back tyres of the cycle. Suppose the original cycle went from left to right. If we draw tangents to the track of the back tyre from left to right, they will intersect the front tyre’s track. The length of these segments will be equal to the distance between two tyres (Fig. 3). But, if we draw tangents to the back tyre’s track from right to left, these tangents will intersect with the front tyre’s track and will have a varied length (Fig. 4). Using this method, we can determine the direction in which the cycle went. In this case, that is left to right.
Next, Kaushik introduced us to Cavalieri’s Principle through different animations. Along with that he briefly explained the Steinmetz Solid. He took the example of a bicylinder, also known as Mou He Fang Gai, literally “two square umbrella” (Fig. 5). Here, the solid is the intersection between two cylinders. Using the Mou He Fang Gai, one can determine the volume of a sphere enclosed by it. Kaushik explained these concepts beautifully and gave us problems to reflect over.
The session was packed with students from grades 7 to 12. There were 15 students from HSLN Global Smart School, 25 students from Sri Sarada Devi Vidya Kendra, and participants who came with their parents simply because they loved maths. The students said they had enjoyed the session and found it a great experience to get the opportunity to think beyond the textbook. We feel that such events are forums for like-minded maths enthusiasts to come together and discuss. Kaushik felt that the session had gone well and wished he had more time to explain the connection between the cycle problem and Cavalieri’s Principle. He was glad to hear good ideas formulated by students and believed students needed to return home without solutions for all problems so that they had food for thought.
Professor Supurna Sinha stated that she was pleased to see young students actively participating and concretising their ideas. Professor Joseph Samuel enjoyed meeting young mathematicians. He was delighted to show students experiments in the labs whenever he could. He also discussed the Maths Circles to spark interest.
Overall, the session was a thought-provoking, fun-filled, and interesting experience for everyone — young and old alike. What was taught was explained appealingly, and the rest was left to be discovered. We can’t wait for the next event!
Mrinalini Biswas, Sharanyaa Biswas, and Shloka Suraj attended MathSpark 2024 held at ICTS on the 21st December, 2024.