Overview

The LUMS Math Circle session on April 15, 2026, took students on an unusual mathematical journey, one that began with a bear at the North Pole and ended in the curved fabric of spacetime. Led by Dr. Rizwan Khalid and Dr. Syed Moeez Hassan, the session invited students to challenge one of geometry’s oldest assumptions: What if space is not flat?

Through puzzles, hands-on activities, and real-world examples, students discovered that when the world curves, even familiar ideas like circles, straight lines, and triangles begin to behave in surprising ways.

Session Structure and Content

The session opened with a playful riddle: a photographer sees a bear to the east, runs north to get a camera, and then photographs the bear by facing south. What colour was the bear?

The answer “white” immediately placed everyone at the North Pole, where directions behave differently than on flat maps. What started as a simple puzzle quickly became a doorway into the world of curved geometry.

From there, students explored the idea of the celestial sphere and learned about geodesics — the shortest paths on curved surfaces. Using maps and flight routes, they saw how airplanes often follow curved-looking paths because, on a sphere, those paths are actually the shortest. A striking discussion on world maps and the distorted size of Africa helped students realize that even the maps they trust can be misleading.

Interactive Activities

The heart of the session was hands-on exploration.

Students first worked on flat surfaces, drawing circles and measuring their circumference and diameter, confirming the familiar ratio of π. They also drew triangles and verified that their angles always added up to 180°.

Then the experiment moved onto spherical surfaces using physical balls. Here, things changed. Students discovered that the circumference-to-diameter ratio was no longer π, and for larger circles, the difference became even more noticeable.

Using string, they traced the shortest paths between points on the sphere and discovered that these paths were parts of great circles — the spherical version of straight lines.

Finally, students built spherical triangles and measured their angles, only to find that the total was always greater than 180°. For many, this was the moment when curved geometry stopped being an idea and became something real.

The session ended by connecting these ideas to Einstein’s view of the universe: space itself can curve, and that curvature shapes how objects move.

Learning Outcomes

By the end of the session, students:

• Understood the difference between flat and curved geometry.

• Explored how π changes on curved surfaces.

• Learned that geodesics replace straight lines on spheres.

• Discovered that triangles on curved surfaces behave differently.

• Saw how these geometric ideas connect to modern physics and spacetime.

Conclusion

From a bear puzzle to Einstein’s universe, the session showed students that geometry is far richer than the lines and shapes of a classroom textbook. By exploring curvature with their own hands, they discovered that changing the shape of space changes the rules of mathematics itself.

Acknowledgments

The LUMS Math Circle team gratefully acknowledges Dr. Rizwan Khalid and Dr. Syed Moeez Hassan for conducting this engaging and thought-provoking session. Special thanks are extended to the Mathematics Department staff for their organizational support and to all participating students for their curiosity and enthusiasm.

Certificates

At the end of the session, certificates of participation were distributed to the students in recognition of their active engagement and successful completion of the activities. 

Here are some highlights from the event: