Celebrating the International Day of Mathematics at LUMS

The International Day of Mathematics (IDM), also known as Pi Day, is a global celebration of the importance of mathematics and its fascinating concepts. On March 14, 2025, the Department of Mathematics at LUMS, alongside the LUMS Students Mathematics Society (LSMS), organised an exciting event to mark the occasion. Dr. Imran Anwar, Associate Professor at LUMS and newly appointed IDM Ambassador for Pakistan, led the celebrations, emphasising the event’s mission to inspire creativity through mathematics. 

The Department of Mathematics at SBASSE had a wonderful time celebrating the International Day of Mathematics. This year’s theme, Mathematics, Art, and Creativity, set the perfect stage for an inspiring talk by Dr. Waqas Ali Azhar on The Beauty of Fractals.

Dr. Azhar took us on a fascinating journey through the mesmerizing world of fractals—mathematical structures that reveal infinite complexity and breathtaking beauty. From the self-replicating patterns of nature to their applications in art, design, and even technology, his talk showcased how mathematics fuels creativity in unexpected ways.

The talk began by acknowledging the brilliant mathematical contributions of renowned artists such as Leonardo da Vinci, Salvador Dalí, and M.C. Escher. A particular highlight was Da Vinci’s illustrations in Divina Proportione, a book by Luca Pacioli that explores the mathematical elegance of the golden ratio. Additionally, Escher’s famous artwork Print Gallery captivated the audience, showcasing the intricate interplay between mathematics and artistic creativity.

Dr. Azhar then explored the intriguing intersection of geometry and infinity, introducing complex phenomena that challenge our intuition. He discussed captivating mathematical structures that exhibit paradoxical properties—such as shapes with an infinite perimeter but a finite area, surfaces with an infinite area yet a finite volume, and sets containing infinitely many points but measuring zero in length. Some of these fascinating examples are illustrated below:

He then introduced the concept of fractals and fractal dimension, beginning with a thought-provoking question: "You’ve seen 2D and 3D shapes, but do you believe a shape could have a dimension of 1.58?" This led to the construction of the famous Sierpiński Triangle, a fractal with precisely that dimension.

To illustrate its formation, he started with an equilateral triangle, connecting the midpoints of its sides to form a smaller, central triangle, which was then removed. This process left three smaller triangles, each undergoing the same transformation recursively, continuing indefinitely. Through this iterative method, the audience witnessed how simple geometric rules can create infinitely intricate and self-replicating patterns.

Another fascinating aspect of his talk was the exploration of multiple ways to construct the Sierpiński Triangle. One particularly exciting method was the Chaos Game.

Dr. Azhar began by marking three points on the board—labeled Red, Green, and Below—to form an equilateral triangle. He then placed a random point, called the seed, inside the triangle. Inviting audience participation, he asked them to randomly choose one of the three colors. Each time a color was chosen, the seed was moved halfway toward the corresponding point, forming a new seed. This process was repeated over and over, and to everyone’s surprise, the seemingly random movements gradually revealed the unmistakable shape of the Sierpiński Triangle.

This demonstration beautifully illustrated how order can emerge from randomness, showcasing the deep connection between chaos, patterns, and mathematical structure.  Here is a link to this wonderful simulation.

Just when everyone thought the excitement had peaked, Dr. Azhar had yet another surprise in store. He introduced an unexpected yet captivating way to create fractals—using toothpicks!

He demonstrated how simple rules could generate intricate fractal patterns by strategically placing toothpicks in a recursive manner. But the most astonishing part of this section was the unexpected connection between these toothpick fractals and Mathematical Biology. Dr. Azhar explained how similar fractal structures emerge in the study of disease spread, providing valuable insights into how infections propagate through networks.

A simulation of this can be found here

For those who attended, we hope the session sparked new insights and appreciation for the harmony between math and art. If you missed it, we encourage you to explore the world of fractals—you’ll be amazed at how they appear everywhere, from galaxies to leaves!