For Pi Day, my students and I explored our campus (Aga Khan University) to see if we could find viewpoints that were mathematically intriguing. I wanted us to begin to think about mathematics for our course on The Philosophy of Mathematics. In 10 teams of 4-5, my students took some amazing shots. They also wrote reflection-pieces for their photographs. I’m delighted to share their works with their permission on my blog.
Two teams wrote on the mathematical relation of ‘Isomorphism’, which we will later use to prove that some infinities are larger than others.
Poetic Isomorphism
Team 3: Sarir Ahmad, Zakir Ali, Mehtab Ali Khan, Amjad Ali Shah, and Faizan Amir
Team 3: This image is captured at the lake side of Aga Khan University Karachi, where the reflection of University Centre (UC) building dances gracefully upon the gentle ripples of water, presenting a poetic exhibition of Isomorphism. The term Isomorphism is derived from “iso” meaning same and “morphism” meaning different, portrays the idea of elements being both alike and distinct simultaneously. Here, the reflection of the building appears identical to the real structure yet is fundamentally disparate as it is not the building itself rather a mirroring of its essence upon water’s surface. This dual nature of being same and different at the same time, through different angles, showcase the concept of isomorphism.
An Isomorphic Expression of Itself
Team 7: Shaharyar Amin, Arsalan Danish, Atiya Kiyani, Karim Ullah, and Suhrab Wali
Team 7: [The photograph] represents a one-to-one correspondence (or mapping) between two sets that preserves binary relationships between their elements. [Isomorphism] serves as a powerful tool to study one to one mathematical relationship between two things. Their historical impact reverberates across various mathematical disciplines, making them a fundamental concept in algebraic reasoning. Because an isomorphism preserves some structural element of a set or mathematical group, it is sometimes used to transfer a complex set onto a simpler or more well-known set in order to determine the original set’s properties.
One of the teams expressed the Golden Ratio in their photograph with a beautiful shot of a staircase. This was taken at the Faculty of Arts and Science (UC Building, top floors):
Golden Spiral: A Journey Within and Beyond
Team 5: Sana Hameed, Tanzeel Ali Khan, Jafar Uddin, Saad Waheed, and Syed Ali Arsalan. (Independently, Team 1 also captured a spiral staircase. Team 1: Kamil Ahmad, Wajaht Ali, Zaynub Aly, Sahar Mubeen, Shahzad Rahim.)
Team 5: Humanity has always been in search of patterns and one such pattern that we are surrounded by is the Golden Ratio – also known as the Divine Proportion. It is a mathematical reality that is everywhere, from the phalanges of our fingers to the pillars of the Parthenon. For this picture, as the staircase spirals downwards, it resembles the Fibonacci sequence in three-dimensional space, where the sizes of the steps and their distance from the center approximate successive Fibonacci numbers, whose ratio converges to
the Golden Ratio.
[The] image and its potential alignment with the Golden Ratio might evoke the Rationalist perspective that the universe is ordered and comprehensible, with innate patterns that can be discerned through reason. The staircase serves as a metaphor for intellectual ascent or descent—philosophical or scientific progress—reflecting the Rationalist belief in the power of the human mind to unlock the secrets of the world. The philosophical significance lies in the staircase’s silent testimony to the idea that even in human architecture and design, there is a resonance with the deeper, universal patterns that govern the cosmos. This alignment with the Golden Ratio is not only a tribute to human ingenuity but also a recognition of the underlying order that guides both our creativity and the natural world.
Here are some more team photographs from the Mathematics Scavenger Hunt:
Nature’s Lesson in Unity
Team 6: Rubica Shah, Kamran Abid, Safwan Diar Ali, Muhammad Faheem, and Shah Zaib.
Echoes of Perfection: Reflections in Harmony (Symmetries)
Team 8: Kulzoom Zaman, Zia Ullah, Didar Ali, Azhar Uddin, and Mohammad Irshad
Waves of Time (Trigonometric Functions)
Team 4: Ayesha Munshi, Zeeshan Karim, Abdus Salam, Anayah Shahzad, and Rifat Jahan.
Doubt me if I Cannot Fix it Properly (Tessellation)
Team 10: Mahpara Karim, Naveera Taj, Ali Hasnain, and Ujala Hussain
The Tap of Infinity
Team 2: Aleena Allawala, Samrin Alam, Shan Muhammad, Anees Murtaza Wali, and Saleem Raza
philosoyousuf 