Möbius Strip
We are delighted to share the captivating highlights from the recent "Möbius strip" session, held on October 27, 2023. This exceptional event was graced by the presence of the esteemed visiting foreign faculty, Professor Abdelghani Zeghib from CNRS, and the ever enthusiastic Dr. Waqas Ali Azhar, an Adjunct Faculty member at LUMS.
Professor Zeghib, a distinguished researcher known for his close association with the "Preparatory School" in Algeria, a training ground for exceptionally talented students in Olympiads and mathematical endeavors, brought a unique and enlightening perspective to the session.
The exploration commenced with the intriguing concept of dimensions. Professor Ghani initiated the journey by illustrating a line segment, extending it to a complete line on the board, and posing a fascinating question: Can one traverse from the segment to the outer line without touching the endpoints of the segment? Initially, the unanimous response was "no." However, the brilliance of a student's suggestion to leap over one of the endpoints cracked open the door to a profound discussion on dimensions. Professor Ghani explained that to achieve this feat, an extra dimension was required, introducing the audience to the world of two dimensions.
Continuing on this enthralling path, Professor Ghani drew a 2-D square on the board and posed another question: Could one move from inside the square to its outside without touching the boundary? This time, students, having encountered similar scenarios before, confidently responded, "Yes, but we need one more dimension." This experience provided an intuitive understanding of the concept of dimensions. Professor Ghani further delved into Einstein's groundbreaking use of four dimensions in his general theory of relativity.
Fueled by the excitement of the students, Professor Ghani delved into the 5th postulate of Euclidean geometry, revealing that it took mathematicians 17 centuries to demonstrate its unprovability. This revelation elicited a wide range of expressions, from surprise to delight.
Did you know?
Einstein's theory of relativity is a remarkable idea that helps us understand how gravity works in our universe. It's like this: Imagine you have a big, stretchy trampoline. When you put something heavy, like a bowling ball, in the middle of the trampoline, it creates a dip in the fabric. Now, if you roll a smaller ball on the trampoline nearby, it will follow a curved path and eventually roll into the dip.
In space, things work similarly according to Einstein's theory. Instead of a trampoline, we have something called spacetime, which combines space (the three dimensions we're familiar with: length, width, and height) and time (the past, present, and future). Massive objects, like the Earth or the Sun, create a kind of dent in this spacetime around them. This "dent" is what we experience as gravity.
So, when we talk about four dimensions, we mean adding time as the fourth dimension to the three we already know. It's like having a complete description of where and when things happen. These four dimensions of spacetime are all connected, and the dents created by massive objects affect how everything moves around in the universe. It's a bit like the trampoline, but on a cosmic scale, and it helps explain the way our world and the cosmos work together in a fascinating way.
Dr. Waqas joined the discourse, rephrasing the question: "Given only two dimensions separated by a line, with one point above and one below the line, could one join them with a continuous curve without crossing the line?" The resounding "No" that followed, due to the absence of an extra dimension, paved the way for Professor Ghani to unveil the famous theorem in mathematics known as the Intermediate Value Property, leaving the students in awe of this renowned result in calculus.
The session continued with an exploration of how a line can divide a plane into two components. Dr. Waqas introduced the concept of gluing in mathematics, explaining that equivalent points on an object can be identified and then glued together to create a new object. An example was provided, where students envisioned gluing the endpoints of a line segment. They swiftly grasped the process and realized it would transform into a circle, but then came the curveball: "But this is a 2D object, isn't it?" This question left the students in a state of shock as Dr. Waqas elucidated how removing one point from the line segment separates it into two components, whereas the same isn't true for a circle. This intriguing property of glued objects continued to captivate the students.
The session progressed with the construction of the Möbius strip, achieved by twisting and gluing opposite sides of a rectangle. A moment of cinematic joy occurred as the room briefly plunged into darkness, with a clip from "Avengers: End Game" being shown, featuring Tony Stark working on time travel while modeling our universe as a Möbius strip.
The instructors then guided the participants through a hands-on activity, where they crafted cylinders and Möbius strips using paper and cut them along the center lines. The revelation that a cylinder splits into two components while the Möbius strip remains a single component was nothing short of magical. Participants also experimented with gluing:
cylinders with cylinders
cylinders with Möbius strips
Möbius strips with Möbius strips
revealing surprising outcomes when cut along the center lines. It was a delightful moment for all, as some students found themselves with interlinked heart shapes after cutting the strips. Some clicks from these activities:
The session concluded on a high note, with Professor Abdelghani Zeghib receiving a memento from Dr. Haniya Azam, an Assistant Professor in the Math department at LUMS, and the participants receiving certificates of participation from Professor Ghani.
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