Exploring the Fibonacci-Pythagoras Link: A Journey in Pattern Recognition
We are excited to share highlights from our inaugural LUMS Math Circle session of the year, titled "Exploring the Fibonacci-Pythagoras Link: A Journey in Pattern Recognition", led by Dr. Imran Anwar and Dr. Waqas Ali Azhar. This interactive session focused on the fascinating relationship between the Fibonacci sequence and Pythagorean triplets, blending pattern recognition with mathematical discovery.
The session kicked off with engaging puzzles on pattern recognition. Dr. Imran and Dr. Waqas challenged students with intriguing problems, such as rabbit population growth and car parking scenarios. After solving these puzzles, the instructors asked, “Do you notice any patterns in the sequences?” The room buzzed with excitement as students confidently responded, "Yes!" It was the perfect moment to introduce the Fibonacci sequence—where each number is the sum of the two preceding ones—discovered by the Italian mathematician Leonardo Fibonacci, also known as Leonardo of Pisa, around 1170 AD.
Dr. Imran then posed a classic question: “What is the most important theorem about right-angled triangles that you know?” Almost unanimously, the answer was, "The Pythagorean Theorem." Building on this, Dr. Imran explained how the well-known (3, 4, 5) triangle satisfies the theorem and introduced the concept of Pythagorean triplets—sets of three integers that fit the formula. Dr. Waqas followed up with an intriguing challenge: “Can you find all such triplets? Or even more exciting, can you find them using the Fibonacci sequence?”
This question took students by surprise, as the Fibonacci sequence and Pythagorean triplets seemed unrelated. The instructors then unveiled the algorithm of Fibonacci squares, a method that allows students to construct all primitive Pythagorean triplets. It was a moment of discovery, as students realized the hidden connection, with expressions of surprise and delight filling the room. Dr. Imran further highlighted that Fibonacci squares can also generate all possible reduced fractions in the interval (0,1), deepening the students' appreciation for the power of mathematical patterns.
Another memorable moment came when the instructors asked, “How can you determine if the corner of a wall forms a right angle?” They demonstrated a clever technique using Pythagorean triplets: if you measure 3 and 4 units along two directions, and the distance between those points is exactly 5 units, then the angle is a perfect right angle.
The session concluded with a certificate distribution ceremony, celebrating the participants’ enthusiasm and learning. We extend our heartfelt thanks to the instructors, and the dedicated volunteers for making this event a success.
Special thanks to Ms. Noreen Sohail and Mr. Qamar Hussain for their continued organizational support.
Here are some highlights from the event:
