By Sriram U (Grade 10) Sep 2024
Math is everywhere in the universe – from the patterns in snowflakes to the orbits of planets. Many objects and phenomena that we see around us can be represented mathematically using equations. For example, when a soccer player kicks a ball, it follows a parabolic path. The distance the ball travels and its maximum height can be calculated using mathematical equations.
[Image 1] Source: Google DeepMind
You may have seen your Pediatrician using “Growth Charts” to forecast the height of a kid when they are of a certain age. These charts can be modeled by an exponential equation.
[Image 2] Source: CDC Growth Chart developed by National Center for Health Statistics
In this blog, we’ll explore how graphs can be used to visualize mathematical relationships and to see the world around us in a new perspective.
Shapes and Objects on Graphs
The simplest graph is a line. Lines show the relationship between two things that change at a constant rate. For example, if you are tracking your height over time, you could plot it using a simple linear plot. You might be thinking the Growth Chart in Image A doesn’t look linear at all. You’re correct. This is because humans don’t grow at a constant rate. We typically have growth spurts during infant years and teenage years. You can notice that in the CDC Growth Chart from 0-10 months age where the growth is exponential; and it tends to become linear from ages 1-3 years.
Combine a bunch of lines and you get various shapes like squares, rectangles, and triangles. These are all 2-dimensional visualizations whereas in the real-world what we encounter is 3-D. For example, the Egyptian Pyramids, Amazon shipping boxes (cubes and cuboids) shipping containers, houses, books, etc. You can also get creative here venturing into polygons like stars, pentagons, and hexagons.
Image generated using Google Gemini
Circles can also be graphed using equations. Though it looks a bit complicated, it’s just math magic and lots of fun! The 3-d visualization of a circle would be a sphere. Look a-round and you’ll find plenty of spherical objects all the way from the planets, Sun, Moon to soccer balls, basket balls, tennis balls, fruits, bubbles, and beads.
Another shape that you’ll frequently encounter are the parabolas. These are U-shaped curves typically used to model things like the path of a ball, the shape of a bridge, projectile motions, etc.
Image: A graphical image (literally) created using a series of equations and visualized using Desmos – an online graphing tool
All sorts of shapes can be represented using mathematical equations. And, graphs are a great tool to visualize these equations. Graphs are widely used, in the real-world, in everything from science to economics. Next time you see a graph, remember that it’s not just a bunch of lines and dots – it’s a window into a world of possibilities. And, if someone asks you to make a graph, remember it’s not just an assignment – it’s a way to tell a visual story.
The Nerd Corner
Math and Physics combined with the latest advancements in Artificial Intelligence has transformed spectator/audience experience and teams’ performance in sports – Thanks to AI-driven sports analysis and enhancements to game analysis, player performance analysis, corner kick patterns and adjustments. If you are interested to learn more about this, check out this blog from Google DeepMind.
[Image 3] Source: Google DeepMind blog on TacticAI
(A) How corner kick situations are converted to a graph representation. Each player is treated as a node in a graph. A graph neural network operates over this graph updating each node’s representation using message passing.
(B) How TacticAI processes a given corner kick. All four possible combinations of reflections are applied to the corner, and fed to the core TacticAI model. They interact to compute the final player representations, which can be used to
predict outcomes.
