Imagine drawing two parallel lines on paper. They never meet — right? But what if we pretend they meet “at infinity”? This simple idea leads to a new geometric world called the projective plane, where shapes behave more symmetrically and geometric rules become cleaner and more elegant.
In this world, we can study curves defined by equations. Here’s a fascinating question:
If a curve is described by an equation with only rational numbers, can we find points on the curve whose coordinates are also rational numbers?
For circles, parabolas, and other conic curves, something magical happens: if you can find just one rational point, then there is a simple geometric trick — using straight lines — to find infinitely many more. Everything falls neatly into place.
But when we try the same trick on curves of degree three (cubics), the method fails.The solutions no longer follow a simple pattern. New surprises emerge, hinting at hidden structures and rules waiting to be discovered.
This talk is a journey from familiar shapes in school geometry to the rich and mysterious world of higher-degree curves. It shows how a small change in the degree of a curve can completely transform the story, and why exploring these patterns keeps mathematicians curious even today.
Prerequisite: Basic familiarity with high school algebra and geometry — points, lines, and simple curves. No advanced mathematics is required; curiosity and imagination are enough!
