The simplest geometric idea is the point, and then the line, the plane, and the solid. Shapes like circles, squares, rectangles, and triangles are flat, and we can think of them as being parts of a plane, flat like a drawing. Shapes like spheres, cubes, and pyramids are solid, and we can think of them as being part of the whole universe – as indeed any real object is.
When people work with geometric shapes, there are some things they often want to know about them. We’d like to know how big they are (their area or their volume), and how long their edge is or how big their surface is (their perimeter or their surface area). People like the Babylonians and the Egyptians and the Chinese figured out a lot of ways of calculating the size of different shapes when you only know a few things about that shape. We still use their methods today.
But how can we be sure that our ways of figuring the size of shapes are always going to work? What if they work for most triangles, but not for all triangles? Greek mathematicians like Euclid and Pythagoras developed the idea of a proof, where you could start from a few ideas that seemed sure to be true, and use logic to show that if those first rules, or axioms, were true, then the more complicated methods based on those rules also had to be true – not just sometimes, but every time.
Finally, we’d like to know how our ideas about shapes relate to our ideas about numbers. Can we describe shapes using only numbers? Can we describe a numerical equation using only shapes? Well yes, we can. We call this part of mathematics algebra.