How did Plato map four levels of knowledge onto one line?

In Plato's Republic, Socrates asks his companion Glaucon to picture something deceptively simple: a single line, cut into two unequal pieces, with each of those pieces then cut again in the same proportion. That one image became one of the most compact maps of knowledge in philosophy, because Plato uses the four resulting segments to rank everything a mind can know, from least real to most real.

The bottom two segments cover the visible world. The very lowest piece is images: shadows, reflections in water, anything that is just a copy of a copy. Plato calls the kind of knowing that deals with these mere conjecture. The next segment up covers actual physical things, the animals, plants and objects that cast those shadows and reflections, which Plato calls belief. Together, conjecture and belief make up what he calls opinion, our ordinary, everyday grip on a changing world.

The top two segments cover the intelligible world, the world of ideas rather than things. The lower of these is mathematical reasoning: thinking about abstract objects like perfect geometric lines, which are not the same as the imperfect lines you might draw on paper. The highest and final segment is pure philosophical understanding, reached by reason alone through careful argument, not resting on any assumption but climbing toward the most fundamental principle of all.

The point of the whole structure is a single comparison: a shadow is to the object that casts it what the whole physical world is to the world of Ideas, or Forms. Each step up the line is more real and more true than the one below, and Plato goes as far as denying that the changing world of the lowest level can be known at all. Understanding, for him, isn't one thing: it's a ladder, topped by knowledge of the Form of the Good.