Footnote

If we've imagined the schema properly with Plato, everything material in the world is formed of the four bodies in various arrangements-- i.e., as earth (cube), air (octahedron), fire (pyramid), and water (icosahedron) combine to appear as this material substance or that. And while, on the face of things, we have enough geometric form and number to satisfy even the most devout Pythagorean, Plato is more than that. His triangles not only manifest as the substrate of materiality but also manifest an ideality which transcend the limits of numeric ratio.

According to Karl Popper's hypothesis (Popper 1998, p257), Plato may have supposed he'd discovered the secret of mathematics by finding a bridge between irrational numbers and geometric form by 'squaring' the circle. This bridge was only shown to initiates, way back when, and no one left a record, so we have no way of knowing about it except through Popper's conjecture. But Popper demonstrates that Plato's square (composed of four isosceles triangles) and Plato's equilateral (composed of six 'perfect' scalenes) can be used to construct a circle and a commensurate rectilinear figure the area of which equals the area of the circle.

While that's not exactly 'squaring' the circle, the rectilinear figure embodies the important ratio known as Pi. In effect, the geometrical construction demonstrates that the square root of 2 plus the square root of 3 equals Pi. Or so Plato may have thought, having only fractions with whole number numerator and demoninator for calculation. The proper calculation requires the use of decimals, the utility of which only became known centuries later. With decimal fractions, we can see that Pi equals 3.14159... while the square root of 2 plus the square root of 3 equals 3.14626... That's close enough to appear identical in the lens of ancient mathematics, and Plato believed he beheld square roots commensurable with pure form, i.e., the circle as symbolized in Pi.