Imagine slicing a cone in half, along a horizontal line, and separating the two halves. We would then expose two new surfaces: the bottom of the top half (call this ‘a’), and the top of the bottom half (call this ‘b’). Now ask yourself: is the surface area of a equal in size to that of b? You might think that it can’t be: a cone has sloping sides and it tapers towards the top; if the surfaces didn’t get decreasingly smaller from bottom to top, then we wouldn’t have a cone here, but a cylinder. However, if the surfaces do have differently-sized areas, then we don’t have something with smoothly sloping sides. Rather, if stacked the surfaces together and zoomed in on our view, we’d see that we had a figure with stepped sides – something more like a pyramid.
The ancient Greek philosopher Democritus (c.460–c.370 BC) used this idea to argue that there are smallest units of space (and by ‘space’ here we mean the dimensions that we live in, rather than the universe outside of our planet). To see this, now imagine that instead of a solid object, we are separating in half a geometrical cone figure marked out in a region of space. Now, supposing we accept that we must be able to have something at least like diagonal lines through the space we occupy, then the surfaces of a and b must differ, meaning that if we were again to zoom in, we would find pyramid steps to the figure’s sides. But since we are supposing that this cone is a mere region of space, these ‘steps’ but be smallest units of the spatial dimension. This analogy might help: think about drawing a diagonal line on Microsoft Paint. If you zoom in, you’ll see that the line has a zig-zag shape. This is because the screen on which you draw the line has smallest units of space for the ‘paint’ to fill. The thought experiment can likewise show us that space is grainy, possessing smallest units for us to fill. Quite a surprising conclusion to reach just from thinking about a humble cone! | |||