I read the following on a page about atomic orbitals (p and d orbitals in particular) which seem 2 me like 3d lemniscates (figures 8 or ∞ rotated about an axis of symmetry to form tear drop pairs or toruses. http://www.chemguide.co.uk/atoms/properties/atomorbs.html Taking chemistry further: If you imagine a horizontal plane through the nucleus, with one lobe of the (p) orbital above the plane and the other beneath it, there is a zero probability of finding the electron on that plane. So how does the electron get from one lobe to the other if it can never pass through the plane of the nucleus? At this introductory level you just have to accept that it does! If you want to find out more, read about the wave nature of electrons. I can visualizably understand that 4 spatial dimensions allow for a 3d object 2b rotated around a plane w/o intersecting it just like 3 dimensions allow a 2d object 2b rotated around a line w/o intersecting it. I've got no idea of whether that has anything to do w/ explaining the above statement about not crossing the plane of the nucleus. I'm wanting a qualitative, visualizable understanding if at all possible I've also read that ψ^2 gives probability density clouds for electron position and I contemplate that this perhaps is what necessitated that schrodinger's wave function ψ be complex valued. I can rudimentarily visualize the x (real + imaginary) and y (real plus imaginary) in terms of circles and hyperbolas. Can anybody help me out with understanding the wave nature of particles - mathematical rigor and symbolic logic are not my aptitudes - but I like to think I'm capable of understanding this stuff.