 Hello, this is Professor Steven Nashaband. I'm here to help you out a little bit with looking at the nodal structure on pi subsystems and conjugated polyenes. So that's quite a mouthful. And here I'm just let me kind of lay this out for you. So we have a hydrocarbon here. It's got a bunch of carbon-carbon double bonds. It's all actually called deca-pentine. And what I want to show you first is you can see how it's a planar molecule. It's all flat like this. And if I want to look at the molecular orbitals on it, I go here and just going to start off at the bottom and kind of point out to you that they kind of come in two families, one which looks kind of like this, where all the electron density seems to be kind of in the plane, like that one and that one. But here we get our first instance of one that is going to have what we call pi symmetry. You can see it has reflection at symmetry across the plane of the molecule. And second, it doesn't have any transverse nodes, no nodes at all moving in the up and down direction here. And let's see if we can find some more. That was the first one. I'm still moving up, still moving up. Now this is the next one. This is the next pi subsystem molecular orbital. And you can see it has a node right down the middle. And this one has also pi subsystem and has two nodes and three nodes. Finally, we get up to the homo, which has four nodes and transverse nodes. And now we get up to the lumo, no electrons in this one. But you can see if there were electrons, there would be this many five nodes running transverse along the molecule. So the idea is that on all of these electrons are in this pi subsystem, electrons are definitely delocalized across the entire length of the molecule. And we have this idea of the more nodes in this transverse direction that sort of indicates that the electron is moving faster and faster in the horizontal direction. Or as I go down here in the structure, slower and slower because we're getting fewer and fewer nodes.