Note: This is going to be extremely laggy and take much longer than it would on Turbowarp. I would highly recommend you go to Turbowarp in fact. Instructions: Press space to render the Hypersphere, the sliders are pretty self explanatory, however I would suggest leaving them unchanged the first time you render it.
Explanation: Alright so it goes without saying that this looks like a sphere, however as I will explain it is drastically different. First of all I'll talk about how I know this is actually accurate. So first things first, how do you make a sphere? By having different sized circles extending from the center, a sphere is made up of multiple circles. However this sphere, due to it having an infinite amount of edges looks like a black blob. So next you need a lighting engine, in this case how far away each circle in the sphere is from a certain point, moving around the certain point can make it look like the sphere is rotation, surprisingly enough. So that's how you make a sphere, to make a hypersphere it's a little more complicated, as you are using spheres instead of circles, and a few other things are noticeable different. As you can imagine 4D lighting exactly the same as 3D lighting, however it is referring to the lighting of each sphere instead of each circle. Unless I'm mistaken this works flawlessly, however if I am mistaken I would enjoy if you would point it out in the comments.