Single-Variable calculus is commonly taught from a graphic standpoint, using the concepts of area and tangent lines to describe the world of integrals and derivatives. However, when we move to multivariable calculus, most students ask the question: what does a 4D surface look like? Without this idea, it is difficult to draw the connection between techniques learned in elementary calculus and techniques in the 4th dimension and beyond. We certainly cannot pull out a hypersphere from our textbooks. Here's one method:
This technique picks a single point on the 4D hypersphere's surface and then calculates the null space (i.e. the set of vectors that is perpendicular to a vector coming from the origin to the chosen point). Then the entire hypersphere is projected onto the null space. In our 3D realm, one puts on a pair of panoramic goggles and then stands on a single point on a 3D sphere, examining his/her surroundings to get a 2D image.
This is a computational nightmare, so in order to make the solution presentable in real-time, one has to select certain cases (shown in the slide-show) and then interpolate the projections in between.
Visualizing a 4D hypersphere { x=Cos[s], y=Sin[s], z=Cos[t], w=Sin[t] }s= [0, 360°] ; t=[0, 360°] s= [0, 360°] ; t=0°If you cannot see the slide show, click here.
| s= [0, 360°] ; t=60°If you cannot see the slide show, click here.
| s= [0, 360°] ; t=120°If you cannot see the slide show, click here.
| s= [0, 360°] ; t=180°If you cannot see the slide show, click here.
| s= [0, 360°] ; t=240°If you cannot see the slide show, click here.
| s= [0, 360°] ; t=300°If you cannot see the slide show, click here.
|
posted by Jeffrey Wang
|
|
