FLATLAND's sphere intersection scenario, explored for four dimmensions I recently finished this wonderful new vintage edition of FLATLAND.
http://amzn.com/918775116X
In 1884, Edwin Abbott wrote this strange and enchanting novella called FLATLAND, in which a square who lives in a two-dimensional world comes to comprehend the existence of a third dimension but is unable to persuade his compatriots of his discovery. Through the book, Abbott skewered hierarchical Victorian values while simul¬taneously giving a glimpse of the mathematics of higher dimensions.
There is in one place a very interesting description of how the FLATLAND people would observe the intersection of a sphere through their flatland - namely, like the bing bang of a point which expands as a circle and then contracts again to a point and disappears.
My question is twofold:
(1) I am searching very excitedly literature - and I  mean both fiction and academic mathematical references - where the same concept is being described for a three dimensional world (geometry). What are scenarios for a a three dimmensional world that undergoes the intersection of a "hyper"- sphere of four dimmensions?
(2) Beyond references, any mathematical "intuition" (including even sketched ideas) for scenarios would be also of interest to me?
I would be very thankful for your insights.
Thanks
 A: I have thought a lot about the subject a number of years ago, some of these things are actually pretty easy to show.
I don't have any references, however with some calculations you can come up with what it looks like when you intersect certain fourth-dimensional shapes with a three dimensional space.
The ones I can do without relying on intuition are simple ones, like the fourth-dimensional equivalent of a sphere or a cone. I'll demonstrate the sphere now.
Let w represent a new spacial direction orthogonal to the x, y, and z axis in the usual Cartesian coordinate plane.
The equation for the 4th dimensional equivalent of a sphere centered at the origin with a radius of 1 is this:
$$
x^2+y^2+z^2+w^2=1
$$
It's easy to imagine this shape intersecting with a 3 dimensional plane where w=0 as this shape passes through, but the actual math appears awkward. Instead, let's look from the perspective of this 4th dimensional shape. Imagine a 3 dimensional space is moving through it at a rate of 1 unit per second. Let's start this motion when w=-1 and end it at 1, so the entire ordeal takes 2 seconds to complete. The shape when w=-1 is as follows.
$$
x^2+y^2+z^2+(-1)^2=1\\
x^2+y^2+z^2+1=1\\
x^2+y^2+z^2=0
$$
The only solution to the above equation is: $$(0,0,0)$$ Or, simply a single mathematical point in 3 dimensional space.
When w=1, the same situation presents itself. Let's examine what happens when w is a constant between -1 and 1.
$$
x^2+y^2+z^2+w^2=1\\
$$
Move constants to one side.
$$
1-w^2=x^2+y^2+z^2\\
$$
It might be hard to see, but if w is seen as a constant this is the equation of a 3-dimensional sphere. The radius of this sphere starts at 0 when w=-1 (so it starts as a mathematical point,) and increases to a maximum of 1 when w=0. When the center of the 4-dimensional sphere passes through the 3-dimensional space, the intersection reaches its maximum radius and decreases again. When w=1, the intersection reduces to a point again, and vanishes when w>1.
You can discover more about how this shape changes as it passes through by performing analysis on the above equations. If the math is too cumbersome, just try to think about what happens on a 3d-2d level and scale it up by 1 dimension. The result is effectively the same.
