Drawing spheres on the surface of a hypercube

I could write another description of what the higher dimensions are, but personally I "A Wrinkle in Time" to describe that to me when I was 8 years old. Whatever that book left out was filled in by years of video games and movies to fill in the gaps. You know what the first dimension is, and the second and the third. You can't picture anything higher than that, but that's not a problem because the best of us can't. I'm here to talk about a much crazier, much wilder idea.

The problem with describing anything higher than 3 dimensions is that the task of imagining what a 4 dimensional cube or sphere looks like and this is impossible. Physically adding an extra dimension for the sake of modeling the 4th dimension is like blaming penguins for me being bad at analogies. This doesn't have to be the case though, because it's easier to imagine what a pyramid looks like on the surface of a hyper-sphere or a hyper-cube. This is because the surface of a 4th dimensional object has 3 dimensions. Allow to me to explain.

A cube is a 6 sided object with 12 equal edges. What makes it a cube is the fact that it has height, length, and width. Each side, or face, has only length and width thereby making it a 2 dimensional plane. A cube is a solid object bounded by a 6 equal squares. A cube is an easy place to start because it can be given to a child along with a Sharpie and they can draw all manner of shapes on its surface. Then Euclidean geometry can be explained to the child because they need to grow up sometime and understand how basic shapes can be modeled by numbers. All the shapes will fall under the basic rules: squares will have angles that add up to 360^o and areas equal to their length times their height, triangles have angles that add up to 180^o and have areas equal to its height times its base.

Anyone went the extra step for geometry knows that the properties of these shapes change when drawn along a curved surface, like a sphere. A sphere is a solid object bounded by equal circles, which means that a line is no longer straight, a line becomes curved. All lines are great circles and all great circles intersect. A great circle is the equator for simple reference. These lines can still draw shapes on a sphere it's just that the properties of the shapes change. A triangle angles are equal to 180^o plus the area of the triangle. This makes transferring shapes from the surface of a globe to a flat map hard since the shapes are not the same.

A hyper-sphere is a catchall term referring to a sphere with more than 3 dimensions. A 3-sphere is a 4 dimensional sphere, bounded by 3 dimensional circles. All great circles on a 3-sphere are great spheres. Much like we can draw squares, circles and triangles on the surface of a ball, we could draw cubes, spheres and pyramids on the surface of a 3-sphere. A cube on the surface of hyper-dimensional object would have 12 equal curved edges, or better yet it would be the intersection of six spheres touching. As long as the distance of each edge remains the same, then distortion of the cube would be the same no matter where it is along the surface of the hyper-sphere. Along a hyper-hyperbola however, its proportions would change depending on its location along the 3 dimensional curve. Sometimes it would have finite volume, other times it would have infinite volume.

I'm stopping here for today. I've thought too hard about this for too long which feels like prolonged exposure to hallucinogenics. My advice is to picture distorted boxes, because as soon as you try to imagine yourself standing on the surface of a 4D sphere it brings up a lot of weird questions. Does a 4th dimensional light source create a 3rd dimensional shadow? Am I a 3D shadow of my 4th dimensional self? Don't think about it, it's too much.