In Chapter 11 it mostly deals with spheres, circles, and hyperspheres. If it is a space of one dimension the 1-sphere consists of two points at a given distance on each side of a center point. The 2-sphere is the circle, and the 3-sphere is what is commonly called a sphere. Everything after three dimensions those are called hyperspheres.
A circles Cartesian formula is a^2 + b^2 + c^2 = r^2 (r stands for the radius). A spheres formula is a^2+b^2+c^2=r^2. A four sphere equation is a^2+b^2+c^2+d^2=r^2. And the ladder of Euclidean hyperspaces goes up so on. A circles surface is a line of one dimension.Also mentioned in the chapter was that circles would diminish. An example used was that if a flatlander started to paint the surface of a sphere which he lived on, if he extended the paint outward in ever widening circles, he would reach a halfway point at which the circles would begin to diminish, with himself on the inside, and he would eventually paint himself inside a spot.
A spheres surface is two dimensional and a 4-spheres surface is 3-dimensional. Einstein proposed that the surface of a 4-sphere is a model of the cosmos that is unbounded but finite at the same time. Another example Einstein suggested was that if a spaceship left the earth and traveled far enough in any direction , the spaceship would return to earth eventually. Also mentioned in the chapter was that circles would diminish.
Many hyperspheres are just what one expect by analogy with lower-order spheres. Spheres rotate around the center line, a circle rotates around a central point and a 4-sphere rotates around a center plane. A circle on a line is just a line segment. Hyperspheres are impossible to see. Their properties can be studied by a simple extension of analytic geometry to more than three coordinates.
The main concept of this chapter is that it describes hyperspheres, circles, and spheres. And it explains the radius, dimensions and edges of the spheres, circles, and hyperspheres.
From my first reference "The Thirteen Spheres Problem" by A.J. Wasserman was very interesting to me. One thing that I wish the author would've included in the chapter was when he talked about Leech and his sketch. Leech in 1956 drew a sketch of an elegant proof that was presented. I feel like the sketch would've been nice to see in the chapter. One thing that the author omitted from the chapter that I'm happy about was that the explanation of the kissing number. I found it difficult to understand. The kissing number k(n) is the highest number of equal non overlapping spheres in R(n) that tough another sphere of the same size. In three dimensions the kissing number problem is asking how many white billiard balls can kiss(tough) a black ball.
Curves for a Tighter Fit by Ivers Peterson was easy to understand for me. One thing that the author didn't include that I'm happy about was when they talked about the eight-dimensional space. It said "they ponder, for instance, the most efficient way of packing the eight-dimensional equivalents of ordinary spheres into an eight-dimensional space." I didn't like it because I'm not much of a big fan of dimensions higher than four. One thing I wished the author would've included was the problem of filling a large shipping container with identical ball bearings. Spheres don't fit together as neatly as cubes. No matter how cleverly you arrange the balls, about one-quarter of the space in the carton or in any other container tightly packed with identical balls, will remain unoccupied. I found that problem very interesting because I haven't thought about that before and it was an interesting fact to me.
My final reference was Sphere Packings, Lattices, and Groups by J.H. Conway and N.J.A. Sloane was a reference I had to re-read and research a bit. Something that I wished the author would've included in the chapter was when the reference brought up Bounds for codes and sphere packing in chapter 9. I wish he would've included it because I wanted to know more about sphere packing. I'm glad he didn't include the spherical codes that are constructed from binary codes. I thought that would've been a little difficult to understand.
In a chapter that was strenuous at times, you did an impeccable job with defining exactly what a hypercube is. Also, the examples you gave were pretty good with the visualization of a hypersphere. At the end of the day though, I think we can all agree that we're not fans of dimensions higher than four.
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