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Hyperbolic and Spherical Geometry

Logic_on_rails
Posts: 2,445
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2/1/2012 1:51:03 AM
Posted: 4 years ago
Hopefully this is placed in the right section.

I've been reading a little bit about these forms of geometry recently and have found the topic interesting. Obviously, Euclidean geometry is simple enough, and spherical geometry is easy enough to understand.

I'm having some difficulty on some points of hyperbolic geometry though. For example, here is something that I'm struggling to comprehend in general (I quote from the book):

"However, distances in the hyperbolic disc are not defined in the usual way, and become larger, relative to normal distance, as you approach the boundary. Indeed, they become so much larger that the boundary is, despite appearances, infinitely far from the centre."

I understand the concept and implications of the statement. However, why the concept should exist beyond a theoretical sense is confusing (the book tells me not to think like that, but I'm quite lost on this) .

This was clearly terribly worded, however, I'd like to try and get a discussion on hyperbolic geometry going (I'll be away in 3 days and be gone for 2, so I might not be able to reply to every post unfortunately) .
"Tis not in mortals to command success
But we"ll do more, Sempronius, we"ll deserve it
Ren
Posts: 7,102
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2/1/2012 9:08:47 AM
Posted: 4 years ago
At 2/1/2012 1:51:03 AM, Logic_on_rails wrote:
Hopefully this is placed in the right section.

I've been reading a little bit about these forms of geometry recently and have found the topic interesting. Obviously, Euclidean geometry is simple enough, and spherical geometry is easy enough to understand.

I'm having some difficulty on some points of hyperbolic geometry though. For example, here is something that I'm struggling to comprehend in general (I quote from the book):

"However, distances in the hyperbolic disc are not defined in the usual way, and become larger, relative to normal distance, as you approach the boundary. Indeed, they become so much larger that the boundary is, despite appearances, infinitely far from the centre."

I understand the concept and implications of the statement. However, why the concept should exist beyond a theoretical sense is confusing (the book tells me not to think like that, but I'm quite lost on this) .

This was clearly terribly worded, however, I'd like to try and get a discussion on hyperbolic geometry going (I'll be away in 3 days and be gone for 2, so I might not be able to reply to every post unfortunately) .

Okay. Man, this is the hardest question I've seen on this forum. We're approaching the edges of my mathematical comprehension as it currently stands.

Okay. This is how I understand it.

Hyperbolic geometry is such that it can express extremely complex and abstract shapes as a mathematical statement. In this way, it can translate as a graphical image that can literally warp in space, bending backward or twisting forward or spinning in a spiral literally into infinity. See, such things don't exist in Euclidean geometry, because it abides by the Fifth Postulate, which essentially states that two lines converging on one another will meet. So, constructing a mathematical statement for the image would see if you looked at yourself in a mirror that is directly facing another mirror behind you would be impossible. The statement would result in a value, no matter how large, that suggests an "end," when in fact, there is no "end" in that image. It continues on forever.

I think it's telling you not to attempt to try to imagine it in reality, because it's not meant to exist in reality. The only real application I can see for such translations is communication with computers. Using hyperbolic geometry, we can mathematically construct complicated shapes that are such that they have the organic and beautifully complex appearance of reality.

I guess, to transcend into three dimensional space from two dimensional space, one must assume a dimension higher than three dimensional space. My exposure to hyperbolic geometry is due to my studies in higher dimensions postulated in topography mathematics (I'm a physics addict). In that regard, it doesn't exist beyond the theoretical sense.
Logic_on_rails
Posts: 2,445
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2/3/2012 12:14:24 AM
Posted: 4 years ago
At 2/1/2012 9:13:35 AM, Ren wrote:
Whoa, wtf, you're 15??

Firstly, thank you for your other reply, it was quite useful.

On the age point, I think that one can be able to be intellectually interested in many topics and be able to effectively reason even if the brain might not be perfectly developed. What age does limit is one's knowledge due to time constraints (lifespan) on one's reading. I couldn't possibly be well versed, but I'd hope I can converse with reason and such behind my words.
"Tis not in mortals to command success
But we"ll do more, Sempronius, we"ll deserve it