Flirting with Infinity

Topology is the branch of mathematics (geometry, to be more precise) that deals with the quantification, study, and manipulation of surfaces, spaces, and the deformation thereof.


Put another way, topology is the study of how surfaces and spaces can be manipulated and deformed while maintaining their intrinsic properties, and what differentiates various spaces from one another.

For instance, in topology, a sphere and a box are identical because their surface points are all "connected" to each other in the same way. You can deform a sphere into a box.

However, a sphere and a donut are different. You cannot deform a sphere into a donut without tearing or joining surfaces, changing the way the surface points are connected.

Likewise, a donut can be deformed into a coffee cup, but not into a kettle.

Therefore, the science of topology is concerned with the characteristics of spaces and surfaces that define and differentiate the different types of spaces and surfaces. For instance, if the actual set of points in a 3-D space that define a sphere is not necessary for defining the type of topology that the sphere is, then what does mathematically differentiate a sphere/box from a donut/coffee cup or kettle?

Talking about the topology of space is a discussion about the "shape" of space in its 3-D projection. Is space closed and spherical/oblong/rectangular/toroidal? Is space flat? Is space saddle-shaped and hyperbolic like a saddle?

Whatever the shape of the 3-D projection of space - even if space is perfectly flat - local geometries are curved by the presence of mass and energy.

Thank you very much. I got a lot more out of your post than from the other texts I've been going through.

I agree with the discussion that in order to talk about infinite you must talk about a 4th dimension. Based on the expanding of the universe, Space is infinite because it is constantly expanding. At any one point in time, the universe is finite and can be measured if we possessed the necessary tools in order to do so, but by the time it is measured, it is not the same size anymore.

This is how it is looked at for electrons, by the time the position is found it is already in a new position. So really infinite only exists because of an extra dimension and can only exist because the universe is infinite.

I'm probably correct to assert that wormholes and black holes are properties of local curvatures due to mass effect. Topological properties do not depend on mass or any other measurement. They are characteristics of the layout of space.

Different possible topologies of the universe can, I believe, make the ideas of wormholes and white holes impossible.

I start getting a little fuzzy at this point though. I haven't spent much time thinking about that, mostly because the energy that it would take to create even a small wormhole in ideal conditions is so astronomically vast that it's impossible.

Your argument is rather complex. I don't understand it very well.
Please tell me what these mean(it's your signature):

"Teneo haud fines finium"

"Veritas et Aequitas, Haud misericordia."

Just going back to the start of this thread - I'm curious. Why would the angle disappear at all? As I see it, the only way the angle could become zero degrees is if the length of the stretch causes the width of the upper and lower lines of the triangle to intermingle as one. So what if the all of the three lines comprising the visible triangle actually reduced in width proportionate to the increase to the length? By this means, an angle would always remain, which makes sense given that all of the lines were joined to begin with, and would therefore remain joined for infinity. It seems to me that in terms of angle size, the theory is that infinity would do away with the angle altogether, but is this really the case? Why can't the angle simply reduce for infinity also, but never actually completely disappear which, logically, it wouldn't given that the two opposing ends of the triangle are joined to begin with and will forever remain that way.

I dont get what youre saying. The fact that it can never come full circle and arrive where it started.

Isnt that precisely what makes it infinite?

To suggest that you can and do come full circle would suggest that the circumference is measurable.

Yes, and if you can't come full circle it's not a circle.