Have a little fun with IFUU, IFSBB & IFBM...

[Billy.G]

Hi all. I’m not sure if this is the right place to put this but here’s some entertaining (?) mental exercises for you (see pdf). My apologies to those who’s first language isn’t English [esp if you’re wanting to try & translate it ] – and to those for whom it is.
Due to my personal / domestic circumstances it may take me some while to respond to any questions – though, as I say, this is meant to be more along the lines of mental exercise. I hope you have fun. Billy.G.

 

[HowToBe]

I took a quick look at your PDF. I find this sort of thinking fun, although I just had time to look at the first page. The sort of problems you are dealing with so far as I've read seem to be in the realm of "topology". My introduction to the term was in a book called "Chaos: A History" which I rather enjoyed. I would criticize that your model doesn't seem defined specifically enough to say that the inside of the "tennis ball" is finite and the outside is infinite. You describe the material that fills and surrounds the ball as being infinitely ductile and elastic. I can imagine the "material/field" inside the ball as being infinite yet bounded. Maybe the ball has an infinite size? Or maybe an infinitely elastic material can be infinitely compressed? I'm not sure "finite but unbounded" would be the best description here, therefore. When I think of that phrase, I think first of "looped" spaces such as a toroidal universe.

Actually, the tennis ball example reminded me of this video I saw a number of years ago illustrating Mobius Transforms:

http://<iframe width="560" height="...O4" frameborder="0" allowfullscreen></iframe>

Around the time I found that video, I was musing about how the edge of an infinitely large circle and a line could be the same thing. This short video illustrates my thinking if you pay attention to the way the circle becomes a line just before it inverts:

http://<iframe width="560" height="...jM" frameborder="0" allowfullscreen></iframe>

Finally, here's a thread in which I was engaging in similar "play":
https://cassiopaea.org/forum/index.php/topic,22678.msg245691.html#msg245691

The relationship between the finite and the infinite really is a mystery, as they just don't seem to mix well, like oil and water. But you run into such interesting things when you try anyway (like emulsions!). I think the first video above gives a big clue when it "introduces another dimension" to explain the mathematical transformations. Thanks for posting!

 

[Billy.G]

Thanks for your reply. The videos are very interesting - food for thought. This is something of a new experience for me, as I only rarely have anyone to discuss this sort of thing with, and I've never shared any of my notes at all. Having been lead here with a 'breadcrumb' trail i figured it was 'time' that they were 'stress tested' and also to see if what I had written was understandable to others. I'm not a mathematician, more a self taught engineer. It does look like I'm trying to do maths with prose, & I had wondered if all this was describable with maths. I think 'yes' is the answer. I does annoy me intensely that this sort of thing is, or has been throughout my life, such an insular subject reserved for those who know the 'sacred jargon/language' of advanced maths but I now see that's just how the consortium (and above) wants it. Was it Nils Bohr who said something along the lines that if 'physical' theories couldn't be understood in plain language what's the point. On your points about my 'model' ; i didn't specify that the inside of the tennis ball was finite as it seems an obvious thing - to me. I added an 'infinite' outside as an afterthought to show that the 'outside' becomes 'encapsulated' in the turning inside out maneuver showing that, it too, was also finite. I described the balls 'filling' as infinitely ductile purely to indicate that it could be 'stretched' to any degree without snapping. In page two I do indeed compress the, now inside out, ball infinitely to create an singularity (or get rid of it entirely) and hence remove the boundary & arrive at an unbounded finite space that appears infinite. Part of the point I'm trying to get across is that the term infinite may just be a red herring and that apparently infinite things may just be a 'trick' of 'opposing mirrors' or feedback.
Lastly can you check the link to you post as nothing happens when i click on it. If you can give me title i can run a search. Thanks again.

 

[HowToBe]

Amusingly, I got my formatting "inside out" when I tried to post the link.
Topic: Understanding The Seven-Spoked Wheel

This page has some interesting info as well:
https://math.stackexchange.com/questions/82220/a-circle-with-infinite-radius-is-a-line

Contrast the discussion there with this one on the same website in which there is much less openness to the idea:

https://math.stackexchange.com/questions/708634/is-an-infinite-line-the-same-thing-as-an-infinite-circle

Years ago I recall doing a search about this idea regarding lines and circles, and found a forum discussion where the person asking this question was basically shouted down by people saying "infinity is a concept, not a number". In a way that's fair, because in math you must be specific - a lot of the reason for the difficult jargon is the need to be extremely accurate and specific to communicate about mathematics in an efficient way. On the other hand, you can't crush every creative jump in thought that appears "wrong" if you want new and useful concepts to emerge.

 

[Billy.G]

Thanks. The second youtube clip you posted has had my little grey cells in overdrive (we don't seem to have a 'lightbulb' emoji) Before doing anything about that I'm determined to get 'deconstructing time' finished. I like your thinking on the line to circle. As my first piece of real networking here this is quite enjoyable.

 

[HowToBe]

Here's a link I opened in the course of my search-engine-ing earlier but didn't look at until just now. It is very readable and cool.
http://www.askamathematician.com/2011/04/q-is-the-edge-of-a-circle-with-an-infinite-radius-curved-or-straight/

excerpt (my bold):

Old school topologists get very excited about this stuff.

Say you have two lines on a plane. They’ll always intersect at exactly one point, unless they’re parallel in which case they’ll never intersect at all. But the Greek Geometers, back in the day, didn’t like that; they wanted a more universal theorem. So they included the “line at infinity” with their plane, and created the “projective plane”. In so doing they created a new space where every pair of straight lines intersect at one point, no matter what.

What this searching has highlighted for me is that you get different results depending on what kind of environment you assume you are working in, such as the "projective plane" vs. The Riemann Sphere (two different models of surfaces on which you might be graphing your lines and circles - different ways the grid might behave at infinity). The following comment by the author of the article illustrates some of my thoughts better than I can:

The projective plane is one of a few different ways of looking at the situation (there’s no “correct” one). Having a center at infinity dictates the direction of the line, so two “circles” with different centers on the line at infinity would be two lines that cut across the plane at different angles. They’d intersect at one point (somewhere in the plane).
Circles and lines are different in the ordinary “Cartesian” plane, nearly the same in the projective plane, and exactly the same in the “compactified” plane (the Riemann sphere). It’s just a matter of picking your topological poison.
I prefer the projective plane because it’s easier to picture. Most mathematicians prefer the Riemann sphere (a point at infinity instead of a line) because it makes complex (as in “square root of -1”) mathematics easier. But, you know, whatever works.

Mathematics seems to be largely about about picking a set of assumptions (crafting an environment - a physics in a sense) and then seeing what sort of patterns emerge when you set up situations and follow the assumptions to their conclusion. Then, I suppose the mathematical truths of the highest order are those which are true in greatest number of mathematical universes, or which must apply to all reasonable universes.