Circle, Square, and Infinity

[shintashi]

A while ago, I fonud myself between degrees and having a lot of spare time to do some studying. The two subjects I decided to crack down on were Physics and Mathematics, and using the former to be somewhat of an instruction in the latter.

 

Years ago I worked on formulating a hypothesis about the difference in n-dimensional manifolds between what is required for a circle/sphere and what is required for a polygon or polyhedron. I noticed that while squares really did require four lines and two dimensions to exist, a circle was really just one line along curved space and had more in common with a mobius strip than a square.

 

While working with some translations on various math texts including three books by Euler and several essays on people such as Gauss and Descartes, and naturally a lot of research into the history of Zero and Berkeley's retort on fluxions, I decided to get back into the research I put off back then.

 

First of all, I propose that a circle is jsut a line. If a circle is just a line, then it is only necessary for a point plus a vector to generate. If space is curved, then a circle a perfectly straight line, while space curves around an imaginary point out to a given radius.

 

I also noticed that a sphere couldn't in its construction be truly 3 dimensional because it was too simple to construct. A sphere is not two circles overlapping in a 360 degree pattern, but instead, a 360 degree curve of space acompanied by a 180 degree turn at a right angle.If you set an ordinary coin, such as a penny into a spin, you may notice the optical illusion of a sphere forming.

 

This is somewhat connected to the idea of why waves travel in sine functions. At first this seems entirely ridiculous for a motion to go up then change direction and head down, then, defying all logic the thing turns itself about and moves back up again.

 

Many times I looked at a wheel, such as a bicycle reflector moving across a plane generating the frequency but recently I'ts occured to me, that if the curve is not seen as a snake winding through a medium, but instead seen as a series of derivative points being emitted as linear shocks, it would then appear to be a wave, but instead would be a series slightly out of sync kind of like how a caramel machine moves back and forth to deposit candy on a wafer bar.

 

Anyway, that's not the main focus. What I was really thinking about was something a bit different, and the concept of Epsilon, or to say, a number slightly bigger than 0 but still basically infinitely small.

 

If we are to establish that a circle is exactly equal distance from its center on all sides, then we are to understand that a circle with a radius of zero is identical to a point. We are also to understand that when the radius is at 90 degrees to the circumference, the circumference is to be infinite.

 

If the circumference is at 90 degrees to the radius, which is a straight line, incedentally, then it must be a line, even though before it was a curve. Being a line requires a dimension in the same way being a square requires two.

 

this leads me to believe that if the radius is set to zero, we have a point and to infinity we have a line. If these two aspects are true, then how many dimensions are really used in the creation of a circle or sphere? It also seems obvious that collapsing a sphere's radius at any point would yield a point in three dimensional space, but it seems to me that without a second vector function it is highly probable that instead of completely eliminating the curved space you could actually just reduce the sphare back into a circle, or potentially, a line. I then began to wonder if a computation using complex numbers would better illustrate a way of reducing or generating a sphere.

 

This, admittedly, I have yet to resolve, although it seems I discovered something also odd.

 

There is no such thing as a radius measured as epsilon, for it will either be zero or it will be an integer, however,

 

There is no such thing as a polygon or polyhedron that having its most central points converge toward a point will ever reach a zero radius on all points. Instead, some proportion must always be epsilon.

 

If you begin with something as rudimentary as a point and a square, you will find this to be true. For only at zero and infinity do circles and squares converge, but at epsilon does a circle begin, and at epsilon does a square end.

[LaurieAG]

I also noticed that a sphere couldn't in its construction be truly 3 dimensional because it was too simple to construct. A sphere is not two circles overlapping in a 360 degree pattern, but instead, a 360 degree curve of space acompanied by a 180 degree turn at a right angle.If you set an ordinary coin, such as a penny into a spin, you may notice the optical illusion of a sphere forming.

...

Anyway, that's not the main focus. What I was really thinking about was something a bit different, and the concept of Epsilon, or to say, a number slightly bigger than 0 but still basically infinitely small.

...

If you begin with something as rudimentary as a point and a square, you will find this to be true. For only at zero and infinity do circles and squares converge, but at epsilon does a circle begin, and at epsilon does a square end.

 

Hello shintashi, this might extend your line of thinking.

 

If an electron/photon is a point going around in a circular orbit very quickly it appears as a sphere. In essense it can be modelled as a point and a vector if you center your observations on the single point.

 

As soon as you construct anything by combining two such point objects they gain a separation distance that cannot be reduced without materially effecting the construct.

[Turtle]

A while ago, I fonud myself between degrees and having a lot of spare time to do some studying. The two subjects I decided to crack down on were Physics and Mathematics, and using the former to be somewhat of an instruction in the latter. ...

 

 

There is no such thing as a polygon or polyhedron that having its most central points converge toward a point will ever reach a zero radius on all points. Instead, some proportion must always be epsilon.

 

If you begin with something as rudimentary as a point and a square, you will find this to be true. For only at zero and infinity do circles and squares converge, but at epsilon does a circle begin, and at epsilon does a square end.

 

Given nothing but minimal spheres (your epsilon), then one is left with a close-packing problem. If you are as serious about your study as you imply, I would like to suggest you read Buckminster Fuller's seminal work, Synergetics: Explorations in the Geometry of Thinking . Here's the link to an online version >>> R. Buckminster Fuller's Synergetics

 

Here's a link to a diagram I feel is germane to your discussion, i.e. the equi-lateral/equi-angular triangle is the better polygon than the square for minimalizing a metric. Note that no polygon except the triangle is 'automatically' equi-angular if it's equi-lateral and vice versa. >> 100.00 SYNERGY

 

Infinitely consumed by Fuller's geometry,

Turtle

:naughty: :)

[shintashi]

Mr. Bucky Ball?

 

I'm always happy to find links to free math resources. This afternoon my companion acquired a copy of Penrose's road to Reality per request. She went for a Guide on Quantum Physics herself (she's a biochemistry major with an insatiable curiosity for related fields). either way, the concept of a triangle seems ideal for demonstrating the points between geometric surface, epsilon and ratio.

 

As for the rotating electron comparison, it reminds me someone of string theory theory.

 

Thanks to everyone for replying - I'm always open to new possibilities.