The Circle

[i blame blame]

Its a matter of Perspective.
The following illustration is a side view of a "perfect circle"!


___________________

But a perfect circle shouldn't have any thickness.

[i blame blame]

(yes, circles can have 'points', and 'points' can be circles!)

Mathematically, circles DO have points, but points can't be circles.

[i blame blame]

There cannot ever be 1/3 decimally.

So just go ahead and use a numbering system of base 3, if this bugs you so.

[nameless]

^^^
Is there a point to your niggling comments re; my posts?
Obviously you missed (or ignored) my point.

[i blame blame]

^^^
Is there a point to your niggling comments re; my posts?

I intended to point out that a numbering system of base 10 is completely arbitrary and only determined by culture. Perhaps this system gained dominance throughout most of the world because we have 10 fingers and 10 toes. Any rational number could be represented by a single digit if you chose an appropriate base to do so.

Obviously you missed (or ignored) my point.

I missed it.

[nameless]

I intended to point out that a numbering system of base 10 is completely arbitrary and only determined by culture. Perhaps this system gained dominance throughout most of the world because we have 10 fingers and 10 toes. Any rational number could be represented by a single digit if you chose an appropriate base to do so.

I tend to agree with that, but I think that the term 'arbitrary' needs examination and, in light of modern thought and science, redefinition.

Obviously you missed (or ignored) my point.

I missed it.

The notion of 'imperfection' is Perspective based; perceivable to/as certain Perspectives. Oddly enough, people who perceive (egoically) imperfection would most often describe themselves as 'imperfect', yet the perceived 'imperfect' Perspective assumes to define 'perfection'.
Of course, all perceived definitions of 'perfection' sum-total to the complete definition of 'perfection', whether of a 'circle' (which the wagons do when attacked, and if successful, can be defined as a 'perfect circle'... context.
One can certainly define a 'perfect circle' as not existing, but, within context, everything exists!

[bytesplicer]

Sorry to resurrect an old thread, just thought I'd join my fellow circle lovers!

Quick thought (or pen and paper) experiment. Say you have a circle (or sphere) of a particular area/volume, 1 for the sake of argument. Sticking with 2d for now, the surface area (circumference) and radius are easy enough to calculate from the area of 1. Now, say you want to increase the surface area of this system, with the constraint that the area must remain constant. How can this be achieved? If a way can be found, can it be applied iteratively, thus increasing the surface area without limit, while still retaining a constant area? I can think of one way of doing this (assuming I didn't bork the maths or logic, which is a distinct possibility) which is quite interesting in itself, though there may be others.

[Richard Baron]

Now, say you want to increase the surface area of this system, with the constraint that the area must remain constant. How can this be achieved?

I am not sure exactly what is meant here. You seem to be saying that you want to increase and keep constant the same thing. But subject to that doubt, is your method the one that is used to generate the Von Neumann Paradox, which is a 2-dimensional version of the Banach-Tarski Paradox?

[bytesplicer]

No, there are similarities, but there is no paradox involved here, at least not from a geometrical point of view. Those paradoxes are a lot more interesting (and complicated) too! Not trying to increase one thing while keeping it constant, here I'm thinking about increasing the surface area (circumference) without changing the area. This is possibly a variation or a puzzle under another name, not quite sure, and it also relates to fractals. The method involved in those paradoxes does give a big clue though

Note, the 'trick' seems to apply to shapes other than circles/spheres, it's more a general property of geometric constructs, but in applying the trick iteratively there seems to be advantages and more elegance (as is usual) when spheres/circles are involved. I find it interesting, though still may have fluffed the basic maths, part of the reason I'm fielding the question is I suck at maths!

[Richard Baron]

Ah, now I think I understand. I was thrown by your use of the term "surface area" for circumference. The circumference is a length, not an area.

So is the idea to keep the circle shape at a certain level of resolution, but to make the boundary wiggle, so as to make it longer, with the wiggles being small enough that they are only detectable at a higher level of resolution? If so, I guess that the main problem is to define the distinction between levels of resolution in such a way as to make the result interesting. I do not, for example, think that trading on the difference between rationals and reals would give you a straightforward route to the result, both because any circle (on a co-ordinate system) is going to include points with irrational co-ordinates, and because there are infinitely many different rational numbers in the gap between any two reals.

Or perhaps you do something else?

[bytesplicer]

Yeah, apologies for the confusing terminology, started writing about spheres then changed my tack to circles. What I really meant by surface area was a boundary, a length in 2d or an area in 3d. Not much help to someone trying to answer, lucky for me you got the gist!

That's not what I had in mind, but again very similar. I'd read about this technique before, and it's very interesting in itself, and seems to belong to a similar class with regards to increasing boundary while fixing area. Do you know if this particular field has a name? I couldn't find much on it, particular interested in what some of these structures look like over time.

My own variation is stupidly simple in comparison, and maybe a bit of a cheat. Simply divide the circle (or in general, polygon, or solid in 3d) into 2 or more smaller circles. Doesn't matter if they're disjoint or touching (though overlapping seems difficult to calculate). While the total area will still sum to 1, the total boundary seems to grow exponentially as you get more and smaller pieces. If you add a constraint about arrangement of the circles, say symmetrical around the starting circle, over time you'll get a larger circle made from the smaller ones, as if you'd just answered the original question by saying 'expand the circle and make it less dense'. Sort of two answers in one.

This seems a natural product of geometry, if you divide a shape you'll end up with more boundary. You could divide the sphere into pie slices and get more, and this leads me to the main point. Which answer, including your own, is 'best' in regards to the increase (the fastest increase in boundary)? There seems to be quite a few different ways to do it, say you had a smart computer and asked it to solve this, what shape would it start with, and what would it do to that shape to solve the problem? What other properties do different approaches have? How do wiggly boundaries compare with division? What kind of problems, you describe issues with a wiggly solution, are there similar issues with division. With the circles method you get to deal with only circles at every stage, whereas what you describe requires using different components (say, alternating semi-circles or triangles inside and outside the boundary, thus balancing area while increasing surface area). Also, with using circles, you get nice gaps into which smaller circles can 'grow', providing indefinite 'inward' expansion. Trying this with say, squares also yields increased surface area but seems a lot more 'messy'. In my example, if using circles and you set a constraint that new circles 'grow' adjacent to their parents, and only grow so as not to overlap, I can imagine you'd get some amazingly complicated structures and emergent behaviour.

I'm thinking along the line of what 'rules', if any, govern this, relating to the initial shape and how you modify it. In the way that a circle represents the minimum boundary for a given area, is there a shape that produces a maximum increase in boundary as you modify it, and how does this vary with the choice of how you divide the shape. Are there variations that could pack into or even tile the entire plane at infinity. Also, how would these techniques change with higher dimensions? Could you use a finite 4d 'area' to generate an infinite 3d volume (assuming the boundary of a 4d shape is a volume). What happens if you iterate rules across dimensions, say using the infinite volume from 4d to generate infinite area in 3d, then using that to generate infinite circumference in 2d. Can you work in reverse, using lower dimensions to generate higher, and if so, do you always end up with less? 1d has me completely stumped!

Another variation I can think of involves using the circumference from one circle as the diameter of another (and vice-versa) resulting in a 'stacking' of circles (or spheres). Not quite the same problem, but you get an interesting pattern of distances between the nested circles, which seems to look the same at all resolutions.

My final thought, are there choices of shapes or rules that can decrease the boundary, working in reverse (while keeping the area constant). Can't seem to get my head round that one, but maybe it's easy.

Sorry, lots of questions and thoughts, thinking out loud. I find the whole area quite fascinating. My feeling is that circles/spheres/nspheres may have some mathematical 'advantages' with regard to this kind of expansion, but just speculation. No joy in finding anything online about this kind of problem.

Oh, and sorry for long post, still trying to get my verbiage under control!

[i blame blame]

I intended to point out that a numbering system of base 10 is completely arbitrary and only determined by culture. Perhaps this system gained dominance throughout most of the world because we have 10 fingers and 10 toes. Any rational number could be represented by a single digit if you chose an appropriate base to do so.

I tend to agree with that, but I think that the term 'arbitrary' needs examination and, in light of modern thought and science, redefinition.

Oh also, you can represent irrational numbers with a single digit. This is often done when dealing with multiples, fractions trigonometric operations or a combination thereof of π.

Obviously you missed (or ignored) my point.

I missed it.

The notion of 'imperfection' is Perspective based; perceivable to/as certain Perspectives. Oddly enough, people who perceive (egoically) imperfection would most often describe themselves as 'imperfect', yet the perceived 'imperfect' Perspective assumes to define 'perfection'.
Of course, all perceived definitions of 'perfection' sum-total to the complete definition of 'perfection', whether of a 'circle' (which the wagons do when attacked, and if successful, can be defined as a 'perfect circle'... context.
One can certainly define a 'perfect circle' as not existing, but, within context, everything exists![/quote]I see. In the context of mathematics a circle should have 0 thickness.

[Richard Baron]

Hello Bytesplicer

It sounds as though what you propose is, or is very close to, fractals generated by iterated function systems.

Yes, you can certainly have an arbitrarily long boundary enclosing an area that is less than a given finite number. An obvious example is the Mandelbrot set. I think that you have to be a bit careful about notions of infinity and of determinate area, though. We don't just have the usual need to define infinity and limits carefully. We also have the twist that a fractal's Hausdorff dimension exceeds its topological dimension - so in a certain sense, the boundary of the beetle-shape that is the Mandelbrot set doesn't have a length.

Another element that I detect in your thought is the point that some shapes get you more area for a given boundary length than others. If you want to minimize the boundary for a given area, draw a single circle. Any other shape (such as two circles, or a rectangle) will need more boundary for the same area.

[bytesplicer]

Hi Richard

My thought about the circle representing a minimum boundary is what led me to wonder whether iterating the division (as per my last post) would also represent the minimum rate of increase in boundary, or perversely, whether it could be a maximum. Was thinking along the lines of fractals, and got slightly confused, because, while (to me) the circle appears almost to be 'the ultimate fractal' for many reasons, it doesn't quite seem to be considered so because of its regularity. The concept of a zero-length (or null length?) boundary is also something I didn't consider at all, and has got my brain buzzing

Thanks very much for your reply, just from the terms you included I've got a lot of references and related topics to read up on now, dipped my toes into a big pond! Gulp!

[i blame blame]

I found it !

I found it !



A Perfect Circle


http://www.collegehumor.com/video:1735158

Impressive.