When I was in the later grades of Grammar school; I had become interested in polygons. During that period, I had remembered an old Sesame Street segment where Ernie asks Bert how many sides a circle had: One? None? Well, trying to draw polygons, and realizing that the more sides you add, the more it looks like a circle; I figured it must really besides! In technical terms, a circle can be viewed as the “limiting case” of a regular polygon with a fixed radius. As the number of sides approaches infinity, the length of the sides decreases towards zero. The angles approach 180°, and the figure’s sides smooth out to a perfect circle! Notice, they approach the values of 0 or 180. They don’t reach it, since you can’t actually reach infinity (as if it were a single point anyway). So these values are called asymptotes.
On the other hand, if we now fix the length of sides, the radius now approaches infinity instead, and the adjacent sides flatten out to an infinite straight line tiled with the original lengthed line segments (the “Sides”). This has been called an apeirogon. (It is the shape of the new “number line” I propose above with zero as the common starting and ending point, but the line still extending infinitely both ways!) Just like “pentagon” means “five sides”, and hexagon” means “six sides” (“-gon” actually refers to the number of angles, as it will always equal the number of sides in 2D geometry. “gon” is actually directly akin to the word “knee”); “apeiro-” is the prefix for infinity. It basically means “without perimeter”.

Since this term is used only for the straight line case where the length of sides was fixed; I tried to come up with other terms in a math discussion for the fixed radius “circle” version, such as “closed apeirogon”, or “achanegon“; “achanes” being another Greek term for infinity (http://www.kypros.org/cgi-bin/lexicon); this one meaning “roofless” (according to this site http://www.quantavolution.org/vol_13/firenotblown_16.htm; this was the ancient terms used by Sophocles, Fragment 1030, and Kerenyi, The Gods of the Greeks, p. 270 regarding the dancing floor at The labyrinth at Knosos).

We end up with a paradox; because the circle, and the endless straight line are supposed to be the same object! What has happened, is that the straight line, in which the lengths of the sides remained fixed is an infinitessimal projection of the perimeter of the circle! The sides in the “circle” projection had shrunk down to zero in length, remember! This makes sense, as every line segment is considered to be composed of an infinite number of points, which are zero in length. It has to be, as mathematically, there is no limit to division of length. Take any object with a width, or area, those lengths can always be halved, tenthed, hundredth, etc. Take any space between objects, and it can always be halved, etc.* A polygon is defined by how many of these straight lines meet at angles. The circle’s perimeter does not have straight lines meeting at angles, but is rather a smooth curve, still defined as a set of points. So all you have to do is consider the circle’s “points” as its “sides”. If you blow one of these sides up by a magnitude of infinity, you will see it in a series of line segments at 180° with an infinite radius! An infinite number of these line segments still occupies a single point on the circle!

In connection with this, you may wonder how a straight line composed of line segments at 180° can stretch out to a curved, closed figure. Well, just as I just said, that infinite line is really only one single point on the circle. Since the radius is fixed in the circle, then at any given angle, the same radius length from the center, you will find a point, which will magnify to an infinite line. The entire set of these infinite points is the circle! While the lines segments are “next to each other” (adjacent) at 180°; the individual points on the circle are not “next to each other”. If you take one point, of zero length, and place it “next to” another point, also of zero length, they won’t sit beside one another; but occupy the same spot! (Unless you have some amount of space between them). It would take an infinite number of them to reach the “next” place; and even that is undefinable, as they do not fill any space for there to be any “next place”. “Next to” nothing is still nothing! Only when magnified infinitely, do the points become line segments that can lie “next to” each other.

It is the fixed radius that determines that at any angle; even thousandths or millionths of a degree, and smaller, there will be one of these “points” that magnify to an infinite line. In other words, a finite radius in an ∞-gon forces the object into a finite convex hull (surface closed around a point) whose perimeter is set by the radius. The finiteness of this perimeter forces the line segments and even the infinite line they make up, down to points, in order to “fit”. These will form the entire set of points that same radius length from the center, and this is the definition of a circle! This paradox is the result of messing around with zeroes and infinities in the same equations!

(full width table back on original article site until I figure how to change the margins here):

 polygon name n (Number of sides) C (angle at center: 360/n) A (interior angle of polygon: (180-c) S (length of sides) if r=1 (2 cos A/2) r (radius) if s=1 (.5/cos A/2) common “shape” name trigon 3 120° 60° 1.732 0.57735 triangle tetragon 4 90° 90° 1.414 0.7071 square pentagon 5 72° 108° 1.17557 0.85065 hexagon 6 60° 120° 1 1 octagon 8 45° 135° 0.765 1.30656 decagon 10 36° 144° 0.618 1.618 hecatontagon 100 3.6° 176.4° 0.06282 15.918 hecatonogdocontagon 180 2° 178° 0.0349048 28.649 triacosiahexacontagon 360 1° 179° 0.017453 57.296 chiliagon 1000 .36° 179.64° .006283 159.155 megagon 1,000,000 .036° 179.999964° .00000628318 159,915.5077 gigagon 1,000,000,000 .000036° 179.999999964° .00000000000000628318 159,155,077.52 teragon 1,000,000,000,000 .000000036° 179.999999999964° .00000000000000000628318 159155077524.438 [“Great Zero”]-agon) 69720375229712477164533808935312303556800 .0000000000000000000000000000000000000051634° 179.9999999999999999999999999999999999999948365° .0000000000000000000000000000000000000000901 11096307049865062522983129408834xxxxxxxxxxx.x apeirogon ∞ 0° 180° 0 ∞ circle

Everything with n as 1000 and above, with a radius that fits on a page, will look like perfect circles. n=100, 180, and 360 will look like circles with bumpy perimeters. 1000 and one million, the “sides” might be smaller than the ink dots making up the circle. The next two, the sides would be smaller than light waves. For the largest finite number; “great zero”, the sides would be smaller than the strings that make up the fabric of space.* Magnify one of the sides to 1 inch (or centimeter, etc) , you will see what looks like an infinite straight line tiled with that side and its adjacent neighbors. Eventually, you will move along the perimeter in sizeable numbers of degrees, but the more sides, the longer you will have to travel to get to the next measurable “dot” or degree. A chiliagon whose sides were one inch long, would be have a radius of about 26½ feet, and be 53 feet in diameter! A megagon whose sides were one inch, would have a radius of 2½ miles, and be 5 miles wide. A gigagon with one inch sides would be just over 5000 miles wide (a little bigger than the planet Mars), and a teragon, 5 million miles. With 69 tridecillion sides, it would be almost 30 trillion light years wide; thousands of times larger than the known universe! A single degree would be wider than the universe.

Starting from the center (also beyond the visible universe), in any angle you travel in, you will find what looks like an infinite series of line segments, (and keep in mind, one inch each!). If you tried to travel along this line, you would basically never be able to get to the point where the radius one degree; or even a thousandth of a degree, away intersects the line. It’s just too far away for us to reach with the limitation of space and time. And this with a number that is still technically finite! The “[googol]gon” would even be bigger than that. (Or the sides smaller with a 1 inch radius). Similar figures for the teragon in the table, but with around 100 0’s or 9’s. One inch sides producing a radius and diameter trillions of trillions times larger than the universe, and one inch radii producing sides trillions of trillions times smaller than the smallest hypothetical particles of the universe. These are even ‘more than perfect’, perfect circles, since the smallest particles making up any circle we draw would not even be that small! The chiliagon for all of our practical usages, is a perfect circle. You probably would only barely be able to feel the angles or make out the ends of the one inch sides of the 53 foot model. They would all be so slight as to make a smooth curve. Higher than that, it will be indistinguishable from a circle in our perception.

So, infinity is the extreme case where our everyday logic breaks down, and you have an infinite series of infinite straight lines at every infinitessimal point. You would never be able to get to the next point traveling on one of these lines. You would never leave the radius at the angle you started from. They would not even be “too far”. In this case, two points separated by any degree would really not even be connected to each other. There would be an infinite amount of space between them. In mathematical terms, the infinite line tiled with countable line segments is designated with an infinite quantity known as “aleph 0”. However, when we “zoom out” to the whole circle, we are actually making a “quantum leap” from aleph 0 to another infinite quantity called simply “C” (for “continuum”). Even though they are both infinite, C is actually “bigger” than aleph 0. While aleph 0’s units are countable, C’s “units” (the points), are not countable, because of the infinite density of points described above. There is no way to even begin assigning numbers to each point to “count’ them. Hence, the paradox is the result of leaping between two types of infinity. One infinity we “approach” forever, by adding countable units, and the other, where we have presumed to have “reached” infinity; yet have made an infinite “leap” to do so.

The circle is made to be infinity, rather than simply assigning it as 1,000 or 1,000,000 sides, because it is only a hypothetical concept. One million sides is basically indistinguishable from one billion sides, or larger than that, or all of the numbers inbetween. The figure is already at its limiting shape, and it’s only the scale it is constructed on, and the perception of those measuring it, that make them possibly different. So we might as well say that the perfect circle is the entire set of points a given radius from the center, which hypothetically is infinite, even if we can’t measure anything that small or large.

Likewise, by extension, a sphere is an achanehedron (infinite number of points for its “faces”). A hypersphere is an achanechoron, and a hyperhypersphere is an achanetesseron. (“hedron” means “seat”, for the “faces” of 3D closed figures. Other extended geometrical terms and their logical extensions you can find used online: “choron” means “room” —connected to “choreography“, for the 3D “cells” of a 4D polytope; and -tesseron is for the 4D hypercells of a 5D polytope. “-tope” is the n-D collective suffix for facets in any dimension. “facet” is the collective term for the “sides” that bound closed figures in any dimension: lines segments in 2D; “faces” or plane segments in 3D; 3D “cells” in 4D, etc. ).

*If string theory is true, there may actually be a “smallest length” called the “Planck length” (10-33 cm). This is so, because the theory proposes the “fabric” of space as itself consisting of loops of string that are that length. Smaller that that, the concept of “space” breaks down, and there is no longer any medium to make any measurements on. Still, numerically, for the sake of this discussion, we can conceive of a tenth of that size; 10-34 cm.

1. I have proven that the circle is a polygon with 2^55 sides
Send an email and I email my work that prove the theory.
Thanks

• I had been thinking over the months what to make of this, and it sounds interesting, but if you want, you can post the basic idea here.

• I have new theory in calculus to share.
Send an email and I will email my work file to prove it with no charge.
Thank you,
Giuseppe
This is my email gstagno31@gmail.com

2. I had to make an account, because this topic has, rather suddenly, been racking my brain, and I had come to a similar conclusion that after adding many sides the number becomes infinitesimal, making it near zero, in which lies the paradox. Thanks for sharing!

• As I say for erictb I say for Theo.
Thank you,
Giuseppe

3. Hi erictb
The basic idea is: Arc Theory.
In drawing isosceles triangles on the circular sector,it has shown the area between
the Arc and the Side; all the triangles are varying in size smaller near the arc
until they cover the total area.
The small Arc length at this stage is equal at the small Side being best for completing
the total length of the Arc;Therefore,at this state the small Side is used
to calculate the length of the Arc.
Thank you,
Giuseppe Stagno

4. (also toward the end has an embedded video link to another one that explains “uncountability”).

In my view, numbers like “googolplex” are virtually useless. If there aren’t even that many particles in the universe, (and the number itself in base 10 can’t even fit in the universe), then it’s for all purposes “infinity” or “part of” infinity, since as pointed out, infinity is really a “kind” of number than an integer. They are basically as hypothetical as “infinity” itself!
But then “what about googolplex + 1?”, (to prove it is technically finite). On the scale of googolplex, “1” basically becomes meaningless. It’s virtually infinitessimal. It basically is like a unit of “uncountable infinity”. This is what I used to illustrate the “quantum leap” from an finite length side apeirogon to a finite radius circle. So from 1 to googolplex is also for all good purposes a quantum leap, and there is virtually no difference between googolplex and googolplex + 1 or even googolplex², or googolplexgoogolplex (which would then be substantially larger). They are all well out of the range of any possible quantities, and so to our everyday scale, are all like the “alephs” and “omegas” the video discusses.