Tini Circle Groups (TCG)

[Epsilon=One]

Tini Circle Groups (TCG)

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This post is in memoriam: René Descartes [1596-1650]

Tini is a neologism/acronym for Tangent, Infinity Integer.

A Tini Circle Group (TCG) can begin with a circle that has a radius with a value of any Natural integer. The circles can diminish without end with radii of Natural integer (NI) curvatures. (Curvature is the reciprocal of the radius; i.e. One divided by the radius, "1/r.") The circles can be internally or externally tangent.

Thus, it is hueristically illustrated that: there is never "space" that can be "empty."

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Brunardot Groups (BG)
Tini Cirt 1h with Tini Cirts: 2:1s, 3:1s, 4:1s, ad infinitum
Integer Curvatures from
One (The Elliptical Constant) to the Infinite




Integer values
for the letters A thru G, above,
can be generated by any Natural integer.

All circular curvatures (reciprocal of the radius) are Natural integers.

For every Natural integer, there is at least one, and often many arrays, that are all never ending. Can you calculate the Natural integer arrays and all their branches (corollaries)?

All curvatures are a simple algebraic function of the preceding array’s Natural integer curvature.

There are two categories of Tini Circle Groups (TCG): symmetrical and asymmetrical.

There are three types of asymmetrical Tini Circle Groups (TCG): single, dual, and hylotron.

All four of the various groups can be inserted within any circle of any other category or type of TCG.

All TCGs (of external circles) are uniquely described by the largest two circles of the group; and, when necessary, an alpha character designation is added for the type.

The term for uniquely defining a Tini Circle Group is Tini Cirt (Tangent Infinity Circle Term). An example of a Tini Cirt is: 8:5a, which describes the large outer circle's integer curvature as 8; and the largest inner circle as 13 (8 + 5); and, the category is asymmetrical..

All the following circles’ integers, to Infinity, are set by the first two circles’ curvatures; and, all integer circle curvatures are calculated with simple, algebraic arrays.

Of course, internal tangent circles are independent of the outer arrays.

Thus, internally and externally, every circle can have every space to Infinity filled with a smaller circle that has an integer for its curvature.

One must ask: Why always Natural integers when the complex equation involves four variables to factor and square roots that must be divided?

Then, imagine that the symbolic packed circles are
spheroids in the manner of pulsing Pulsoids ...!

Why ?
And, again,
Why ?
And, again,
Why ?
Over and over,
until . . . Infinity.

"Truth lies with simplicity,
which is Nature's signature."

The basis of Tini Circle Groups is René Descartes' formula for Tangent Circles: See: Tangent Circles

©Copyright 2005-2008 by Brunardot. All rights reserved.
Terms: Dialogue21.com, Brunardot, and Pulsoid Theory must be cited.

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[Midgar21]

Curvature

I was under the impression that curvature is expressed by 1/r, with r being the circle's radius. In any case, for the pictures you posted, I'm assuming that every circle here has a radius smaller than 1, since that would be the only way for a circle to have a natural integer as its curvature (e.x., for a circle to have a curvature of 8, its radius must be 1/8, since 1/[1/8] = 8). But then if these circles thus have radii smaller than values of one, then how can epsilon exist?

[Epsilon=One]

Each circle has a radius that is the multiple of a common denominator.

I was under the impression that curvature is expressed by 1/r, with r being the circle's radius.

You are correct. If I stated otherwise, please indicate where and I will make the correction.

In any case, for the pictures you posted, I'm assuming that every circle here has a radius smaller than 1, since that would be the only way for a circle to have a natural integer as its curvature (e.x., for a circle to have a curvature of 8, its radius must be 1/8, since 1/[1/8] = 8). But then if these circles thus have radii smaller than values of one, then how can epsilon exist?

You are arguing that a value for one has been expressed, which according to Gödel, is not possible. (Of course, I disagree with Gödel.) Nevertheless, my argument is that each circle has a radius that is the multiple of a common denominator.

And, also of course, the circles of a Tini Circle Group symbolize a Natural spheroid, a Pulsoid, which is an ellipsoid; and all ellipsoids are defined by the Elliptical Constant (EC). That is to say: the “common denominator” of all TCGs is the EC, which is the heuristic symbolization of the Conceptual Unit (CU).

[Epsilon=One]

I found the error . . . in the original post and corrected it.

You are correct. If I stated otherwise, please indicate where and I will make the correction

Thanks. I found the error concerning diameter vs. radius . . . in the original post and corrected it.

Amazing how once an error is made, the writer can never see it.

[Midgar21]

Re: Each circle has a radius that is the multiple of a common denominator.

Nevertheless, my argument is that each circle has a radius that is the multiple of a common denominator.

I'm not quite sure I understand. Could you perhaps illustrate what you mean using one of the diagrams?

[Epsilon=One]

There is a deep, common relationship among the Tini Circle Group integers . . .

I'm not quite sure I understand. Could you perhaps illustrate what you mean using one of the diagrams?

Yes.

There is a deep, common relationship among the Tini Circle Group integers such that by changing any curvature integer by a single integer, or any value, the entire array collapses.

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In the above diagram, I have arbitrarily chosen the adjacent, tangent curvatures of 156, 182, 254, and 1268, near the top center.

The highest relationship is that when each of the selected integer numbers is squared (24,336, 33,124, 64,516, and 1,607,824) they total exactly 1,729,800.

And, when the same selected numbers are added (156 + 182 + 254 + 1268) they equal 1,860.

If 1,860 is squared (3,459,600) and divided by 2, it is exactly the same integer 1,729,800.

The same process will yield equal integer values, regardless which four tangent circles you select. And, arrays can be constructed such that the smallest integer value of an array can be any integer.

This demonstrates a complex, unique, integer, relationship between tangent circles (special ellipses) that extends to the Natural arrangement of fundamental quanta, which have an outer ellipsoidal "envelope" that becomes spherical after a relatively few pulses and subsequent compression.

The "Natural arrangement" of quanta is such that there is never any "empty" space.

"Space" quanta (the Dyosphere) and fundamental, Intrinsic time (FIT) redefines Einstein's space-time.

[ste]

Re: Tini Circle Groups (TCG)

Your example Tini Cirt 84:172s looks quite similar to your third (moving) signature image. How are these arrays of circles related to Pulsoids and Resoloids?

[Epsilon=One]

Space is “packed” with compressed Pulsoids.

Your example Tini Cirt 84:172s looks quite similar to your third (moving) signature image.

I do not understand exactly what this comment concerns. ??? You may be comparing heuristic symbolism to a Natural phenomenon. The symbolism is related to all fundamental geometry of motion.

How are these arrays of circles related to Pulsoids and Resoloids?

These arrays have little to do with Resoloids other than that Resoloids approach being circular in cross-section and each have integer relationships.

Pulsoids have a more direct relationship to the arrays because of an analogous spheroidal geometry with an integer relationship of its major diameter to its salient structural geometry. Pulsoids are the “dark matter” “packed” quanta of space.

There is no empty “space” and thus, space is “packed” with compressed Pulsoids as heuristically demonstrated with Tini Circle Groups (TCG).