BANACH-TARSKI PARADOX & THE REAL TRINARY LOGIC

MATRIX OF THE FIBONACCI, GEOMETRIC,

& PERFECT NUMBER SERIES

“A ball can be decomposed into a finite number of point sets and reassembled into two balls identical to the original.”

“The **Banach–Tarski paradox** is a
theorem in set theoretic geometry which states that a solid ball in
3-dimensional space can be split into a finite number of non-overlapping pieces,
which can then be put back together in a different way to yield *two*
identical copies of the original ball. The reassembly process involves only
moving the pieces around and rotating them, without changing their shape.
However, the pieces themselves are complicated: they are not usual solids but
infinite scatterings of points. A stronger form of the theorem implies that
given any two 'reasonable' objects (such as a small ball and a huge ball),
either one can be reassembled into the other. This is often stated colloquially
as 'a pea can be chopped up and reassembled into the Sun'. . . .

“The reason the Banach–Tarski theorem is called a paradox is because it contradicts basic geometric intuition. 'Doubling the ball' by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations preserve the volume, but the volume is doubled in the end.

“Unlike most theorems in geometry,
this result depends in a critical way on the axiom of choice in set theory. This
axiom allows for the construction of nonmeasurable sets, collections of points
that do not have a volume in the ordinary sense and require an uncountably
infinite number of arbitrary choices to specify. . . .^{2}

“It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another. . . .

“Using the
Banach–Tarski paradox, it is possible to obtain *k* copies of a ball in the
Euclidean *n*-space from one, for any integers *n* ≥ 3 and *k* ≥
1, i.e. a ball can be cut into *k* pieces so that each of them is
equidecomposable to a ball of the same size as the original.”

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1 =
Infinitely Dense Sphere, K = Keeper, M = Decompose & Reassemble, MM = M
Iterated Once**

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1,
0, , 1 = Real
Trinary Logic Respective Digit Analogues**^{3}

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et cetera, ad infinitum.

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Compare the Iterative
Sequential Summing of the Real Trinary Logic Geometric Series, Matrix of the
Numerical Geometric and Perfect Number Series**

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(For a full exposition available on this web
click on the image)**

et cetera, ad infinitum.

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Compare the Real Trinary
Logic Sequence Appearing as the Central “Trunk” of the Fibonacci Rabbit Tree
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**
(For a full exposition available on this web
click on the image)**

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**

et cetera, ad infinitum.

**Notes**

** ^{
1} Complete
text **available at:
http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox