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Take the number $\pi = 3.14159265359...$

Since $\pi$ is an infinite, non-repeating decimal, strings of decimals can repeat, and it would defy the definition of $\pi$ if we say that such a string of length $n$ repeats infinitely many times in sequence, is it a contradiction to state that other patterns or structures, outside of in-sequence ordering, of repeating sequences arise from the number?

For example, say that the $n$th string of length $k$ of $\pi$ is called $a$ and denoted as: $$...a_1a_2a_3...a_{k-2}a_{k-1}a_k...$$

Is it possible that the structure (I think I'm using this term loosely) of $a$ is isomorphic to the structure of $b$ such that $b$ is of length $k$ and denoted as:

$$...b_1b_2b_3...b_{k-2}b_{k-1}b_k...$$

If so, does this constitute a contradiction to the definition of infinitely non-repeating?

David Farmilant
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    I'm confused how group theory or isomorphisms are relevant here. What groups are you comparing? And what is the "structure of $a$ and $b$"? Also, being an "infinite, non-repeating decimal" isn't parts of the definition of $\pi$. $\pi$ is (usually) defined as the ratio between a circle's circumference and its diameter, and from this definition it can be proven that it is irrational. – Brian61354270 Mar 06 '20 at 03:38
  • I'm not sure how to argue for the use of those two topics, so I removed them. My understanding of the topic is based more in intuition and that can be isolating. The way I'm thinking of $a$ and $b$ is that they represent some value or maybe a matrix representation or projection from a higher dimension which is in and of itself a pattern. Does it make sense to ask if irrationality is linear? – David Farmilant Mar 06 '20 at 04:04

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