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add static images to 4.appendix

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queue-miscreant 2025-03-05 03:15:21 -06:00
parent ca20316aba
commit b71e12a9ab
3 changed files with 23 additions and 2 deletions
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posts/polycount/4/appendix/cendree_divmod_fractal.png Normal file
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posts/polycount/4/appendix/index.qmd
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@ -186,7 +186,11 @@ cendree2DivModCycleExpansion = take 11 $
putStrLn . unlines . map show $ cendree2DivModCycleExpansion putStrLn . unlines . map show $ cendree2DivModCycleExpansion
``` ```
We get the same series if we expand 2 directly (which we can also use to check its validity): Note that in this case, we're truncating the alternating series!
This means that naively evaluating the series as before will not give the correct value.
To check the validity of this series, we can check that we get the same series before the truncated elements
by expanding 2 directly:
```{haskell} ```{haskell}
cendree2DivMod15Steps = map (pairEval (sqrt 3 + 1)) $ cendree2DivMod15Steps = map (pairEval (sqrt 3 + 1)) $
@ -194,7 +198,8 @@ cendree2DivMod15Steps = map (pairEval (sqrt 3 + 1)) $
putStrLn . unlines . map show $ cendree2DivMod15Steps putStrLn . unlines . map show $ cendree2DivMod15Steps
``` ```
We can *also* use this to get a series for negative one, a number which has a terminating expansion Up to the carry head, the series are the same.
We can do the same thing to get a series for negative one, a number which has a terminating expansion
in the balanced alphabet. in the balanced alphabet.
```{haskell} ```{haskell}
@ -253,7 +258,23 @@ Within these two clusters, the rightmost portion of them appears to be the same
If you try switching between the *κ*-adic options, you can even see the smaller If you try switching between the *κ*-adic options, you can even see the smaller
and larger shapes changing in the same way as one another. and larger shapes changing in the same way as one another.
If you prefer not to use JavaScript, I also prepared a [Python script](./kadic.py) to run locally[^1].
You can trace out images from this version can be seen below:
:::: {.row .centered}
::: {layout-ncol="2"}
![`quotRem`](./cendree_quotrem_fractal.png)
![`divMod`](./cendree_divmod_fractal.png)
:::
Clusters of *κ*-adics, with self-similar patterns boxed in red.
::::
This is actually great news -- if you switch between the *κ*-adics and the "random binary" option, This is actually great news -- if you switch between the *κ*-adics and the "random binary" option,
you can see that the latter option tends to the same pattern as the 2-adics. you can see that the latter option tends to the same pattern as the 2-adics.
Thus, even if the expansions for the integers are individually chaotic, together they possess a Thus, even if the expansions for the integers are individually chaotic, together they possess a
much different structure than pure randomness. much different structure than pure randomness.
[^1]: You will also need the [`divMod`](./cendree_DivMod_count_1024_256_digits.csv)
and [`quotRem`](./cendree_QuotRem_count_1024_256_digits.csv) data files.