fix typos in 4.appendix
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@ -233,7 +233,7 @@ Perhaps it is still unconvincing that expanding the integers in this way gives s
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In fact, since the expansions are in binary or (balanced) ternary, the integers should just
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In fact, since the expansions are in binary or (balanced) ternary, the integers should just
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be a subset of the 2-adics or 3-adics.
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be a subset of the 2-adics or 3-adics.
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Still, wanted to see what these numbers actually "look" like, so I whipped up an interactive diagram.
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Still, I wanted to see what these numbers actually "look" like, so I whipped up an interactive diagram.
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You should definitely see [this page](/interactive/adic/) for more information, but
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You should definitely see [this page](/interactive/adic/) for more information, but
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the gist is that *p*-adics can be sent into the complex plane in a fractal-like way.
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the gist is that *p*-adics can be sent into the complex plane in a fractal-like way.
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@ -258,10 +258,15 @@ Within these two clusters, the rightmost portion of them appears to be the same
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If you try switching between the *κ*-adic options, you can even see the smaller
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If you try switching between the *κ*-adic options, you can even see the smaller
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and larger shapes changing in the same way as one another.
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and larger shapes changing in the same way as one another.
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If you prefer not to use JavaScript, I also prepared a [Python script](./kadic.py) to run locally[^1].
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This is actually great news -- if you switch between the *κ*-adics and the "random binary" option,
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You can trace out images from this version can be seen below:
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you can see that the latter option tends to the same pattern as the 2-adics.
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Thus, even if the expansions for the integers are individually chaotic, together they possess a
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much different structure than pure randomness.
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:::: {.row .centered}
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If you prefer not to use JavaScript, I also prepared a [Python script](./kadic.py) using Matplotlib.[^1]
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Here are a couple of screenshots from the script, which demonstrates the self-similarity mentioned above.
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:::: {.row}
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::: {layout-ncol="2"}
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::: {layout-ncol="2"}
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@ -271,10 +276,5 @@ You can trace out images from this version can be seen below:
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Clusters of *κ*-adics, with self-similar patterns boxed in red.
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Clusters of *κ*-adics, with self-similar patterns boxed in red.
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::::
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::::
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This is actually great news -- if you switch between the *κ*-adics and the "random binary" option,
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you can see that the latter option tends to the same pattern as the 2-adics.
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Thus, even if the expansions for the integers are individually chaotic, together they possess a
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much different structure than pure randomness.
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[^1]: You will also need the [`divMod`](./cendree_DivMod_count_1024_256_digits.csv)
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[^1]: You will also need the [`divMod`](./cendree_DivMod_count_1024_256_digits.csv)
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and [`quotRem`](./cendree_QuotRem_count_1024_256_digits.csv) data files.
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and [`quotRem`](./cendree_QuotRem_count_1024_256_digits.csv) data files.
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@ -23,9 +23,9 @@ import { expansions as oldExpansions } with { base as base } from "../../../../i
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expansionsOrAdics = baseSelector == "b-adic"
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expansionsOrAdics = baseSelector == "b-adic"
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? oldExpansions
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? oldExpansions
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: baseSelector == "κ-adic, balanced"
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: baseSelector == "κ-adic, balanced"
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? adicExpansionsDivMod
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? adicExpansionsQuotRem
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: baseSelector == "κ-adic, binary"
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: baseSelector == "κ-adic, binary"
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? adicExpansionsQuotRem
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? adicExpansionsDivMod
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: d3.range(adicExpansionsQuotRem.length / 10).map(() => d3.range(15).map(() => +(Math.random() > 0.5)))
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: d3.range(adicExpansionsQuotRem.length / 10).map(() => d3.range(15).map(() => +(Math.random() > 0.5)))
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import { plot } with {
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import { plot } with {
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