108 lines
3.7 KiB
Plaintext
108 lines
3.7 KiB
Plaintext
---
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title: "Complex Embedding of *p*-adics"
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description: |
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Visualizing fractals generated by sending *p*-adics to the complex plane.
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format:
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html:
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html-math-method: katex
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categories:
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- interactive
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- algebra
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---
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```{ojs}
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//| echo: false
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{{< include ./showAdic.ojs >}}
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```
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About
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-----
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The Wikipedia article on the [*p*-adic valuation](https://en.wikipedia.org/wiki/P-adic_valuation)
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contains [a figure](https://commons.wikimedia.org/wiki/File:2adic12480.svg) whose description
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provides a way to map *p*-adics into the complex numbers[^1].
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The gist is to construct a Fourier series over truncations of numbers.
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Each term of the series is weighted by a geometrically decreasing coefficient *c*.
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$$
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[...d_2 d_1 d_0]_p \mapsto e^{2\pi i [d_0] / p}
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+ c e^{2\pi i [d_1 d_0] / p^2}
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+ c^2 e^{2\pi i [d_2 d_1 d_0] / p^2}
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+ ... \\
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f_N(d; p) = \sum_{n = 0}^N c^n e^{2\pi i \cdot [d_{n:0}]_p / p^{n + 1}}
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$$
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Assuming the first term dominates, this can be interpreted as placing numbers evenly
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around the unit circle according to their one's place.
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Then, we offset each by smaller circles, each centered on the last, using more and more digits.
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This produces a fractal pattern that looks like a wheel with *p* spokes.
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At each point on where the spokes meet the rim, there is another smaller wheel with *p* spokes,
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ad infinitum.
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This is somewhat visible in the Wikimedia diagram.
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All of the odd numbers are at the left side of the diagram, since the leading term of the series for them
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is negative one.
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The odd numbers are further split into those of the forms $4k + 1$ and $4k + 3$.
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Controls
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--------
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Each of the inputs corresponds to something from the above formula.
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- *b*, the base of the expansions.
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- Each point is the expansion of an integer in base *b*,
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which corresponds to a sequence of digits (*d* in the above formula).
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- *p*, the base used in the embedding.
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- The same *p* that appears in the above formula.
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- Truncations of digit sequences ($d_{n:0}$) are interpreted as strings in base *p*,
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then divided by $p^{n+1}$.
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- *c*, the geometric constant.
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- Smaller *c* means more tightly packed points.
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Note that only 1024 points are calculated, and that only fifteen terms of the series are used ($N = 15$).
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Suggestions
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-----------
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### Invariance of *b*
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Set *b* and *p* to 2 and *c* to 0.55.
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Slowly increase *p* to 3.
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The pattern should largely remain the same, but appear to be rotated around.
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Increasing *p* further shears the pattern into an indistinct line of points.
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This shows that the embedding base *does* matter, but the way the expansions are constructed
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gives rise to the pattern.
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### Small Changes
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With *c* sufficiently small (about 0.3), and *p* fixed at 2, changing *b* keeps the diagram
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almost the same.
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This is because many numbers get sent to the same place as others.
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For example, when only considering the first term of the series, "1" and "3" are both mapped to
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$e^{\pi i} = e^{3 \pi i}$
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If you make *c* larger, you can notice that points disappear from the case when *p* is 2.
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If the plot used a massive number of points, this difference wouldn't be as visible.
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### Blooming flower, in reverse
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Try cranking *b* and *p* up to 10 and set *c* to something around 0.2 so that you can
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clearly see a decagon made of decagons
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Then, strobe *p* down to its minimum value.
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Notice the larger decagon and smaller decagons collapse into nonagons when *p* hits 9.
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### Lesser Bases
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Set *b* to 2 and *p* to 1.1, the lowest it can go.
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Moving *c* back and forth, you should be able to see an interesting fractal.
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[^1]: Taken from the paper "Fractal geometry for images of continuous embeddings of p-adic
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numbers and solenoids into Euclidean spaces" (DOI: 10.1007/BF02073866).
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