---

---

```{ojs}
{{< include ./showAdic.ojs >}}
```

About
-----

The Wikipedia article on the [*p*-adic valuation](https://en.wikipedia.org/wiki/P-adic_valuation)
  contains [a figure](https://commons.wikimedia.org/wiki/File:2adic12480.svg) whose description
  provides a way to map *p*-adics into the complex numbers[^1].
The gist is to construct a Fourier series over truncations of numbers.
Each term of the series is weighted by a geometrically decreasing coefficient *c*.

$$
[...d_2 d_1 d_0]_p \mapsto e^{2\pi i [d_0] / p}
  + c e^{2\pi i [d_1 d_0] / p^2}
  + c^2 e^{2\pi i [d_2 d_1 d_0] / p^2}
  + ... \\
  f_N(d; p) = \sum_{n = 0}^N c^n e^{2\pi i \cdot [d_{n:0}]_p / p^{n + 1}}
$$

Assuming the first term dominates, one way of interpreting this is that we place numbers
  around the unit circle according to their one's place.
Then, we offset each by smaller circles, each centered on the last, using more and more digits.
This produces a fractal pattern that looks like a wheel with *p* spokes.
At each point on where the spokes meet the rim, there is another smaller wheel with *p* spokes,
  ad infinitum.

This is somewhat visible in the Wikimedia diagram.
All of the odd numbers are at the left side of the diagram, since the leading term of the series for them
  is negative one.
The odd numbers are further split into those of the forms $4k + 1$ and $4k + 3$.

Each of the inputs corresponds to something from the above formula.

- *b*, the base of the expansions.
    - Integers are converted to their representations in base *b*,
      which are sequences of digits (*d* in the above formula).
- *p*, the base used in the embedding.
    - The same *p* that appears in the above formula.
    - Truncations of digit sequences ($d_{n:0}$) are interpreted as strings in base *p*,
      then divided by $p^{n+1}$.
- *c*, the geometric constant.
    - Smaller *c* means more tightly packed points.

Additionally, instead of using an integer base, you can also use either *κ*-adic representation.
If using an integer base, only one thousand twenty four (1024) points will be calculated.
Note also that only fifteen terms of the series are used ($N = 15$).


[^1]: Taken from the paper "Fractal geometry for images of continuous embeddings of p-adic
    numbers and solenoids into Euclidean spaces" (DOI: 10.1007/BF02073866).
