From 9786cb0f614c5dded71ea1ee3b9b55367c9bcb1a Mon Sep 17 00:00:00 2001
From: queue-miscreant <cubebert549@gmail.com>
Date: Sun, 2 Mar 2025 01:08:50 -0600
Subject: [PATCH] add interactive page for p-adics

---
 interactive/p-adics/index.qmd    | 56 ++++++++++++++++++++++++++
 interactive/p-adics/showAdic.ojs | 69 ++++++++++++++++++++++++++++++++
 2 files changed, 125 insertions(+)
 create mode 100644 interactive/p-adics/index.qmd
 create mode 100644 interactive/p-adics/showAdic.ojs

diff --git a/interactive/p-adics/index.qmd b/interactive/p-adics/index.qmd
new file mode 100644
index 0000000..0b378fd
--- /dev/null
+++ b/interactive/p-adics/index.qmd
@@ -0,0 +1,56 @@
+---
+
+---
+
+```{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).
diff --git a/interactive/p-adics/showAdic.ojs b/interactive/p-adics/showAdic.ojs
new file mode 100644
index 0000000..5fd545c
--- /dev/null
+++ b/interactive/p-adics/showAdic.ojs
@@ -0,0 +1,69 @@
+// Convert x into its expansion in base b, as an array of integers (ascending powers of b)
+toBase = (b, x) => {
+  if (x == 0) { return [0] }
+
+  let ret = [];
+  while (x > 0) {
+    ret.push(x % b);
+    x = Math.floor(x / b);
+  }
+  return ret;
+};
+
+// Interpret xs as an expansion in base b (ascending powers of b)
+fromBase = (b, xs) => {
+  let plval = 1
+  return xs.reduce((a, x) => {
+    let new_a = x * plval;
+    plval *= b;
+    return a + new_a;
+  }, 0)
+}
+
+// Map an expansion to a complex number (as a 2-array).
+// The expansion is interpreted in base `b`.
+// `n` terms of the series are used, with geometric ratio `c`
+adicAngles = (expansion, b, n, c) => {
+  return d3.range(n).reduce((a, i) => {
+    let angle = fromBase(b, expansion.slice(null, i + 1)) / (b ** (i + 1));
+    let x = c**i * Math.cos(2 * Math.PI * angle);
+    let y = c**i * Math.sin(2 * Math.PI * angle);
+    return [a[0] + x, a[1] + y];
+  }, [0, 0]);
+}
+
+pointCount = 1024;
+expansions = d3.range(pointCount).map((x) => toBase(base, x));
+
+embedding = expansions.map(
+  (x) => adicAngles(x, embedBase, 15, geometric)
+);
+
+viewof base = Inputs.range([2, 10], {
+  value: 2,
+  step: 1,
+  label: "Base of expansions (b)",
+});
+
+viewof embedBase = Inputs.range([2, 10], {
+  value: 2,
+  step: 0.2,
+  label: "Embedding base (p)",
+});
+
+viewof geometric = Inputs.range([0.005, 0.995], {
+  value: 0.9,
+  step: 0.005,
+  label: "Geometric ratio (c)",
+});
+
+plot = Plot.plot({
+  grid: true,
+  inset: 10,
+  // aspectRatio: 1,
+  width: 640,
+  height: 480,
+  marks: [
+    Plot.dot(embedding, { r: 1 })
+  ]
+});