revisions for interactive.p-adics

interactive/p-adics/index.qmd

@@ -1,8 +1,18 @@
 ---
-
+title: "Complex Embedding of *p*-adics"
+description: |
+  Visualizing fractals generated by sending *p*-adics to the complex plane.
+format:
+  html:
+    html-math-method: katex
+categories:
+  - interactive
+  - algebra
 ---
 
 ```{ojs}
+//| echo: false
+
 {{< include ./showAdic.ojs >}}
 ```
 
@@ -23,7 +33,7 @@ $$
   f_N(d; p) = \sum_{n = 0}^N c^n e^{2\pi i \cdot [d_{n:0}]_p / p^{n + 1}}
 $$
 
-Assuming the first term dominates, one way of interpreting this is that we place numbers
+Assuming the first term dominates, this can be interpreted as placing numbers evenly
   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.
@@ -35,11 +45,15 @@ All of the odd numbers are at the left side of the diagram, since the leading te
   is negative one.
 The odd numbers are further split into those of the forms $4k + 1$ and $4k + 3$.
 
+
+Controls
+--------
+
 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*,
+    - Each point is the expansion of an integer in base *b*,
-      which are sequences of digits (*d* in the above formula).
+      which corresponds to a sequence 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*,
@@ -47,9 +61,46 @@ Each of the inputs corresponds to something from the above formula.
 - *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.
+Note that only 1024 points are calculated, and that only fifteen terms of the series are used ($N = 15$).
-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 ($N = 15$).
+
+Suggestions
+-----------
+
+### Invariance of *b*
+
+Set *b* and *p* to 2 and *c* to 0.55.
+Slowly increase *p* to 3.
+The pattern should largely remain the same, but appear to be rotated around.
+Increasing *p* further shears the pattern into an indistinct line of points.
+This shows that the embedding base *does* matter, but the way the expansions are constructed
+  gives rise to the pattern.
+
+
+### Small Changes
+
+With *c* sufficiently small (about 0.3), and *p* fixed at 2, changing *b* keeps the diagram
+  almost the same.
+This is because many numbers get sent to the same place as others.
+For example, when only considering the first term of the series, "1" and "3" are both mapped to
+  $e^{\pi i} = e^{3 \pi i}$
+
+If you make *c* larger, you can notice that points disappear from the case when *p* is 2.
+If the plot used a massive number of points, this difference wouldn't be as visible.
+
+
+### Blooming flower, in reverse
+
+Try cranking *b* and *p* up to 10 and set *c* to something around 0.2 so that you can
+  clearly see a decagon made of decagons
+Then, strobe *p* down to its minimum value.
+Notice the larger decagon and smaller decagons collapse into nonagons when *p* hits 9.
+
+
+### Lesser Bases
+
+Set *b* to 2 and *p* to 1.1, the lowest it can go.
+Moving *c* back and forth, you should be able to see an interesting fractal.
 
 
 [^1]: Taken from the paper "Fractal geometry for images of continuous embeddings of p-adic

interactive/p-adics/showAdic.ojs

@@ -45,9 +45,9 @@ viewof base = Inputs.range([2, 10], {
   label: "Base of expansions (b)",
 });
 
-viewof embedBase = Inputs.range([2, 10], {
+viewof embedBase = Inputs.range([1.1, 10], {
   value: 2,
-  step: 0.2,
+  step: 0.1,
   label: "Embedding base (p)",
 });