add interactive section to 4.appendix

posts/polycount/4/appendix/index.qmd

@@ -9,7 +9,8 @@ date: "2021-02-09"
 date-modified: "2025-02-12"
 categories:
   - algebra
-  - python
+  - haskell
+  - interactive
 execute:
   eval: false
 ---
@@ -30,7 +31,7 @@ A more unusual consequence of this is that despite the initial choice of alphabe
   [negative numerals could be cleared](../#all-positive)
   from the expansion by using an extra-greedy borrow.
 
-These choices actually only differ in only one significant way: the choice of integer division function.
+There is only one significant difference in how the two are implemented: the choice of integer division function.
 In Haskell, there exists both `quotRem` and `divMod`, with the two disagreeing on negative remainders:
 
 ::: {.row layout-ncol="2"}
@@ -43,13 +44,13 @@ divMod (-27) 5
 ```
 :::
 
-We can factor our choice out of our two digit-wide carry function by presenting it as an argument:
+We can factor our choice out of our two digit-wide carry function by passing it as an argument:
 
 ```{haskell}
 -- Widened carry of a particular repeated amount
 -- i.e., for carry qr 2, the carry is 22 = 100
-carry2' qr b = carry' []
+carry2 qr b = carry2' []
-  where carry' zs (x:y:z:xs)
+  where carry2' zs (x:y:z:xs)
           | q == 0    = carry' (x:zs) (y:z:xs)  -- try carrying at a higher place value
           | otherwise = foldl (flip (:)) ys zs  -- carry here
             where ys     = r : y-x+r : z+q : xs
@@ -61,8 +62,8 @@ We happen to know the root for $\langle 2, 2|$ is $\sqrt 3 + 1$, so using an app
   we can check that we get a roughly constant value by evaluating at each step.
 
 ```{haskell}
-{-| layout-ncol: 2 -}
+-- | layout-ncol: 2
-{-| code-fold: true -}
+-- | code-fold: true
 
 import IHaskell.Display
 
@@ -74,8 +75,8 @@ pairEval x = (,) <*> hornerEval x . map fromIntegral
 -- Directly expand the integer `n`, using the `carry` showing each step for `count` steps
 expandSteps count carry n = take count $ iterate carry $ n:replicate count 0
 
-cendree4QuotRem10Steps = map (pairEval (sqrt 3 + 1)) $ expandSteps 10 (carry2' quotRem 2) 4
+cendree4QuotRem10Steps = map (pairEval (sqrt 3 + 1)) $ expandSteps 10 (carry2 quotRem 2) 4
-cendree4DivMod10Steps = map (pairEval (sqrt 3 + 1)) $ expandSteps 10 (carry2' divMod 2) 4
+cendree4DivMod10Steps = map (pairEval (sqrt 3 + 1)) $ expandSteps 10 (carry2 divMod 2) 4
 
 markdown "`quotRem`"
 markdown "`divMod`"
@@ -101,14 +102,14 @@ Since the `divMod` implementation clears negative numbers from expansions, we ca
 The result is another chaotic series:
 
 ```{haskell}
-cendree2DivModCycleExpansion = take 11 $ iterate (carry2' divMod 2) $ take 15 $ 0:0:cycle [1,-1]
+cendree2DivModCycleExpansion = take 11 $ iterate (carry2 divMod 2) $ take 15 $ 0:0:cycle [1,-1]
 putStrLn . unlines . map show $ cendree2DivModCycleExpansion
 ```
 
 We get the same series if we expand 2 directly (which we can also use to check its validity):
 
 ```{haskell}
-cendree2DivMod15Steps = map (pairEval (sqrt 3 + 1)) $ expandSteps 15 (carry2' divMod 2) 2
+cendree2DivMod15Steps = map (pairEval (sqrt 3 + 1)) $ expandSteps 15 (carry2 divMod 2) 2
 putStrLn . unlines . map show $ cendree2DivMod15Steps
 ```
 
@@ -116,7 +117,7 @@ We can *also* use this to get a series for negative one, a number which has a te
   in the balanced alphabet.
 
 ```{haskell}
-cendreeNeg1DivMod15Steps = map (pairEval (sqrt 3 + 1)) $ expandSteps 15 (carry2' divMod 2) (-1)
+cendreeNeg1DivMod15Steps = map (pairEval (sqrt 3 + 1)) $ expandSteps 15 (carry2 divMod 2) (-1)
 putStrLn . unlines . map show $ cendreeNeg1DivMod15Steps
 ```
 
@@ -126,7 +127,7 @@ If we take the last iterate of this, increment the zeroth place value, and apply
 
 ```{haskell}
 cendree0FromNeg1DivMod = (\(x:xs) -> (x + 1):xs) $ snd $ last cendreeNeg1DivMod15Steps
-cendree0IncrementSteps = map (pairEval (sqrt 3 + 1)) $ take 15 $ iterate (carry2' divMod 2) cendree0FromNeg1DivMod
+cendree0IncrementSteps = map (pairEval (sqrt 3 + 1)) $ take 15 $ iterate (carry2 divMod 2) cendree0FromNeg1DivMod
 
 putStrLn . unlines . map show $ cendree0IncrementSteps
 ```
@@ -139,27 +140,34 @@ Actually demonstrating this and proving it is left as an exercise.
 Are these really -adic?
 -----------------------
 
-Perhaps it is still unconvincing that expanding the integers in this way gives something that is
+Perhaps it is still unconvincing that expanding the integers in this way gives something
   indeed related to *p*-adics.
-The Wikipedia article on the [*p*-adic valuation](https://en.wikipedia.org/wiki/P-adic_valuation)
+In fact, since the expansions are in binary or (balanced) ternary, the integers should just
-  contains [a figure](https://commons.wikimedia.org/wiki/File:2adic12480.svg) whose description
+  be a subset of the 2-adics or 3-adics.
-  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*.
 
-$$
+Still, wanted to see what these numbers actually "look" like, so I whipped up an interactive diagram.
-[...d_2 d_1 d_0]_p \mapsto e^{2\pi i [d_0] / p}
+You should definitely see [this page](/interactive/adic/) for more information, but
-  + c e^{2\pi i [d_1 d_0] / p^2}
+  the gist is that *p*-adics can be sent into the complex plane in a fractal-like way.
-  + c^2 e^{2\pi i [d_2 d_1 d_0] / p^2}
-  + ... \\
-  f(d; p) = \sum_{n = 0}^N c^n e^{2\pi i \cdot [d_{n:0}] / p^{n + 1}}
-$$
 
-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 wheel with *p* spokes, ad infinitum.
 
-[^1]: Taken from the paper "Fractal geometry for images of continuous embeddings of p-adic
+```{ojs}
-    numbers and solenoids into Euclidean spaces" (DOI: 10.1007/BF02073866).
+// | echo: false
+
+{{< include ./showAdicWithKappa.ojs >}}
+```
+
+First, notice that with $b = 2$ and $p = 2$, switching between the "b-adic" option
+  and the "*κ*-adic" option appears cause some points to appear and disappear.
+This is easiest to see when $c \approx 0.5$.
+
+Next, notice that when plotting the *κ*-adics, some self-similarity different from the 2- and 3-adics
+  can be observed for $c \approx 0.75$.
+There appear to be four clusters of points, with the topmost and rightmost appearing to be
+  similar to one another.
+Within these two clusters, the rightmost portion of them appears to be the same shape
+  as the larger figure.
+
+This is actually great news -- if you switch between the *κ*-adics and the "random binary" option,
+  you can see that the latter option tends to the same pattern as the 2-adics.
+Thus, even if the expansions for the integers are individually chaotic, together they possess a
+  much different structure than pure randomness.

posts/polycount/4/appendix/showAdicWithKappa.ojs

@@ -0,0 +1,66 @@
+/*
+FileAttachments:
+  ./cendree_DivMod_count_1024_256_digits.csv: ./cendree_DivMod_count_1024_256_digits.csv
+  ./cendree_QuotRem_count_1024_256_digits.csv: ./cendree_QuotRem_count_1024_256_digits.csv
+*/
+
+// Import expansions from file.
+// Odd numbers are injected by replacing the first entry of each row with "1"
+asIntegers = (x) => {
+  let xs = x.split("\n").map((y) => y.split(",").map((z) => +z))
+  return [...xs, ...(xs.map((ys) => ys.with(0, 1)))]
+};
+
+adicExpansionsDivMod = FileAttachment(
+  "./cendree_DivMod_count_1024_256_digits.csv"
+).text().then(asIntegers);
+adicExpansionsQuotRem = FileAttachment(
+  "./cendree_QuotRem_count_1024_256_digits.csv"
+).text().then(asIntegers);
+
+import { expansions as oldExpansions } with { base as base } from "../../../../interactive/p-adics/showAdic.ojs";
+
+expansionsOrAdics = baseSelector == "b-adic"
+  ? oldExpansions
+  : baseSelector == "κ-adic, balanced"
+    ? adicExpansionsDivMod
+    : baseSelector == "κ-adic, binary"
+      ? adicExpansionsQuotRem
+      : d3.range(adicExpansionsQuotRem.length / 10).map(() => d3.range(15).map(() => +(Math.random() > 0.5)))
+
+import { plot } with {
+  expansionsOrAdics as expansions,
+  embedBase as embedBase,
+  geometric as geometric,
+} from "../../../../interactive/p-adics/showAdic.ojs";
+
+viewof baseSelector = Inputs.radio([
+  "b-adic",
+  "κ-adic, balanced",
+  "κ-adic, binary",
+  "Random Binary",
+], {
+  value: "b-adic",
+  label: "Expansions",
+});
+
+viewof base = Inputs.range([2, 5], {
+  value: 2,
+  step: 1,
+  label: "Base of expansions (b)",
+  disabled: baseSelector != "b-adic",
+});
+
+viewof embedBase = Inputs.range([2, 5], {
+  value: 2,
+  step: 0.1,
+  label: "Embedding base (p)",
+});
+
+viewof geometric = Inputs.range([0.005, 0.995], {
+  value: 0.9,
+  step: 0.005,
+  label: "Geometric ratio (c)",
+});
+
+plot

posts/polycount/4/cendree.hs

@@ -1,3 +1,5 @@
+import Data.List (intercalate)
+
 -- Widened borrow of a particular repeated amount
 -- i.e., for borrow2 qr 2, the borrow is 22 = 100
 -- `qr` is an integer division function returning the quotient and remainder.
@@ -65,7 +67,11 @@ data QRMethod = QuotRem | DivMod deriving Show
 qrMethod QuotRem = quotRem
 qrMethod DivMod = divMod
 
-cendreeEvens :: QRMethod -> Int -> Int -> IO ()
+cendreeEvens' :: QRMethod -> Int -> Int -> [[Int]]
-cendreeEvens qr m n = writeFile fn $ unlines $ take m $ map show $ evensQR qr' n 2 $ truncadicQR qr' (n + 2) 2 $ 2:repeat 0
+cendreeEvens' qr m n = take m $ evensQR qr' n 2 $ truncadicQR qr' (n + 2) 2 $ 2:repeat 0
   where qr' = qrMethod qr
-        fn = "cendree_" ++ show qr ++ "_count_" ++ show m ++ "_" ++ show n ++ "_digits.txt"
+
+cendreeEvens :: QRMethod -> Int -> Int -> IO ()
+cendreeEvens qr m n = writeFile fn $ display $ cendreeEvens' qr m n
+  where display = unlines  . map (intercalate "," . map show)
+        fn = "cendree_" ++ show qr ++ "_count_" ++ show m ++ "_" ++ show n ++ "_digits.csv"