[math/fractals/dragon/complex-bits] First cut

Author: harold

src/math/fractals/dragon/complex_bits.clj

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+^{:kindly/hide-code true
+  :clay {:title  "Dragon Curve Fractal - Complex & Bits"
+         :quarto {:type     :post
+                  :author   [:harold]
+                  :date     "2025-09-01"
+                  :description "A familiar image is re-discovered in a sequence of complex numbers and the binary representation of the natural numbers."
+                  :image    "dragon-curve.png"
+                  :category :math
+                  :tags     [:fractals :dragon :curve :complex :imaginary :binary]
+                  :keywords [:fractals :dragon :curve :complex :imaginary :binary]}}}
+(ns math.fractals.dragon.complex-bits
+  (:require [clojure.string :as s]
+            [fastmath.complex :as c]
+            [scicloj.kindly.v4.kind :as kind]))
+
+;; ### Part 1: Complex Numbers
+
+;; Clojure's [fastmath](https://generateme.github.io/fastmath/notebooks/notebooks/complex_quaternion/index.html) library provides [complex numbers](https://en.wikipedia.org/wiki/Complex_number).
+
+;; For example, if we let `a=b=1`, then
+
+^:kindly/hide-code
+(kind/tex "a+bi=1+1i")
+
+
+(c/complex 1 1)
+
+;; Here's a fun fact about powers of one more than the imaginary unit.
+
+^:kindly/hide-code
+(kind/tex "(1+i)^k=\\sqrt{2}^ke^{ik\\pi/4}")
+
+;; One does not need to be Euler to sense spinning,
+
+;; and `k` from `(0 1 2 3 ...)` gives a sequence of points.
+
+(def pts
+  (for [k (range)]
+    (c/pow (c/complex 1 1)
+           (c/complex k 0))))
+
+;; The first few of which are...
+
+(take 10 pts)
+
+;; I plot, therefore I am:
+
+(kind/hiccup
+ (into [:svg {:width "256" :height "256" :viewBox "-24 -24 48 48"}
+        [:rect {:x -24 :y -24 :width 48 :height 48 :fill :#f8f8f8}]]
+       (concat (for [[x y] (take 10 pts)]
+                 [:line {:x1 0 :y1 0 :x2 x :y2 y :stroke :#222}])
+               (for [[x y] (take 10 pts)]
+                 [:circle {:cx x :cy y :r 1.5 :fill :#88f}]))))
+
+;; Spinning indeed.
+
+;; ---
+
+;; ### Part 2: Binary Bits
+
+;; In computers, numbers are made of bits.
+
+(defn bit
+  [n pos]
+  (bit-and (int (/ n (Math/pow 2 pos))) 1))
+
+(for [pos (reverse (range 8))]
+  (bit 42 pos))
+
+;; Hidden in the bits is a draconiform structure, we use them to carefully sum the `pts`.
+
+(defn dragon-by-bits
+  ([n] (dragon-by-bits n (count (Integer/toBinaryString n))))
+  ([n pos]
+   (letfn [(a [n pos]
+             ;; We move according to pts, occasionally turning
+             (if (zero? (bit n pos))
+               c/ZERO
+               (let [turn? (if (zero? (bit n (inc pos)))
+                             c/ONE
+                             c/I)
+                     pt (nth pts pos)] ;; (1+i)^pos
+                 (c/mult turn? pt))))
+           (m [n pos]
+             ;; When the bits flip, we turn
+             (if (= (bit n pos) (bit n (inc pos)))
+               c/ONE
+               c/I))]
+     (if (>= pos 0)
+       (c/add (a n pos)
+              (c/mult (m n pos)
+                      (dragon-by-bits n (dec pos))))
+       c/ZERO))))
+
+;; ---
+
+;; ### Part 3: A Glimpse
+
+;; > "It does not do to leave a live dragon out of your calculations, if you live near him."
+;; >
+;; > -- J.R.R. Tolkien
+
+(let [dragon-pts (for [n (range 512)] (dragon-by-bits n))
+      furthest (* 1.15 (apply max (map Math/abs (flatten dragon-pts))))
+      [l t w h] [(- furthest) (- furthest) (* 2 furthest) (* 2 furthest)]]
+  (kind/hiccup
+   [:svg {:width "512" :height "512" :viewBox (s/join " " [l t w h])
+          :shape-rendering :crispEdges}
+    [:rect {:x l :y t :width w :height h :fill :#f8f8f8}]
+    [:path {:d (reduce (fn [eax [x y]]
+                         (str eax " L" x " " y))
+                       (let [[x y] (first dragon-pts)]
+                         (str "M" x " " y))
+                       (rest dragon-pts))
+            :vector-effect "non-scaling-stroke"
+            :stroke :#222
+            :fill :none}]]))
+
+;; Inspired by this [lovely rosettacode gnuplot code](https://rosettacode.org/wiki/Dragon_curve#Version_#1.).