mostly finish making polycount.cell1 renderable

polycount/cell1/.gitignore

@@ -0,0 +1 @@
+*.mp4

polycount/cell1/carry2d/expansion.py

@@ -0,0 +1,96 @@
+from sympy.abc import x, y
+import numpy as np
+
+from .carry import Carry
+
+
+def add(
+    expansion: np.ndarray, carry: Carry, val: int = 1, center: tuple[int, int] = (0, 0)
+):
+    """
+    Add `val` to a position in an `expansion` designated the `center`, then apply `carry`.
+    """
+    expansion[center] += val
+    carry.apply(expansion)
+    return expansion
+
+
+def times(expansion: np.ndarray, carry: Carry, val: int):
+    """
+    Multiply all digits in an `expansion` by `val`, then apply `carry`.
+    """
+    expansion *= val
+    carry.apply(expansion)
+    return expansion
+
+
+def poly_from_array(
+    array: np.ndarray, center: tuple[int, int] = (0, 0), x_var=x, y_var=y
+):
+    """
+    Convert a numpy array to a polynomial in two variables.
+    Note that indices are row-major in Numpy, so center is in the form (y, x).
+    """
+    ret = 0
+    for i, row in enumerate(array):
+        for j, val in enumerate(row):
+            ret += val * (x_var ** (i - center[1]) * y_var ** (j - center[0]))
+    return ret
+
+
+def _first_nonzero(arr: np.ndarray, axis: int):
+    """
+    Find the index of the first nonzero entry of `arr` along `axis`.
+    Modified from https://stackoverflow.com/questions/47269390
+    """
+    mask = arr != 0
+    return min(
+        np.where(mask.any(axis=axis), mask.argmax(axis=axis), arr.shape[axis] + 1)
+    )
+
+
+def latex_polynumber(
+    arr: np.ndarray, center: tuple[int, int] | None = None, show_zero: bool = False
+):
+    """
+    Convert a 2D Numpy array into a LaTeX array representing a polynomial in two variables.
+    Vertical lines and `\\hline`s are placed to indicate the transition between negative
+        and nonnegative powers, similar to a typical fractional separator (decimal point).
+
+    By default, this assumes the constant term is in position (0, 0) of the array.
+    It may be alternatively supplied by `center`, which designates `arr[center]` as the constant term.
+    Note that indices are row-major in Numpy, so `center` is in the form (y, x).
+
+    `show_zero` defaults to False, which causes entries in `arr` which are 0 to not be drawn.
+    When true, all 0 entries will be drawn.
+    """
+    upper_left = _first_nonzero(arr, 0), _first_nonzero(arr, 1)
+    lower_right = (
+        len(arr) - _first_nonzero(np.flip(arr, 0), 0) - 1,
+        len(arr[1]) - _first_nonzero(np.flip(arr, 1), 1) - 1,
+    )
+
+    center_left, center_top = center or (0, 0)
+    if center is not None:
+        # this does not need offset, since we iterate over this range
+        center_top = center[0]
+        # but this does for the array environment argument
+        center_left = center[1] - upper_left[1]
+
+    num_columns = lower_right[1] - upper_left[1]
+    column_layout = ("c" * center_left) + "|" + ("c" * (num_columns - center_left))
+
+    # build array output
+    ret = "\\begin{array}{" + column_layout + "}"
+    for i in range(upper_left[0], lower_right[0] + 1):
+        if i == center_top:
+            ret += " \\hline "
+        ret += " & ".join(
+            [
+                str(arr[i, j] if arr[i, j] != 0 or show_zero else "")
+                for j in range(upper_left[1], lower_right[1] + 1)
+            ]
+        )
+        # next row
+        ret += " \\\\ "
+    return ret + "\\end{array}"

polycount/cell1/index.qmd

@@ -1,20 +1,21 @@
 ---
 title: "Counting in 2D: Lines, Leaves, and Sand"
+draft: true
 format:
   html:
     html-math-method: katex
 date: "2021-02-23"
-date-modified: "2025-2-14"
+date-modified: "2025-2-20"
 jupyter: python3
 categories:
   - algebra
   - python
 execute:
-    eval: false
+    echo: false
 ---
 
 ```{python}
-#| echo: false
+from pathlib import Path
 
 import sympy
 from sympy.abc import x, y
@@ -40,7 +41,8 @@ from carry2d.expansion import (
 
 
 Previously, I've written about positional systems and so-called "[polynomial counting](../1)".
-Basically, this generalizes familiar integer bases like two and ten into less familiar irrational ones like the golden ratio.
+Basically, this generalizes familiar integer bases like two and ten into less familiar irrational ones
+  like the golden ratio.
 However, all systems I presented have something in common: they all use polynomials of a single variable.
 Does it make sense to count in two variables?
 
@@ -259,21 +261,23 @@ An additional property of this system is that the number of "1"s that appear in
 These two properties make it a sort of hybrid between unary and binary, bridging their
   binary expansion and its [Hamming weight](https://en.wikipedia.org/wiki/Hamming_weight).
 
-This notation is very heavy, but fortunately lends itself well visualization.
+This notation is very heavy, but fortunately lends itself well to visualization.
 Each integer's expansion is more or less an image, so we can arrange each as frames in a video:
 
 ```{python}
-#| code-fold: true
+if not Path("./count_xy2.mp4").exists():
+  animate_carry_count(
+      carry = Carry([
+        [-2, 1],
+        [ 1, 0]
+      ]),
+      frames=list(range(1024))
+  ).save(
+    "./count_xy2.mp4",
+    fps=60
+  )
 
-animate_carry_count(
+Video("./count_xy2.mp4")
-    carry = Carry([
-      [-2, 1],
-      [ 1, 0]
-    ]),
-    frames=list(range(200))
-).save("count_xy2.mp4")
-
-Video("count_xy2.mp4")
 ```
 
 Starting at the expansion of twelve, the furthest extent is not in the first column or row.
@@ -289,32 +293,45 @@ Fortunately, this is a simple enough system that we can scalar-multiply base exp
 Ascending the powers of *n* in this manner for the carries where *n* = 3 and 4:
 
 ```{python}
-#| code-fold: true
 #| layout-ncol: 2
 
-animate_carry_count(
+# Run the animations until we overflow in the standard 100x100 range
-    carry = Carry([
+if not Path("./count_xy3.mp4").exists():
-      [-3, 1],
+  try:
-      [ 1, 0]
+    animate_carry_count(
-    ]),
+        carry = Carry([
-    operation="multiply",
+          [-3, 1],
-    op_val=3,
+          [ 1, 0]
-    frames=list(range(200))
+        ]),
-).save("count_xy3.mp4")
+        operation="multiply",
+        op_val=3,
+        frames=list(range(200))
+    ).save("count_xy3.mp4")
+  except ValueError:
+    pass
 
-animate_carry_count(
+if not Path("./count_xy4.mp4").exists():
-    carry = Carry([
+  try:
-      [-4, 1],
+    animate_carry_count(
-      [ 1, 0]
+        carry = Carry([
-    ]),
+          [-4, 1],
-    operation="multiply",
+          [ 1, 0]
-    op_val=4,
+        ]),
-    frames=list(range(200)),
+        operation="multiply",
-).save("count_xy4.mp4")
+        op_val=4,
+        frames=list(range(200)),
+    ).save("count_xy4.mp4")
+  except ValueError:
+    pass
 
 display(
-  Video("count_xy3.mp4"),
+  Video("./count_xy3.mp4"),
-  Video("count_xy4.mp4")
+  Video("./count_xy4.mp4")
+)
+
+display_latex(
+  Latex("$$x + y = 3$$"),
+  Latex("$$x + y = 4$$"),
 )
 ```
 
@@ -342,10 +359,9 @@ Taking *x* or *y* to higher powers will dilate expansions along that axis.
 
 The carry is not the only curve which exists in a system, as each integer is also a polynomial unto itself.
 Adding a constant term will not change anything unless it causes carries to occur.
-Starting at the carry digit, some of the curves made by the carry $x + y = 2$ are shown below.
+Starting at the carry digit, some of the *incremented curves* made by the carry $x + y = 2$ are shown below.
 
 ```{python}
-#| code-fold: true
 #| layout: [[3, 3], [2, 2, 2]]
 
 expansion = np.zeros((50, 50), dtype=np.int64)
@@ -386,12 +402,11 @@ This curve happens to intersect with the *x* and *y* axes, which we can interpre
 For dibinary, these are the aforementioned "typical binary expansions".
 We can also evaluate the polynomial at $(x, y) = (1, 1)$, a point on the carry curve,
   to compute its digital root, which is also equal to the integer it represents.
-
 This underlies a key difference from the 1D case.
+1D systems are special because the carry can be equated with a base (or bases), which
+  is a discrete, zero-dimensional point (one dimension less) on a number line.
 In higher dimensions, there is no proper "base" we can plug in to "test" that an expansion
   is valid like we can with the golden ratio in phinary.
-1D systems are special in this regard: the carry can be equated with a base (or bases), which
-  is a discrete zero-dimensional point on a number line.
 
 
 Casting the Line
@@ -449,8 +464,6 @@ The expansion in the *y* column looks trivially correct, since $y^2 = 9$ when $x
 We can check the base of the expansion of nine ("2100") by factoring a polynomial:
 
 ```{python}
-#| code-fold: true
-
 nine_x2y3 = 2*x**3 + x**2 - 9
 
 display_latex(
@@ -469,32 +482,38 @@ This transposes the carry (and hence the expansion).
 For good measure, let's see what it looks like when we count in the $n = 3$ and $n = 4$ cases.
 
 ```{python}
-#| code-fold: true
 #| layout-ncol: 2
 
-animate_carry_count(
+if not Path("./count_x2y3.mp4").exists():
-    carry = Carry([
+  animate_carry_count(
-      [-3, 2],
+      carry = Carry([
-      [ 1, 0]
+        [-3, 2],
-    ]),
+        [ 1, 0]
-    operation="add",
+      ]),
-    op_val=3,
+      operation="add",
-    frames=list(range(200))
+      op_val=3,
-).save("count_x2y3.mp4")
+      frames=list(range(200))
+  ).save("./count_x2y3.mp4")
 
-animate_carry_count(
+if not Path("./count_x3y4.mp4").exists():
-    carry = Carry([
+  animate_carry_count(
-      [-4, 3],
+      carry = Carry([
-      [ 1, 0]
+        [-4, 3],
-    ]),
+        [ 1, 0]
-    operation="add",
+      ]),
-    op_val=4,
+      operation="add",
-    frames=list(range(200)),
+      op_val=4,
-).save("count_x3y4.mp4")
+      frames=list(range(200)),
+  ).save("./count_x3y4.mp4")
 
 display(
-  Video("count_x2y3.mp4"),
+  Video("./count_x2y3.mp4"),
-  Video("count_x3y4.mp4")
+  Video("./count_x3y4.mp4"),
+)
+
+display_latex(
+  Latex("$$2x + y = 3$$"),
+  Latex("$$3x + y = 4$$"),
 )
 ```
 
@@ -512,12 +531,11 @@ The binary alphabet is the smallest possible, so the minimal shrinkage is in dib
 
 ### Incremented Curves
 
-Below are the incremented curves for $n = 3$.
+Below are some incremented curves for $n = 3$.
 These curves show an interesting phenomenon: an extra line is present in in the expansion of six.
 This line turns into a spike at twelve, then at fifteen, it seems to grow to encompass the point at infinity.
 
 ```{python}
-#| code-fold: true
 #| layout-ncol: 2
 
 x2y3 = Carry([
@@ -525,17 +543,15 @@ x2y3 = Carry([
   [ 1, 0]
 ])
 
-expansion = np.zeros((50, 50), dtype=np.int64)
-last_i = 0
 for i in [3, 6, 12, 15]:
-  carry_add(expansion, x2y3, i - last_i)
+  expansion = np.zeros((50, 50), dtype=np.int64)
+  carry_add(expansion, x2y3, i)
   plot_implicit(
     poly_from_array(expansion) - i,
     (x, -10, 10),
     (y, -10, 10),
     title=f"{i}",
   )
-  last_i = i
 ```
 
 
@@ -543,77 +559,13 @@ Leaves in the Mirror
 --------------------
 
 Algebraic curves are not my forte.
-However, I believe it would be a good idea to briefly examine some classical examples.
+Still, we can very briefly look at some classical examples from the perspective of carries.
-
-All Fermat curves $x^n + y^n = 1$ are not suitable as carries, since their digital root is unbounded.
-Scaling up the curve, as by $x^n + y^n = 2$ will just degenerate back to the previous examples with lines,
-  but spaced out by zeros between entries.
-
-The [folium of Descartes](https://en.wikipedia.org/wiki/Folium_of_Descartes) is another
-  classical curve that played a role in the development of calculus, described by the equation:
-
-$$
-\begin{gather*}
-  x^3 + y^3 - 3axy = 0
-  \\ \\
-  \begin{array}{|c} \hline
-    0 & 0 & 0 & 1 \\
-    0 & -3a \\
-    0 & \\
-    1
-  \end{array}
-\end{gather*}
-$$
-
-The position of the 3*a* term may be confusing.
-As the negative coefficient, we must place it atop the digit we are carrying, meaning that
-  expansions will propagate toward negative powers of *x* and *y* as well as positive.
-While the shape of the curve is different from an ordinary line, the coefficients are arranged
-  too similarly; for $a = 2/3$ (so the curve passes through the point (1, 1)) incrementing appears as:
-
-```{python}
-#| echo: false
-#| layout-ncol: 2
-
-folium = Carry([
-  [ 0, 0, 0, 1],
-  [ 0,-2, 0, 0],
-  [ 0, 0, 0, 0],
-  [ 1, 0, 0, 0],
-])
-dims = (50, 50)
-center = (dims[0] // 2, dims[1] // 2)
-two_folium = carry_add(np.zeros(dims, dtype=np.int64), folium, 2, center=center)
-four_folium = carry_add(np.zeros(dims, dtype=np.int64), folium, 4, center=center)
-
-display_latex(
-  Latex(
-    r"\begin{gather*}"
-    + "2: &"
-    + latex_polynumber(two_folium, center=center, show_zero=True) + r"\\"
-    + r"\\ \\ 4: &"
-    + latex_polynumber(four_folium, center=center, show_zero=True) + r"\\"
-    + r"\end{gather*}"
-  )
-)
-
-animate_carry_count(
-    carry=folium,
-    operation="add",
-    op_val=4,
-    frames=list(range(200)),
-).save("count_folium.mp4")
-
-display(Video("count_folium.mp4"))
-```
-
-Clearly, this also tends back toward dibinary, but with the yellow digits spaced out more.
 
 
-### Discounting Curves
+### Quadratic Curves
 
-The interpretation of the carry as a curve is secondary, so I'd like to disqualify
+Unfortunately, the most common quadratic curves are largely uninteresting.
-  some archetypal quadratic objects.
+Briefly, here are some comments about some the simplest cases:
 
 - Unit circle ($x^2 + y^2 - 1 = 0$) and hyperbola ($\pm x^2 \mp y^2 - 1 = 0$)
   - The sum of coefficients is positive, which means that expansions are unbounded.
@@ -632,11 +584,81 @@ Other polynomials in *x* and *y* may produce rotations or dilations of these cur
 What truly matters is that one cell affects relatively close cells in a certain way.
 
 
+### Higher-Order Curves
+
+All unit Fermat curves $x^n + y^n = 1$ are not suitable as carries, since their digital root is unbounded.
+Scaling up the curve, for example to $x^n + y^n = 2$, will just degenerate back to the previous
+  examples with lines, but spaced out by zeros between entries.
+
+The [folium of Descartes](https://en.wikipedia.org/wiki/Folium_of_Descartes) is another
+  classical curve that played a role in the development of calculus, described by the equation:
+
+$$
+\begin{gather*}
+  x^3 + y^3 - 3axy = 0
+  \\ \\
+  \begin{array}{|c} \hline
+    0 & 0 & 0 & 1 \\
+    0 & -3a \\
+    0 & \\
+    1
+  \end{array}
+\end{gather*}
+$$
+
+As written above, the position of the 3*a* term may be confusing.
+As the negative term in an explicit carry, we must place it atop the digit we are carrying,
+  meaning that expansions will propagate toward negative powers of *x* and *y* as well as positive.
+While the shape of the curve is different from an ordinary line, the coefficients are arranged
+  too similarly; for $a = 2/3$ (so the curve passes through the point $(1, 1)$) incrementing appears as:
+
+```{python}
+#| layout-ncol: 2
+
+folium_carry = Carry([
+  [ 0, 0, 0, 1],
+  [ 0,-2, 0, 0],
+  [ 0, 0, 0, 0],
+  [ 1, 0, 0, 0],
+])
+folium_dims = (50, 50)
+folium_center = (folium_dims[0] // 2, folium_dims[1] // 2)
+folium_two = carry_add(np.zeros(folium_dims, dtype=np.int64), folium_carry, 2, center=folium_center)
+folium_four = carry_add(np.zeros(folium_dims, dtype=np.int64), folium_carry, 4, center=folium_center)
+
+display_latex(
+  Latex(
+    r"\begin{gather*}"
+    + "2: &"
+    + latex_polynumber(folium_two, center=folium_center, show_zero=True) + r"\\"
+    + r"\\ \\ 4: &"
+    + latex_polynumber(folium_four, center=folium_center, show_zero=True) + r"\\"
+    + r"\end{gather*}"
+  )
+)
+
+if not Path("./count_folium.mp4").exists():
+  try:
+      animate_carry_count(
+          carry=folium_carry,
+          operation="add",
+          op_val=4,
+          frames=list(range(200)),
+      ).save("./count_folium.mp4")
+  except ValueError:
+    pass
+
+display(Video("./count_folium.mp4"))
+```
+
+Clearly, this also tends back toward dibinary, but with the yellow digits spaced out more.
+
+
 Laplace's Sandstorm
 -------------------
 
 As stated previously, polynomials naturally multiply by convolution.
-In this context, a form which frequently arises is an approximation of the 2D Laplacian.
+In the context of this operation, the discrete 2D Laplacian is frequently used
 While commonly presented as a matrix, it has very little to do with the machinery of
   linear algebra which normally empowers matrices.
 
@@ -659,8 +681,6 @@ $$
 $$
 
 ```{python}
-#| echo: false
-
 plot_implicit(x + y + x*y**2 + y*x**2 - 4*x*y, (x, -10, 10), (y, -10, 10))
 ```
 :::
@@ -718,21 +738,21 @@ As a cellular automaton, it is rather popular as a coding challenge
 These videos discuss toppling the initial value, but do not point out the analogy to polynomial expansions.
 
 ```{python}
-#| code-fold: true
 #| layout-ncol: 2
 
-animate_carry_count(
+if not Path("./count_x2y3.mp4").exists():
-    carry = Carry([
+  animate_carry_count(
-      [ 0, 1, 0],
+      carry = Carry([
-      [ 1,-4, 1],
+        [ 0, 1, 0],
-      [ 0, 1, 0],
+        [ 1,-4, 1],
-    ]),
+        [ 0, 1, 0],
-    operation="add",
+      ]),
-    op_val=4,
+      operation="add",
-    frames=list(range(200)),
+      op_val=4,
-).save("count_laplace.mp4")
+      frames=list(range(200)),
+  ).save("./count_laplace.mp4")
 
-Video("count_laplace.mp4")
+Video("./count_laplace.mp4")
 ```
 
 The shapes produced by this pattern are well-known, with larger numbers appearing as fractals.

polycount/cell1/todo.txt

@@ -0,0 +1,5 @@
+mp4s saved do not get copied or linked over to the rendering folder
+
+center figures
+
+make sure to object-fit: scale