undraft sand-2; minor edits

posts/polycount/sand-1/index.qmd

@@ -10,6 +10,7 @@ date-modified: "2025-02-20"
 jupyter: python3
 categories:
   - algebra
+  - sandpile
   - python
 execute:
     echo: false

posts/polycount/sand-2/index.qmd

@@ -1,6 +1,7 @@
 ---
-title: "Counting in 2D 2: Reorienting Polynomials"
-draft: true
+title: "Counting in 2D, Part 2: Reorienting Polynomials"
+description: |
+  Geometric properties of certain 2D systems and attempts to classify them.
 format:
   html:
     html-math-method: katex
@@ -9,15 +10,28 @@ date-modified: "2025-02-23"
 jupyter: python3
 categories:
   - algebra
+  - geometry
+  - sandpile
   - python
 execute:
     echo: false
 ---
 
+<style>
+  .cell-output-display .figure {
+    text-align: center;
+  }
+
+  .figure-img {
+    max-width: 512px;
+    object-fit: contain;
+    height: 100%;
+  }
+</style>
+
 ```{python}
 from pathlib import Path
 
-from IPython.display import Video, display
 from matplotlib import pyplot as plt
 import numpy as np
 
@@ -139,11 +153,9 @@ if not Path("./count_balanced_folium.mp4").exists():
     ).save("count_balanced_folium.mp4")
   except ValueError:
     pass
-
-display(
-  Video("./count_balanced_folium.mp4"),
-)
 ```
+
+{{< video ./count_balanced_folium.mp4 >}}
 :::
 
 
@@ -183,11 +195,9 @@ if not Path("./count_half_laplacian.mp4").exists():
     ).save("count_half_laplacian.mp4")
   except ValueError:
     pass
-
-display(
-  Video("./count_half_laplacian.mp4"),
-)
 ```
+
+{{< video ./count_half_laplacian.mp4 >}}
 :::
 
 Fortunately, this is the case.
@@ -477,9 +487,8 @@ if not Path("./count_triangle_carries.mp4").exists():
     ).save("./count_triangle_carries.mp4")
   except ValueError:
     pass
-
-Video("./count_triangle_carries.mp4")
 ```
+{{< video ./count_triangle_carries.mp4 >}}
 
 Counting in each of the $x^n + y^n + \ldots$ systems
 :::
@@ -492,7 +501,7 @@ It has somehow exceeded a threshold that the centered one has not.
 Incidentally, the L-shapes in the implicit carry (which I noted can be interchanged) never turn up,
   and the "initial rule" is useless.
 Phinary, the simplest implicit case in one dimension, still requires the initial rule to go
-  from the expansion of 2 (10.01) to 3 (100.01).
+  from the expansion of two (10.01) to three (100.01).
 
 
 Isosceles Rotations
@@ -536,13 +545,12 @@ if not Path("./count_bad_rotated_triangle.mp4").exists():
   except ValueError:
     pass
 
-display(
-  Video("./count_bad_rotated_triangle.mp4"),
-)
 ```
+
+{{< video ./count_bad_rotated_triangle.mp4 >}}
 :::
 
-But there are only even numbers in the first component of each vector
+But there are only even numbers in the first component of each vector.
 Therefore, the carry below will produce the same pattern.
 
 ::: {layout-ncol="2"}
@@ -575,11 +583,9 @@ if not Path("./count_bad_rotated_triangle_reduced.mp4").exists():
     ).save("./count_bad_rotated_triangle_reduced.mp4")
   except ValueError:
     pass
-
-display(
-  Video("./count_bad_rotated_triangle_reduced.mp4"),
-)
 ```
+
+{{< video ./count_bad_rotated_triangle_reduced.mp4 >}}
 :::
 
 Neither of these are similar to the original.
@@ -654,11 +660,9 @@ if not Path("./count_good_rotated_triangle.mp4").exists():
     ).save("./count_good_rotated_triangle.mp4")
   except ValueError:
     pass
-
-display(
-  Video("./count_good_rotated_triangle.mp4"),
-)
 ```
+
+{{< video ./count_good_rotated_triangle.mp4 >}}
 :::
 
 The counting video looks very promising, but can the polynomial factored?
@@ -727,11 +731,9 @@ if not Path("./count_good_rotated_triangle_reduced.mp4").exists():
     ).save("./count_good_rotated_triangle_reduced.mp4")
   except ValueError:
     pass
-
-display(
-  Video("./count_good_rotated_triangle_reduced.mp4"),
-)
 ```
+
+{{< video ./count_good_rotated_triangle_reduced.mp4 >}}
 :::
 
 All of these similar systems is share something in common: the pseudo-base is located at
@@ -753,11 +755,11 @@ No matter if we multiply or divide by *x* or *y*, the polynomial retains its gen
   as well as whether or not it is irreducible.
 The above method shows that rotational symmetry also exists for polynomials beyond interchanging *x* and *y*.
 
-Carries seem to possess a symmetry which polynomials do not: the ability to contract and expand bases,
+For integers, carries possess a symmetry which polynomials do not: the ability to contract and expand bases,
   meaning that in some sense, $P(x) \sim P(x^k)$ for any *k*.
 This preserves the counting system despite expansions and contractions which pay no regard for reducibility.
 This is also the reason that systems like base-$\sqrt 2$ are so uninteresting:
-  for integers, it's just spaced out binary.
+  for integers, it just amounts to spaced-out binary.
 
 Three questions that immediately come to mind are: