revisions to pentagons.2

posts/pentagons/1/index.qmd

@@ -53,6 +53,14 @@ class PolyData:
   vertex_count: int
   edge_count: int
 
+  @classmethod
+  def from_hexagons(cls, hexagon_count: int):
+    return cls(
+      hexagon_count=hexagon_count,
+      vertex_count=2*hexagon_count + 20,
+      edge_count=3*hexagon_count + 30,
+    )
+
 
 @dataclass
 class GoldbergData:
@@ -151,6 +159,11 @@ Thus, by asserting that the figure is composed of pentagons and hexagons,
 This is the case even if there are no hexagons, in which case we are describing a dodecahedron
   (which, incidentally, has 20 vertices).
 
+Consequently, we get an expression for the number of vertices and edges in terms of the number of hexagons.
+
+- $3V = 6F_6 + 5 F_5 = 6F_6 + 60 = 3V \implies V = 2F_6 + 20$
+- $2V = 6F_6 + 5 F_5 \implies E = 3F_6 + 30$
+
 
 ### Duality
 
@@ -417,7 +430,7 @@ $$
 \end{align*}
 $$
 
-Giving an equal share of triangles to each pentagon as before, each has $\|(a, b)\|$ triangles.
+Giving an equal share of triangles to each pentagon as before, each has $\|a + bu\|$ triangles.
 
 
 ### Triangles to Hexagons
@@ -455,8 +468,8 @@ Beyond Soccer Balls
 -------------------
 
 The three cases of paths as described above are also used to delineate classes of Goldberg polyhedra.
-Only class I and II polyhedra produce figures which are mirror-symmetric.
-Class III polyhedra come in chiral pairs, due to the choice of a clockwise turn in their construction.
+Only class I and II polyhedra produce figures which are mirror-symmetric ($I_h$, which contains $I$).
+Class III polyhedra lack this; they come in chiral pairs due to the choice of a clockwise turn in their construction.
 
 
 ### Basic Conway Operators
@@ -630,12 +643,7 @@ goldberg_classes = {
 def poly_from_dodecahedral_goldberg_parameter(parameter: tuple[int, int]):
   a, b = parameter
   hexagon_count = 10*(a*a + a*b + b*b - 1)
-  vertex_count = (60 + 6*hexagon_count) // 3
-  return PolyData(
-    hexagon_count=hexagon_count,
-    vertex_count=vertex_count,
-    edge_count=(vertex_count * 3) // 2,
-  )
+  return PolyData.from_hexagons(hexagon_count)
 
 
 def goldberg_table_row(data: GoldbergData) -> list[str]:
@@ -655,7 +663,7 @@ def goldberg_table_row(data: GoldbergData) -> list[str]:
 
 Markdown(tabulate(
   [
-    ([class_] if i == 0 else [""]) + goldberg_table_row(item)
+    ([class_ if i == 0 else ""]) + goldberg_table_row(item)
     for class_, items in goldberg_classes.items()
     for i, item in enumerate(items)
   ],

posts/pentagons/2/goldberg_triangles.py

@@ -0,0 +1 @@
+../1/goldberg_triangles.py