(b,c), chirality, and why the blueprint is the name

$T = b^2 + bc + c^2$ keeps showing up, so what are $b$ and $c$? They’re a walk. A small, bossy walk that decides everything about the ball before you’ve folded a single unit. Stand on one of the 12 corner vertices, step $b$ units along a row of triangles, turn 60°, step $c$ units. Where you land is the next corner. That single instruction, the vector $(b,c)$, pins down the whole ball: its size (through $T$), its twist, and its handedness.

The $T$ formula isn’t arbitrary either. It’s the law of cosines on a 60° triangular lattice: two steps of length $b$ and $c$ meeting at the lattice’s 120° angle give a squared separation of $b^2 + bc + c^2$.¹1 The $+bc$ cross-term is the $\cos(120°) = -\tfrac{1}{2}$ contribution, doubled, the signature of a 60° lattice rather than a square one. So “how far apart are the corners” and “how many triangles fit between them” are literally the same computation.

The walk also sorts every ball into three classes:

• Class I: $c = 0$. The walk runs straight along a lattice row. Aligned, no twist.
• Class II: $b = c$. A symmetric diagonal walk. Twisted, but still mirror-symmetric.
• Class III: $b \neq c$, both nonzero. The walk is lopsided, and the ball becomes chiral: it exists in distinct left- and right-handed versions you genuinely cannot fold into one another. The 210-unit ball is the smallest chiral one, meaning a paper ball can have a handedness, the way your hands do, or a seashell, or a strand of DNA.

There are three naming systems, and it’s worth knowing which to reach for. Descriptive names just count faces (60 kite faces gives “deltoidal hexecontahedron”) and tell you nothing about construction. Conway notation names the recipe: seed letters T/C/O/D/I and operators applied right-to-left like function composition, so kD is “kis dodecahedron” (the pentakis dodecahedron) and tI is “truncate the icosahedron” (a soccer ball).²2 Conway operators include k (kis), t (truncate), d (dual), a (ambo), s (snub). A fun consequence: dkD is a truncated icosahedron again. kD and tI are duals of each other. But for geodesic balls specifically there’s Coxeter’s notation, $\{3,5+\}(b,c)$: triangular faces (the 3), arranged around icosahedral 5-fold vertices (the 5+), subdivided by the vector $(b,c)$.

And that’s the quiet payoff of the whole rabbit hole:

• $30 = \{3,5+\}(1,0)$
• $90 = \{3,5+\}(1,1)$
• $210 = \{3,5+\}(2,1)$ (chiral)
• $270 = \{3,5+\}(3,0)$

The $(b,c)$ I’d been chasing for its own sake, the little walk that fixes size, twist, and handedness, is literally the name. The “real object” I kept saying was lurking behind the unit count is exactly what Coxeter notation writes down. The blueprint and the name are the same string. I went hunting for the real object behind the number and it had been wearing the number the whole time.