In Conway, Burgiel, and Goodman-Strauss's The Symmetries of Things, every symmetry feature on a surface has a fixed price in the Two-Dollar Orbifold Shop. When the total bill is strictly less than $2.00, the pattern lives on the sphere S², and your change from $2.00—the orbifold Euler characteristic χorb = 2 − Cost = 2 / |G|—tells you the exact number of fundamental domains |G| = 2 / (2 − Cost) that tile the globe. Click any row or symbol preset below to inspect its 3D spherical tiling, great-circle mirrors, gyration poles, and folded orbifold receipt.
n$(n−1)/n
Gyration · Cone Point
An n-fold rotation pole not lying on any mirror plane.
2 → $1/2 · 3 → $2/3 · 4 → $3/4 · 5 → $4/5
*$1
Kaleidoscope · Mirror Boundary
Opens a boundary loop of mirror reflections on the folded orbifold.
* alone (bilateral mirror) → $1.00
*n$(n−1)/(2n)
Corner Point · Mirror Crossing
Two great-circle mirrors crossing at angle π/n (half the cost of a cone).
*2 → $1/4 · *3 → $1/3 · *4 → $3/8 · *5 → $2/5
×$1
Miracle · Cross-Cap
A glide-reflection or antipodal inversion path with no mirror line.
× alone (antipodal S₂) → $1.00
< $2|G| = 2/(2−Cost)
The Spherical Magic Theorem
Every spherical group costs $2 − $2/|G|. Your change from $2 is χorb = 2/|G|.
7 polyhedral + 7 axial families = all 14
Filter:
Sphere
Symbol
Group, Notations & Common Examples
Magic Theorem Cost
Change χ & |G|
Why 4 Symbols Under $2.00 Fail to Tile the Sphere ("Bad Orbifolds")
Every symbol below has a Magic Theorem cost strictly under $2.00, yet none of them is the quotient of S² by a group of isometries. Click any card below to visualize why a single pole or unequal poles cannot close up consistently on a sphere.
N (e.g. 3)Cost: $2/3 (< $2)
The Teardrop (Lone Gyration)
Every 3D rotation axis pierces the sphere in two antipodal poles. You cannot have a 3-fold cone at the North Pole without a matching 3-fold cone at the South Pole (33).
pq (e.g. 23)Cost: $1/2 + $2/3 = $7/6
Spindle / Unequal Poles (p ≠ q)
A single rotation axis cannot spin by 180° at the North Pole and 120° at the South Pole—the northern and southern hemispheres would tear apart along the equator.
*N (e.g. *3)Cost: $1 + $1/3 = $4/3
Teardrop Mirror (Lone Corner)
Two great-circle mirrors on a sphere always intersect at two antipodal corners (a lune *NN), never at a single corner.
*pq (e.g. *23)Cost: $1 + $1/4 + $1/3 = $19/12
Unequal Mirror Lune (p ≠ q)
Two great circles cross at the exact same dihedral angle at both antipodal intersections, so their corner orders must be equal (*pp).