Sphere Grids

A globe is covered in a grid: horizontal lines of latitude and vertical lines of longitude. A finite sphere grid does the same thing with finite circles, giving us a way to talk about poles, equators, rows, and columns using the S1 vocabulary we already know.

Lesson Goal

Learn to read a finite sphere as a stack of latitude circles indexed by a longitude position, and see why each latitude row is itself just a finite circle from S1.

The Idea

Slice a globe into horizontal rings. Each ring is a circle. Stack the rings from the south pole up to the north pole:

A latitude picks which ring you are on (which row).
A longitude picks where you are around that ring — and going around a ring is exactly rotation in a finite circle C_n.
The poles are the two rings that have shrunk to a single point.

So a sphere grid is not a brand-new object. It is finitely many copies of S1 (the latitude rings) threaded onto a vertical axis. Everything you learned about rotation, coils, and period in S1 still works along each row.

Worked Example

Build a small grid with m = 3 latitude rings, each a circle C_4 (4 longitude steps):

row 0 (south):  • • • •      ← a copy of C_4
row 1 (middle): • • • •      ← a copy of C_4
row 2 (north):  • • • •      ← a copy of C_4

Rotate longitude by one step on the middle row: 0 → 1 → 2 → 3 → 0. That is ordinary C_4 rotation; the latitude index never changes. To move between rows you change latitude, not longitude. Reusing S1 here is the whole point: the grid inherits the period and orbit facts of the circle on every row.

Common Mistake

A finite sphere grid is a learning scaffold, not a claim that continuous sphere geometry has been formalized. The certified facts are the finite cell counts and the row-is-a-circle structure recorded in the cards below. See What “Proved” Means Here for the boundary.

Checkpoint

On a sphere grid, which coordinate behaves exactly like an S1 rotation, and which coordinate do you change to climb from the equator toward a pole?

Source Trail

Paper source: S2 Sphere Grids And Latitude Coils