The shape of space
Start with a plain question: can space be flat and still finite? Build a flat box whose faces join, find the circles it would draw on the microwave sky, learn to tell a real match from luck, then recompute the search and read the Planck limits yourself.
L1 · Concept
Can a flat universe be finite?
Yes. Space can be flat everywhere and still close on itself like a box whose faces join. Planck found no repeated circles, so any such box is very large.
L1 · Chapter
How the three-torus makes flat space finite
Glue opposite faces of a cube and straight lines still obey Euclid, yet space has finite volume and every object has a lattice of copies.
L2 · Chapter
Why a small universe draws circles on the sky
Two copies of the last-scattering sphere cross in a circle. Derive its angular radius, its multiplicity and why it vanishes when the box exceeds the view.
L3 · Chapter
Telling a real circle match from a lucky one
Define the circle statistic, derive its spread for unrelated circles, see how noise lowers a true match, and set a threshold from false pairs.
L3 · Method
Recompute matched circles in a toy three-torus
Run one standard-library Python script on saved inputs, compare true circle pairs with random false matches, and read four failure cases.
L3 · Paper guide
How to read Planck's cosmic topology limits
Read Planck 2015 XVIII figures 11 and 12 and the 0.97 limit, then see what the COMPACT preprint says those limits leave open.