---
title: "The shape of space"
description: "Learn how a flat universe can be finite, why a small one draws matched circles on the microwave sky, and how Planck limits the box."
canonical_url: "https://sandboxphysics.com/stories/cosmic-topology/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/cosmic-topology/"
last_modified: "2026-11-14"
---

# The shape of space

Sandbox Physics · 2026-11-12

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.

1. L1 · Concept

   ## [Can a flat universe be finite?](https://sandboxphysics.com/stories/cosmic-topology/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.

2. L1 · Chapter

   ## [How the three-torus makes flat space finite](https://sandboxphysics.com/stories/cosmic-topology/flat-but-finite-three-torus/)

   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.

3. L2 · Chapter

   ## [Why a small universe draws circles on the sky](https://sandboxphysics.com/stories/cosmic-topology/why-a-small-universe-draws-circles/)

   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.

4. L3 · Chapter

   ## [Telling a real circle match from a lucky one](https://sandboxphysics.com/stories/cosmic-topology/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.

5. L3 · Method

   ## [Recompute matched circles in a toy three-torus](https://sandboxphysics.com/stories/cosmic-topology/recompute-matched-circles-in-a-toy-torus/)

   Run one standard-library Python script on saved inputs, compare true circle pairs with random false matches, and read four failure cases.

6. L3 · Paper guide

   ## [How to read Planck's cosmic topology limits](https://sandboxphysics.com/stories/cosmic-topology/reading-planck-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.