---
title: "Flat Torus Universe and Matched Circles Interactive Lab | Sandbox Physics"
description: "Can a flat universe still be finite? Yes. Flatness is a local property, and a flat space can still close on itself like a video-game screen whose edges join."
canonical_url: "https://sandboxphysics.com/labs/flat-torus-matched-circles/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/flat-torus-matched-circles/"
last_modified: "2026-11-14"
---

# Flat Torus Universe and Matched Circles

**Can a flat universe still be finite?** Yes. Flatness is a local property, and a flat space can still close on itself like a video-game screen whose edges join. If such a box were smaller than the visible universe, the microwave sky would repeat the same pattern along pairs of circles. Planck found no such circles, so any box must be about as large as the visible universe or larger.

Resize a flat periodic box and watch the same sky spots return as matched circle pairs, then compare each match with chance.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

This synthetic teaching sky uses one flat cubic three-torus with box edge L, measured in units of the comoving distance to last scattering. A Gaussian random field with a Gaussian-smoothed spectrum is made exactly periodic on a grid of 64 cells per edge and read on a thin last-scattering sphere. Optional white noise is independent in each one-degree sky cell. The model omits curvature, the other compact spaces, the integrated Sachs-Wolfe and Doppler terms, the thickness of last scattering, masks, foregrounds and beams, and it uses no Planck or WMAP map. The chance level is the largest match among unrelated axes in this one sky, not a calibrated false-detection probability. The box is a teaching example, not a measured shape of our Universe.

## The same sky, read along circles

Synthetic sky, not a Planck map

### Whole-sky map with the predicted circles

### Temperature around each circle

### Circle search against chance

How the circles and the match are computed

$$
\cos\alpha=\frac{|\mathbf g|}{2\chi_{\mathrm{rec}}},\qquad S(\phi_*)=\frac{2\langle T_A(\psi)\,T_B(\psi+\phi_*)\rangle}{\langle T_A^2+T_B^2\rangle}
$$

Each lattice translation shorter than twice the last-scattering distance gives one pair of circles. The search compares back-to-back circles on 96 evenly spread axes and the short lattice axes, at every rotation. The grey chance level uses only blind axes more than eight degrees from any clone axis.

[Cornish, Spergel and Starkman (1998), circles in the sky](https://arxiv.org/abs/astro-ph/9801212v1) · [Planck 2015 results XVIII, background geometry and topology](https://arxiv.org/abs/1502.01593v2)

[Continue with the course on the shape of space](https://sandboxphysics.com/stories/cosmic-topology/)

## Flat Torus Universe and Matched Circles

**Background** Measurements say space is very nearly flat. Flatness describes geometry near every point; it does not decide whether space goes on forever or joins back onto itself.

**Why it matters** In a flat three-torus, walking out of one face of a box brings you back through the opposite face. If the box is smaller than the sphere from which the microwave background reaches us, some light reaches us twice and the sky repeats itself.

- **One-sentence intuition**: A copy of the observer at distance less than twice the last-scattering distance makes a pair of circles with identical temperatures. A box larger than that leaves no circles, and the sky alone cannot tell it from infinite flat space.

### Core mathematical model

#### Copies of the observer

$$
\mathbf x\sim\mathbf x+L\mathbf n,\qquad \mathbf n\in\mathbb Z^3,\qquad \mathbf g=L\mathbf n
$$

Points that differ by a whole number of box edges are the same point. In the covering space, a copy of the observer sits at every lattice point. Distances are in units of the comoving distance to last scattering.

#### Where two spheres meet

$$
\cos\alpha=\frac{|\mathbf g|}{2\chi_{\mathrm{rec}}},\qquad |\mathbf g|<2\chi_{\mathrm{rec}}
$$

Our last-scattering sphere and the sphere of a nearby copy meet on a circle. We see it twice, around the direction of that copy and around the opposite direction, with the same angular radius.

#### Comparing two circles

$$
S(\phi_*)=\frac{2\langle T_A(\psi)\,T_B(\psi+\phi_*)\rangle}{\langle T_A^2+T_B^2\rangle}
$$

The statistic equals one for identical circles and scatters about zero for unrelated ones. Taking the largest value over many rotations and axes also lifts it by chance, so a match must clear the chance level of the same sky.

### Common difficulties

#### Flat does not mean infinite

**Typical misconception** If space is flat, it must go on forever.

**Better mental model** Curvature is measured locally. A flat three-torus has zero curvature everywhere and a finite volume. Local measurements cannot tell the two apart; only light that has travelled around the space can.

### Run the experiment

1. 01

   #### Find the repeated spot

   Keep the default box and watch the two photon paths. Before looking at the data row, predict how many circle pairs a box of edge 1.2 makes.

   **What to observe:** Light from one spot reaches you from two directions. The six nearest copies give three pairs at 53.1 degrees and the twelve diagonal copies give six pairs at 31.9 degrees.

2. 02

   #### Grow the box past the limit

   Drag the box corner outward. Predict the edge length at which the last pair disappears, then check it in the scan chart.

   **What to observe:** The circles shrink and vanish when the edge reaches two last-scattering distances. Beyond that, the best match found sits on the chance level everywhere.

3. 03

   #### Add noise

   Choose Noise hides the match. Compare the box-axis match with the chance level, then save the record.

   **What to observe:** Noise as strong as the signal roughly halves the match. The pair is still physically there, but in this sky it no longer stands above chance.

Physical conclusion

Read a missing circle signal as a lower bound on the size of space, not as proof that space is infinite.

## Related experiments and articles

The shape of space · Reading path

### From a flat box to matched circles and Planck limits

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Can a flat universe be finite?**](https://sandboxphysics.com/stories/cosmic-topology/can-a-flat-universe-be-finite/)
2. [Article **How the three-torus makes flat space finite**](https://sandboxphysics.com/stories/cosmic-topology/flat-but-finite-three-torus/)
3. [Article **Why a small universe draws circles on the sky**](https://sandboxphysics.com/stories/cosmic-topology/why-a-small-universe-draws-circles/)
4. [Article **Telling a real circle match from a lucky one**](https://sandboxphysics.com/stories/cosmic-topology/telling-a-real-circle-match-from-a-lucky-one/)
5. [Article **Recompute matched circles in a toy three-torus**](https://sandboxphysics.com/stories/cosmic-topology/recompute-matched-circles-in-a-toy-torus/)
6. [Article **How to read Planck's cosmic topology limits**](https://sandboxphysics.com/stories/cosmic-topology/reading-planck-topology-limits/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)