# Grid-cell topology — Quotient out repetition; expose the underlying organization

Gardner et al. · Nature 2022 · Figure 2a–b

[Publication](https://www.nature.com/articles/s41586-021-04268-7) · [Original image source](https://media.springernature.com/full/springer-static/image/art%3A10.1038%2Fs41586-021-04268-7/MediaObjects/41586_2021_4268_Fig2_HTML.png)

## Inspect the actual images before designing

- [focused figure](../../assets/calibration/a4bc70a9-torus-a.png) — SHA-256 `9d3a6904f224744001271b61fa7175247a8f4aa120ccfc6f41f7630114175024`. Publisher panels a and b cropped separately, preserving their internal arrangements. Full figure available in the viewer.
- [focused figure](../../assets/calibration/42be2c29-torus-b.png) — SHA-256 `978e809b7e1460eef0b4fe149e4288c98e600116018797ecb0e73dde328c992d`. Publisher panels a and b cropped separately, preserving their internal arrangements. Full figure available in the viewer.
- [full figure](../../assets/calibration/1663f848-torus2.png) — SHA-256 `903d931ad3003e4b2c59a7227151fd6104fb6f81b28df959a89e8f2930b52ddc`. Full published figure

Reading a caption or this guide is not image inspection. Open at least two approved reference images at readable size and record their hashes.

**Use when:** Periodic or symmetric repetitions obscure a simpler state representation.

## See the mechanism

1. In panel a, follow how opposite edges of the unwrapped domain correspond on the torus.
2. In b, compare repeated firing fields in physical space with compact toroidal hotspots.
3. The same activity is organized differently because the coordinates identify periodic states.

**Mechanism:** Grid-cell population activity has two periodic coordinates; physical positions can revisit the same population phase.

**Visual construction:** Represent periodicity on a torus and its unwrapped domain. Show physical and toroidal firing maps side by side.

**What the eye understands:** The scattered repetitions are expressions of one compact periodic organization—not unrelated peaks.

**Why an ordinary plot is weaker:** A physical heatmap shows repetition. The new coordinates reveal the shared structure behind it.

## Learn this visual style, then adapt it

**Observe:** The torus and its unwrapped domain make periodic identifications visible, beside physical and toroidal activity maps.

**Apply:** Show the original and transformed coordinate systems together, including their identification rule. Track one state through the mapping; use geometry only for the structure established by analysis.

Record `composition_observation`, `encoding_observation`, `style_observation`, and `planned_application` for this image. Specify visible layout, persistent geometry, selective emphasis, annotation placement, color roles and whitespace—not just “clean” or “beautiful”.

## Transfer into the project

**Replace the objects:** Grid phase → your periodic state; physical repetitions → equivalent instances; toroidal coordinates → justified identified variables.

**Keep the relationship:** State exactly which positions are identified. Keep inferred state topology distinct from physical anatomy.

1. Show the complicated representation alongside the transformed one.
2. Make the edge identifications or symmetry action explicit.
3. Track a single state's identity across the two representations.

**Acceptance test:** Does the coordinate change preserve the relation claimed? Support topology with analysis, not the appearance of a projection.

**Do not copy literally:** A torus-shaped rendering does not establish toroidal topology.

**Scientific boundary:** This is a topology of population activity, not an anatomical doughnut. It does not by itself prove a specific biophysical circuit.

## Attribution and rights

MechanismFigures design interpretation; not a quotation or endorsement by the paper authors.

Original authors/publishers retain image rights. The repository MIT license covers code/commentary, not the figures. Article Rights and permissions states CC BY 4.0; reviewed figure credit lines contain no separate restriction. Original and focused assets are separately labeled.

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