# The Sense of Place: Grid Cells in the Brain and the Transcendental Number e

arXiv: https://arxiv.org/abs/1304.0031 · HTML: https://mlsanity.com/paper/1304.0031/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Mar 2013.

## Built on by (most cited first, 12 of 12)

- [Self-Supervised Learning of Representations for Space Generates Multi-Modular Grid Cells](https://mlsanity.com/paper/2311.02316/) (2023)
- [On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding](https://mlsanity.com/paper/2405.16865/) (2024)
- [Range, not Independence, Drives Modularity in Biologically Inspired Representations](https://mlsanity.com/paper/2410.06232/) (2024)
- [Understanding Neural Coding on Latent Manifolds by Sharing Features and Dividing Ensembles](https://mlsanity.com/paper/2210.03155/) (2022)
- [Emergence of Grid-like Representations by Training Recurrent Networks with Conformal Normalization](https://mlsanity.com/paper/2310.19192/) (2023)
- [REMI: Reconstructing Episodic Memory During Internally Driven Path Planning](https://mlsanity.com/paper/2507.02064/) (2025)
- [A Computational Theory of Learning Flexible Reward-Seeking Behavior with Place Cells](https://mlsanity.com/paper/2204.11843/) (2022)
- [Determinantal Point Process Attention Over Grid Cell Code Supports Out of Distribution Generalization](https://mlsanity.com/paper/2305.18417/) (2023)
- [GridPE: A Grid Cell-Inspired Unified Position Embedding for Arbitrary-Dimensional Spaces](https://mlsanity.com/paper/2406.07049/) (2024)
- [A Grid Cell-Inspired Structured Vector Algebra for Cognitive Maps](https://mlsanity.com/paper/2503.08608/) (2025)
- [The Metric Slingshot: Navigational Reuse as Width-Optimal Structural Decoupling in Continual Learning](https://mlsanity.com/paper/2603.15412/) (2026)
- [How does Chain of Thought decompose complex tasks?](https://mlsanity.com/paper/2604.08872/) (2026)

## Abstract (arXiv)

Grid cells in the brain respond when an animal occupies a periodic lattice of "grid fields" during spatial navigation. The grid scale varies along the dorso-ventral axis of the entorhinal cortex. We propose that the grid system minimizes the number of neurons required to encode location with a given resolution. We derive several predictions that match recent experiments: (i) grid scales follow a geometric progression, (ii) the ratio between adjacent grid scales is the square root of e for idealized neurons, and robustly lies in the range 1.4-1.7 for realistic neurons, (iii) the scale ratio varies modestly within and between animals, (iv) the ratio between grid scale and individual grid field widths at that scale also lies in this range, (v) grid fields lie on a triangular lattice. The theory also predicts the optimal grids in one and three dimensions, and the total number of discrete scales.