Definition

The torus $T^2 = S^1 \times S^1$ is the Cartesian product of two circles, parametrised by a pair of angles $(\theta_1,\theta_2) \in [0,2\pi)^2$, each taken modulo $2\pi$ independently. A standard embedding in $\mathbb{R}^3$ (radii $R > r > 0$):

$x = (R + r\cos\theta_2)\cos\theta_1, \quad y = (R+r\cos\theta_2)\sin\theta_1, \quad z = r\sin\theta_2$

More generally, in a $d$-dimensional space, $T^2$ can appear as two orthogonal 2D circular subspaces occupied independently: any pair $(\theta_1,\theta_2)$ that occurs on one circle can co-occur with any angle on the other.

Intuition

The surface of a donut. Walking along one direction (around the tube) returns you to your start after wrapping once; walking along the other direction (around the hole) returns you after wrapping the other way — and the two walks don’t interfere with each other. Two clock faces running independently, glued into one joint state space.

Properties

• $H_1(T^2) = \mathbb{Z}^2$. Two independent, non-contractible loops (go around one way; go around the other way), neither reducible to a multiple of the other. This is the discriminating invariant against a bare circle ($H_1 = \mathbb{Z}$, one loop): the Künneth formula gives $H_1(S^1\times S^1) \cong \big(H_1(S^1)\otimes H_0(S^1)\big) \oplus \big(H_0(S^1)\otimes H_1(S^1)\big) \cong \mathbb{Z}^2$.
• Flat (zero Gaussian curvature) intrinsically, despite looking curved when embedded in $\mathbb{R}^3$ — the standard embedding’s curvature is purely extrinsic. $T^2$ is the unique closed, orientable, genus-1 surface, and $\chi(T^2) = 0$ (Euler characteristic), consistent with Gauss–Bonnet ($\iint_{T^2} K \, dA = 2\pi \chi(T^2) = 0$).
• A strictly stronger claim than a circle. Finding one cyclic variable is $S^1$-evidence; the torus specifically requires two independent, orthogonal cyclic coordinates coexisting in the same joint space. Two separate, unrelated circles found in two separate experiments do not establish a torus — see Circle Exercise 5 for why independence must be checked directly, not inferred from two individually-confirmed circles.
• Universal cover is $\mathbb{R}^2$. $T^2 = \mathbb{R}^2 / \mathbb{Z}^2$ — the plane modulo integer translations in two independent directions, i.e. a square (or any fundamental parallelogram) with opposite edges identified.
• Fundamental group $\pi_1(T^2) = \mathbb{Z}^2$, abelian — unlike the fundamental group of surfaces of genus $\geq 2$, which is non-abelian.

Relative to the “disc” (vector-addition) case

A rank-2 linear projection of $T^2$‘s standard 4-coordinate embedding $(\cos\theta_1,\sin\theta_1,\cos\theta_2,\sin\theta_2) \mapsto (\cos\theta_1+\cos\theta_2,\sin\theta_1+\sin\theta_2)$ collapses the torus onto a filled, contractible 2D region (Betti numbers $(1,0,0)$, like a disc — no independent loop survives the projection) rather than a genuine $T^2$ ($(1,2,1)$). Moisescu-Pareja, McCracken, Wiltzer, Létourneau, Daniels, Precup & Love (2025) prove that in single-hidden-layer networks trained on modular addition $(a+b) \bmod n$, which of the two occurs is governed entirely by whether the two per-input phase variables in a “simple neuron” model are perfectly correlated (disc) or independent (torus) — see Theorem 4.1 in Theoretical / Analytical‘s Persistent homology (Betti number analysis) entry for the empirical validation.

Key evidence

The same 2025 paper resolves an apparent counter-example to the universality hypothesis: Zhong et al. (2023) had reported that “Clock” (trainable-attention) and “Pizza” (uniform-attention) networks trained on identical modular-addition data learn geometrically distinct circuits. Using PCA and persistent homology (Betti-number distributions) across 703 independently trained single-hidden-layer networks (MLP-Add, MLP-Concat, and both attention variants), the authors find Clock and Pizza are in fact nearly indistinguishable (MMD scores 0.018-0.024 between them) and both learn the disc/vector-addition manifold, not two different structures — while a third architecture (MLP-Concat) learns the genuine torus, later linearly projected down to the same disc/circle en route to the output logits. This is entirely a toy-model, synthetic-modular-arithmetic result (no pretrained language model is studied) — see modular-addition-torus-universality.

A second, independent line of evidence comes from biologically-inspired path-integration RNNs rather than language-model toy tasks. Xu, Gao, Zhang, Wei & Wu (2024) prove that a specific training-time mechanism (“conformal normalization,” which rescales the RNN’s input velocity so local neural-state displacement is proportional to physical displacement regardless of heading direction) forces the population-activity manifold to be a compact connected abelian Lie group — hence topologically a torus — and that conformal isometry further restricts the induced 2D lattice to be hexagonal rather than square, matching the hexagonal grid cells observed in the mammalian entorhinal cortex. Ablating the mechanism at training time eliminates the hexagonal structure. See conformal-normalization-provably-yields-toroidal-hexagonal-grid-cell-representations.

An earlier study using an ordinary (non-conformal) continuous-attractor path-integration RNN found that hexagonal gridness score and toroidal-manifold membership are not the same thing in practice: Schøyen, Pettersen, Holzhausen, Fyhn, Malthe-Sørenssen & Lepperød (2023) identify a cell cluster whose autocorrelogram point cloud forms a torus across multiple trained environments, distinct from the subset of cells with high hexagonal gridness score, and show by targeted ablation that only torus-cluster membership — not gridness score — is causally load-bearing for path-integration accuracy. See toroidal-cell-cluster-not-gridness-score-is-causally-load-bearing-for-path-integration.

Pellegrino & Chadwick (2025, NeurIPS 2025) derive a Riemannian pullback metric directly from a real task-trained continuous-time RNN’s own dynamics (rather than from a static point cloud) on a sequential working-memory task, and show the resulting hyper-torus’s Gaussian curvature is genuinely non-flat — switching sign at different points in time — while the pulled-back metric’s eigenvalues collapse toward zero as the network retrieves different stored memories, demonstrating the torus’s own intrinsic shape dynamically warps during task execution rather than remaining the fixed flat torus this node’s other evidence describes. See task-trained-rnns-state-space-manifold-including-a-hyper-torus-in-a-working-memory-task-develops-measured-non-flat-sign-changing-gaussian-curvature-and-eigenvalue-collapse-as-the-network-executes-its-task.

Foundational grid-cell evidence (pre-conformal-normalization)

Two earlier, foundational papers established the empirical phenomenon that Xu, Gao, Zhang, Wei & Wu (2024, above) later explained mechanistically. Cueva & Wei (2018) train an ordinary vanilla RNN end-to-end on a path-integration task (estimating 2D position from velocity in square and triangular arenas), with no architectural imposition of periodicity, and find hexagonal grid-cell-like periodic tuning (plus border and band cells) emerges in unit activity, measured via gridness score on spatial autocorrelograms — purely descriptive, no causal intervention. See cueva-wei-2018-hexagonal-grid-cell-like-periodic-spatial-tuning-quantified-by-gridness-score-emerges-in-a-trained-vanilla-rnn-with-no-architectural-imposition-of-periodicity. Banino et al. (2018) train an LSTM path-integration module feeding a deep RL navigation agent, similarly finding gridness-scored hexagonal and border-vector-cell-like tuning, and additionally show a causal result: ablating the highest-gridness-scoring units significantly degrades navigation performance, while ablating heterogeneous non-grid units does not. See banino-etal-2018-grid-cell-like-codes-emerge-in-a-trained-lstm-path-integration-module-supporting-vector-based-shortcuts-in-a-navigating-rl-agent-and-ablating-the-highest-gridness-units-degrades-navigation. A third foundational model, the Tolman-Eichenbaum Machine (Whittington, Muller, Mark, Chen, Barry, Burgess & Behrens 2020), trained on a broader range of spatial and non-spatial relational-memory tasks, develops the same measured hexagonal-periodicity grid units alongside band, border, and object-vector cells, plus hippocampal-like place cells whose remapping between distinct environments follows a structured, non-random pattern matching recorded biological data — extending the grid-code cluster beyond pure spatial path-integration into general relational-structure learning. See whittington-etal-2020-the-tolman-eichenbaum-machine-develops-real-hexagonal-grid-cells-band-cells-border-cells-and-remapping-place-cells-when-trained-on-spatial-and-relational-tasks.

Exercises

Base

1. Write down explicit coordinates for two points on $T^2$ that differ only in $\theta_1$ (same $\theta_2$), and explain why they lie on the “same” small circle in the standard $\mathbb{R}^3$ embedding.

2. Why is $T^1$ (as in, the 1-fold product $S^1$ alone) just the circle, and not something new?

Middle

3. Using the Künneth formula $H_1(A\times B) \cong (H_1(A)\otimes H_0(B)) \oplus (H_0(A)\otimes H_1(B))$, confirm $H_1(T^2) = \mathbb{Z}^2$ from $H_0(S^1)=\mathbb{Z}$, $H_1(S^1)=\mathbb{Z}$.

4. Suppose activations for two candidate cyclic variables are collected and each, considered alone, traces a clean circle under PCA. Describe a concrete statistical test (not just a visual one) for whether the joint configuration is a genuine $T^2$ rather than a 1-dimensional diagonal subset of it (recall Circle Exercise 5).

Pro

5. Prove that $T^2 = \mathbb{R}^2/\mathbb{Z}^2$ (the plane modulo the integer lattice) is homeomorphic to $S^1\times S^1$.

6. A model’s activations for two cyclic variables live in a joint 4D space and are confirmed (by the independence test of Exercise 4) to realize a genuine product structure with $H_1=\mathbb{Z}^2$. A skeptic argues this is still not “one torus” but “two circles that happen to be independent,” and that the distinction is merely semantic. Give a precise mathematical sense in which they are wrong — i.e. a property the joint configuration has that the disjoint pair does not.