5 From Euler Characteristic to Euler Characteristic Surfaces
“In mathematics the art of proposing a question must be held of higher value than solving it.” – Georg Cantor
This chapter builds a hierarchy of topological descriptors, each motivated by the limitations of the previous one. We begin with a single integer summary of a simplicial complex, and end with a surface-valued descriptor capable of characterising dynamical systems.
| Step | Object | Description |
|---|---|---|
| 1 | \chi | Single number — topological fingerprint |
| 2 | \chi(t) | Euler Characteristic Curve (ECC) — \chi tracked across a filtration |
| 3 | \operatorname{ECT} | ECC for every direction — complete shape descriptor |
| 4 | \chi(r, t) | Euler Characteristic Surface (ECS) — ECC across scale r and time t |
| 5 | d(\chi_1, \chi_2) | Euler Metric — distance between two ECSs |
5.1 Euler Characteristic
Given a simplicial complex K, we want a single number that captures its topology. Recall that we can count the simplices of each dimension — vertices, edges, triangles, and so on. The Euler Characteristic combines these counts into one integer via an alternating sum.
For a simplicial complex K with C_k simplices of dimension k, the Euler Characteristic is defined as: \chi(K) = \sum_k (-1)^k C_k = C_0 - C_1 + C_2 - C_3 + \cdots
The classical instance of this formula is Euler’s formula for convex polyhedra: for any convex polyhedron with V vertices, E edges, and F faces, V - E + F = 2. For a tetrahedron, V = 4, E = 6, F = 4, and indeed 4 - 6 + 4 = 2.
5.1.1 Connection to Betti Numbers
The Euler Characteristic is not merely a count — it is deeply connected to the homology of K.
\chi(K) = \sum_k (-1)^k \beta_k = \beta_0 - \beta_1 + \beta_2 - \cdots where \beta_k = \operatorname{rank}(H_k(K;\mathbb{Z})) is the k-th Betti number.
Recall the topological meaning of each Betti number:
- \beta_0: number of connected components
- \beta_1: number of independent loops
- \beta_2: number of enclosed voids
The Euler Characteristic compresses the full homological fingerprint into a single integer. It is lossy — many different combinations of Betti numbers yield the same \chi — but it is extremely cheap to compute, requiring no boundary matrix reduction.
\chi is a topological invariant: if two spaces are homeomorphic, they have the same Euler Characteristic.
| Space | \chi |
|---|---|
| Sphere S^2 | 2 |
| Torus T^2 | 0 |
| Klein bottle K^2 | 0 |
Exercise 5.1
- Compute \chi for a simplicial complex with C_0 = 5 vertices, C_1 = 6 edges, and C_2 = 2 filled triangles. Verify your answer using the Euler–Poincaré formula by computing all Betti numbers explicitly.
- Add a 6th edge to the above complex that creates a new loop. How do \beta_1 and \chi change?
- Show that the torus T^2 and the Klein bottle K^2 have the same Euler Characteristic but are not homeomorphic. What invariant distinguishes them?
5.2 The Euler Characteristic Curve (ECC)
The examples above show that \chi alone can fail to distinguish spaces. But recall that in TDA we do not study a fixed complex — we study a filtration, a nested sequence of complexes parameterised by a scale. We can therefore ask: how does \chi evolve as the filtration grows?
For a filtration K(t_0) \subseteq K(t_1) \subseteq \cdots \subseteq K(t_n), the Euler Characteristic Curve (ECC) is the function: \chi(t) = \sum_k (-1)^k C_k(t) = \sum_k (-1)^k \beta_k(t) that tracks \chi as a function of the filtration parameter t.
The ECC is a step function: it jumps at each t where a new simplex enters the filtration. Each jump records a topological event — a feature is born (a new component, loop, or void appears) or dies (components merge, loops fill in).
The ECC has several useful properties:
- It is a complete summary of the Betti curves: \chi(t) = \beta_0(t) - \beta_1(t) + \beta_2(t) - \cdots
- It is computationally cheap: count simplices by dimension at each filtration level, \mathcal{O}(n) in practice
- No boundary matrix reduction is required
- It is lossy relative to the full persistence diagram, but often sufficient for applications
Exercise 5.2 Take 5 points \{a, b, c, d, e\} in the plane with all pairwise distances known. Compute the ECC for the following filtration of a Rips complex:
| t | Event | C_0,\, C_1,\, C_2 | \chi(t) |
|---|---|---|---|
| 0 | 5 isolated vertices | 5,\, 0,\, 0 | |
| 1 | Add edges ab, bc, cd | 3,\, 3,\, 0 | |
| 2 | Add edge ac, fill triangle abc | 2,\, 3,\, 1 | |
| 3 | Add remaining edges, merge all components | — |
- Complete the \chi(t) column.
- Sketch the ECC as a step function over t.
- Identify which steps correspond to a feature birth and which to a feature death.
5.3 The Euler Characteristic Transform (ECT)
A single ECC depends on the direction used to build the filtration (for example, a height function). Different directions reveal different geometric features of the shape. The ECT resolves this by collecting all possible ECCs simultaneously.
Let M \subset \mathbb{R}^n be a compact, triangulable subset and let v \in S^{n-1} be a unit direction vector. For each v, define the height function f_v(x) = v \cdot x and the corresponding sublevel set filtration. The Euler Characteristic Transform (ECT) is the map: \operatorname{ECT}(M) : S^{n-1} \times \mathbb{R} \to \mathbb{Z}, \qquad (v,\, t) \mapsto \chi\!\left(M \cap \{x : v \cdot x \leq t\}\right)
The following theorem is the key theoretical result that justifies using the ECT as a shape descriptor.
The ECT is injective on the class of compact, triangulable subsets of \mathbb{R}^n: M_1 \ne M_2 \implies \operatorname{ECT}(M_1) \ne \operatorname{ECT}(M_2). In other words, the ECT completely characterises shape — it is a sufficient statistic.
Proof sketch. For any two distinct vertices of M, one can construct a direction v that separates them via the height function. Injectivity at the vertex level then propagates to the full complex by induction on skeleton dimension. \square
In practice, one approximates the ECT by sampling K directions uniformly from S^{n-1} (typically K = 100–1000) and computing the ECC over a discretised filtration grid for each direction. The resulting ECCs are concatenated into a single feature vector suitable for use in machine learning pipelines.
Exercise 5.3
- Let M be a filled triangle in \mathbb{R}^2. Compute \operatorname{ECT}(M)(v, t) for the direction v = (1, 0) (left-to-right sweep) as a function of t.
- Now compute \operatorname{ECT}(M)(v, t) for v = (0, 1) (bottom-to-top sweep). How do the two ECCs differ?
- Explain intuitively why a single direction is not sufficient to distinguish all compact triangulable subsets of \mathbb{R}^2.
5.4 Euler Characteristic Surfaces (ECS)
The ECS extends the ECC to dynamical systems by adding a second axis: spatial scale r. Rather than a curve in t alone, we obtain a surface over (r, t).
5.4.1 The Scale Parameter
For a binary image or point cloud observed at time t, the r-neighbourhood expansion S_r = \{x : d(x, S) \leq r\} is homotopically equivalent to the Čech complex at scale r. Running this expansion for increasing r is exactly the filtration we already know, now repeated independently at each time step t.
For a binary image time series, the Euler Characteristic Surface (ECS) is: \chi(r, t) = N_b(r, t) - N_w(r, t) where N_b(r, t) is the number of connected foreground clusters and N_w(r, t) is the number of connected background regions at scale r and time t.
This is a direct consequence of the 2D Euler formula \chi = \beta_0 - \beta_1 for binary images. No boundary matrix reduction is required.
5.4.2 The Euler Metric
To compare two dynamical systems, we measure the L^2 distance between their ECSs.
For two Euler Characteristic Surfaces \chi_1 and \chi_2 defined over scales r \in [0, R] and times t \in [0, T]: d(\chi_1, \chi_2) = \|\chi_1 - \chi_2\|_2 = \left(\int_0^R \int_0^T \bigl(\chi_1(r,t) - \chi_2(r,t)\bigr)^2\, dr\, dt\right)^{\!1/2}
A large Euler Metric between two consecutive time snapshots indicates a topological transition; a small value indicates a stable regime.
5.4.3 Stability Theorems
The ECS is not only computationally convenient — it comes with rigorous stability guarantees connecting it to full persistent homology.
\|\chi(K(r,t^*)) - \chi(L(r,t^*))\|_1 \leq 2\sum_n W_1\!\left(\operatorname{PD}_n(K),\, \operatorname{PD}_n(L)\right) The ECS 1-metric at any fixed time t^* is bounded above by the sum of 1-Wasserstein distances between the corresponding persistence diagrams.
For a uniformly continuous dynamical system with M data points, a position perturbation of size \varepsilon leads to an ECS change bounded by: \tfrac{1}{3}M(M+1)(M+2)\,\varepsilon
Exercise 5.4
- Explain why \chi(r, t) = N_b(r, t) - N_w(r, t) corresponds to \sum_k (-1)^k \beta_k on a 2D binary image. Work through the 2D case of the Euler–Poincaré formula explicitly.
- In the drying droplet experiment (Roy et al. 2020), \chi(r, t) becomes negative during an intermediate phase. What does a negative \chi imply about the relative sizes of \beta_0 and \beta_1? What physical event causes this?
- The Euler Metric is identical whether computed via coarse-graining or Alpha complexes. Why is this important for practitioners working with different data modalities?
- Could the ECS or Euler Metric serve as an order parameter for detecting phase transitions? Compare with a classical order parameter such as mean particle velocity in the Vicsek model, identifying at least one advantage and one limitation.
5.5 ECS vs Persistent Homology
| Feature | Persistence Diagrams | ECS + Euler Metric |
|---|---|---|
| Information | Complete homology at all scales | Compressed: \sum(-1)^k \beta_k |
| Computation | \mathcal{O}(n^{2.37}) boundary matrix reduction | \mathcal{O}(n + R \cdot T) |
| Time series | Requires extensions (vineyards, zigzag) | Native spatio-temporal format |
| Statistics | Requires Wasserstein/bottleneck metrics | Hilbert space — standard L^2 tools apply |
| Stability | Bottleneck stability theorem | Bounded by PH; temporal perturbation bound |
Prefer Persistent Homology when you need full multi-scale homological detail, when comparing shapes by birth/death structure, or when datasets are small enough that computational cost is not a concern.
Prefer ECC / ECT / ECS when comparing many time series, when you need to average or regress on topological summaries, when working with dynamical systems that have a native time axis, or when detecting topological transitions and regime changes.
5.6 References
Roy, Mitra, Dutta et al. (2020). Physics of Fluids 32, 123310. ECS applied to drying droplets — introduces the framework and level-curve visualisation.
Roy, Mitra, Dutta et al. (2023). Physics of Fluids 35, 083305. Euler metric and validation on fluid systems — eggbeater flow and experimental droplets.
Roy et al. (2025). La Matematica 4:735–763. Mathematical foundations: stability theorems, Vicsek model, time-series framework.
Turner, Mukherjee, Boyer (2014). Foundations of Computational Mathematics. ECT injectivity theorem.