Study faces across dimensions with fast topological summaries. View graphs, exports, examples, and instant metrics. Understand complex structure through counts, ranks, and consistent reports.
Enter simplex counts for each dimension. Turn on boundary ranks only when you know the ranks of the boundary maps.
This sample shows a connected complex with one independent 1-cycle and internally consistent Euler-Poincaré values.
| Complex | f₀ | f₁ | f₂ | f₃ | r₁ | r₂ | r₃ | χ | β₀ | β₁ | β₂ | β₃ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sample Cycle-Rich Complex | 6 | 10 | 5 | 1 | 5 | 4 | 1 | 0 | 1 | 1 | 0 | 0 |
| Meaning | Vertices | Edges | Triangles | Tetrahedra | rank ∂₁ | rank ∂₂ | rank ∂₃ | Euler | Components | 1-cycles | 2-voids | 3-cavities |
The calculator combines simplex counts and optional boundary ranks to describe combinatorial and homological structure.
Dimension d = max{k | f_k > 0} Total simplices T = Σ f_k Euler characteristic χ = Σ (-1)^k f_k Reduced Euler characteristic χ̃ = χ - 1, for non-empty complexes Betti numbers: β₀ = f₀ - r₁ β_k = f_k - r_k - r_{k+1}, for 1 ≤ k ≤ 5 β₆ = f₆ - r₆ Occupancy ratio ρ_k = f_k / C(f₀, k+1)Here, fₖ is the number of k-dimensional simplices and rₖ is the rank of boundary map ∂ₖ. When ranks are accurate, the calculator checks the Euler-Poincaré identity by comparing χ with Σ(-1)ᵏβₖ.
It computes dimension, total simplices, Euler characteristic, reduced Euler characteristic, simplex occupancy ratios, and optional Betti numbers when boundary ranks are supplied.
An f-vector lists simplex counts by dimension. For example, f₀ counts vertices, f₁ counts edges, f₂ counts triangles, and higher entries count higher-dimensional simplices.
Betti numbers need boundary map rank information, not only simplex counts. Without those ranks, the calculator can still compute combinatorial summaries but not reliable homology values.
Euler characteristic is an alternating sum of simplex counts. It summarizes global structure and helps verify consistency when compared against alternating Betti numbers.
Warnings appear when counts exceed combinatorial limits, boundary ranks exceed feasible matrix rank, or computed Betti numbers become negative. These usually signal inconsistent inputs.
Occupancy compares the observed number of k-simplices with the maximum possible count C(f₀, k+1) using the same number of vertices.
Yes. The form includes simplex counts through dimension six, which supports many educational, combinatorial, and computational topology workflows.
Use CSV for spreadsheets, numerical review, and software pipelines. Use PDF for reports, handouts, audits, and sharing a fixed calculation snapshot.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.