Counting Holes in All Dimensions

Algebraic Topology · Axiom Academy

INTRO Counting Holes in All Dimensions Visual exploration of 0-dimensional, 1-dimensional, 2-dimensional holes and beyond Step 1: Dimension 0 — Counting Components The 0th homology group H₀ counts connected components. Use the slider to change the number of separate pieces and watch how H₀ responds. Step 2: Dimension 1 — Counting Loops The 1st homology group H₁ counts independent 1-dimensional holes — loops you can walk around. Click the button to add loops and see H₁ grow. Step 3: Dimension 2 — Counting Voids The 2nd homology group H₂ counts 2-dimensional "voids" — enclosed hollow spaces. These are harder to visualize in 2D, but crucial for understanding 3D shapes. Homology doesn't stop at dimension 2. For any space X, there's an infinite sequence of homology groups detecting holes in every dimension.

This is the written version of the interactive lesson above. See the full Algebraic Topology course.