Surfaces and their Euler Characteristic (χ)

Qualitative visualization: each panel shows a closed, orientable surface with genus g. The formula χ = 2 − 2g holds.
Sphere (g=0)
χ = 2
b₀−b₁+b₂ = 1−0+1=2
Isosurface of a metaball: sphere topology.
Torus (g=1)
χ = 0
b₀−b₁+b₂ = 1−2+1=0
Ring of metaballs forming a "donut".
Double Torus (g=2)
χ = −2
b₀−b₁+b₂ = 1−4+1=−2
Two connected rings in one piece ⇒ two handles.
Surface with 3 holes (g=3)
χ = −4
b₀−b₁+b₂ = 1−6+1=−4
Three connected handles (pretzel) via metaball field.