THE GENUS count the holes — the number no stretching can change
The genus of a surface is, intuitively, its number of handles or holes: a sphere has genus 0, a torus (donut) genus 1, a two-holed pretzel genus 2. It is a topological invariant — no amount of stretching, denting, or bending changes it, because you cannot add or remove a hole without tearing. It is tied to the Euler characteristic by χ = 2 − 2g: compute V − E + F from any triangulation of the surface and the genus falls out, the same every time. The genus is the classic answer to “what stays the same when everything else can move” — the founding question of topology.
THE TECHNIQUE χ = 2 − 2g; genus from any triangulation's V−E+F
The Euler characteristic of a torus (χ=0 from any triangulation), turned into its genus. The demo reads off the number of holes: live demo
HISTORY & CREDIT Riemann, 1857
“A coffee mug and a donut are different shapes.” — topologically they are the same: one hole each, genus 1. The genus ignores every difference a continuous deformation can make and keeps only what tearing would cost. cited
1857 · Bernhard Riemann — introduced this invariant as the connectivity (Zusammenhang) of a Riemann surface; the name genus (Geschlecht) was coined by Clebsch in 1865. relation · χ = 2 − 2g ties genus to the Euler characteristic (Euler 1758, Poincaré). now · the primary invariant classifying closed orientable surfaces — the mug and the donut are one.
Triangulate a surface however you like, form V−E+F, and out comes the genus — the same number every time. The holes are what deformation cannot touch. Riemann 1857
RECOMMEND FOR I-13 genus from χ, computed
On the canonical compiler, a torus has χ=0, so g = (2 − χ)/2 = 1 — one hole:
$ i13 run inv_genus.i13 # g = (2 - chi)/2, torus
chi_torus = 0
genus = 1
Recommend: the genus is a topological invariant computed by an integer arithmetic step — the cleanest kind for i13. Whatever triangulation you feed it, V−E+F gives the same χ, and (2−χ)/2 gives the same genus; the invariance is baked into the topology, and i13 just does the arithmetic. It pairs with Gauss–Bonnet (dart 281): that dart got χ from geometry, this one turns χ into the count of holes. Both are quantities no continuous transformation can move — the batch's theme, in its purest topological form.