A carbon atom’s orbitals don’t bond in their raw shapes. Instead one round s-orbital and some dumbbell p-orbitals BLEND into a set of identical hybrid arms, spaced as far apart as they can get — that’s why methane is a perfect tetrahedron at 109.47° and graphene a flat hexagon at 120°. The mixing recipe sets the shape of matter. Slide from sp to sp² to sp³.
Hybridization mixes an atom’s valence s and p orbitals into equivalent hybrid orbitals that point as far apart as possible (minimising electron repulsion, VSEPR): sp = 2 arms at 180° (linear), sp² = 3 arms at 120° (trigonal planar), sp³ = 4 arms at 109.47° (tetrahedral). One s + n p orbitals give n+1 identical hybrids; the recipe fixes the molecular geometry (linear CO₂, flat graphene, tetrahedral methane/diamond). A fail-loud self-check throws unless sp³ = 109.47°, sp² = 120° and sp = 180°. ◆ real quantum chemistry, node-verified.
The idealised VSEPR angles for equivalent hybrids (exact for symmetric cases); lone pairs and different substituents bend the real angles (water is 104.5°, not 109.47) — the mix-orbitals-into-equal-arms rule and its symmetric angles are the exact backbone.