Normal surface

Each normal disk is either a triangle which cuts off a vertex of the tetrahedron, or a quadrilateral which separates pairs of vertices.

This is either an octagon that separates pairs of vertices, or an annulus that connects two triangles and/or quadrilaterals by a tube.

The concept of normal surfaces is due to Hellmuth Kneser, who utilized it in his proof of the prime decomposition theorem for 3-manifolds.

The notion of spun normal surface is due to Bill Thurston.

Regina is software that enumerates normal and almost-normal surfaces in triangulated 3-manifolds, implementing Rubinstein's 3-sphere recognition algorithm, among other functionalities.

A normal surface intersects a tetrahedron in (possibly many) triangles (see above left) and quadrilaterals (see above right)
An example of an octagon and annulus piece in an almost normal surface