Editing Tetrahedron (section)

== Subdivision and similarity classes ==
Tetrahedra subdivision is a process used in computational geometry and 3D modeling to divide a tetrahedron into several smaller tetrahedra. This process enhances the complexity and detail of tetrahedral meshes, which is particularly beneficial in numerical simulations, finite element analysis, and computer graphics. One of the commonly used subdivision methods is the '''Longest Edge Bisection (LEB)''', which identifies the longest edge of the tetrahedron and bisects it at its midpoint, generating two new, smaller tetrahedra. When this process is repeated multiple times, bisecting all the tetrahedra generated in each previous iteration, the process is called iterative LEB. 

A '''similarity class''' is the set of tetrahedra with the same geometric shape, regardless of their specific position, orientation, and scale. So, any two tetrahedra belonging to the same similarity class may be transformed to each other by an affine transformation. The outcome of having a limited number of similarity classes in iterative subdivision methods is significant for computational modeling and simulation. It reduces the variability in the shapes and sizes of generated tetrahedra, preventing the formation of highly irregular elements that could compromise simulation results.

The iterative LEB of the regular tetrahedron has been shown to produce only 8 similarity classes. Furthermore, in the case of nearly equilateral tetrahedra where their two longest edges are not connected to each other, and the ratio between their longest and their shortest edge is less than or equal to <math>\sqrt{3/2}</math>, the iterated LEB produces no more than 37 similarity classes.<ref>{{Cite journal |last1=Trujillo-Pino |first1=Agustín |last2=Suárez |first2=Jose Pablo |last3=Padrón |first3=Miguel A. |date=2024 |title=Finite number of similarity classes in Longest Edge Bisection of nearly equilateral tetrahedra |journal=Applied Mathematics and Computation |volume=472 |pages=128631 |doi=10.1016/j.amc.2024.128631 |issn=0096-3003|doi-access=free |hdl=10553/129894 |hdl-access=free }}</ref>