The geometry of three dimensional space, despite its obvious importance, is a sadly neglected topic in modern
mathematics. One of the reasons is that the topic is rather awkward and difficult when approached with the standard tools of metrical affine and spherical geometry.
In this talk we show that ideas of rational trigonometry, both in the planar and spherical (elliptic) setting, open new doors to our understanding, supported by fascinating algebraic relations.
We will start with a quick linear algebra presentation of rational trigonometry, introduce the beautiful and remarkable
ZOME construction system, and then tackle the trigonometry of a tetrahedron---the fundamental object in three dimensional geometry.