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A simple way to enumerate some types of groups due to J. H. Conway (Zwillinger 1995).

A geometry constructed without reference to measurement. The only primitive concepts are those of points and intermediacy. There are 10 axioms underlying ordered geometry.

A nonsingular linear map A:R^n->R^n is orientation-reversing if det(A)<0.

A bicupola in which the bases are in the same orientation.

A birotunda in which the bases are in the same orientation.

Coordinates defined by an orthocentric system.

A cupolarotunda in which the bases are in the same orientation.

Two (or more) intersecting curves that are perpendicular at their intersection are said to be orthogonal.

Two or more lines or line segments which are perpendicular are said to be orthogonal.

Two subspaces S_1 and S_2 of R^n are said to be orthogonal if the dot product v_1·v_2=0 for all vectors v_1 in S_1 and all v_2 in S_2.