Arithmetic-definability of geometrically-defined arithmetic concepts
For purposes of discussion take arithmetic to be the study of the [ positive real] numbers, sequences of numbers, etc. and take geometry to be the study of geometric objects: points, lines, polygons, and the like. Some arithmetic properties are initially defined by reference to geometric objects. For example, call a triple x,y, and z of numbers triangular if and only if there is a triangle whose sides are in the triple ratio x:y:z.
We ask for examples of geometrically defined arithmetic concepts (1) that are known to be arithmetically definable, (2) that are known to be not arithmetically definable, and (3) that are not known to be arithmetically definable and not known to be not arithmetically definable.
For example the concept of triangularity of number triples is a geometrically defined arithmetic concepts that is known to be arithmetically definable. It is known that a triple x,y, and z of numbers is triangular iff x+y>z , y+z>x, and z+x>y .