How can a direction be measured or specified? Without a coordinate system, this task seems impossible. Although a given coordinate system can specify a mean of measuring and hence quantify directions, ambiguities can still arise since there are distinctly two topologically nonequivalent coordinate systems: the right-handed system (RHS) and the left-handed system (LHS) or the clockwise (CW) and the counterclockwise (CCW) systems. Conventional wisdom tends to prefer the RHS or the CW system. Their uses do not create any orientation difficulty for all macroscopic systems: planetary motions of both centralized revolutions and axial rotations, binary stars motions, or galactic spiral motions. However, in the microscopic quantum reality of local infinitesimal motions, both RHS and LHS or CW and CCW must be considered altogether to coexist in space-time in order to formulate a gauge theory of direction.

Direction becomes a quantum gauge wherever and whenever it remains unchanged under certain local Abelian gauge transformation using both the operations of multiplication or of addition or at most only scalar multiples of the same two fundamental gauges of the same physical order and the same physical dimension. These are the H-pluses and H-minuses of different physical orders and different physical dimensions. But regardless of their physical orders all directional gauges are intrinsically one dimensional. However, it appears that only second order directional gauges can be extracted from second order nonsingular matrices embedded in a sieve of Diophantus. Fundamentally these H-pluses and H-minuses are symmetric singular Hadamard matrices of different physical orders but all of only two physical dimensions. The eight intrinsic one dimensional directional gauges are simply the eight properties of directional invariance. Their continuous transformations create the one physical dimension of Hopf topology, two physical dimensions of Möbius topology, and the three physical dimensions of Klein topology. A complete description of matter, energy, and squares of energy exemplified by all the elementary particles of high energy physics can be achieved without using four or higher dimensional topologies.