
Since these fields are discrete, an algorithm examines each
possible permutation of P^{n}, with the form
Z_{2},...,Z_{2}, where
Z_{2} = ±1, P = 2, and n
= 3, 6, 12 and verifies that all the axioms and the theorems are
valid within these groups and various subgroups. First, each element is
multiplied by itself to verify theorem 4.1. Next,
each pair of element a,b of the groups,
G_{f}8, G_{f}64, and
G_{f}4096, are multiplied together to check that
the result ab and ba are equal, and are also in these groups. This proves
axioms A1 and A8. Continuing in this
manner, the axioms, A2, A3,
A4, A5, A6, and
A7, are also verified for these groups and semigroups,
thus proving these axioms. Again, using the result ab as a pole point,
P, the algorithm finds the subset of all the orthogonal
point pairs over the hypercomplex field, which are perpendicular to each
other and form the group
G_{f}'64XG_{f}64. The property of points
being perpendicular is very important because we define the property of
handedness with three perpendicular base vectors. Two of these
perpendicular vectors are identified, and their product defines a third
perpendicular vector, the pole point, P. This orthogonal vector triplet
forms the complex that models handedness. With this vector triplet, a
geodesic line is also defined. By evaluating specific bits in the pole's
array, (0,1,2,3,4,5), the direction of the
geodesic line is determined by the sense of its pole. When the product is
negative, a counterclockwise rotation is determined, which models
lefthandedness. When the product of bits two and five are positive, a
clockwise rotation is determined, which models righthandedness. 
