Split-quaternion - Profile

Profile

The subalgebras of P may be seen by first noting the nature of the subspace {z i + x j + y k : x,y,z ∈ R}. Let

r(θ) = j cos θ + k sin θ

The parameters z and r(θ) are the basis of a cylindrical coordinate system in the subspace. Parameter θ denotes azimuth. Next let a denote any real number and consider the coquaternions

p(a, r) = i sinh a + r cosh a

v(a, r) = i cosh a + r sinh a.

These are the equilateral-hyperboloidal coordinates described by Alexander Macfarlane.

Next, form three foundational sets in the vector-subspace of the ring:

E = { r ∈ P: r = r(θ), 0 ≤ θ < 2 π}

J = {p(a, r) ∈ P: a ∈ R, r ∈ E}, hyperboloid of one sheet

I = {v(a, r) ∈ P: a ∈ R, r ∈ E}, hyperboloid of two sheets.

Now it is easy to verify that

{q ∈ P: q2 = + 1} = J ∪ {1, −1}

and that

{q ∈ P: q2 = −1} = I.

These set equalities mean that when p ∈ J then the plane

{x + yp: x, y ∈ R} = D_p

is a subring of P that is isomorphic to the plane of split-complex numbers just as when v is in I then

{x + yv: x, y ∈ R} = C_v

is a planar subring of P that is isomorphic to the ordinary complex plane C.

Note that for every r ∈ E, (r + i)2 = 0 = (r − i)2 so that r + i and r − i are nilpotents. The plane N = {x + y(r + i): x, y ∈ R} is a subring of P that is isomorphic to the dual numbers. Since every coquaternion must lie in a D_p, a C_v, or an N plane, these planes profile P. For example, the unit quasi-sphere

SU(1, 1) = {q ∈ P: qq* = 1}

consists of the "unit circles" in the constituent planes of P: In D_p it is a unit hyperbola, in N the "unit circle" is a pair of parallel lines, while in C_v it is indeed a circle (though it appears elliptical due to v-stretching).These ellipse/circles found in each C_v are like the illusion of the Rubin vase which "presents the viewer with a mental choice of two interpretations, each of which is valid".