A motor or engine is the means to apply acceleration, and the motor plane is the geometric algebra to account for velocity changes (i.e. acceleration) in the electronic view of space and time. The alternative complex number plane D = { z = x + y j : x, y ∈ R } with j² = +1 has been called the motor plane since the days of W.K. Clifford.

Henrik Lorentz, Albert Einstein, and Hermann Minkowski, whether they knew it or not, applied motor plane geometic algebra. Emil Borel published Space and Time in 1926 (London, Blackie). To see his geometric vision of "hyperbolic rotation" refer to his essay "Introduction Geometrique a Quelques Theories Physiques" which is now available on-line from Cornell University. It expanded on his note the previous year (1913) in Comptes Rendues Acad. Sci. Paris

But these works of the founders of relativity and of Emil Borel fall short of algebraic transformation presentation. For that there are merely hints before everything opened up with the evolution of linear algebra, topology, group theory and differential geometry.

In the 19th Century James Cockle introduced the motor plane when he announced the tessarines (which contain a motor j and a plane x + y j ). He discussed the zero divisors 1+j and 1-j . More importantly, he noted the first hyperbolic versor

According to William Kingdon Clifford, the motor plane D provides the arithmetic of angular velocity addition. In a paper published in 1882 among his mathematical works he develops the concept of addition of "rotors". He uses the Greek letter omega to represent a motor and proceeds to make some of the same observations as Cockle about the idempotents in the motor plane. These observations are found on page 394 of his Mathematical Works edited by Tucker.

In the 20th Century the motor plane had expositors in Argentina, Russia, Germany, Mexico, Canada, & USA. These writers are university mathematics professors who view D as a abstract ring, not merely the skeleton of spacetime theory. Indeed, in science it is important to distinguish a mathematical model from the object whose pattern is being abstracted.
The widely circulated College Mathematical Journal(26:268-80) printed Garret Sobczyk's "The Hyperbolic Number Plane" in 1995. He teaches at Universidad de las Americas in Puebla, Mexico. The paper presents a reasonable demonstration of the basics as well as an application in solving the reduced cubic equation x³+ 3 a x + b = 0 .
Digging deeper into the 20th Century shows that in 1935 the motor plane, and the function theory f:D → D, was described by J.C. Vigneax and A. Durañona y Vedia at Universidad Nacional de La Plata, República Argentina. Their take-off point is Emil Borel's description of a "hyperbolic rotation about a point". It is interesting that both James Cockle and the Argentines use j to represent the motor that generates the plane D. The lessons from La Plata are found in Contribución a las Ciencias Físicas y Matemáticas, the University's journal of dissemination. It includes the use of D to elucidate the d'Alembert PDE solution mentioned above.

Consider now together a Russian and a German contribution to motor plane theory. They are both textbooks with scant allusion to special relativity but detailed consideration of a variety of complex planes, starting with the ordinary one C , then the motor plane D, and on to the dual numbers
N = { z = x + y n : x, y ∈ R } where n²= 0. The latter atypical complex plane N is frequently needed to complete ideas suggested by C and D .
1)Complex Numbers in Geometry(1963,1968) I.M. Yaglom, Moscow
2)Vorlesungen über Geometrie der Algebren(1973), W. Benz, Bochum
These text use C,D, and N to form projective lines and homography groups. They both make extensive use of cross ratios. In the case of Yaglom's text one has the benefit of an appendix on non-Euclidean geometries in the plane, while Benz includes a large bibliography.

The symbol D corresponds to "Double Numbers", Yaglom's term for the motor plane. Benz calls D something like "the atypical complex numbers". But both Yaglom and Benz work with the Cockle convention to represent the basic motor by j and a point in the plane by x + yj.
In on-line encyclopedias D is called the split-complex plane. Another tendency (Vignaux, Sobczyk, quadratic forms) is to use the word "hyperbolic", a term introduced by Apollonius and heavily exploited, especially recently.

We see that a century and a half have passed since James Cockle's real tessarines first embodied the plane D. Though the development of algebra D became welcome as an example of a commutative, associative ring with zero-divisors, its exploitation for the physics of space and time has probably driven popular interest. As other contexts are brought forward to use D's modelling power, we can expect the purely mathematical appreciation to grow.

As D is a spacetime ring of two dimensions, one can ask if there are spacetime rings of four dimensions that contain D. The answer, that two were discussed by Alexander Macfarlane in 1900, has been prepared for students. You can read about his hyperbolic quaternions and about his comments on the "exspherical" breakdown of motors and rotors in the coquaternions before the International Congress of Mathematicians in Paris.

The equation zz* = 1 is commonly connected to the idea of a circle. This locus is the ordinary circle in the case of the complex plane C, but on the plane D the locus is a unit hyperbola. In D there is the further locus zz* = −1 which is a counter-circle in the sense that it balances zz* = 1 as a counter-weight balances a load.

If z = x + t j , write Re(z) = x for the real part of z. Now we assume x corresponds to length in units of 30 cm. and t corresponds to time in units of nanoseconds.

Definition: Two points z and w are D-orthogonal if Re(w z*) = 0. We write z ⊥ w
Exercise: For every point z on the counter-circle there is a hyperbolic angle a such that z ⊥exp(a j).
In application to the kinematics of relativity, the counter circle is called the hodograph.

Definitions: Given w in D to represent the event (here, now) and given a z on the hodograph, the timeline through w with speed z is { t z + w : t ∈ R}. For every hodograph point z there is a relation S on D given by pSq when (p-q) ⊥ z which we read as "p and q are simultaneous with respect to z".
Exercise: Show that the relation S is transitive: pSq and qSr imply pSr.

For every hodograph point z, by the previous exercise, there is a rhombus {0, z, exp(aj), (1+j)exp(a) }. The points on the side between 0 and exp(aj) are simultaneous with respect to z, as are the points on the side between z and (1+j)exp(a). The timelines 0 + tz and exp(aj) + tz include two sides of the rhombus; they can be viewed as the worldlines of experimenters measuring light speed along the line y = x in D, the diagonal of all spacetime rhombi regardless of the hyperbolic angle parameter a. These experimenters inevitably agree on the same space-time ratio c = 1 as the speed of light because their understanding of simultaneity puts their separation at √(cosh 2a) for both the time and space variables.

So the spacetime rhombi grow very slim and long as parameter a grows large but actually the area stays the same as the original square (a=0) in the rest frame: one unit. You can verify this assertion by noting that the short diagonal of a spacetime rhombus has length √(2) (cosh a - sinh a) = √(2) exp(-a) while the long diagonal has length √(2) exp(a). (Four congruent right triangles make up the rhombus.)