Introduction to Algebraic Motors

Motor language
Before mathematics, consider the language context of the term "motor": Motive power is the first meaning of motor. Power-driven equipment evolved in stages through the human and animal traction, river-flow, steam, electrical and internal combustion phases. The electro-chemical “fuelcell reaction”:
2H2+ O2→ 2 ( H2O )+ energy
is still in commercial start-up (2009), though this reaction promises clean driving power (assuming hydrogen is cleanly produced, say by photovolaic-powered electrolysis). In transportation, the drivers were “engines” in the time of steam, then “motors” for electrical propulsion, and commonly for cars. But formally, the “internal combustion engine” (i.c.e.) harkens back to the steam-age word. In electronics, a voltage difference is refered to as an electromotive force, an E.M.F. Similarly, in physiology the "motor neurons" are those that control the power of locomotion. Moreover, the concept of "motivation" is key to many branches of psychology.
The word “motor” evolved into common English usage not only in applied science as stated, but also in mathematical writing. Clearly it forms a variant to "rotor", an interpretation of the square-root minus one operator i which introduces a 90 degree counter-clockwise rotation when acting on the complex number plane C.
One of the most incisive expressions came from Mario Pieri: he used the concept of "a motion" to replace the congruence concept underlying geometry since Euclid. Pieri's convention follows the "transformation" approach in modern geometry promoted by Felix Klein, Sophus Lie, Isaak Yaglom and other writers.

Source of algebraic motors
With James Cockle's answers to the quaternions, and the introduction of Cayley's matrix multiplication in the 1850's, the notion or concept of "number" required extension:

Exercise: Compute the squares of the following matrices:

some matrices whose square is the identity

Such matrices were called motors in the nineteenth century by W.K. Clifford and Alexander MacAulay. Their definitive property   m2 = 1   ( identity matrix ) makes m a motor, but   m = 1   and   m = − 1   are considered excluded so that motors did not exist in algebra until matrices came to play.

These algebraic entities contribute to the description of the geometric concept of hyperbolic angle: The hyperbolic angle is the independent variable of the transcendental functions sinh (hyperbolic sine) and cosh (hyperbolic cosine) which are so useful in mathematical models. In fact, attention to the basic concepts of "motor" and "hyperbolic angle" serves to provide these models with a descriptive foundation.
Gestation of the Algebraic Motor

First we introduce the concepts of hyperbolic sector and hyperbolic angle.
The area   L under y = 1/x   and between   x = 1   and a variable  x is log x  , where the base is   e = 2.171828...  
Since any triangle with altitude on xy = 1 has area 1/2, one can increase   L   by ½ , then decrease by ½ , leaving the resulting area   L = log x .

hyperbolic sector

hyperbolic trigonometry OF = √2 cosh L
PF = √2 sinh L

parametrized hyperbolic circle T = cosh L + j sinh L = exp ( jL)
parametrized analogue unit "circle"

The relation of the logarithm function to the hyperbola   y = 1/x   lead Leonard Euler to call it "hyperbolic logarithm" (1748). Earlier (1647), Gregoire de Saint-Vincent had demonstrated the quadrature of the hyperbola sector in the Belgium. In the words of David Eugene Smith, in his book History of Mathematics (1923), "the quadrature of the hyperbola is referred to its asymptotes" and "as the area increased in arithmetic progression, the abscissas increased in geometric series". A.A. de Sarasa, in 1649, put together logarithms, as they were understood, with Saint-Vincent's quadrature (area computation).
From the modern point of view, the exponential series splitting into cosh x + sinh x = exp(x) ,

and the splitting of   { jn }n ∈ N  =   { 1 , j } when   j   is a motor leads to

exp(jL) = cosh L + j sinh L.

Thus anyone contemplating functions from the power series viewpoint will run into the motor idea when interested in the parametrized hyperbola
{(cosh L , sinh L ): L ∈ R }.

The discoverer of the first motor was James Cockle in 1848; he wrote of a “new imaginary in algebra” for the London-Edinburgh-Dublin Philosophical Magazine. In a follow-up article he noted that with a motor one has

( 1+j ) ( 1-j )  =  1-1+j-j = 0

This equality showed the first zero-divisors, a case where non-zero quantities multiply with product zero. Unfortunately, James Cockle called this a “product of impossibles”, showing he confused what was impossible before, with what he had enabled through the use of tessarines.

Exercises on Motors

  1. Show that there are no motors in the conventional complex plane C
  2. Show that there are no motors in Hamilton's quaternion ring H .
  3. Find the motors in Cockle's ring of tessarines T.
  4. Is this matrix a motor: row 1 (1,0), row 2 (0,-1) ?
  5. Show that the Pauli matrices are algebraic motors.
  6. Find the motors in the biquaternion ring B.  Hint: B is isomorphic to M(2,C).
    form of a biquaternion
    show that     w = -z = ± √(1 − xy)   where   x,y ∈ C .

  7. Let D = { z = x + y j : x,y ∈R } where j is a motor. Show that (D, + , • ) is an algebraic ring and find its group of units U.
  8. Show that V = { exp(aj) : a∈R } is a sub-group of U and find the quotient U/V.
  9. For z = x + yj in D, write Re(z) = x for the "real part of z". For w, z in D, suppose the rays from the origin to w and z have reciprocal slopes. Show that Re(z w*) = 0. Show that this algebraic condition leads to ordinary perpendicularity in the ordinary complex number plane C. When w and z are in D and the condition holds, we say that they and their rays are hyperbolic-orthogonal.
  10. Confirm that the hyperbolic versor concept corresponds to the splitting of the exponential series mentioned above.
  11. The algebraic motor is an analytic tool for geometry. Compare the ideas of synthetic spacetime with those you have developed with motors.

Ortho-rhombus

Take S = {(0,0), (1,0), (0,1), (1,1)} as the fundamental square in the Cartesian plane.
In the motor plane S = { 0, 1, j, 1+j}. Let u = exp(aj) = cosh a + j sinh a , then
uS = {0, u, ju, (1+j)exp a}. The parameter a determines the rotation u = exp(aj) and hence uS is the hyperbolically-rotated quadrilateral. It is easily verified that the sides of the quadrilateral are hyperbolic-orthogonal to each other at vertices 1 and j, and that the sides have constant unit hyperbolic length, regardless of the value of parameter a. Thus the name ortho-rhombus for this quadrilateral.
Any such ortho-rhombus can be taken as an “observer” at rapidity a , whose frame of reference may be brought to Euclidean squareness by hyperbolic rotation with v = exp(−aj) = 1/u because S = v(uS). The principle of relativity can be formulated with the premise that any ortho-rhombus is a suitable frame of reference for inertial physics.

For more on Plane D

Sir James Cockle: Geometric Algebrist