The limitations have surely inhibited progress in biomechanics [7], [8] (especially for surgical, prosthetic and therapeutic purposes), progress in robotics, space geodesy, and other subjects. In particular, library and ONLINE searches of "rotation" provide references to subjects cited above which relate to Swiss mathematician Leonhard Euler (1707-1783) or Italian mathematician Geralamo Cardano (1501-76) [11], [12]. Few know that Cardano invented the universal joint (used today in our automobiles) and that Cardano was known by contemporaries as a physician, suggesting that "Cardan angles" (as they are labeled -- deflating this Italian creator as "French") may have originated from his studies of human joints (e.g. ball-in-socket : balls rotating about in sockets). And existence of "Cardan angles" suggests that Euler might, knowingly or unknowingly, have derived the "Euler angles" from those of Cardano. A great mystery in current histories of mathematics, physics, engineering, medicine, and biomechanics -- to be inflated.
(Clifford Truesdell, in writing of "A Program toward Rediscovering the Rational Mechanics of the Age of Reason" [18], declares -- with his italics -- that "rational mechanics is a part of mathematics". If we accept his thesis -- as the author is so inclined -- then deflated knowledge about rotation et al in mechanics is primarily due to negligence in mathematics.)
Below, we connect some of these "pieces", perhaps motivating others to "connect more of the dots" and unfurl inflating. (Please note that, after first deriving these results, we encountered a book [3] stating that the "eigenvalues" of a "rotation operator" are "1, eiq , eiq ", without diplaying the derivation and without indicating its implications.)
Perhaps we should heed the warning of the late Carl Sagan (1934-96): "We've arranged a global civilization in which the most crucial elements ... profoundly depend on science and technology. We have also arranged things so that no one understands science and technology. This is a prescription for disaster. We might get away with it for a while, but sooner or later this combustible mixture of ignorance and power will blow up in our faces ....." [15].
2. CHARACTERISTIC PROPERTIES OF LINEAR OPERATORS. A frequent representation of a linear operator is by a matrix, M or Mrc , where the subscripts denote the number of rows and columns of the matrix. However, deflations arise even here. For example, the derivation cited below may not been attempted because of limitation to linear operators over, say, R2. Thus, given [19] matrix u:
æ 0 - 1 ö
ç ÷
è 1 0 ø
"Then u is the counterclockwise rotation of the plane
through 90 degrees. The corresponding polynomial is X2 + 1 = 0, which has no roots in
R. Thus possesses no eigenvalues." (But others [3] think differently.)To leave the matter here is comparable to declaring that the diagonal of a square has no numeric description, motivating Platonists to declare "geometry incompatible with arithmetic". Their thesis that "motion is geometry set to time" inhibited arithmetic development of mechanics (except by Islamic scientists) -- a deflation which stymied Western science and technology for 2000 years and supported slavery. (Galileo Galilei (1564-1642) was denounced as "a Pythagorean" for writing of a numeric "law of falling bodies".) Sagan's warning [11] suggests that we have not yet learned lessons of "The Dark Ages".
Given M 
M22
:
æ m11 m12 ö
ç ÷
è m21 m22 ø
I I22:
æ 1 0 ö
ç ÷
è 0 1 ø
then we obtain matrix, M - lI, for real
number l:
THEOREM: Real number l
is a characteristic value (a.k.a. proper value, eigenvalue) or root
of matrix M iff matrix M - l
I is singular. That is, if its characteristic polynomial equation has the
form: l2 - (m11 + m22
)l + (m11a22 - m12m21.
) = 0.
(1) yields the characteristic equation system:
æ m11 - l m12ö
ç ÷
è m21 m22 - lø
3. CHARACTERISTIC PROPERTIES OF ROTATIONS IN TWO-DIMENSIONS. Let M be
the matrix for rotating coordinates (x,y) by angle q
. The characteristic matrix equation is:
æ cos q sin q ö æ x ö æ x ö
ç ÷ ç ÷ = l ç ÷ (1)
è - sin q cos q ø è y ø è y ø
x cos q + y sin q = lx
- x sin q + y cos q = ly
Substituting these values in the characteristic polynomial
equation, l2 - (m11 + m
22)l + (m11m22 - m12m21
) = 0, we have: l2 - (2 cos
q) l + (cos2 q
+ sin2 q) = l2
- (2 cos q)l + 1 = 0.
This quadratic equation has the two distinct
solutions: l = cos q +
i sin q, and l =
cos q - i sin q , for
i = (- 1)1/2 -- the two characteric values of the ROTATION OPERATOR.
Substituting these values for l in (2) yields:
If, in rotational eigenvalue l = cosq + i sin q,
we successively substitute q = 0,
p/2, p, 3p/2, the four cardinal
points of the compass become the rectangular axes for the complex-plane, which is also
what we find below. (W. R. Hamilton (1805-1865) knew that the complex function is the
2-D rotation operator, motivating his research which resulted in the quaternion 3-D
rotation operator. Again, this recalls the "deflation of rotation". Although the
literature of mathematics and physics is sprinkled with comments about the questionable value of
quaternions and about its recondite nature, yet a few comments in the literature note
that Euler's parameters are quaternions -- and this information has been
disseminated widely ONLINE [20].) More that is implicit in the rotation operator.
The literature notes the usefulness of the formula of
Abraham de Moivre (1667-173): (cos q + i sin
q)n = cos nq + i sin nq. Although listed as prior to the Euler Formula, the de Moivre
Formula can be derived from it: (ej)(ek) = e
j+k. (The history of this relation seems absent from, or "sparse" in, the
literature. More "deflation".)
The Euler Formula invokes enlightening connections
with Multivector Theory5 (a.k.a. Geometric Algebra, Clifford Algebra) with
eiq as a SPINOR, rotating a
vector5 (providing the most efficient and accurate method for integrating the
equations of motion, especially in celestial mechanics6, 17) -- thus, also
implicit in the rotation operator. This especially indicates "deflation" in standard
literature of rotation, since multivectors provide explication in a fragment of a
line where standard methods require several lines or a page. (Note: The "complex number" -- due
also to Cardano [12], [16] -- is the simplest form of a spinor.)
Two matrices, A and B are equivalent [10] iff, for two other nonsingular (i.e.
invertible) matrices, C and D, B = CAD. If CD = I (the identity matrix), then
we have B = D-1AD, a collineatory (similarity)
transformation, with the equivalents as transforms of each other. (A
similarity transformation can reduce a matrix to a diagonal form.) We can easily show that
R is collineartory with itself, i.e. a collineartory operator
.
A real orthogonal transformation is also congruent
[10], which means that it "can reduce a quadratic form to a sum or squares" although "the
reduction is by no means unique"7, but becomes unique if a congruent transformation
is also orthogonal.
Furthermore, orthogonal transformations are "principal
axis transformations ... used in the problem of reducing a conic to principal axes and in finding
the principal axes or a rotating body, or in reducing kinetic and potential energy to sums of
squares terms. The eigenvectors are frequently called normal coordinates in these cases"
[10].
An orthogonal operator over a complex field is
"unitary". To encompass the above properties, theoretically and in application, the rotation
operatory should be treated as a degenerate unitary operator. If, decades ago, the
rotation operator had been so treated , the surprise evoked by the need from complex
numbers in quantics might not have arisen, and progress in quantics might have
been more rapid.
An orthogonal operator provides an easy test for
"vectorhood". Given an alleged vector, transform it by an orthogonal operator. If it
preserves length and direction, then it is a vector10.
German mathematician Felix Klein (1849-1923), in
characterizing geometries via associated groups, showed that all properties of
Euclidean geometry can be subsumed under congruence-invariance of the Euclidean Group,
consisting of translation, rotation, and reflection.
Each of these transformations is an isometry
[2]: a length-preserving map. Furthermore, every isometry can be formed by
concatenating these three isometries [2] -- although isometries exist that does not
contain one of these.
However, according to Biedenharn and Louck [3[, Bachmann
showed that Euclidean geometry can be derived solely from reflection [1], [3]. His book
has apparently never been translated from German and is little known (B!). So it is not
surprising that apparently no one has shown how to derive Klein's Euclidean Group from a
reflection group .
On the other hand, we should remember Euler's
demonstration that every motion can be reduced to a rotation followed by a translation.
Also, French mathematician E. Cartan (1869-1951) showed this reduction can be simplified to a
reflection [6].
Biedenharm and Louck [3] simplify the Euclidean Group
by displaying translation and rotation as "successive reflections in two planes". For
translation, the planes are parallel; for rotation, the distinct planes
intersect.
Group Theory has motivated the characterization of the
most advanced portions of physics in terms of conservation laws [14], crowned by proof of
German-American mathematician, Emmy Noether [13] (1882-1935), that these principles can be
derived from symmetries 13 of the Action Principle formulated by French
mathematician, Joseph Lagrange (1736-1813). More rotation inflation.
Attention to polarization motivated physists to
consider "degrees of freedom" not only in "outer space" (with coordinated positions) but
also in "inner space". This trend continued with treatment of many physical properties
(spin, stong isospin, weak isospin, etc.) of inner spaces.
Another interesting aspect of polarization was that
the polarization axis can be rotated, "bringing in" the rotational mathematics
discussed above. Mathematical physicists then began to characterize, even derive, inner
space properties in terms of, or from, a rotation operator14.
However, it has, perhaps, not been noted (B?) that associated with two transformations of the
Euclidean Group are two conservation laws: linear momentum associated with
translation and angular momentum associated with rotation. But none has be associated
with the other transformation, reflection, which (says Birnbaum and Louck2) E.
W. Bachmann [1] and Cartan[6] show to be more basic.
One may imagine a "bimomentum" associated with reflection
, which may also "explain" the spin property of quantics. Certainly, reflection
and the spinor transformations of German Hermann Grassmann (1809-1897), British Hamilton
and William Clifford (1845-1879), and French Olinde Rodrigues (1799-1851) are all represented by
the same rotation operator:
However, this yields the same linear and characteristic
properties as those in two-dimensions. And, since much of the deflation involves
this case, it very much suggests rotation inflation could abet progress.
Consider, now, the "improper" operator, which
achieves "the same rotation [as above], followed by a reflection in the X-Y plane" [7]. For q = 0
x cos q + y sin q = lx
- x sin q + y cos q = ly
Substituting our two solutions yields
(since u/i = - iu) the two characteristic vectors:
æ ix ö æ x ö
è y ø, è iy ø (3),
These solution complex functions are part of "Euler's
Formula": eiq = cos q
+ sin q; e- iq
= cos q - sin q. And these
apparently were derived from the MacLaurin Series for ex, for
cos q, and for sin q. (At least, this is the usual presentation of Euler's Formula in
the literature.) The above derivation shows that Euler's Formula is implicit in the rotation
operator. And it shows that, just as mensuration must be developed over the real
number system, so rotation and similar transformations must be developed over
the complex number system, as the following results further indicate.
4. LINEAR PROPERTIES OF THE TWO-DIMENSIONAL ROTATION OPERATOR: R. A
matrix represents an orthogonal operator [10] iff it is invariant under
multiplication (in either order) by its transpose and inverse. The transpose of Rrc is Rt = Rcr. We can easily
find, that its inverse, R-1, is equivalent to its
transpose, so that (involutorily) R = (Rt)t =
(R-1)- 1 = (Rt)- 1 = (R- 1)t =
R, hence, R is an orthogonal operator.
5. ROTATION IN GROUP THEORY. It is well known that every discrete group is a subgroup of a
permutation group. Although rarely (if ever) noted, every continuous group is analogous
or homologous to a rotation group14. (More inflation.)
6. "INNER SPACE" ROTATION. In 1808, French engineer Etienne-Louis Malus [21] (1775-1812)
discovered the polarization of light in a rhomb of Icelandic spar (and created the label
"polarization"). Augustin Fresnel (1788-1824) decided that this fitted the notion of light as
transverse wave motion -- differing from the longitudinal wave motion of sound and water
-- thereby preparing for stereochemistry and the electromagnetic equations of
Scot James Clerk Maxwell (1831-1879) which challenged German physicist, Heinrich Hertz
(1857-1894), to discover "radio".
æ cos 1/2 q sin 1/2 q ö
ç ÷
è - sin 1/2 q cos 1/2 q ø
The interconecting of reflection, translation, reflection
described in Sec. 4 might provide a model for interconnecting bimomentum, linear
momentum, angular momentum.
7. CHARACTERISTIC PROPERTIES OF ROTATION IN THREE-DIMENSIONS. The "setup" here is:
æ cos q sin q 0 ö
ç ÷
ç - sin q cos q 0 ÷
ç ÷
è 0 0 1 ø
æ cos q sin q 0 ö
ç ÷
ç - sin q cos q 0 ÷
ç ÷
è 0 0 - 1 øThis improper operator is also orthogonal [7], providING another reason for deriving rotation (and its applications) from reflection . It also suggests that "experimenting" with different values for the basic form may invoke enlightening and useful variations in many, many research realms.
8. TRANSFORMATIONS VIA EULER ANGLES AND "CARDAN" ANGLES [4]. Some of the above results facilitate these explications. We label these transformations by script font: respectively, E and C.
Euler Transformation for azimuth angle, f; polar angle, f; body angle, y : E = EfE qE y. Ef is matrix for R with f replacing q . E f is the inverse of the matrix for R, with row ordering 1, 2, 3 becoming 3, 1, 2. Ey is matrix for R with y replacing q.
Cardano Transformation: C = C fCq Cy. These are the same as those for Euler, except that the middle subtransformation does not resort to the inverse of R.
The products are tediously obtained [4]. The characteristic values and vectors of the subtransformations obviously follow from those above. The characteristic value of the total transformation -- for each case -- may be succinctly written: ef+ q+f. But the literature seems more flamboyant.
9. CHARACTERISTIC PROPERTIES OF ROTATION IN FOUR-DIMENSIONAL MINKOWSKI SPACE. For a wave propagating along the x-axis9, we have x2 - (ct)8 = 0, x'2 - (ct')2 = 0 , for c the vacuum velocity of light and t for time. Let t = it, t' = it', for i = (-1)1/2 . The resulting Lorentz Transformation (in The Special Theory of Relativity) results (similarly to above citations about "inner space" properties) from a rotation matrix intermediary derived from that in two-dimensional "ordinary space-time", via the substitutions y -> ct, cos q -> cosh f, sinh q -> sinh f, that is,
æ cosh f sinh f ö
ç ÷
è - sinh f cosh f øCalculations similar to those above yield the characteristic values of this rotation matrix: l = cosh f + sinh f = e f, and l = cosh f - sinh f= e- f . This is a homologue of Euler's formula (stated above). this and (e f)n = enf yields a homologue of de Moivre's Formula: (cosh f + sinh f)n = cosh nf + sinh n f. And this section clearly displays more "B" and more rotation inflation.
10. CONCLUSIONS. It should be clear from this that, currently, "the history of rotation mathematics and physics with applications in medicine, biomechanics, space geodesy, etc." is very inadequately written and should be a challenge to readers of this magazine to inflate.
(The widower author -- survivor, since 2000, of a life-long victim of infant poliomielitus -- thought, early in their 52-year marriage, that his training in mathematics and physics might contribute to prosthestic help for his dear wife, but found this -- for some of the above reasons -- impossible, yet believes these hopes could yet be realized for others.)
Those who KNOW seldom CARE.
Those who CARE rarely KNOW.
When KNOW meets CARE
And teams with DARE
Then MT. CONSTIPATION will blow.(jh)
1. E. Bachmann, Aufbau der Geometrie aus dem Spiegelungsgegriff, Springer, Berlin, 1959.
2. Benjamin Baumschlag and Bruce Chandler, Schaum's Outline Series Theory and Problems of Group Theory, McGraw-Hill Publishing Company, New YOrk, 1968.
3. L, C. Biedenharn and J. Louck, Angular Momentum in Quantum Physics: Theory and Application, ENCYCLOPEDIA OF MATHEMATICS and Its Applications, V. 8, Ed. Gian-Carlo Rota, 1981.
4. Ted Clay Bradbury, Mathematical Methods with Application to Problems in the Physical Sciences, John Wiley & Sons, New York, 1984.
5. John Hays, "Algebra Is Arithmetic Backwards for Inheritance, Fiduciaries, Measures, Optimization, Topology, History, Life, & All 'Backwards' Projects", algfront.htm with hyperlink, "Arithmetic Redux", http://.../.../redux.htm .
6. David Hestenes, New Foundations of Mechanics, Kluwer Academic Publishers, Dordrecht, 1986.
7. David Hestenes, "Reaching and Neurogeometry", Neural Networks, 7, No. 2, pp. 65-77,, Elsevier Science, Ltd., New York, 1994. Also, http://modelingnts.la.asu.edu/InvarBK2,pdf
8. David Hestenes, "Saccadic and Compensatory Eye Movements", Neural Networks, 7, No. 1, pp. 79-88,, Elsevier Science, Ltd., New York, 1994. Also, http://modelingnts.la.asu.edu/InvarBK1,pdf
9. M. S. Longair, Theoretical Concepts in Physics, pp. 262-4, Cambridge University Press, Cambridge, England, 1984.
10. Henry Margenau and George Moseley Murphy, The Mathematics of Physics and Chemistry, D. Van Nostrand, Inc., Princeton, 1943.
11. Oystein Ore, Cardano, The Gambling Scholar, Dover Publications Inc., New York, 1953.
12. Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness, Oxford University Press, Oxford, 1994.
13. Constance Reid and Herman Weyl, Hilbert, With an Appreciation of Hilbert's Work by Herman Weyl, Springer-Verlag, Berline, 1972.
14. Joe Rosen, Symmetry Discovered, Cambridge University Press, Cambridge, 1970.
15. Carl Sagan, The Demon-Haunted World - Science as a Candle in the Dark, Random House, N3w York, 1997.
16. David Eugene Smith, A Source Book in Mathematics V. 1, Dover Publications, Inc., New York, 1959.
17. E. L. Stiefel and G. Schiefele, Linear and Regular Cellestial Mechanics, Springer-Verlag, N.Y., (1971).
18. Clifford Truesdell, Essays in the History of Mechanics, Springer-Verlag, New York, 1968.
19. Seth Warner, Modern Algebra, Dover Publications, Inc., New York, 1965.
20. Eric Weisstein,Euler Parameters, http://mathworld.wolfram.com/EulerParameter.htm.
21. Sir Edmund Whittaker, A History of Theories of Aether and Electricity, V. I, Thomas Nelson & Sons, Ltd., London, 1951.