======================================================================
=                       History of quaternions                       =
======================================================================

                            Introduction
======================================================================
In mathematics, quaternions are a non-commutative number system that
extends the complex numbers. Quaternions and their applications to
rotations were first described in print by Olinde Rodrigues in all but
name in 1840, but independently discovered by Irish mathematician Sir
William Rowan Hamilton in 1843 and applied to mechanics in
three-dimensional space. They find uses in both theoretical and
applied mathematics, in particular for calculations involving
three-dimensional rotations.


                        Hamilton's discovery
======================================================================
In 1843, Hamilton knew that the complex numbers could be viewed as
points in a plane and that they could be added and multiplied together
using certain geometric operations.  Hamilton sought to find a way to
do the same for points in space. Points in space can be represented by
their coordinates, which are triples of numbers and have an obvious
addition, but Hamilton had difficulty defining the appropriate
multiplication.

According to a letter Hamilton wrote later to his son Archibald:
Every morning in the early part of October 1843, on my coming down to
breakfast, your brother William Edwin and yourself used to ask me:
"Well, Papa, can you multiply triples?"  Whereto I was always obliged
to reply, with a sad shake of the head, "No, I can only add and
subtract them."

On October 16, 1843, Hamilton and his wife took a walk along the Royal
Canal in Dublin. While they walked across Brougham Bridge (now Broom
Bridge), a solution suddenly occurred to him. While he could not
"multiply triples", he saw a way to do so for 'quadruples'. By using
three of the numbers in the quadruple as the points of a coordinate in
space, Hamilton could represent points in space by his new system of
numbers. He then carved the basic rules for multiplication into the
bridge:
:i^2 = j^2 = k^2 = ijk = -1.\,

Hamilton called a quadruple with these rules of multiplication a
'quaternion', and he devoted the remainder of his life to studying and
teaching them. From 1844 to 1850 Philosophical Magazine communicated
Hamilton's exposition of quaternions. In 1853 he issued 'Lectures on
Quaternions', a comprehensive treatise that also described
biquaternions. The facility of the algebra in expressing geometric
relationships led to broad acceptance of the method, several
compositions by other authors, and stimulation of applied algebra
generally. As mathematical terminology has grown since that time, and
usage of some terms has changed, the traditional expressions are
referred to classical Hamiltonian quaternions.


                             Precursors
======================================================================
Hamilton's innovation consisted of expressing quaternions as an
algebra over R. The formulae for the multiplication of quaternions are
implicit in the four squares formula devised by Leonhard Euler in
1748; Olinde Rodrigues applied this formula to representing rotations
in 1840.


                              Response
======================================================================
The special claims of quaternions as the algebra of four-dimensional
space were challenged by James Cockle with his exhibits in 1848 and
1849 of tessarines and coquaternions as alternatives. Nevertheless,
these new algebras from Cockle were, in fact, to be found inside
Hamilton�s biquaternions. From Italy, in 1858 Giusto Bellavitis
responded to connect Hamilton�s vector theory with his theory of
equipollences of directed line segments.

Jules Hoüel led the response from France in 1874 with a textbook on
the elements of quaternions. To ease the study of versors, he
introduced "biradials" to designate great circle arcs on the sphere.
Then the quaternion algebra provided the foundation for spherical
trigonometry  introduced in chapter 9. Hoüel replaced Hamilton�s basis
vectors i,j,k with i1,   i2, and i3.
The variety of typefaces (fonts) available led Hoüel to another
notational innovation: 'A' designates a point, 'a' and \mathrm{a} are
algebraic quantities, and in the equation for a quaternion
:\mathcal{ A} = \cos \alpha + \mathbf{A} \sin \alpha ,
\mathbf{A} is a vector and α is an angle. This style of quaternion
exposition was perpetuated by Charles-Ange Laisant and Alexander
Macfarlane.

William K. Clifford expanded the types of biquaternions, and explored
elliptic space, a geometry in which the points can be viewed as
versors. Fascination with quaternions began before the language of set
theory and mathematical structures was available. In fact, there was
little mathematical notation before the Formulario mathematico. The
quaternions stimulated these advances: For example, the idea of a
vector space borrowed Hamilton�s term but changed its meaning. Under
the modern understanding, any quaternion is a vector in
four-dimensional space. (Hamilton�s vectors lie in the subspace with
scalar part zero.)

Since quaternions demand their readers to imagine four dimensions,
there is a metaphysical aspect to their invocation. Quaternions are a
philosophical object. Setting quaternions before freshmen students of
engineering asks too much. Yet the utility of dot products and cross
products in three-dimensional space, for illustration of processes,
calls for the uses of these operations which are cut out of the
quaternion product. Thus Willard Gibbs and Oliver Heaviside made this
accommodation, for pragmatism, to avoid the distracting
superstructure.

For mathematicians the quaternion structure became familiar and lost
its status as something mathematically interesting. Thus in England,
when Arthur Buchheim prepared a paper on biquaternions, it was
published in the American Journal of Mathematics since some novelty in
the subject lingered there.  Research turned to hypercomplex numbers
more generally. For instance, Thomas Kirkman and Arthur Cayley
considered the number of equations between basis vectors would be
necessary to determine a unique system. The wide interest that
quaternions aroused around the world resulted in the Quaternion
Society. In contemporary mathematics, the division ring of quaternions
exemplifies an algebra over a field.


                       Principal publications
======================================================================
* 1853 'Lectures on Quaternions'
* 1866 'Elements of Quaternions'
* 1873 'Elementary Treatise' by Peter Guthrie Tait
* 1874 Jules Hoüel: '�léments de la Théorie des Quaternions'
* 1878 Abbott Lawrence Lowell:  Quadrics: Harvard dissertation:
* 1882 Tait and Philip Kelland: 'Introduction with Examples'
* 1885 Arthur Buchheim:  Biquaternions
* 1887 Valentin Balbin: (Spanish) 'Elementos de Calculo de los
Cuaterniones', Buenos Aires
* 1899 Charles Jasper Joly: 'Elements' vol 1, vol 2 1901
* 1901 Vector Analysis by Willard Gibbs and Edwin Bidwell Wilson
(quaternion ideas without quaternions)
* 1904 Cargill Gilston Knott: third edition of Kelland and Tait's
textbook
* 1904 'Bibliography' prepared for the Quaternion Society by Alexander
Macfarlane
* 1905 C.J. Joly's 'Manual for Quaternions'
* 1940 Julian Coolidge in 'A History of Geometrical Methods', page
261, uses the coordinate-free methods of Hamilton's operators and
cites A. L. Lawrence's work at Harvard. Coolidge uses these operators
on dual quaternions to describe screw displacement in kinematics.


                             Octonions
======================================================================
Octonions were developed independently by Arthur Cayley in 1845  and
John T. Graves, a friend of Hamilton's. Graves had interested Hamilton
in algebra, and responded to his discovery of quaternions with "If
with your alchemy you can make three pounds of gold [the three
imaginary units], why should you stop there?"

Two months after Hamilton's discovery of quaternions, Graves wrote
Hamilton on December 26, 1843 presenting a kind of double quaternion
that is called an 'octonion', and showed that they were what we now
call a normed division algebra; Graves called them 'octaves'. Hamilton
needed a way to distinguish between two different types of double
quaternions, the associative biquaternions and the octaves.  He spoke
about them to the Royal Irish Society and credited his friend Graves
for the discovery of the second type of double quaternion. observed in
reply that they were not associative, which may have been the
invention of the concept. He also promised to get Graves' work
published, but did little about it; Cayley, working independently of
Graves, but inspired by Hamilton's publication of his own work,
published on octonions in March 1845 - as an appendix to a paper on a
different subject. Hamilton was stung into protesting Graves' priority
in discovery, if not publication; nevertheless, octonions are known by
the name Cayley gave them - or as 'Cayley numbers'.

The major deduction from the existence of octonions was the eight
squares theorem, which follows directly from the product rule from
octonions, had also been previously discovered as a purely algebraic
identity, by Carl Ferdinand Degen in 1818. This sum-of-squares
identity is characteristic of composition algebra, a feature of
complex numbers, quaternions, and octonions.


                         Mathematical uses
======================================================================
Quaternions continued to be a well-studied 'mathematical' structure in
the twentieth century, as the third term in the Cayley-Dickson
construction of hypercomplex number systems over the reals, followed
by the octonions and the sedenions; they are also useful tool in
number theory, particularly in the study of the representation of
numbers as sums of squares. The group of eight basic unit quaternions,
positive and negative, the quaternion group, is also the simplest
non-commutative Sylow group.

The study of integral quaternions began with Rudolf Lipschitz in 1886,
whose system was  later simplified by Leonard Eugene Dickson; but the
modern system was published by Adolf Hurwitz in 1919. The difference
between them consists of which quaternions are accounted integral:
Lipschitz included only those quaternions with integral coordinates,
but Hurwitz added those quaternions 'all four' of whose coordinates
are half-integers. Both systems are closed under subtraction and
multiplication, and are therefore rings, but Lipschitz's system does
not permit unique factorization, while Hurwitz's does.


                      Quaternions as rotations
======================================================================
Quaternions are a concise  method of representing the automorphisms of
three- and four-dimensional spaces. They have the technical advantage
that unit quaternions form the simply connected cover of the space of
three-dimensional rotations.

For this reason, quaternions are used in computer graphics, control
theory, robotics, signal processing, attitude control, physics,
bioinformatics, and orbital mechanics. For example, it is common for
spacecraft attitude-control systems to be commanded in terms of
quaternions. 'Tomb Raider' (1996) is often cited as the first
mass-market computer game to have used quaternions to achieve smooth
3D rotation. Quaternions have received another boost from number
theory because of their relation to quadratic forms.


                              Memorial
======================================================================
Since 1989, the Department of Mathematics of the National University
of Ireland, Maynooth has organized a pilgrimage, where scientists
(including physicists Murray Gell-Mann in 2002, Steven Weinberg in
2005, Frank Wilczek in 2007, and mathematician Andrew Wiles in 2003)
take a walk from Dunsink Observatory to the Royal Canal bridge where,
unfortunately, no trace of Hamilton's carving remains.


                             References
======================================================================
*
* G. H. Hardy and E. M. Wright, 'Introduction to Number Theory'. Many
editions.
* Johannes C. Familton (2015) [https://arxiv.org/abs/1504.04885
Quaternions: A History of Complex Non-commutative Rotation Groups in
Theoretical Physics], Ph.D. thesis in Columbia University Department
of Mathematics Education.


License
=========
All content on Gopherpedia comes from Wikipedia, and is licensed under CC-BY-SA
License URL: http://creativecommons.org/licenses/by-sa/3.0/
Original Article: http://en.wikipedia.org/wiki/History_of_quaternions

You are viewing proxied material from sdf.org. The copyright of proxied material belongs to its original authors. Any comments or complaints in relation to proxied material should be directed to the original authors of the content concerned. Please see the disclaimer for more details.