G. Sobczyk: Conformal Mappings in Geometric Algebra, AMS Notices 59(2), 264-273


Notices of the AMS Volume 59, Number 2, DOI: http://dx.doi.org/10.1090/noti793
by Garret Sobczyk, who is professor of mathematics at Universidad de Las Américas-Puebla.
His email address is: garret_sobczyk_at_yahoo.com.

Download full paper: http://www.ams.org/notices/201202/rtx120200264p.pdf

In 1878 William Kingdon Clifford wrote down
the rules for his geometric algebra, also
known as Clifford algebra. We argue in this
paper that in doing so he laid down the
groundwork that is profoundly altering the
language used by the mathematical community to
express geometrical ideas. In the real estate business
everyone knows that what is most important
is location. We demonstrate here that in the business
of mathematics what is most important to
the clear and concise expression of geometrical
ideas is notation. In the words of Bertrand Russell,

…A good notation has a subtlety
and suggestiveness which at times
make it seem almost like a live
teacher.

Heinrich Hertz expressed much the same thought
when he said,

One cannot escape the feeling
that these mathematical formulae
have an independent existence and
an intelligence of their own, that
they are wiser than we are, wiser
even than their discoverers, that
we get more out of them than we
originally put into them.

The development of the real and complex number
systems represents a hard-won milestone in
the robust history of mathematics over many centuries
and many different civilizations [5], [29].
Without it mathematics could progress only haltingly,
as is evident fromthe history ofmathematics
and even the terminology that we use today. Negative
numbers were referred to by Rene Descartes
(1596–1650) as “fictitious”, and “imaginary” numbers
were held up to even greater ridicule, though
they were first conceived as early as Heron of

Fig. Herman Gunther Grassmann (1809–1877) was
a high school teacher. His far-reaching
Ausdehnungslehre, “Theory of extension”, laid
the groundwork for the development of the
exterior or outer product of vectors. William
Rowan Hamilton (1805–1865) was an Irish
physicist, astronomer, and mathematician. His
invention of the quaternions as the natural
generalization of the complex numbers of the
plane to three-dimensional space, together
with the ideas of Grassmann, set the stage for
William Kingdon Clifford’s definition of
geometric algebra. William Kingdon Clifford
(1845–1879) was a professor of mathematics
and mechanics at the University College of
London. Tragically, he died at the early age of
34 before he could explore his profound ideas.

Alexandria, the illustrious inventor of the windmill
and steam engine during the first century AD
[18]. What most mathematicians fail to see even
today is that geometric algebra represents the
grand culmination of that process with the completion
of the real number system to include the
concept of direction. Geometric algebra combines
the two silver currents of mathematics, geometry
and algebra, into a single coherent language. As
David Hestenes has eloquently stated,

Algebra without geometry is blind,
geometry without algebra is dumb.

Source: pp. 264 and 265 of http://www.ams.org/notices/201202/rtx120200264p.pdf

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