## This site introduces the first of two printed volumes on the work of Hermann

## Grassmann as it re-emerges from one and a half centuries of isolation.

The books have grown out of an interest in Grassmann's work over the past four decades.
There is something fascinating about the beauty with which the mathematical structures Grassmann
discovered (invented, if you will) describe the physical world, and something also fascinating about how
these beautiful structures have been largely lost to the mainstreams of mathematics and science.
Volume 1: Foundations - develops the algebraic foundations of the structures.
Volume 2: Extensions - will explore some of their extensions to other systems.
### The genesis of Grassmann algebra

### Hermann Günther Grassmann was born in 1809 in Stettin, near the border of Germany and Poland. He was

### only 23 when he discovered the method of adding and multiplying points and vectors which was to become

### the foundation of his Ausdehnungslehre. In 1839 he composed a work on the study of tides entitled Theorie

### der Ebbe und Flut, which was the first work ever to use vectorial methods. In 1844 Grassmann published his

### first Ausdehnungslehre (Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik) and in the same year

### won a prize for an essay which expounded a system satisfying an earlier search by Leibniz for an 'algebra of

### geometry'. Despite these achievements, Grassmann received virtually no recognition.

In 1862 Grassmann re-expounded his ideas from a different viewpoint in a second Ausdehnungslehre (Die
Ausdehnungslehre. Vollständig und in strenger Form). Again the work was met with resounding silence from
the mathematical community, and it was not until the latter part of his life that he received any significant
recognition from his contemporaries. Of these, most significant were J. Willard Gibbs who discovered his
works in 1877 (the year of Grassmann's death), and William Kingdon Clifford who discovered them in depth
about the same time. Both became quite enthusiastic about this new mathematics.
More details on the biography of Grassmann may be found on the Biography page of this site, but for the
most comprehensive biography see Petsche [2009] on the same page.
## The power of Grassmann algebra

The intrinsic power of Grassmann algebra arises from its fundamental product operation, the exterior product.
The exterior product codifies the property of linear dependence, so essential for modern applied mathematics,
directly into the algebra. Simple non-zero elements of the algebra may be viewed as representing constructs
of linearly independent elements. For example, a simple bivector is the exterior product of two vectors; a line
is represented by the exterior product of two points; a plane is represented by the exterior product of three
points.
## The focus of the Grassmann algebra books

The focus of these two volumes is to provide a readable account in modern notation of Grassmann's major
algebraic contributions to mathematics and science in a form accessible to scientists and engineers, students
and professionals alike. Consequently mathematical terminology which does not make an essential
contribution to understanding the basic concepts has been avoided. The only assumption made as to the
reader's background is that they have some familiarity with basic linear algebra.
The focus is also to provide an environment for exploring applications of Grassmann algebra. For general
applications in higher dimensional spaces, computations by hand in any algebra become tedious, indeed
limiting, thus restricting the hypotheses that can be explored. For this reason the volumes are integrated with a
Mathematica package called GrassmannAlgebra. You can read the book without using the package, or you
can use the package to extend the examples in the text, experiment with hypotheses, or explore your own
interests. To get a copy of the GrassmannAlgebra package, see the Contact page.
### Volume 1: Foundations

Volume 1: Foundations comprises six chapters. Chapter 1 provides a brief preparatory overview, introducing
the seminal concepts of each chapter, and solidifying them with simple examples. This chapter (which can be
downloaded from this site) is designed to give you a global appreciation with which better to understand the
detail of the chapters which follow. However, it is independent of those chapters, and may be read as far as
your interest takes you. Chapter 2 discusses the exterior product - the
fundamental product operation of the algebra - and shows how it creates the
suite of linear spaces which form the algebra. Chapter 3 discovers that the
symmetry in this suite leads to another product, the regressive product,
elegantly 'dual' to the exterior product. Equipped with these dual products,
Chapter 4 shows how the algebra can be interpreted geometrically to easily
recreate projective geometry. The construction of the conic shown is an
example. Chapter 5 lays the foundation for introducing a metric by defining
for each element, a partner - its complement. Then Chapter 6 shows how to
combine all of these to define the interior product, a much more general
product than the scalar product. These chapters form the essential core for a
working knowledge of Grassmann algebra and its extensions in the second
volume. They are most profitably read (or at least scanned) sequentially. The
rest is exploration!
### Volume 2: Extensions

Volume 2: Extensions is currently a work in progress. When it is published you will find explorations using
the fundamental theory developed in this volume extended to other topics, for example the generalized
product, and to other branches of mathematics, for example the hypercomplex and Clifford algebras.
### Acknowledgements

Writing these sorts of books also involves a lot of Mathematica coding. In this I would like to especially
acknowledge my great appreciation of David Park's interest in Grassmann algebra and Grassmann calculus
and the thought-provoking conversations which have ensued. Particularly, I would like to acknowledge his
many contributions to the GrassmannAlgebra and GrassmannCalculus software packages. To get a copy of
these packages, see the Contact page.
Above all however, I must acknowledge Hermann Grassmann. His contributions to mathematics and science
put him among the great thinkers of the nineteenth century.
I hope you enjoy exploring this beautiful mathematical system.
John Browne
October, 2012
An introduction to the Algebra of
Hermann Grassmann

John Browne

This website is an introduction to rediscovering and exploring
the Algebra of Hermann Grassmann using the computer
algebra system Mathematica.

“For
I
have
every
confidence
that
the
effort
I
have
applied
to
the
science
reported
upon
here,
which
has
occupied
a
considerable
span
of
my
lifetime
and
demanded
the
most
intense
exertions
of
my
powers,
is
not
to
be
lost.
…
a
time
will
come
when
it
will
be
drawn
forth
from
the
dust
of
oblivion
and
the
ideas
laid
down
here
will
bear
fruit.
…
some
day
these
ideas,
even
if
in
an
altered
form,
will
reappear
and
with
the
passage
of
time
will
participate
in
a
lively
intellectual
exchange.
For
truth
is
eternal,
it
is
divine;
and
no
phase
in
the
development
of
truth,
however
smal
l
the
domain
it
embraces,
can
pass
away
without
a
trace.
It
remains
even
if
the
garments
in
which
feeble
men
clothe
it
fall
into
dust.”
Hermann Grassmann - in the foreword to the Ausdehnungslehre of 1862,
translated by Lloyd Kannenberg.
© John Browne 2012
This site and book are protected by copyright. All or parts may not be copied or disseminated in any way without the permission of the
copyright owner. You may copy, reference or quote small sections of the work as long as due acknowledgement is made.