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  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
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.
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.
An introduction to the Algebra of
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 small 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 2018.
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