Grassmann Algebra

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Grassmann Algebra

What is Grassmann Algebra? Grassmann algebra is a mathematical system which predates vector algebra, and yet is more powerful than it, subsuming and unifying much of the algebra used by engineers and physicists. It has remained relatively unknown since its discovery over 160 years ago, yet is now emerging as a potential mathematical system for describing such diverse applications as robotic manipulators and fundamental physical theories.

Where does its power come from? The intrinsic power of Grassmann algebra arises from its fundamental product operation, the exterior product. The exterior product codifies the property of linear dependence directly into the algebra. Simple non-zero elements of the algebra are products of linearly independent elements. For example, a simple bivector is the exterior product of two independent vectors; a line is represented by the exterior product of two independent points and a plane is represented by the exterior product of three independent points. Exterior products of linearly dependent elements are zero.

The best original source to Grassmann's contributions to mathematics and science is his collected works:

Grassmann, Hermann
Hermann Grassmanns Gesammelte Mathematische und Physikalische Werke.
Teubner, Leipzig. Volume 1 (1896), Volume 2 (1902,1904), Volume 3 (1911).

Grassmann's algebraic writings are predominantly developed in two books which have been recently translated by Lloyd C. Kannenberg:

Grassmann, Hermann
A New Branch of Mathematics: The Ausdehnungslehre of 1844 and other works.
Open Court, Illinois. (1995) ISBN 0-8126-9275-6

Grassmann, Hermann
Extension Theory
American Mathematical Society. London Mathematical Society. (2000) ISBN 0-8218-2031-1

Grassmann algebra is intimately connected with Clifford algebra. David Hestenes has called this broader structure the Geometric Calculus.

Hermann Grassmann

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 (extension theory). 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.

A most readable account of Grassmann's life woven with the lives of other mathematicians instrumental in the development of modern vector analysis is given by Michael Crowe:

Crowe, Michael
A History of Vector Analysis
Notre Dame, London. (1967)

Desmond Fearnley-Sander has made a study of Grassmann's fundamental contributions to algebra.

A brief biography of Grassmann is contained in my book, which is discussed in the next section.

Grassmann Algebra Book

I am at present writing a book on Grassmann algebra called:

Grassmann Algebra: Exploring applications of extended vector algebra with Mathematica

The primary focus of this book is to provide a readable account in modern notation of Grassmann's major algebraic contributions to mathematics and science. I would like it to be accessible to scientists and engineers, students and professionals alike. Consequently I have tried to avoid all mathematical terminology which does not make an essential contribution to understanding the basic concepts. The only assumptions I have made about the reader's background is that they have some familiarity with basic linear algebra.

The secondary focus of this book is to provide an environment for exploring applications of the 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, when the book is completed, it will include a Mathematica package for exploring Grassmann algebra. You can read the book without using the package, or you will be able to use the package to extend the examples in the text, experiment with hypotheses, or explore your own interests.

The incomplete draft for the book was first posted on the web in February 2001, and has not changed since then. The final draft of the book and accompanying software will be posted upon completion in 2008, and will include interactive geometric depictions of Grassmann entities. Manipulating these entities will require Mathematica version 6.

If you are interested in viewing the book in incomplete draft form, check out my Grassmann Algebra book page.

If you would like to discuss anything in the book, or on Grassmann algebra in general, I would like to hear from you; particularly on how you are using it in your own work. My email address is at the bottom of the page.

Mathematica

Mathematica is a powerful system for doing mathematics on a computer. It has an inbuilt programming language ideal for extending its capabilities to other mathematical systems like Grassmann algebra. It also has a sophisticated mathematical typesetting capability. The book Grassmann Algebra uses both. The chapters are typeset by Mathematica in its standard notebook format, making the book interactive in its electronic version with the GrassmannAlgebra package. GrassmannAlgebra can turn an exploration requiring days by hand into one requiring just minutes of computing time.

If you are interested in Mathematica, check out the Wolfram Research site.

Contact Details

John M Browne
http://grassmannalgebra.info
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