*More than 150 years ago - Hermann Grassmann made a prophecy*

*More than 150 years ago - Hermann Grassmann made a prophecy*

*“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**.*

This website is an introduction to rediscovering

and exploring the Algebra of Hermann Grassmann.

The beauty and power of Grassmann algebra was first revealed to me as I explored its workings using a computer and Wolfram Mathematica software. I soon discovered - It is indeed an algebra conceived before its time, a beautiful mathematical system ideally suited to power computing in the modern world.

John Browne, May 2021

**This site introduces the work of Hermann Grassmann as it re-emerges from one and a half centuries of isolation.**

My three published books on Grassmann Algebra have grown out of an interest in Grassmann's work over the past five 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 - First published in October 2012 develops the algebraic foundations of the structures.

Volume 2: Extensions - First published May 2021 explores hypercomplex and associative algebras. It should be read in conjunction with Volume 1.

See BOOKS TAB for more information on where to read or buy John Browne's books.

**The genesis of Grassmann algebra**

**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.*

## This Support Website - Publication history

This website - grassmannalgebra.com is the current support site for my three Grassmann Algebra Books, code downloads for use with Mathematica and reference downloads for a range of support material.

My Grassmann algebra project has been a life-time project dating from my doctoral thesis in the area in 1978. In October 2001 I published an incomplete book draft "Grassmann Algebra: Exploring applications of extended vector algebra with Mathematica" on my university home page. The computations in this draft were done with early versions of Mathematica and draft versions of the *GrassmannAlgebra* package.

WARNING: Some versions of this early draft have found their way on-line as a PDF. this version is incomplete and contains errors and many omissions.

There is no official eBook or PDF version available at this time. Sites offering these as a free download may deliver malicious code.

In February 2007 I moved my home page to an old version of Google Sites: http://grassmannalgebra.info, but no changes were made to the drafts. The 200th anniversary of Grassmann's birth in April 1809 prompted me to release, in 2009, the book and package project in the state it was at that time.

The relevant content of the 2009 heritage site expressly maintained to support the Mathematica versions and explorations prior to the publication of Grassmann Algebra Volume 1 in 2012, can be found on Legacy-download page this site.

**Precedence claims:** The publication of the original draft on the web on October 25, 2001 establishes claim to precedence for any new or original previously unpublished results it may have contained. Subsequent drafts also establish claim to precedence for any subsequent results.

Any reference to elements of this work should follow standard journal publication practice.

John Browne, May 2021

John Browne (1942- 2021) died from the complications of cancer 28 June 2021 - RIP.

**Yet his work lives-on **

*John Browne - exploring another love - Cradle Mountain Wilderness World Heritage Area, Lake St. Clair National Park, Tasmania, Australia.*

**What is Grassmann Algebra?**

**What is Grassmann Algebra?**

Grassmann algebra is a mathematical system which predates vector algebra, and yet is more powerful, subsuming and unifying much of the algebra used by engineers and physicists today and in the foreseeable future. It has remained relatively unknown since its discovery around 1832, 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, 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 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. These properties generate a geometric algebra par excellence.

As John Browne adds in the preface of Grassmann Algebra Volume 2 Extensions: Additionally, its axioms have a special property - they may be dualized to give the axioms for its dual product, the regressive product. Together these enable all the scalar product, inner product, interior product and metric notions that we might expect of an algebra of space.

Every entity of a Grassmann algebra notation possesses a critically important property - its grade. For the most part, Grassmann Algebra Volume 1 studies entities of a single grade. But it is straightforward to extend the algebra to work naturally with entities which are linear combinations of entities of different grades. When we do this, we discover a new capability: new product operations may be defined on the algebra (for example hypercomplex products, geometric products and Clifford products) which do not require any axioms additional to those of the Grassmann algebra. We can therefore include these product operations in the algebra as we did for the regressive, scalar, inner and interior product operations.

The best original source for Grassmann's contributions to science and mathematics 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 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

A bibliography of earlier writers in the Grassmannian tradition is given in Grassmann Algebra Volume 1 or on the Bibliography page at grassmannalgebra.com.

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.

MATHEMATICA

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 application 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.

*EDITORS NOTE: John Browne's original Mathematica package has been enhanced and extended by David Park who worked closely with John in recent years. The enhanced application is called "GrassmannCalculus" and is available on this site at page Code-Download.*

**Volume 1: Foundations** comprises six chapters.

(The entire book is free to read on Google Books) *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.

*Chapter 4* shows how the algebra, equipped with these dual products, 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.

*Chapter 6* then 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**

First published May 2021. In it you will find

explorations using the fundamental theory developed in Volume 1 extended to other topics, for example the generalized product, and to other branches of mathematics, like the hypercomplex and Clifford algebras.

**Acknowledgements**

**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 this (now combined) application, see the page Code-Download, this site.

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, 2020

© Estate J. Browne 2022

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**Precedence claims** The publication of the original draft on the web on October 25, 2001 establishes claim to precedence for any new or original previously unpublished results it may have contained. Subsequent drafts also establish claim to precedence for any subsequent results. Any reference to elements of this work should follow standard journal publication practice.

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