PREFACE

John Browne has spent the last 50 years of his lifetime exploring and rediscovering the beauty and power of Grassmann Algebra. He has ordered and compressed what he has learned into three books now available from Amazon.

His latest and final book just released in 2021 ties it all together and shows in detail how some important linear algebras such as the hypercomplex, geometric and Clifford algebras can be constructed entirely within Grassmann algebra by defining specialized product operations using only the exterior and interior products.

John Browne has struggled with Cancer to get his last book Grassmann Algebra Volume 2: Extensions into print - he writes in the Foreword:

"This book is an abridged draft edition. It is being published now in this form due to circumstances beyond the author's control. There will not be another edition. You will miss some things you might have expected: page numbers in the table of contents, a comprehensive bibliography and references, an index, a well-reviewed text.

This book (Volume 2) follows on from Volume 1 so you may also need Volume 1 for reference. The book shows how the hypercomplex and associative algebras are hidden in Grassmann algebra. It shows how the quaternions and octonions and their split variants, and the geometric and Clifford algebras, are simply Grassmann algebra. It does this by extending its two familiar product operations, the exterior and interior products to define a suite of products together called the generalized Grassmann product. It shows how hypercomplex, geometric and Clifford products may then be defined as linear combinations of generalized Grassmann products in which the scalar coefficients are restricted to unity or negative unity only. This binary variability is sufficient to endow properties to a product operation, for example associativity. The book concludes by finding four associative product operations, two of which being the geometric and Clifford products. But these work on Grassmann entities only, so they are operations of the Grassmann algebra."

All books have been written using Wofram Mathematica because of its sophisticated typesetting capabilities and its ability to check and compute the written word, "explorations that would be intractable by hand take seconds to compute and typeset".

Vale John Browne 28 June 2021