Exploring Grassmann Algebra with Mathematica

Volume 1 of Grassmann Algebra, titled Foundations, delves into the exploration of extended vector algebra using Mathematica. Grassmann algebra extends traditional vector algebra by introducing the exterior product, which allows for the algebraic representation of linear dependence. This extension enables the representation of vectors as higher-grade entities such as bivectors, trivectors, and multivectors.

In addition to the exterior product, Grassmann algebra also introduces the regressive dual known as the regressive product. This pair shares similarities with Boolean duals of union and intersection. By designating one of the elements in the vector space as an origin point, points can be defined. The exterior product then extends these points into higher-grade located entities, from which lines, planes, and multiplanes can be defined. The theorems of Projective Geometry can be expressed as formulae involving these entities and the dual products.

Introducing the (orthogonal) complement operation allows for the extension of the scalar product of vectors to the interior product of multivectors. In this more general case, the result may no longer be a scalar. The concept of vector magnitude is expanded to multivector magnitude. For example, the magnitude of the exterior product of two vectors (a bivector) represents the area of the parallelogram formed by them.

This volume focuses on entities that are the sums of elements of the same grade. Grassmann algebra entities are not limited to the same grade, and the types of products are not constrained to just the exterior, regressive, and interior products. For instance, quaternion algebra can be seen as the Grassmann algebra of scalars and bivectors under a new product operation. Other algebras, including Clifford, geometric, and higher order hypercomplex algebras like the octonions, can be defined similarly.

The exploration of various entities, operations, and algebras will be the central theme of the upcoming volume. The innovative mathematical structures discovered by Hermann Grassmann offer a beautiful description of the physical world, yet they have not received much recognition in mainstream mathematics and science. His seminal work, Ausdehnungslehre, published in 1862, gained recognition later in his life, particularly by Gibbs and Clifford.

David Hestenes' Geometric Algebra has played a crucial role in raising awareness of Grassmann's contributions in recent times. The text aims to make Grassmann's ideas accessible to a wide audience, including scientists, engineers, students, and professionals. Mathematical terminology is kept to a minimum, with a focus on understanding basic concepts. While familiarity with basic linear algebra is helpful, knowledge of Mathematica is not required for grasping Grassmann's elegant ideas.

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