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Application of Grassmann Algebra to Geometry Using Mathematica
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| The Role of Mathematics in Modern Engineering: Proceedings of AEMC '94 |
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Introduction; Grassmann Approach to Geometry; The Mechanism Class; Mechanism Examples; Mechanism Synthesis; Conclusions
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From the Introduction:
Hermann Grassmann is an important figure in the historical development of the vector and tensor calculus. In his book "Die Lineale Ausdehnungslehre" (Grassmann, 1862) Grassmann creates a mathematical language of significant power in its application to geometry.
The word "Ausdehnungslehre" (exterior theory) is indicative of the fact that an element may be "extended" by taking the (exterior) product of it with another. For example, two distinct points may be "extended" into an element defining the straight line joining them by forming their exterior product. This element may be extended into an element defining a plane by multiplying it by a third point exterior to the line, and so on for higher dimensions. The "exteriorness" here is the geometric equivalent of "liner independence". If two elements are not exterior to one another (not independent) then their exterior product is zero.
Grassmann, in the preface to his "Ausdehnungslehre" of 1862 said: "I am aware that the form which I have given the science is imperfect...there will come a time when these ideas, perhaps in a new form, will arise anew and will enter into living communication with contemporary developments" (Grassmann, 1862).
The combination of the elegance of the Ausdehnungslehre with the symbolic processing power of the computer may well be out of those developments.
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