Hyperspace is neither a fiction nor an abstraction. Every interaction we have with our every-day world occurs in high-dimensional spaces of objects and coordinates and momenta. This dynamical hyperspace—also known as phase space—is as real as mathematics, and physics in phase space can be calculated and used to predict complex behavior. Although phase space can extend to thousands of dimensions, our minds are incapable of thinking even in four dimensions—we have no ability to visualize such things. 

Grassmann was convinced that he had discovered a fundamentally new type of mathematics—he actually had.

            Part of the trick of doing physics in high dimensions is having the right tools and symbols with which to work.  For high-dimensional math and physics, one such indispensable tool is Hermann Grassmann’s wedge product. When I first saw the wedge product, probably in some graduate-level dynamics textbook, it struck me as a little cryptic.  It is sort of like a vector product, but not, and it operated on things that had an intimidating name— “forms”. I kept trying to “understand” forms as if they were types of vectors.  After all, under special circumstances, forms and wedges did produce some vector identities.  It was only after I actually stepped back and asked myself how they were constructed that I realized that forms and wedge products were just a simple form of algebra, called exterior algebra. Exterior algebra is an especially useful form of algebra with simple rules.  It goes far beyond vectors while harking back to a time before vectors even existed.

Hermann Grassmann: A Backwater Genius

We are so accustomed to working with oriented objects, like vectors that have a tip and tail, that it is hard to think of a time when that wouldn’t have been natural.  Yet in the mid 1800’s, almost no one was thinking of orientations as a part of geometry, and it took real genius to conceive of oriented elements, how to manipulate them, and how to represent them graphically and mathematically.  At a time when some of the greatest mathematicians lived—Weierstrass, Möbius, Cauchy, Gauss, Hamilton—it turned out to be a high school teacher from a backwater in Prussia who developed the theory for the first time.

            Hermann Grassmann was the son of a high school teacher at the Gymnasium in Stettin, Prussia, (now Szczecin, Poland) and he inherited his father’s position, but at a lower level.  Despite his lack of background and training, he had serious delusions of grandeur, aspiring to teach mathematics at the university in Berlin, even when he was only allowed to teach the younger high school students basic subjects.  Nonetheless, Grassmann embarked on a program to educate himself, attending classes at Berlin in mathematics.  As part of the requirements to be allowed to teach mathematics to the senior high-school students, he had to submit a thesis on an appropriate topic. 

            For years, he had been working on an idea that had originally come from his father about a mathematical theory that could manipulate abstract objects or concepts.  He had taken this vague thought and had slowly developed it into a rigorous mathematical form with symbols and manipulations.  His mind was one of those that could permute endlessly, and he defined and discovered dozens of different ways that objects could be defined and combined, and he wrote them all down in a tome of excessive size and complexity.  When it was time to submit the thesis to the examiners, he had created a broad new system of algebra—at a time when no one recognized what a new algebra even meant, especially not his examiners, who could understand none of it.  Fortunately, Grassmann had been corresponding with the famous German mathematician August Möbius over his ideas, and Möbius was encouraging and supportive, and the examiners accepted his thesis and allowed him to teach the upper class-men at his high school. 

            Encouraged by his success, Grassmann hoped that Möbius would help him climb even higher to teach in Berlin.  Convinced that he had discovered a fundamentally new type of mathematics (he actually had), he decided to publish his thesis as a book under the title Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).  He published it out of his own pocket.  It is some measure of his delusion that he had thousands printed, but he sold almost none, and piles of the books were stored away to be used later as scrap paper. Möbius likewise distanced himself from Grassmann and his obsessive theories. Discouraged, Grassmann turned his back on mathematics, though he later achieved fame in the field of linguistics.  (For more on Grassmann’s ideas and struggle for recognition, see Chapter 4 of Galileo Unbound).

The Odd Identity of Nicholas Bourbaki

If you look up the publication history of the famous French mathematician, Nicholas Bourbaki, you will be amazed to see a publication history that spans from 1935 to 2018 — over 85 years of publications!  But if you look in the obituaries, you will see that he died in 1968.  It’s pretty impressive to still be publishing 50 years after your death.  JRR Tolkein has been doing that regularly, but few others spring to mind.

            Actually, you have been duped!  Nicholas is a fiction, constructed as a hoax by a group of French mathematicians who were simultaneously deadly serious about the need for a rigorous foundation on which to educate the new wave of mathematicians in the mid 20^th century.  The group was formed during a mathematics meeting in 1924, organized by André Weil and joined by Henri Cartan (son of Eli Cartan), Claude Chevalley, Jean Coulomb, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, René de Possel, and Szolem Mandelbrojt (uncle of Benoit Mandelbrot).  They picked the last name of a French general, and Weil’s wife named him Nicholas.  The group began publishing books under this pseudonym in 1935 and has continued until the present time.  While their publications were entirely serious, the group from time to time had fun with mild hoaxes, such as posting his obituary on one occasion and a wedding announcement of his daughter on another. 

            The wedge product symbol took several years to mature.  Eli Cartan’s book on differential forms published in 1945 used brackets to denote the product instead of the wedge. In Chevally’s book of 1946, he does not use the wedge, but uses a small square, and the book  Chevalley wrote in 1951 “Introduction to the Theory of Algebraic Functions of One Variable” still uses a small square.  But in 1954, Chevalley uses the wedge symbol in his book on Spinors.  He refers to his own book of 1951 (which did not use the wedge) and also to the 1943 version of Bourbaki. The few existing copies of the 1943 Algebra by Bourbaki lie in obscure European libraries. The 1973 edition of the book does indeed use the wedge, although I have yet to get my hands on the original 1943 version. Therefore, the wedge symbol seems to have originated with Chevalley sometime between 1951 and 1954 and gained widespread use after that.

Exterior Algebra

Exterior algebra begins with the definition of an operation on elements.  The elements, for example (u, v, w, x, y, z, etc.) are drawn from a vector space in its most abstract form as “tuples”, such that x = [x₁, x₂, x₃, …, x_n] in an n-dimensional space.  On these elements there is an operation called the “wedge product”, the “exterior product”, or the “Grassmann product”.  It is denoted, for example between two elements x and y, as x^y.  It captures the sense of orientation through anti-commutativity, such that

As simple as this definition is, it sets up virtually all later manipulations of vectors and their combinations.  For instance, we can immediately prove (try it yourself) that the wedge product of a vector element with itself equals zero

Once the elements of the vector space have been defined, it is possible to define “forms” on the vector space.  For instance, a 1-form, also known as a vector, is any function

where a, b, c are scalar coefficients.  The wedge product of two 1-forms

yields a 2-form, also known as a bivector.  This specific example makes a direct connection to the cross product in 3-space as

where the unit vectors are mapped onto the 2-forms

Indeed, many of the vector identities of 3-space can be expressed in terms of exterior products, but these are just special cases, and the wedge product is more general.  For instance, while the triple vector cross product is not associative, the wedge product is associative

which can give it an advantage when performing algebra on r-forms.  Expressing the wedge product in terms of vector components

yields the immediate generalization to any number of dimensions (using the Einstein summation convention)

In this way, the wedge product expresses relationships in any number of dimensions.

            A 3-form is constructed as the wedge product of 3 vectors

where the Levi-Civita permuation symbol has been introduced such that

Note that in 3-space there can be no 4-form, because one of the basis elements would be repeated, rendering the product zero.  Therefore, the most general multilinear form for 3-space is

with 2³ = 8 elements: one scalar, three 1-forms, three 2-forms and one 3-form.  In 4-space there are 2⁴ = 16 elements: one scalar, four 1-forms, six 2-forms, four 3-forms and one 4-form.  So, the number of elements rises exponentially with the dimension of the space.

            At this point, we have developed a rich multilinear structure, all based on the simple anti-commutativity of elements x^y = -y^x.  This process is called by another name: a Clifford algebra, named after William Kingdon Clifford (1845-1879), second wrangler at Cambridge and close friend of Arthur Cayley.  But the wedge product is not just algebra—there is also a straightforward geometric interpretation of wedge products that make them useful when extending theories of surfaces and volumes into higher dimensions.

Geometric Interpretation

In Euclidean space, a cross product is related to areas and volumes of paralellapipeds. Wedge products are more general than cross products and they generalize the idea of areas and volumes to higher dimension. As an illustration, an area 2-form is shown in Fig. 1 and a 3-form in Fig. 2.

The wedge product is not limited to 3 dimensions nor to Euclidean spaces. This is the power and the beauty of Grassmann’s invention. It also generalizes naturally to differential geometry of manifolds producing what are called differential forms. When integrating in higher dimensions or on non-Euclidean manifolds, the most appropriate approach is to use wedge products and differential forms, which will be the topic of my next blog on the generalized Stokes’ theorem.

Further Reading

1.         Dieudonné, J., The Tragedy of Grassmann. Séminaire de Philosophie et Mathématiques 1979, fascicule 2, 1-14.

2.         Fearnley-Sander, D., Hermann Grassmann and the Creation of Linear Algegra. American Mathematical Monthly 1979, 86 (10), 809-817.

3.         Nolte, D. D., Galileo Unbound: A Path Across Life, the Universe and Everything. Oxford University Press: 2018.

4.         Vargas, J. G., Differential Geometry for Physicists and Mathematicians: Moving Frames and Differential Forms: From Euclid Past Riemann. 2014; p 1-293.