# A Short History of Hyperspace

Hyperspace by any other name would sound as sweet, conjuring to the mind’s eye images of hypercubes and tesseracts, manifolds and wormholes, Klein bottles and Calabi Yau quintics.  Forget the dimension of time—that may be the most mysterious of all—but consider the extra spatial dimensions that challenge the mind and open the door to dreams of going beyond the bounds of today’s physics.

The geometry of n dimensions studies reality; no one doubts that. Bodies in hyperspace are subject to precise definition, just like bodies in ordinary space; and while we cannot draw pictures of them, we can imagine and study them.

(Poincare 1895)

Here is a short history of hyperspace.  It begins with advances by Möbius and Liouville and Jacobi who never truly realized what they had invented, until Cayley and Grassmann and Riemann made it explicit.  They opened Pandora’s box, and multiple dimensions burst upon the world never to be put back again, giving us today the manifolds of string theory and infinite-dimensional Hilbert spaces.

## August Möbius (1827)

Although he is most famous for the single-surface strip that bears his name, one of the early contributions of August Möbius was the idea of barycentric coordinates [1] , for instance using three coordinates to express the locations of points in a two-dimensional simplex—the triangle. Barycentric coordinates are used routinely today in metallurgy to describe the alloy composition in ternary alloys.

Möbius’ work was one of the first to hint that tuples of numbers could stand in for higher dimensional space, and they were an early example of homogeneous coordinates that could be used for higher-dimensional representations. However, he was too early to use any language of multidimensional geometry.

## Carl Jacobi (1834)

Carl Jacobi was a master at manipulating multiple variables, leading to his development of the theory of matrices. In this context, he came to study (n-1)-fold integrals over multiple continuous-valued variables. From our modern viewpoint, he was evaluating surface integrals of hyperspheres.

In 1834, Jacobi found explicit solutions to these integrals and published them in a paper with the imposing title “De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant; una cum variis theorematis de transformatione et determinatione integralium multiplicium” [2]. The resulting (n-1)-fold integrals are

when the space dimension is even or odd, respectively. These are the surface areas of the manifolds called (n-1)-spheres in n-dimensional space. For instance, the 2-sphere is the ordinary surface 4πr2 of a sphere on our 3D space.

Despite the fact that we recognize these as surface areas of hyperspheres, Jacobi used no geometric language in his paper. He was still too early, and mathematicians had not yet woken up to the analogy of extending spatial dimensions beyond 3D.

## Joseph Liouville (1838)

Joseph Liouville’s name is attached to a theorem that lies at the core of mechanical systems—Liouville’s Theorem that proves that volumes in high-dimensional phase space are incompressible. Surprisingly, Liouville had no conception of high dimensional space, to say nothing of abstract phase space. The story of the convoluted path that led Liouville’s name to be attached to his theorem is told in Chapter 6, “The Tangled Tale of Phase Space”, in Galileo Unbound (Oxford University Press, 2018).

Nonetheless, Liouville did publish a pure-mathematics paper in 1838 in Crelle’s Journal [3] that identified an invariant quantity that stayed constant during the differential change of multiple variables when certain criteria were satisfied. It was only later that Jacobi, as he was developing a new mechanical theory based on William R. Hamilton’s work, realized that the criteria needed for Liouville’s invariant quantity to hold were satisfied by conservative mechanical systems. Even then, neither Liouville nor Jacobi used the language of multidimensional geometry, but that was about to change in a quick succession of papers and books by three mathematicians who, unknown to each other, were all thinking along the same lines.

## Arthur Cayley (1843)

Arthur Cayley was the first to take the bold step to call the emerging geometry of multiple variables to be actual space. His seminal paper “Chapters in the Analytic Theory of n-Dimensions” was published in 1843 in the Philosophical Magazine [4]. Here, for the first time, Cayley recognized that the domain of multiple variables behaved identically to multidimensional space. He used little of the language of geometry in the paper, which was mostly analysis rather than geometry, but his bold declaration for spaces of n-dimensions opened the door to a changing mindset that would soon sweep through geometric reasoning.

## Hermann Grassmann (1844)

Grassmann’s life story, although not overly tragic, was beset by lifelong setbacks and frustrations. He was a mathematician literally 30 years ahead of his time, but because he was merely a high-school teacher, no-one took his ideas seriously.

Somehow, in nearly a complete vacuum, disconnected from the professional mathematicians of his day, he devised an entirely new type of algebra that allowed geometric objects to have orientation. These could be combined in numerous different ways obeying numerous different laws. The simplest elements were just numbers, but these could be extended to arbitrary complexity with arbitrary number of elements. He called his theory a theory of “Extension”, and he self-published a thick and difficult tome that contained all of his ideas [5]. He tried to enlist Möbius to help disseminate his ideas, but even Möbius could not recognize what Grassmann had achieved.

In fact, what Grassmann did achieve was vector algebra of arbitrarily high dimension. Perhaps more impressive for the time is that he actually recognized what he was dealing with. He did not know of Cayley’s work, but independently of Cayley he used geometric language for the first time describing geometric objects in high dimensional spaces. He said, “since this method of formation is theoretically applicable without restriction, I can define systems of arbitrarily high level by this method… geometry goes no further, but abstract science knows no limits.” [6]

Grassman was convinced that he had discovered something astonishing and new, which he had, but no one understood him. After years trying to get mathematicians to listen, he finally gave up, left mathematics behind, and actually achieved some fame within his lifetime in the field of linguistics. There is even a law of diachronic linguistics named after him. For the story of Grassmann’s struggles, see the blog on Grassmann and his Wedge Product .

## Julius Plücker (1846)

Projective geometry sounds like it ought to be a simple topic, like the projective property of perspective art as parallel lines draw together and touch at the vanishing point on the horizon of a painting. But it is far more complex than that, and it provided a separate gateway into the geometry of high dimensions.

A hint of its power comes from homogeneous coordinates of the plane. These are used to find where a point in three dimensions intersects a plane (like the plane of an artist’s canvas). Although the point on the plane is in two dimensions, it take three homogeneous coordinates to locate it. By extension, if a point is located in three dimensions, then it has four homogeneous coordinates, as if the three dimensional point were a projection onto 3D from a 4D space.

These ideas were pursued by Julius Plücker as he extended projective geometry from the work of earlier mathematicians such as Desargues and Möbius. For instance, the barycentric coordinates of Möbius are a form of homogeneous coordinates. What Plücker discovered is that space does not need to be defined by a dense set of points, but a dense set of lines can be used just as well. The set of lines is represented as a four-dimensional manifold. Plücker reported his findings in a book in 1846 [7] and expanded on the concepts of multidimensional spaces published in 1868 [8].

## Ludwig Schläfli (1851)

After Plücker, ideas of multidimensional analysis became more common, and Ludwig Schläfli (1814 – 1895), a professor at the University of Berne in Switzerland, was one of the first to fully explore analytic geometry in higher dimensions. He described multidimsnional points that were located on hyperplanes, and he calculated the angles between intersecting hyperplanes [9]. He also investigated high-dimensional polytopes, from which are derived our modern “Schläfli notation“. However, Schläffli used his own terminology for these objects, emphasizing analytic properties without using the ordinary language of high-dimensional geometry.

## Bernhard Riemann (1854)

The person most responsible for the shift in the mindset that finally accepted the geometry of high-dimensional spaces was Bernhard Riemann. In 1854 at the university in Göttingen he presented his habilitation talk “Über die Hypothesen, welche der Geometrie zu Grunde liegen” (Over the hypotheses on which geometry is founded). A habilitation in Germany was an examination that qualified an academic to be able to advise their own students (somewhat like attaining tenure in US universities).

The habilitation candidate would suggest three topics, and it was usual for the first or second to be picked. Riemann’s three topics were: trigonometric properties of functions (he was the first to rigorously prove the convergence properties of Fourier series), aspects of electromagnetic theory, and a throw-away topic that he added at the last minute on the foundations of geometry (on which he had not actually done any serious work). Gauss was his faculty advisor and picked the third topic. Riemann had to develop the topic in a very short time period, starting from scratch. The effort exhausted him mentally and emotionally, and he had to withdraw temporarily from the university to regain his strength. After returning around Easter, he worked furiously for seven weeks to develop a first draft and then asked Gauss to set the examination date. Gauss initially thought to postpone to the Fall semester, but then at the last minute scheduled the talk for the next day. (For the story of Riemann and Gauss, see Chapter 4 “Geometry on my Mind” in the book Galileo Unbound (Oxford, 2018)).

Riemann gave his lecture on 10 June 1854, and it was a masterpiece. He stripped away all the old notions of space and dimensions and imbued geometry with a metric structure that was fundamentally attached to coordinate transformations. He also showed how any set of coordinates could describe space of any dimension, and he generalized ideas of space to include virtually any ordered set of measurables, whether it was of temperature or color or sound or anything else. Most importantly, his new system made explicit what those before him had alluded to: Jacobi, Grassmann, Plücker and Schläfli. Ideas of Riemannian geometry began to percolate through the mathematics world, expanding into common use after Richard Dedekind edited and published Riemann’s habilitation lecture in 1868 [10].

## George Cantor and Dimension Theory (1878)

In discussions of multidimensional spaces, it is important to step back and ask what is dimension? This question is not as easy to answer as it may seem. In fact, in 1878, George Cantor proved that there is a one-to-one mapping of the plane to the line, making it seem that lines and planes are somehow the same. He was so astonished at his own results that he wrote in a letter to his friend Richard Dedekind “I see it, but I don’t believe it!”. A few decades later, Peano and Hilbert showed how to create area-filling curves so that a single continuous curve can approach any point in the plane arbitrarily closely, again casting shadows of doubt on the robustness of dimension. These questions of dimensionality would not be put to rest until the work by Karl Menger around 1926 when he provided a rigorous definition of topological dimension (see the Blog on the History of Fractals).

## Hermann Minkowski and Spacetime (1908)

Most of the earlier work on multidimensional spaces were mathematical and geometric rather than physical. One of the first examples of physical hyperspace is the spacetime of Hermann Minkowski. Although Einstein and Poincaré had noted how space and time were coupled by the Lorentz equations, they did not take the bold step of recognizing space and time as parts of a single manifold. This step was taken in 1908 [11] by Hermann Minkowski who claimed

“Gentlemen! The views of space and time which I wish to lay before you … They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”Herman Minkowski (1908)

For the story of Einstein and Minkowski, see the Blog on Minkowski’s Spacetime: The Theory that Einstein Overlooked.

## Felix Hausdorff and Fractals (1918)

No story of multiple “integer” dimensions can be complete without mentioning the existence of “fractional” dimensions, also known as fractals. The individual who is most responsible for the concepts and mathematics of fractional dimensions was Felix Hausdorff. Before being compelled to commit suicide by being jewish in Nazi Germany, he was a leading light in the intellectual life of Leipzig, Germany. By day he was a brilliant mathematician, by night he was the author Paul Mongré writing poetry and plays.

In 1918, as the war was ending, he wrote a small book “Dimension and Outer Measure” that established ways to construct sets whose measured dimensions were fractions rather than integers [12]. Benoit Mandelbrot would later popularize these sets as “fractals” in the 1980’s. For the background on a history of fractals, see the Blog A Short History of Fractals.

## The Fifth Dimension of Theodore Kaluza (1921) and Oskar Klein (1926)

The first theoretical steps to develop a theory of a physical hyperspace (in contrast to merely a geometric hyperspace) were taken by Theodore Kaluza at the University of Königsberg in Prussia. He added an additional spatial dimension to Minkowski spacetime as an attempt to unify the forces of gravity with the forces of electromagnetism. Kaluza’s paper was communicated to the journal of the Prussian Academy of Science in 1921 through Einstein who saw the unification principles as a parallel of some of his own attempts [13]. However, Kaluza’s theory was fully classical and did not include the new quantum theory that was developing at that time in the hands of Heisenberg, Bohr and Born.

Oskar Klein was a Swedish physicist who was in the “second wave” of quantum physicists having studied under Bohr. Unaware of Kaluza’s work, Klein developed a quantum theory of a five-dimensional spacetime [14]. For the theory to be self-consistent, it was necessary to roll up the extra dimension into a tight cylinder. This is like a strand a spaghetti—looking at it from far away it looks like a one-dimensional string, but an ant crawling on the spaghetti can move in two dimensions—along the long direction, or looping around it in the short direction called a compact dimension. Klein’s theory was an early attempt at what would later be called string theory. For the historical background on Kaluza and Klein, see the Blog on Oskar Klein.

## John Campbell (1931): Hyperspace in Science Fiction

Art has a long history of shadowing the sciences, and the math and science of hyperspace was no exception. One of the first mentions of hyperspace in science fiction was in the story “Islands in Space’, by John Campbell [15], published in the Amazing Stories quarterly in 1931, where it was used as an extraordinary means of space travel.

In 1951, Isaac Asimov made travel through hyperspace the transportation network that connected the galaxy in his Foundation Trilogy [16].

## John von Neumann and Hilbert Space (1932)

Quantum mechanics had developed rapidly through the 1920’s, but by the early 1930’s it was in need of an overhaul, having outstripped rigorous mathematical underpinnings. These underpinnings were provided by John von Neumann in his 1932 book on quantum theory [17]. This is the book that cemented the Copenhagen interpretation of quantum mechanics, with projection measurements and wave function collapse, while also establishing the formalism of Hilbert space.

Hilbert space is an infinite dimensional vector space of orthogonal eigenfunctions into which any quantum wave function can be decomposed. The physicists of today work and sleep in Hilbert space as their natural environment, often losing sight of its infinite dimensions that don’t seem to bother anyone. Hilbert space is more than a mere geometrical space, but less than a full physical space (like five-dimensional spacetime). Few realize that what is so often ascribed to Hilbert was actually formalized by von Neumann, among his many other accomplishments like stored-program computers and game theory.

## Einstein-Rosen Bridge (1935)

One of the strangest entities inhabiting the theory of spacetime is the Einstein-Rosen Bridge. It is space folded back on itself in a way that punches a short-cut through spacetime. Einstein, working with his collaborator Nathan Rosen at Princeton’s Institute for Advanced Study, published a paper in 1935 that attempted to solve two problems [18]. The first problem was the Schwarzschild singularity at a radius r = 2M/c2 known as the Schwarzschild radius or the Event Horizon. Einstein had a distaste for such singularities in physical theory and viewed them as a problem. The second problem was how to apply the theory of general relativity (GR) to point masses like an electron. Again, the GR solution to an electron blows up at the location of the particle at r = 0.

To eliminate both problems, Einstein and Rosen (ER) began with the Schwarzschild metric in its usual form

where it is easy to see that it “blows up” when r = 2M/c2 as well as at r = 0. ER realized that they could write a new form that bypasses the singularities using the simple coordinate substitution

to yield the “wormhole” metric

It is easy to see that as the new variable u goes from -inf to +inf that this expression never blows up. The reason is simple—it removes the 1/r singularity by replacing it with 1/(r + ε). Such tricks are used routinely today in computational physics to keep computer calculations from getting too large—avoiding the divide-by-zero problem. It is also known as a form of regularization in machine learning applications. But in the hands of Einstein, this simple “bypass” is not just math, it can provide a physical solution.

It is hard to imagine that an article published in the Physical Review, especially one written about a simple variable substitution, would appear on the front page of the New York Times, even appearing “above the fold”, but such was Einstein’s fame this is exactly the response when he and Rosen published their paper. The reason for the interest was because of the interpretation of the new equation—when visualized geometrically, it was like a funnel between two separated Minkowski spaces—in other words, what was named a “wormhole” by John Wheeler in 1957. Even back in 1935, there was some sense that this new property of space might allow untold possibilities, perhaps even a form of travel through such a short cut.

As it turns out, the ER wormhole is not stable—it collapses on itself in an incredibly short time so that not even photons can get through it in time. More recent work on wormholes have shown that it can be stabilized by negative energy density, but ordinary matter cannot have negative energy density. On the other hand, the Casimir effect might have a type of negative energy density, which raises some interesting questions about quantum mechanics and the ER bridge.

## Edward Witten’s 10+1 Dimensions (1995)

A history of hyperspace would not be complete without a mention of string theory and Edward Witten’s unification of the variously different 10-dimensional string theories into 10- or 11-dimensional M-theory. At a string theory conference at USC in 1995 he pointed out that the 5 different string theories of the day were all related through dualities. This observation launched the second superstring revolution that continues today. In this theory, 6 extra spatial dimensions are wrapped up into complex manifolds such as the Calabi-Yau manifold.

## Prospects

There is definitely something wrong with our three-plus-one dimensions of spacetime. We claim that we have achieved the pinnacle of fundamental physics with what is called the Standard Model and the Higgs boson, but dark energy and dark matter loom as giant white elephants in the room. They are giant, gaping, embarrassing and currently unsolved. By some estimates, the fraction of the energy density of the universe comprised of ordinary matter is only 5%. The other 95% is in some form unknown to physics. How can physicists claim to know anything if 95% of everything is in some unknown form?

The answer, perhaps to be uncovered sometime in this century, may be the role of extra dimensions in physical phenomena—probably not in every-day phenomena, and maybe not even in high-energy particles—but in the grand expanse of the cosmos.

By David D. Nolte, Feb. 8, 2023

## Bibliography:

M. Kaku, R. O’Keefe, Hyperspace: A scientific odyssey through parallel universes, time warps, and the tenth dimension.  (Oxford University Press, New York, 1994).

A. N. Kolmogorov, A. P. Yushkevich, Mathematics of the 19th century: Geometry, analytic function theory.  (Birkhäuser Verlag, Basel ; 1996).

## References:

[1] F. Möbius, in Möbius, F. Gesammelte Werke,, D. M. Saendig, Ed. (oHG, Wiesbaden, Germany, 1967), vol. 1, pp. 36-49.

[2] Carl Jacobi, “De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant; una cum variis theorematis de transformatione et determinatione integralium multiplicium” (1834)

[3] J. Liouville, Note sur la théorie de la variation des constantes arbitraires. Liouville Journal 3, 342-349 (1838).

[4] A. Cayley, Chapters in the analytical geometry of n dimensions. Collected Mathematical Papers 1, 317-326, 119-127 (1843).

[5] H. Grassmann, Die lineale Ausdehnungslehre.  (Wiegand, Leipzig, 1844).

[6] H. Grassmann quoted in D. D. Nolte, Galileo Unbound (Oxford University Press, 2018) pg. 105

[7] J. Plücker, System der Geometrie des Raumes in Neuer Analytischer Behandlungsweise, Insbesondere de Flächen Sweiter Ordnung und Klasse Enthaltend.  (Düsseldorf, 1846).

[8] J. Plücker, On a New Geometry of Space (1868).

[9] L. Schläfli, J. H. Graf, Theorie der vielfachen Kontinuität. Neue Denkschriften der Allgemeinen Schweizerischen Gesellschaft für die Gesammten Naturwissenschaften 38. ([s.n.], Zürich, 1901).

[10] B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, Habilitationsvortrag. Göttinger Abhandlung 13,  (1854).

[11] Minkowski, H. (1909). “Raum und Zeit.” Jahresbericht der Deutschen Mathematikier-Vereinigung: 75-88.

[12] Hausdorff, F.(1919).“Dimension und ausseres Mass,”Mathematische Annalen, 79: 157–79.

[13] Kaluza, Theodor (1921). “Zum Unitätsproblem in der Physik”. Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.): 966–972

[14] Klein, O. (1926). “Quantentheorie und fünfdimensionale Relativitätstheorie“. Zeitschrift für Physik. 37 (12): 895

[15] John W. Campbell, Jr. “Islands of Space“, Amazing Stories Quarterly (1931)

[16] Isaac Asimov, Foundation (Gnome Press, 1951)

[17] J. von Neumann, Mathematical Foundations of Quantum Mechanics.  (Princeton University Press, ed. 1996, 1932).

[18] A. Einstein and N. Rosen, “The Particle Problem in the General Theory of Relativity,” Phys. Rev. 48(73) (1935).

# Hermann Grassmann’s Nimble Wedge Product

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 20th 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, …, xn] 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.