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.

August Möbius (1790 – 1868). Image.

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.

Carl Gustav Jacob Jacobi (1804 – 1851)

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).

Joseph Liouville (1809 – 1882)

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.

Facsimile of Liouville’s 1838 paper on invariants

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.

Arthur Cayley (1821 – 1895). Image

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 .

Hermann Grassmann (1809 – 1877).

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].

Julius Plücker (1801 – 1868).

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.

Some of the polytopes studied by Schläfli.

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].

Bernhard Riemann (1826 – 1866). Image.

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).

Area-filling curves by Peano and Hilbert.

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.

Facsimile of Minkowski’s 1908 publication on spacetime.

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.

Felix Hausdorff (1868 – 1942)
Example of a fractal set with embedding dimension DE = 2, topological dimension DT = 1, and fractal dimension DH = 1.585.

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.

The wave equations of Klein-Gordon, Schrödinger and Dirac.

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].

Testez-vous : Isaac Asimov avait-il (entièrement) raison ? - Sciences et  Avenir
Isaac Asimov (1920 – 1992)

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.

John von Neumann (1903 – 1957). Image Credits.

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.

Einstein-Rosen Bridge. Image.

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.

Two-dimensional slice of a six-dimensional Calabi-Yau quintic manifold.


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


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).


[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).

New from Oxford Press: The History of Optical Interferometry (Late Summer 2023)

A Short History of Fractal Dimension

It is second nature to think of integer dimensions:  A line is one dimensional.  A plane is two dimensional. A volume is three dimensional.  A point has no dimensions.

It is harder to think in four dimensions and higher, but even here it is a simple extrapolation of lower dimensions.  Consider the basis vectors spanning a three-dimensional space consisting of the triples of numbers

Then a four dimensional hyperspace is just created by adding a new “tuple” to the list

and so on to 5 and 6 dimensions and on.  Child’s play!

But how do you think of fractional dimensions?  What is a fractional dimension?  For that matter, what is a dimension?  Even the integer dimensions began to unravel when George Cantor showed in 1877 that the line and the plane, which clearly had different “dimensionalities”, both had the same cardinality and could be put into a one-to-one correspondence.  From then onward the concept of dimension had to be rebuilt from the ground up, leading ultimately to fractals.

Here is a short history of fractal dimension, partially excerpted from my history of dynamics in Galileo Unbound (Oxford University Press, 2018) pg. 110 ff.  This blog page presents the history through a set of publications that successively altered how mathematicians thought about curves in spaces, beginning with Karl Weierstrass in 1872.

Karl Weierstrass (1872)

Karl Weierstrass (1815 – 1897) was studying convergence properties of infinite power series in 1872 when he began with a problem that Bernhard Riemann had given to his students some years earlier.  Riemann had asked whether the function

was continuous everywhere but not differentiable.  This simple question about a simple series was surprisingly hard to answer (it was not solved until Hardy provided the proof in 1916 [1]).  Therefore, Weierstrass conceived of a simpler infinite sum that was continuous everywhere and for which he could calculate left and right limits of derivatives at any point.  This function is

where b is a large odd integer and a is positive and less than one.  Weierstrass showed that the left and right derivatives failed to converge to the same value, no matter where he took his point.  In short, he had discovered a function that was continuous everywhere, but had a derivative nowhere [2].  This pathological function, called a “Monster” by Charles Hermite, is now called the Weierstrass function.

Beyond the strange properties that Weierstrass sought, the Weierstrass function would turn out to be a fractal curve (recognized much later by Besicovitch and Ursell in 1937 [3]) with a fractal (Hausdorff) dimension given by

although this was not proven until very recently [4].  An example of the function is shown in Fig. 1 for a = 0.5 and b = 5.  This specific curve has a fractal dimension D = 1.5693.  Notably, this is a number that is greater than 1 dimension (the topological dimension of the curve) but smaller than 2 dimensions (the embedding dimension of the curve).  The curve tends to fill more of the two dimensional plane than a straight line, so its intermediate fractal dimension has an intuitive feel about it.  The more “monstrous” the curve looks, the closer its fractal dimension approaches 2.

Fig. 1  Weierstrass’ “Monster” (1872) with a = 0.5, b = 5.  This continuous function is nowhere differentiable.  It is a fractal with fractal dimension D = 2 + ln(0.5)/ln(5) = 1.5693.

Georg Cantor (1883)

Partially inspired by Weierstrass’ discovery, George Cantor (1845 – 1918) published an example of an unusual ternary set in 1883 in “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (“Foundations of a General Theory of Aggregates”) [5].  The set generates a function (The Cantor Staircase) that has a derivative equal to zero almost everywhere, yet whose area integrates to unity.  It is a striking example of a function that is not equal to the integral of its derivative!  Cantor demonstrated that the size of his set is aleph0 , which is the cardinality of the real numbers.  But whereas the real numbers are uniformly distributed, Cantor’s set is “clumped”.  This clumpiness is an essential feature that distinguishes it from the one-dimensional number line, and it raised important questions about dimensionality. The fractal dimension of the ternary Cantor set is DH = ln(2)/ln(3) = 0.6309.

Fig. 2  The 1883 Cantor set (below) and the Cantor staircase (above, as the indefinite integral over the set).

Giuseppe Peano (1890)

In 1878, in a letter to his friend Richard Dedekind, Cantor showed that there was a one-to-one correspondence between the real numbers and the points in any n-dimensional space.  He was so surprised by his own result that he wrote to Dedekind “I see it, but I don’t believe it.”  The solid concepts of dimension and dimensionality were dissolving before his eyes.  What does it mean to trace the path of a trajectory in an n-dimensional space, if all the points in n dimensions were just numbers on a line?  What could such a trajectory look like?  A graphic example of a plane-filling path was constructed in 1890 by Peano [6], who was a peripatetic mathematician with interests that wandered broadly across the landscape of the mathematical problems of his day—usually ahead of his time.  Only two years after he had axiomatized linear vector spaces [7], Peano constructed a continuous curve that filled space. 

The construction of Peano’s curve proceeds by taking a square and dividing it into 9 equal sub squares.  Lines connect the centers of each of the sub squares.  Then each sub square is divided again into 9 sub squares whose centers are all connected by lines.  At this stage, the original pattern, repeated 9 times, is connected together by 8 links, forming a single curve.  This process is repeated infinitely many times, resulting in a curve that passes through every point of the original plane square.  In this way, a line is made to fill a plane.  Where Cantor had proven abstractly that the cardinality of the real numbers was the same as the points in n-dimensional space, Peano created a specific example.  This was followed quickly by another construction, invented by David Hilbert in 1891, that divided the square into four instead of nine, simplifying the construction, but also showing that such constructions were easily generated.

Fig. 3 Peano’s (1890) and Hilbert’s (1891) plane-filling curves.  When the iterations are taken to infinity, the curves approach every point of two-dimensional space arbitrarily closely, giving them a dimension DH = DE = 2, although their topological dimensions are DT = 1.

Helge von Koch (1904)

The space-filling curves of Peano and Hilbert have the extreme property that a one-dimensional curve approaches every point in a two-dimensional space.  This ability of a one-dimensional trajectory to fill space mirrored the ergodic hypothesis that Boltzmann relied upon as he developed statistical mechanics.  These examples by Peano, Hilbert and Boltzmann inspired searches for continuous curves whose dimensionality similarly exceeded one dimension, yet without filling space.  Weierstrass’ Monster was already one such curve, existing in some dimension greater than one but not filling the plane.  The construction of the Monster required infinite series of harmonic functions, and the resulting curve was single valued on its domain of real numbers. 

An alternative approach was proposed by Helge von Koch (1870—1924), a Swedish mathematician with an interest in number theory.  He suggested in 1904 that a set of straight line segments could be joined together, and then shrunk by a scale factor to act as new segments of the original pattern [8].  The construction of the Koch curve is shown in Fig. 4.  When the process is taken to its limit, it produces a curve, differentiable nowhere, which snakes through two dimensions.  When connected with other identical curves into a hexagon, the curve resembles a snowflake, and the construction is known as “Koch’s Snowflake”. 

The Koch curve begins in generation 1 with N0 = 4 elements.  These are shrunk by a factor of b = 1/3 to become the four elements of the next generation, and so on.  The number of elements varies with the observation scale according to the equation

where D is called the fractal dimension.  In the example of the Koch curve, the fractal dimension is

which is a number less than its embedding dimenion DE = 2.  The fractal is embedded in 2D but has a fractional dimension that is greater than it topological dimension DT = 1.

Fig. 4  Generation of a Koch curve (1904).  The fractal dimension is D = ln(4)/ln(3) = 1.26.  At each stage, four elements are reduced in size by a factor of 3.  The “length” of the curve approaches infinity as the features get smaller and smaller.  But the scaling of the length with size is determined uniquely by the fractal dimension.

Waclaw Sierpinski (1915)

Waclaw Sierpinski (1882 – 1969) was a Polish mathematician studying at the Jagellonian University in Krakow for his doctorate when he came across a theorem that every point in the plane can be defined by a single coordinate.  Intrigued by such an unintuitive result, he dived deep into Cantor’s set theory after he was appointed as a faculty member at the university in Lvov.  He began to construct curves that had more specific properties than the Peano or Hilbert curves, such as a curve that passes through every interior point of a unit square but that encloses an area that is only equal to 5/12 = 0.4167.  Sierpinski became interested in the topological properties of such sets.

Sierpinski considered how to define a curve that was embedded in DE = 2 but that was NOT constructed as a topological dimension DT = 1 curve as the curves of Peano, Hilbert, Koch (and even his own) had been.  To demonstrate this point, he described a construction that began with a topological dimension DT = 2 object, a planar triangle, from which the open set of its central inverted triangle is removed, leaving its boundary points.  The process is continued iteratively to all scales [9].  The resulting point set is shown in Fig. 5 and is called the Sierpinski gasket.  What is left after all the internal triangles are removed is a point set that can be made discontinuous by cutting it at a finite set of points.  This is shown in Fig. 5 by the red circles.  Each circle, no matter the size, cuts the set at three points, making the resulting set discontinuous.  Ten years later, Karl Menger would show that this property of discontinuous cuts determined the topological dimension of the Sierpinski gasket to be DT = 1.  The embedding dimension is of course DE = 2, and the fractal dimension of the Sierpinski gasket is

Fig. 5 The Sierpinski gasket.  The central triangle is removed (leaving its boundary) at each scale.  The pattern is self-similar with a fractal dimension DH = 1.5850.  Unintuitively, it has a topological dimension DT = 1.

Felix Hausdorff (1918)

The work by Cantor, Peano, von Koch and Sierpinski had created a crisis in geometry as mathematicians struggled to rescue concepts of dimensionality.  An important byproduct of that struggle was a much deeper understanding of concepts of space, especially in the hands of Felix Hausdorff. 

Felix Hausdorff (1868 – 1942) was born in Breslau, Prussia, and educated in Leipzig.  In his early years as a doctoral student, and as an assistant professor at Leipzig, he was a practicing mathematician by day and a philosopher and playwright by night, publishing under the pseudonym Paul Mongré.  He was at the University of Bonn working on set theory when the Greek mathematician Constatin Carathéodory published a paper in 1914 that showed how to construct a p-dimensional set in a q-dimensional space [9].  Haussdorff realized that he could apply similar ideas to the Cantor set.  He showed that the outer measure of the Cantor set would go discontinuously from zero to infinity as the fractional dimension increased smoothly.  The critical value where the measure changed its character became known as the Hausdorff dimension [11]. 

For the Cantor ternary set, the Hausdorff dimension is exactly DH = ln(2)/ln(3) = 0.6309.  This value for the dimension is less than the embedding dimension DE = 1 of the support (the real numbers on the interval [0, 1]), but it is also greater than DT = 0 which would hold for a countable number of points on the interval.  The work by Hausdorff became well known in the mathematics community who applied the idea to a broad range of point sets like Weierstrass’s monster and the Koch curve.

It is important to keep a perspective of what Hausdorff’s work meant during which period of time.  For instance, although the curves of Weierstrass, von Koch and Sierpinski were understood to present a challenge to concepts of dimension, it was only after Haussdorff that mathematicians began to think in terms of fractional dimensions and to calculate the fractional dimensions of these earlier point sets.  Despite the fact that Sierpinski created one of the most iconic fractals that we use as an example every day, he was unaware at the time that he was doing so.  His interest was topological—creating a curve for which any cut at any point would create disconnected subsets starting with objects (triangles) with topological dimension DT = 2.  In this way, talking about the early fractal objects tends to be anachronistic, using language to describe them that had not yet been invented at that time.

This perspective is also true for the ideas of topological dimension.  For instance, even Sierpinski was not fully tuned into the problems of defining topological dimension.  It turns out that what he created was a curve of topological dimension DT = 1, but that would only become clear later with the work of the Austrian mathematician Karl Menger.

Karl Menger (1926)

The day that Karl Menger (1902 – 1985) was born, his father, Carl Menger (1840 – 1941) lost his job.  Carl Menger was one of the founders of the famous Viennese school that established the marginalist view of economics.  However, Carl was not married to Karl’s mother, which was frowned upon by polite Vienna society, so he had to relinquish his professorship.  Despite his father’s reduction in status, Karl received an excellent education at a Viennese gymnasium (high school).  Among of his classmates were Wolfgang Pauli (Nobel Prize for Physics in 1945)  and Richard Kuhn (Nobel Prize for Chemistry in 1938).  When Karl began attending the University of Vienna he studied physics, but the mathematics professor Hans Hahn opened his eyes to the fascinating work on analysis that was transforming mathematics at that time, so Karl shifted his studies to mathematical analysis, specifically concerning conceptions of “curves”. 

Menger made important contributions to the history of fractal dimension as well as the history of topological dimension.  In his approach to defining the intrinsic topological dimension of a point set, he described the construction of a point set embedded in three dimensions that had zero volume, an infinite surface area, and a fractal dimension between 2 and 3.  The object is shown in Fig. 6 and is called a Menger “sponge” [12].  The Menger sponge is a fractal with a fractal dimension DH = ln(20)/ln(3) = 2.7268.  The face of the sponge is also known as the Sierpinski carpt.  The fractal dimension of the Sierpinski carpet is DH = ln(8)/ln(3) = 1.8928.

Fig. 6 Menger Sponge. Embedding dimension DE = 3. Fractal dimension DH = ln(20)/ln(3) = 2.7268. Topological dimension DT = 1: all one-dimensional metric spaces can be contained within the Menger sponge point set. Each face is a Sierpinski carpet with fractal dimension DH = ln(8)/ln(3) = 1.8928.

The striking feature of the Menger sponge is its topological dimension.  Menger created a new definition of topological dimension that partially solved the crises created by Cantor when he showed that every point on the unit square can be defined by a single coordinate.  This had put a one dimensional curve in one-to-one correspondence with a two-dimensional plane.  Yet the topology of a 2-dimensional object is clearly different than the topology of a line.  Menger found a simple definition that showed why 2D is different, topologically, than 3D, despite Cantor’s conundrum.  The answer came from the idea of making cuts on a point set and seeing if the cut created disconnected subsets. 

As a simple example, take a 1D line.  The removal of a single point creates two disconnected sub-lines.  The intersection of the cut with the line is 0-dimensional, and Menger showed that this defined the line as 1-dimensional.  Similarly, a line cuts the unit square into to parts.  The intersection of the cut with the plane is 1-dimensional, signifying that the dimension of the plane is 2-dimensional.  In other words, a (n-1) dimensional intersection of the boundary of a small neighborhood with the point set indicates that the point set has a dimension of n.  Generalizing this idea, looking at the Sierpinski gasket in Fig. 5, the boundary of a small circular region, if placed appropriately (as in the figure), intersects the Sierpinski gasket at three points of dimension zero.  Hence, the topological dimension of the Sierpinski gasket is one-dimensional.  Manger was likewise able to show that his sponge also had a topology that was one-dimensional, DT = 1, despite the embedding dimension of DE = 3.  In fact, all 1-dimensional metric spaces can be fit inside a Menger Sponge.

Benoit Mandelbrot (1967)

Benoit Mandelbrot (1924 – 2010) was born in Warsaw and his family emigrated to Paris in 1935.  He attended the Ecole Polytechnique where he studied under Gaston Julia (1893 – 1978) and Paul Levy (1886 – 1971).  Both Julia and Levy made significant contributions to the field of self-similar point sets and made a lasting impression on Mandelbrot.  He went to Cal Tech for a master’s degree in aeronautics and then a PhD in mathematical sciences from the University of Paris.  In 1958 Mandelbrot joined the research staff of the IBM Thomas J. Watson Research Center in Yorktown Heights, New York where he worked for over 35 years on topics of information theory and economics, always with a view of properties of self-similar sets and time series.

In 1967 Mandelbrot published one of his early important papers on the self-similar properties of the coastline of Britain.  He proposed that many natural features had statistical self similarity, which he applied to coastlines.  He published the work as “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension” [13] in Science magazine , where he showed that the length of the coastline diverged with a Hausdorff dimension equal to D = 1.25.  Working at IBM, a world leader in computers, he had ready access to their power as well as their visualization capabilities.  Therefore, he was one of the first to begin exploring the graphical character of self-similar maps and point sets.

During one of his sabbaticals at Harvard University he began exploring the properties of Julia sets (named after his former teacher at the Ecole Polytechnique).  The Julia set is a self-similar point set that is easily visualized in the complex plane (two dimensions).  As Mandelbrot studied the convergence of divergence of infinite series defined by the Julia mapping, he discovered an infinitely nested pattern that was both beautiful and complex.  This has since become known as the Mandelbrot set.

Fig. 7 Mandelbrot set.

Later, in 1975, Mandelbrot coined the term fractal to describe these self-similar point sets, and he began to realize that these types of sets were ubiquitous in nature, ranging from the structure of trees and drainage basins, to the patterns of clouds and mountain landscapes.  He published his highly successful and influential book The Fractal Geometry of Nature in 1982, introducing fractals to the wider public and launching a generation of hobbyists interested in computer-generated fractals.  The rise of fractal geometry coincided with the rise of chaos theory that was aided by the same computing power.  For instance, important geometric structures of chaos theory, known as strange attractors, have fractal geometry. 

By David D. Nolte, Dec. 26, 2020

Appendix:  Box Counting

When confronted by a fractal of unknown structure, one of the simplest methods to find the fractal dimension is through box counting.  This method is shown in Fig. 8.  The fractal set is covered by a set of boxes of size b, and the number of boxes that contain at least one point of the fractal set are counted.  As the boxes are reduced in size, the number of covering boxes increases as 

To be numerically accurate, this method must be iterated over several orders of magnitude.  The number of boxes covering a fractal has this characteristic power law dependence, as shown in Fig. 8, and the fractal dimension is obtained as the slope.

Fig. 8  Calculation of the fractal dimension using box counting.  At each generation, the size of the grid is reduced by a factor of 3.  The number of boxes that contain some part of the fractal curve increases as  , where b is the scale


[1] Hardy, G. (1916). “Weierstrass’s non-differentiable function.” Transactions of the American Mathematical Society 17: 301-325.

[2] Weierstrass, K. (1872). “Uber continuirliche Functionen eines reellen Arguments, die fur keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen.” Communication ri I’Academie Royale des Sciences II: 71-74.

[3] Besicovitch, A. S. and H. D. Ursell (1937). “Sets of fractional dimensions: On dimensional numbers of some continuous curves.” J. London Math. Soc. 1(1): 18-25.

[4] Shen, W. (2018). “Hausdorff dimension of the graphs of the classical Weierstrass functions.” Mathematische Zeitschrift. 289(1–2): 223–266.

[5] Cantor, G. (1883). Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Leipzig, B. G. Teubner.

[6] Peano, G. (1890). “Sur une courbe qui remplit toute une aire plane.” Mathematische Annalen 36: 157-160.

[7] Peano, G. (1888). Calcolo geometrico secundo l’Ausdehnungslehre di H. Grassmann e precedutto dalle operazioni della logica deduttiva. Turin, Fratelli Bocca Editori.

[8] Von Koch, H. (1904). “Sur.une courbe continue sans tangente obtenue par une construction geometrique elementaire.” Arkiv for Mathematik, Astronomi och Fysich 1: 681-704.

[9] Sierpinski, W. (1915). “Sur une courbe dont tout point est un point de ramification.” Comptes Rendus Hebdomadaires des Seances de l’Academie des Sciences de Paris 160: 302-305.

[10] Carathéodory, C. (1914). “Über das lineare Mass von Punktmengen – eine Verallgemeinerung des Längenbegriffs.” Gött. Nachr. IV: 404–406.

[11] Hausdorff, F. (1919). “Dimension und ausseres Mass.” Mathematische Anna/en 79: 157-179.

[12] Menger, Karl (1926), “Allgemeine Räume und Cartesische Räume. I.”, Communications to the Amsterdam Academy of Sciences. English translation reprinted in Edgar, Gerald A., ed. (2004), Classics on fractals, Studies in Nonlinearity, Westview Press. Advanced Book Program, Boulder, CO

[13] B Mandelbrot, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science, 156 3775 (May 5, 1967): 636-638.