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Tuesday, January 31, 2023

A Mathematician's Guided Tour Through Higher Dimensions

The notion of dimension at first seems intuitive. Glancing out the window we might see a crow sitting atop a cramped flagpole experiencing zero dimensions, a robin on a telephone wire constrained to one, a pigeon on the ground free to move in two and an eagle in the air enjoying three.

But as we’ll see, finding an explicit definition for the concept of dimension and pushing its boundaries has proved exceptionally difficult for mathematicians. It’s taken hundreds of years of thought experiments and imaginative comparisons to arrive at our current rigorous understanding of the concept.

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The ancients knew that we live in three dimensions. Aristotle wrote, “Of magnitude that which (extends) one way is a line, that which (extends) two ways is a plane, and that which (extends) three ways a body. And there is no magnitude besides these, because the dimensions are all that there are.”

Yet mathematicians, among others, have enjoyed the mental exercise of imagining more dimensions. What would a fourth dimension—somehow perpendicular to our three—look like?

One popular approach: Suppose our knowable universe is a two-dimensional plane in three-dimensional space. A solid ball hovering above the plane is invisible to us. But if it falls and contacts the plane, a dot appears. As it continues through the plane, a circular disk grows until it reaches its maximum size. It then shrinks and disappears. It is through these cross sections that we see three dimensional shapes.

Similarly, in our familiar three-dimensional universe, if a four-dimensional ball were to pass through it would appear as a point, grow into a solid ball, eventually reach its full radius, then shrink and disappear. This gives us a sense of the four-dimensional shape, but there are other ways of thinking about such figures.

For example, let’s try visualizing the four-dimensional equivalent of a cube, known as a tesseract, by building up to it. If we begin with a point, we can sweep it in one direction to obtain a line segment. When we sweep the segment in a perpendicular direction, we obtain a square. Dragging this square in a third perpendicular direction yields a cube. Likewise, we obtain a tesseract by sweeping the cube in a fourth direction.

Alternatively, just as we can unfold the faces of a cube into six squares, we can unfold the three-dimensional boundary of a tesseract to obtain eight cubes, as Salvador Dalí showcased in his 1954 painting Crucifixion (Corpus Hypercubus).

This all adds up to an intuitive understanding that an abstract space is n-dimensional if there are n degrees of freedom within it (as those birds had), or if it requires n coordinates to describe the location of a point. Yet, as we shall see, mathematicians discovered that dimension is more complex than these simplistic descriptions imply.

The formal study of higher dimensions emerged in the 19th century and became quite sophisticated within decades: A 1911 bibliography contained 1,832 references to the geometry of n dimensions. Perhaps as a consequence, in the late 19th and early 20th centuries, the public became infatuated with the fourth dimension. In 1884, Edwin Abbott wrote the popular satirical novel Flatland, which used two-dimensional beings encountering a character from the third dimension as an analogy to help readers comprehend the fourth dimension. A 1909 Scientific American essay contest entitled “What Is the Fourth Dimension?” received 245 submissions vying for a $500 prize. And many artists, like Pablo Picasso and Marcel Duchamp, incorporated ideas of the fourth dimension into their work.

But during this time, mathematicians realized that the lack of a formal definition for dimension was actually a problem.

Georg Cantor is best known for his discovery that infinity comes in different sizes, or cardinalities. At first Cantor believed that the set of dots in a line segment, a square and a cube must have different cardinalities, just as a line of 10 dots, a 10 × 10 grid of dots and a 10 × 10 × 10 cube of dots have different numbers of dots. However, in 1877 he discovered a one-to-one correspondence between points in a line segment and points in a square (and likewise cubes of all dimensions), showing that they have the same cardinality. Intuitively, he proved that lines, squares and cubes all have the same number of infinitesimally small points, despite their different dimensions. Cantor wrote to Richard Dedekind, “I see it, but I do not believe it.”

Cantor realized this discovery threatened the intuitive idea that n-dimensional space requires n coordinates, because each point in an n-dimensional cube can be uniquely identified by one number from an interval, so that, in a sense, these high-dimensional cubes are equivalent to a one-dimensional line segment. However, as Dedekind pointed out, Cantor’s function was highly discontinuous—it essentially broke apart a line segment into infinitely many parts and reassembled them to form a cube. This is not the behavior we would want for a coordinate system; it would be too disordered to be helpful, like giving buildings in Manhattan unique addresses but assigning them at random.

Then, in 1890, Giuseppe Peano discovered that it is possible to wrap a one-dimensional curve so tightly—and continuously—that it fills every point in a two-dimensional square. This was the first space-filling curve. But Peano’s example was also not a good basis for a coordinate system because the curve intersected itself infinitely many times; returning to the Manhattan analogy, it was like giving some buildings multiple addresses.

These and other surprising examples made it clear that mathematicians needed to prove that dimension is a real notion and that, for instance, n- and m-dimension Euclidean spaces are different in some fundamental way when n ≠ m. This objective became known as the “invariance of dimension” problem.

Finally, in 1912, almost half a century after Cantor’s discovery, and after many failed attempts to prove the invariance of dimension, L.E.J. Brouwer succeeded by employing some methods of his own creation. In essence, he proved that it is impossible to put a higher-dimensional object inside one of smaller dimension, or to place one of smaller dimension into one of larger dimension and fill the entire space, without breaking the object into many pieces, as Cantor did, or allowing it to intersect itself, as Peano did. Moreover, around this time Brouwer and others gave a variety of rigorous definitions, which, for example, could assign dimension inductively based on the fact that the boundaries of balls in n-dimensional space are (n − 1)-dimensional.

Although Brouwer’s work put the notion of dimension on strong mathematical footing, it did not help with our intuition regarding higher-dimensional spaces: Our familiarity with three-dimensional space too easily leads us astray. As Thomas Banchoff wrote, “All of us are slaves to the prejudices of our own dimension.”

Suppose, for instance, we place 2*n* spheres of radius 1 inside an n-dimensional cube with side length 4, and then put another one in the center tangent to them all. As n grows, so does the size of the central sphere—it has a radius of n‾√ − 1. Thus, shockingly, when n ≥ 10 this sphere protrudes beyond the sides of the cube.

The surprising realities of high-dimensional space cause problems in statistics and data analysis, known collectively as the “curse of dimensionality.” The number of sample points required for many statistical techniques goes up exponentially with the dimension. Also, as dimensions increase, points will cluster together less often. Thus, it’s often important to find ways to reduce the dimension of high-dimensional data.

The story of dimension didn’t end with Brouwer. Just a few years afterward, Felix Hausdorff developed a definition of dimension that—generations later—proved essential for modern math. An intuitive way to think about Hausdorff dimension is that if we scale, or magnify, a d-dimensional object uniformly by a factor of k, the size of the object increases by a factor of kd. Suppose we scale a point, a line segment, a square and a cube by a factor of 3. The point does not change size (30 = 1), the segment becomes three times as large (31 = 3), the square becomes nine times as large (32 = 9) and the cube becomes 27 times as large (33 = 27).

One surprising consequence of Hausdorff’s definition is that objects could have non-integer dimensions. Decades later, this turned out to be just what Benoit B. Mandelbrot needed when he asked, “How long is the coast of Britain?” A coastline can be so jagged that it cannot be measured precisely with any ruler—the shorter the ruler, the larger and more precise the measurement. Mandelbrot argued that the Hausdorff dimension provides a way to quantify this jaggedness, and in 1975 he coined the term “fractal” to describe such infinitely complex shapes.

To understand what a non-integer dimension might look like, let’s consider the Koch curve, which is produced iteratively. We begin with a line segment. At each stage we remove the middle third of each segment and replace it with two segments equal in length to the removed segment. Repeat this procedure indefinitely to obtain the Koch curve. Study it closely, and you’ll see it contains four sections that are identical to the whole curve but are one-third the size. So if we scale this curve by a factor of 3, we obtain four copies of the original. This means its Hausdorff dimension, d, satisfies 3*d* = 4. So, d = log3(4) ≈ 1.26. The curve isn’t entirely space-filling, like Peano’s, so it isn’t quite two-dimensional, but it is more than a single one-dimensional line.

Lastly, some readers may be thinking, “Isn’t time the fourth dimension?” Indeed, as the inventor said in H.G. Wells’ 1895 novel The Time Machine, “There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it.” Time as the fourth dimension exploded in the public imagination in 1919, when a solar eclipse allowed scientists to confirm Albert Einstein’s general theory of relativity and the curvature of Hermann Minkowski’s flat four-dimensional space-time. As Minkowski foretold in a 1908 lecture, “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 independent reality.”

Today, mathematicians and others routinely stray outside our comfortable three dimensions. Sometimes this work involves additional physical dimensions, such as those required by string theory, but more often we work abstractly and do not envision actual space. Some investigations are geometric, such as Maryna Viazovska’s 2016 discovery of the most efficient ways of packing spheres in dimensions eight and 24. Sometimes they require non-integer dimensions when fractals are studied in diverse fields such as physics, biology, engineering, finance and image processing. And in this era of “big data,” scientists, governments and corporations build high-dimensional profiles of people, places and things.

Luckily, dimensions don’t need to be fully understood to be enjoyed, by bird and mathematician alike.

Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

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