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“Either this is madness or it is Hell.”
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“‘It’s neither,’ calmly replied the voice of the Sphere, ‘it is Knowledge; it is Three Dimensions: open your eye once again and try to look steadily.’”
This dialogue is from Edwin Abbott’s 1884 novel Flatland. In it, the protagonist, A. Square, who lives in a two-dimensional world, encounters a three-dimensional sphere for the first time. Although ostensibly a satire of Victorian society, the work also contains a mathematical treatise on the fourth dimension. What if we live in a four-dimensional world, but—much like A. Square—have been trapped in a lower dimension?
Many people associate thoughts of higher dimensions with works of science fiction, such as the books of H. P. Lovecraft or blockbusters such as Christopher Nolan’s Interstellar. But the natural sciences have also been studying higher dimensions for centuries.
Mathematicians, for example, investigate which geometric objects could exist in these unimaginable environments and how they can be ordered and measured. In doing so, they have discovered that although many dimensions are relatively easy to analyze, a 4D world specifically poses surprisingly large problems. As mathematician Ciprian Manolescu of Stanford University explains in a Numberphile video, many theorems hold true for all dimensions n except when n = 4: “That’s what makes dimension four so fascinating.”
But why are four dimensions so special? To understand this, one must venture into the abstract field of topology. While topology deals with geometric figures, “details” such as distances or precise curvature play no role. Thus, a sphere with a dent is identical to an ordinary sphere, or a circle is the same as a quadrilateral. In other words, two objects are considered topologically identical if they can be deformed into one another without tearing holes in them or gluing them together at any point.
Even though it sounds complicated, we constantly use topology in our everyday lives without realizing it. For example, we imagine our planet as a sphere, even though Earth, strictly speaking, deviates from a perfectly round shape.
In topology, mathematicians try to reduce objects to their essential properties. This allows them, for example, to make statements about an entire class of objects without having to examine each one individually. One famous example is that, topologically speaking, a doughnut and a cup of coffee with a simple ring handle are the same: they each have exactly one hole.
As long as you stay in one, two or three dimensions, everything is fine. But problems begin in four dimensions. The deformations you can make to transform two shapes into one another can become more complex. If you want to mold one smooth shape into another, sharp corners and edges can suddenly appear during the process. That’s like saying that in order to transform a circle into an oval, you first need to reshape it like a star. This kind of change is unnecessary in one, two or three dimensions, but in four dimensions, it’s sometimes unavoidable.
This phenomenon has led to the existence of two types of equality in topology: First, two objects are generally topologically equal if they can be deformed into each other—regardless of how the process occurs. There is also a stricter form of equality, however: two figures are what mathematicians call “diffeomorphic” if they can be deformed into each other smoothly, without ever creating corners or edges. Thus, in higher dimensions, there are objects that are topologically equal but not diffeomorphic—that is, they cannot be transformed into each other smoothly.
This is a general characteristic of higher dimensions. But four dimensions remain a special case. If you consider a space Rnspanned by n real numbers in each dimension (that is, something like an n-dimensional coordinate system), then this space is always unique for all dimensions n except four. In other words, all spaces that are topologically equivalent to the n-dimensional space Rn are also diffeomorphic to it. This means that one will not encounter vertices and edges when transforming one space into the other. In 1981 mathematician Michael Freedman discovered that the 4D space R4 is an exception. In fact, there are infinitely many 4D figures that can be transformed into the 4D R4 in different ways without being diffeomorphic: all of them exhibit a different kind of pattern of vertices and edges during the transformation.
This makes 4D space a very strange place. But it’s not just 4D space itself that’s strange—so are 4D figures.
Imagine you lived in a five-dimensional world and wanted to sort 4D surfaces: Which ones are diffeomorphic to each other? What classes of surfaces are there?
To understand this, it’s easier to start with our familiar 3D world. There, we can examine 2D surfaces, such as spherical surfaces or a torus (a doughnut shape). As it turns out, all (closed) 2D surfaces can be divided into only three categories, as Henri Poincaré proved as early as 1907: they are either equivalent (diffeomorphic) to a spherical surface, to connected doughnuts, or to connected projective surfaces (which include, for example, the Klein bottle). No matter how complicated a 2D figure may appear, it can always be transformed into one of the three categories—and in a straightforward manner.
If one were to live in the fourth dimension and consider 3D surfaces, eight different categories would emerge: every 3D surface can be reduced to eight basic shapes. William Thurston had suggested this in 1982, but this so-called geometrization of 3-manifolds was only proved in 2003 by Grigori Perelman, who, in doing so, incidentally proved the Poincaré conjecture that every 3D surface without a hole can be deformed into a 3D spherical surface. The result shows that a classification scheme also exists for 3D surfaces.
When turning to higher dimensions and examining surfaces in five or six dimensions or more, things become more difficult. How can such complex objects be categorized?
Mathematicians often use the so-called Whitney trick: Imagine throwing a lasso around a shape and observing its behavior as it contracts to determine if the surface has holes. This method distinguishes a sphere from a torus. Although every closed loop on a sphere contracts to a point, this is not the case with a torus, which has a hole in it.
This approach also works very well for surfaces with more than four dimensions. When a lasso is contracted, a circular area, a disk, is formed. To determine the type of surface, all possible types of resulting disks must be examined separately. The lassoed areas can overlap, however—which poses a problem from a mathematical perspective. For n > 4, the additional dimensions can be used to separate two disks. This is similar to the process of separating two intersecting lines in a plane: moving left-right or up-down is ineffective. Only a third spatial dimension allows the lines to be separated by depth. The same principle applies in five dimensions to two 2D disks. In this way, the Whitney trick can be used to determine which surfaces with a dimension of five or more are diffeomorphic to each other.
So all categories of 2D, 3D, 5D and higher-dimensional surfaces have been found. And, of course, the fourth dimension causes problems once again because, as it turns out, it’s impossible to classify diffeomorphic 4D objects—this world is utter chaos!
The 4D world also holds a secret that Manolescu in his video says is perhaps the most significant problem in topology: Can every 4D surface without a hole be diffeomorphically deformed into a 4D spherical surface? This is the smooth Poincaré conjecture. The ordinary Poincaré conjecture has already been proved for all dimensions n but only in the general topological sense: this allows deformations that produce vertices and edges.
Topologists are therefore interested in how the n-dimensional Poincaré conjecture turns out when only diffeomorphic deformations are permitted. They have now found an answer for every spatial dimension—except for n = 4. The conjecture does not hold in seven dimensions, for example, in which there are 28 different versions of a spherical surface, so-called exotic spheres: that is, 28 different figures without a hole that can only be transformed into one another using vertices and edges.
The number of exotic spheres can also be calculated for all other dimensions, and this number can sometimes be very large. So far, no object has been found in four dimensions that does not have a hole and is not diffeomorphic to a sphere—but it also has not yet been proved that none exist.
Many mathematicians assume that exotic 4D spheres exist. After all, infinitely many different versions of 4D R⁴ space already exist. But perhaps the fourth dimension will surprise us in this case as well.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.