Another major advance in mathematics in the nineteenth century that also played a pivotal role in the advances of twentieth century physics was the development of geometry in higher dimensions. Ever since Descartes created analytic geometry in the seventeenth century, Euclid’s geometry in the Elements could be synthesized with algebra in such a way that mathematicians found rather quickly geometric objects that required more than three spatial dimensions to describe. Moreover, the calculus as created by Newton and Leibniz could be generalized and applied to such geometric objects to enable a deeper understanding of their nature. All of this culminated in the nineteenth century with the work of Bernhard Riemann who developed a general theory of geometry that unified the algebraic and analytic features explored since the seventeenth century. This view of geometry required mathematicians to unshackle their senses in order to comprehend those features of geometric objects that resided in four, or five, or even higher dimensions. Breaking free of our senses in order to comprehend a reality that completely transcends our three- dimensional world requires thinking by analogy to translate our experiences to those of a creature who inhabits such a world of higher dimensions. This development inspired Edwin Abbot Abbot to pen the book Flatland in 1884, which describes two-dimensional creatures who live in a planar world and tells the tale of one such creature (a square) who is paid a visit by a three-dimensional creature (a sphere). What unfolds is the attempts of the sphere to explain his nature to square, who can only experience a world of two dimensions. One of Abbot’s aims in this novel is to give the reader a window into the then recent exploration by mathematicians into the nature of higher dimensions, which illustrates the idea of transcendence in both its scientific and religious sense.To see how analogy can give insight into higher dimensions, consider the following sequence of geometric figures:
The progression from left to right portrays the notion of cube in the appropriate dimension:
- A point is a zero-dimensional cube.
- A line segment is a one-dimensional cube formed by dragging the point one unit to the right along a one-dimensional axial direction.
- A square is a two-dimensional cube formed by dragging the line segment one unit along a second direction perpendicular to the first axial direction.
- A cube is a three-dimensional cube formed by dragging the square segment one unit along a third axial direction perpendicular to both the first and second axial directions.
- A hypercube (or tesseract), then, is a four-dimensional cube formed by dragging the cube segment one unit along a hypothetical fourth axial direction perpendicular to each of the first, second, and third axial directions.
Since we cannot experience such a fourth axial direction, we are left with only a conceptual description about how we may form a hypercube based on how lower dimensional cubes are formed. Properties of the hypercube are then extracted through this analogy.
Using analogy to visualize and analyze higher dimensional objects is one of the main ways mathematicians can translate such objects to our realm of experience for study. This process is necessary as visualizing higher dimensional objects in their actual form is extremely difficult. By imagining how such objects can be constructed from lower dimensional objects, as described above, we may surmise the properties of objects in dimensions four, five, and higher in an inductive way by developing objects from dimensions one, two, and three. This is the method of analogy that is so beautifully described by Abbott in Flatland, particularly in Square’s encounter with the three-dimensional Sphere. As Sphere entered into Flatland, Square perceived a dot which became a circle whose radius grew until it reached Sphere’s radial length, and then shrunk back down until the circle reached a point and then disappeared, as illustrated below. By analogy, one way to imagine a hypersphere in four dimensions is by its appearance as it passes through our physical three-dimensional space.
Initially we would see a point and then, like a balloon, we see a small sphere that inflates until it expands to a sphere of radial length equal to that of the hypersphere and then deflates back down to a point.
Another way to visualize the hypercube through analogy is to consider the cube through the following sequence of perspectives:
Here the cube is to be viewed by focusing on the red back face as the viewer comes from the side of the cube, moving until facing the back face directly through the front face. This last perspective of the cube (on the right) can also be viewed in a two-dimensional fashion, as an outer square and an inner square with nearby corners connected by line segments. This way of viewing a cube two-dimensionally is often called a projection or shadow of the cube. By analogy, we may consider a similar perspective of a hypercube: our three-dimensional view of the cube that focuses on the back face through the front face translates to a four-dimensional view of the hypercube that focuses on the “back cube” through the “front cube.” The resulting projection from four-space to three-space translates “inside/ outside square” for the two-dimensional projection of a three-dimensional cube to “inside/outside cube” for the three-dimensional projection of a four-dimensional cube and appears as follows:
In a similar way, the other Platonic solids possess four-dimensional analogs (see below). Notice the remarkable symmetry and intricacy in the hyper-dodecahedron and the hyper-icosahedron. The simple visual symmetry of these images is a great example of beauty in mathematics.
Now, higher dimensional geometry, as noted before, has found its way into scientists’ efforts to understand the universe in the twentieth century through the developments of general relativity theory and quantum physics. One recent way such higher dimensions have entered physical theories is in cosmology and the effort to describe the large-scale structure of the universe. To get a feel for how these higher dimensions are contemplated, here is another exercise in analogy. Consider the disk:
This is a two-dimensional object. If we consider this disk as viewed edge-on in space and push the center downward, we get a bowl:
Stretching the rim of the bowl to touch a point above the bowl forms a sphere:
Now consider a square with the sides oriented as follows:
Connecting the opposing sides results in a cylinder:
Gluing the top and bottom together gives an inner tube shape:
The resulting shape is called a torus:
Thus, by simply gluing the edges of a square together, we can get a shape that looks completely different. Notice that from the perspective of the square, traveling to one edge transports the traveler back to the opposite edge, which accounts for the two perpendicular circular directions on the torus. Imagine living as an ant on the square: every time you departed the left edge you’d appear on the right edge. The same thing happens seamlessly on the torus.
Consider next a similar square in which only the left and right sides have arrows oppositely directed. The process of identifying those sides can be seen as follows:
The resulting glued object is called a Möbius band. Such an object possesses the feature of being “unoriented” in that, in contrast with a cylinder, it fails to have a distinct inside and outside, so it is considered one-sided. This can be seen in this rendering of a Möbius band:
Finally, consider the square with edges oriented as follows:
Gluing opposite edges in the way that aligns the arrows yields an unoriented surface known as the Klein bottle:
The unorientabilty of this surface prevents it from having both an inside and an outside: an ant can crawl from the inside to the outside without reaching an edge. The picture is deceptive. To view it in three dimensions, as above, requires the neck of the bottle to pierce the body. Four spatial dimensions are required to give a proper depiction. This is the sort of hidden reality that excited me as a student; simply matching up edges on a square can yield a bizarre shape that can’t be portrayed in three dimensions.
We can make note of three features of the surfaces the sphere, the torus, and the Klein bottle. They are:
- Locally two-dimensional: Because we formed them by gluing the edges of a flat square, the surface looks flat and two-dimensional when focusing on any point up close.
- Closed: While the square has an edge or boundary, after gluing the edges the boundary disappears making it edgeless or closed.
- Embedded in higher dimensions: Even though these surfaces are two-dimensional up close, being closed forces them to have three or even four spatial dimensions.
A similar view can be given to the description of our universe. Cosmologists develop models of the cosmos based upon general relativity and supported by astronomical observations. Among them there are some models that are geometrically closed. We know from our own experience that the world is locally three-dimensional. What are the possible geometric descriptions of such a closed three-dimensional structure? By analogy, instead of starting with a square and selecting rules for gluing the outer edge, start with a cube:
Imagining the interior to be our universe, we may consider opposite faces glued according to variations on the gluing rules we contemplated for the square. Performing this gluing for all three pairs of opposing faces gives a closed, locally three-dimensional object which, because of gluing all opposing faces, requires embedding in more than three spatial dimensions observations to properly exist. Note, as with the torus, that viewing beyond a face brings one’s visual field back into the cube through the opposite side, right behind the viewer (you could see the back of your own head).
It should be noted, that other polyhedra can be considered when forming models of the universe. For example, based on astronomical Jeff Weeks has proposed in The Shape of Space that gluing opposing faces of dodecahedron gives a good closed model of our universe:
This geometric object is called a Siefert-Weber manifold.
For a mathematician, the exploration of higher dimensions need not end at four dimensions. For example, in our discussion of Platonic solids, we identified the only five that exist in three-dimensional space. In four-dimensional space, we indicated there is a hyper version of each of the five of the Platonic solids. Are there others? The definition of hyper-Platonic solids does not necessarily exclude other possibilities and, in fact, there is one more, called the 24-cell, whose projection into three-dimensional space is displayed here:
The definition of Platonic solids and hyper-Platonic solids can be generalized to five, six, seven, and beyond to the notion of regular polytope. Can a similar classification be given to such objects in such higher dimensions, or do things become overly complex? Well, the notions of hyper-tetrahedron, hyper-cube, and hyper-octahedron persist easily to every dimension.The amazing thing is that in each dimension of five or higher, the only regular polytopes are the appropriate analogs of the hyper-tetrahedron, hyper-cube, and hyper-octahedron. Here we have a wondrous example of the treasures that can be found within the mathematician’s imagination. In the expectation that higher dimensions imply higher complexity, which in general is true, the high order symmetry of regular polytopes restricts its possibilities to just the most basic types. Unfortunately, or fortunately, geometric objects in higher dimensions generally can take on a variety of complex and exotic features for which any specific assertions that can be declared by a mathematician regarding them often require a list presuppositions in order to get a firm grasp upon their nature. However, in the case of Platonic solids, the use of analogy and symmetry as a means of discerning these geometric objects leads not only to the types of simplicity and unexpected connections that give a breathtaking beauty to our understanding of higher dimensions; it also provides a way of initiating the search for deeper connections as it relates to more general geometric objects in any dimensions by first considering them as a suitably general form of polytope.
The Splendor of Creation: Beauty and the Glory of God
By taking geometric objects and their symmetries as the source of examples of beauty in mathematics, my aim is to offer a sense of how mathematics instills a sense of awe in mathematicians and scientists as they explore the deep inner workings of physical space. Moreover, one can glean from such explorations a sense that physical space is not required to be the way it is. From the viewpoint of mathematics, there is a wealth of possibility for how space can be woven together to give a geometry for the fabric of the cosmos. From such a vantage point, one can easily see the universe as a creation—a creation intended to produce wonder in participants with whom the Creator desires a relationship. Some of these participants may be enraptured by the equations they are contemplating, which disclose in geometric designs the impress of a divine author at the root of our entire existence. Herein is beauty found: to see the presence of the Creator revealed in the designs and relations eloquently articulated through the equations of the physicist or mathematician. Furthermore, in the expressions of the mathematician’s world, the colors of creation’s possibilities can be discerned to be among those on the Creator’s palette, perhaps as seen before the brush has even touched the canvas.
Among the ways that beauty finds its presence within a mathematical discourse are the unexpected connections revealed within the physical makeup of reality, the pleasing encounter with and fruitful productivity from symmetric relations, and the contemplation of transcendent realities within higher dimensions surmised through the power of analogy. Each of these can elicit awe from the mathematician, the scientist, the pastor, and the parishioner alike as they examine the nature of space in its geometric forms. That we may contemplate the ways reality both is and could be is a source of great mystery. If one is willing to step back to take it all in, it can inspire a sense of awe and a consideration of the possibility of a divine author to all that there is—perhaps leading the one contemplating to respond in the most profound fashion: Glory!