There is a very real geometric object, realizable within the relativistic geometry of our universe, which has the properties of a sphere in four dimensions (a “4-hypersphere”); what does it look like? (*)
Note (to professionals in relevant fields): the text below presents no new or unknown idea in physics or mathematics. The reason for writing it is that the author has struggled to learn the answers to questions such as this one for years, finding no explicit answer in published texts. I am sure I was simply unlucky in coming across the right publication (and the gentle reader may e-mail me the reference that treats precisely this topic, in case they are aware of one), but now that I think I know the answer I wish to share it with the interested reader of this page. Another reason for writing a web page for this topic is that a true 4D-sphere should be shown in animation, and this is what this page does; printed material must necessarily depict everything in still drawings. If you plan to contact me, please read this paragraph at the end of this text.
Since I was in high school I remember trying to visualize 4-dimensional objects. I managed to draw what I thought was a 4D-cube, and then a 4D-tetrahedron, while the case of a 4D-sphere seemed to be too easy to even talk about. It was only a few years later, while in college, when I realized that my early high school 4D-creations were simply the 4D equivalents of 3D objects rendered in four-dimensional Euclidean space. Except, there is a problem: the geometry of our universe is not Euclidean. Around 70 years before my high school adventures Einstein had proposed that the true geometry of our world is not Euclidean, but hyperbolic, having at least four dimensions: the three familiar spatial ones, plus the fourth one which is time. Although a four-dimensional Euclidean geometry with time as the fourth dimension was already known since Galileo Galilei’s time, it was Einstein who showed that the fourth dimension, time, is essentially different from the other three dimensions.(*) Therefore, my early creations were unrealistic.(*) And yet, real 4D-objects have to be constructible, if the relativistic geometry is real. What do they look like?
Before actually drawing a 4D-sphere, let us make two important observations. First, since time is the fourth dimension, every object with four dimensions has to appear as moving with respect to a fixed frame of reference (say, this page). How it moves is a matter that defines its shape in the fourth dimension, but that it moves is without doubt -- else it is merely a familiar “static” 3D-object. And second, since we want to draw a 4D-sphere, we have to generalize the definition of the familiar 3D-sphere in four dimensions. Let us do this now.
Now, a distance in either two or three dimensions is something we are very familiar with. What is distance in four dimensions, though? And what does a point in four dimensions look like?
Let us answer this last question first. A 4D-point must be an entity that includes three spatial coordinates (x,y,z), plus a fourth one, which gives the time t during which the 3D-point (x,y,z) “occurs”. Hence, instead of a 4D-point we will be talking about an event with coordinates (x,y,z,t). Think of an event as a familiar point (x,y,z) which “blinks” once: instantly goes on and off, at time t at that location.
OK, so far our 4D-points, or events, have nothing to do with relativity -- and they were utilized also in the same way by Galilean 4D geometry. Now we’ll define the distance between two events.
The familiar formula that gives the distance d between two 3D-points (x1,y1,z1) and (x2,y2,z2) is:
To simplify the above formula we’ll replace expressions such as (x1-x2) simply by x. (If the reader has difficulty seeing a single coordinate as standing for a difference of values, they may think of d as giving the distance of point (x,y,z) from the origin (0,0,0)). Thus, the formula is simplified to this one:
This formula generalizes easily to four dimensions if we remember to subtract the square of time under the square root, rather than add it to the other three squares. This is because space and time are related, according to relativity, as follows:
s = c t
where c is the speed of light. This equation can be understood as follows: suppose an event happens at one end-point of a space-interval s; at that moment, a photon starts from this end-point traveling along s, and the moment it reaches the other end-point a second event happens there. The equation says that every such interval of space s is equivalent to the time t it takes for light to traverse it. If we square both sides of the above equation it becomes: s2 = c2t2, which can also be written as: s2 - c2t2 = 0. Taking the square root on both sides we get: , which says that the 4D-distance between two events such as the described ones is zero. This gives a hint for why we need to subtract time from space. Accordingly, the formula that gives the distance of two events in four dimensions is:
Now we need to recall our earlier definition of a 4D-sphere: it is the locus of events (x,y,z,t) that allow d in the above formula to retain a constant value. Therefore, d plays the role of “radius” in our 4D-sphere.
To draw our 4D-sphere we need to depict a number of 3D-spheres as they change in time so that their 3D-points (x,y,z) along with time t satisfy the previous equation. What could the radius of those 3D-spheres be, as a function of time? We can find this relation very easily if we rewrite the previous equation as follows:
This last equation is the one that tells us how the 3D-spheres (projections of the 4D-sphere in 3D-space) that we want to draw depend on time: the left-hand side of the last equation is the 3D-radius of those spheres, while the right-hand side is a function of t that employs two constants: the predetermined radius d of our 4D-sphere, and the speed of light c. We can denote the left-hand side of this equation by r(t), and plot its values against time:
The above graph shows how the radius of the “time-sliced” (or: projected) spheres in 3D-space changes as a function of time: initially at time 0 the radius starts from a fixed value d (same d as in our formulas, which we arbitrarily decided to be equal to 1) and changes very slowly, but soon it accelerates and reaches “asymptotically” a nearly constant speed. (We also assumed c = 1 in our graph, to be able to see a sensible curve, rather than c = 299,792.458 km/sec, which is the actual speed of light.) This curve is called a hyperbola, and now it becomes evident why relativistic (read: real-world) geometry is said to be a hyperbolic geometry. We are ready now to plot our 3D-spheres the radius of which changes in time according to the above graph. The whole thing that you see below, as it changes in time, is supposed to be depicting a 4D-sphere:
All right. These four seconds of animation (Fig. 2) were arguably what this page is all about. Still, what you see is as incomplete as any rendering of a higher-dimensionality object is on the 2D-plane. So some observations are in order.
First, the “movie” in the image above presumably starts at time equal to “negative infinity”, at which point the time-sliced 3D-sphere has an infinitely large radius. The sphere keeps shrinking until its radius is within the bounds of our box and becomes visible. The shrinking continues up to a minimum at time t = 0, at which point the radius of the time-sliced 3D-sphere is equal to d. Then it starts growing, with time t taking on positive values, exceeding the borders of our box and disappearing from view,(*) and keeps growing forever while time runs to “positive infinity”.
Second, although the 3D-spheres were plotted so that their radius does follow the curve shown in Fig. 1, hence, decelerating its shrinking from an initial near-linear speed and then accelerating until it reaches a near-linear speed again, this behavior can hardly be observed in Fig. 2. The reason is that the curved part of the hyperbola in Fig. 1 happens too fast in Fig. 2, and only at the last few frames before and after reaching the minimum radius d. The reader has to use their imagination to “sense” the non-linear part.
Third, I need to comment on the speed of our animation. If the length of the side of the white box in Fig. 2 is taken to be what it seems to be, namely, a few centimeters, then our animation does not depict a 4D-sphere at all! The reason is that the speed of light is not merely a few centimeters per second, but nearly 30,000,000,000 (thirty American billion) centimeters per second. I used c = 1 in both Fig. 1 & Fig. 2, which corresponds to a very slow speed. Had I used the true value for c, our spheres in Fig. 2 would literally implode and explode at nearly the speed of light! To get an idea about the shape (hence: speed) of the hyperbola for higher values of c, let’s draw again the graph in Fig. 1, this time for c equal to 10 (instead of 1).
We see that with a mere increase of c from 1 to 10 we get a rather sharp hyperbola (therefore, our spheres would shrink and grow rather fast). It’s clearly beyond my visualization abilities to think what the corresponding cases would be for the true c (the speed of light). I can understand such things only as numbers.
Another way to think about what the animation in Fig. 2 shows is this: suppose the white area represented by the box in Fig. 2 is not just a few centimeters, but hundreds of thousands of kilometers wide. In other words, our box can show an area of space engulfing the Earth, the Moon, and a little more. In this way, by increasing enormously the area our box represents, one could think that the speed of shrinking and expansion of the sphere that we observe in Fig. 2 could really be close to the speed of light. Can this idea be right?
Not quite. See, when we look through large expanses of space, the relativistic nature of our universe becomes evident. In our case, the photons from the front portions of the 3D-spheres would arrive to our retinas earlier than the photons from the back parts. (That’s still true if the space covered by the box is small, but the difference is negligible.) This would result in seeing distorted 3D-spheres, with their front parts being ahead of the back parts in implosion and explosion. The larger the 3D-space involved, the worse the distortion.
If you plan to contact me to help me answer the above questions, please note the following. Many people have contacted me over the years during which this page appears on the web, thinking they know the answers. Invariably, I am referred to web pages or to drawings that deal with the 4th dimension in Euclidean geometry. If you plan to do this, please understand that I am not interested in Euclidean, but in relativistic geometry. The Euclidean 4D-cube, for example, has been known since the ancient times. Here is a drawing of it (thanks to Josi Mason for an initial sketch):
Figure 4: 4D Euclidean cube, known since antiquity. The translated 3D-cube is highlighted within it
The 4D Euclidean cube can be constructed by translating a familiar 3D-cube in an imagined fourth dimension. In antiquity, this 4D Euclidean cube was a mere curiosity (and the adjective “Euclidean” was not attached to it, since nothing non-Euclidean was known.) In our times it has been employed in computer science as a possible topology for a network of parallel processors, where every vertex of it is occupied by a single processor. But my question above concerns its relativistic analogue. Since the relativistic 4D-sphere is an open shape (as opposed to its Euclidean analogue, the 4D Euclidean sphere, which is a closed shape), I expect that the 4D relativistic cube should also be an open shape. Similarly, when I ask about volume, the reader should note that the volume of the 4D Euclidean sphere is well known and easily computable by means of familiar integration (see the formula for the nD-sphere at this footnote(*)). But relativistic geometry has a different metric (its formula is given above) and integration with such a metric uses Legendre integrals, which I am not familiar with. So, I appreciate your paying attention to all this information in case you plan to contact me.
The Best Answer that I Received for the Above Questions
People have kindly responded to my request for answers above, some of them very knowledgeable on the subject. The most comprehensive answer that I have received came from Dr. Eric Hartman, in 2005. For years, I was determined to go through it and understand it thoroughly. Finally I realized I will never find the time to do so, so instead of ignoring his kind offer and the time and effort he invested, I prefer to include it below verbatim, and leave it to the reader to go through it, if interested. Warning: it is long, and does require some familiarity with calculus and relativity.
Footnotes (clicking on (^), on the left, brings back to the text)
(^) I am indebted to Maricarmen Martinez, who helped me with the derivation of the formulas as well as with the Mathematica animation.
(^) Note that Euclidean, Galilean, and Minkowskian geometries are all different from each other.
(^) Unrealistic, but not unrealizable. A Euclidean 4D-cube is called a 4-hypercube, is known since antiquity, and has been utilized by computer scientists as one of possible structures for a network topology. See this added paragraph of the present text for a drawing of it.
(^) It should actually not disappear from view: we should be able to see its far wall from the inside, receding to infinity. I did not consider it necessary to depict that part of the animation, especially given the already large size of the GIF file.
(^) 4D-spheres are bounded in 4D-space by definition: there is a 4D-sphere that includes them, namely, their own self. It is their projections on any subspace that includes time as one of its dimensions that have the hyperbolic character, hence, appear unbounded in that subspace.
(^) The volume of the n-dimensional sphere of radius r in Euclidean space is given by the following formula:
V(n, r) = a r n
where a = (1 / (n/2)!) π (n/2) if n is even, and a = ((2 (n+1)/2) / n!!) π ((n-1)/2) if n is odd, where n!! is the product of the odd numbers less than or equal to n.
Einstein, Albert (1961). Relativity: the Special and the General Theory. Crown Trade Paperbacks: New York, 1961.
All graphs and plots on this page were made with Mathematica