While it is now common to accept time as a fourth dimension in a combined space-time, it is perhaps less common to attempt visualizing additional spatial dimensions. One of the most interesting writers on the topic was Charles Hinton, most of whose works were printed before Einstein’s annus mirabilis of 1905. Hinton’s writing inspires one to figure out how a four, five or higher-dimensional object would appear to our three-dimensional eyes.
One way visually to present a four-dimensional cube, or tesseract, is to define the corners of the object in four-dimensional space, perform four-dimensional rotations upon the corner points, and then ‘flatten’ the four-dimensional grid into a perspective in two dimensions of a three-dimensional cross section. This is essentially how the tesseract animation on Wikipedia is generated. While this is most certainly mathematically accurate, is is less useful for visualizing how a solid four-dimensional cube would appear to us, since we would only see the outer surface.
Following the lead of Hinton, the best way to understand the transition to higher-dimensional objects is by thinking about how a truly two-dimensional being would perceive our three-dimensional world. A two-dimensional being would be restricted to a plane with only width and depth, but no height. If a three-dimensional object were pushed through the plane of the two-dimensional being, it would only see the cross-section of its plane with what to us is an object with a third degree of extension.
For example, from the perspective of the two-dimensional being, a three-dimensional sphere pushed through a two-dimensional plane would appear as a point when the sphere touched the plane, grow into a circle with the same radius as the sphere, then shrink back into a point. A cube pushed through the plane, starting from a position sitting flat upon one side, would suddenly appear as a square, and remain the same square until suddenly vanishing as the cube passed out of the plane. If the cube had started tipped up on one corner, however, it would appear as a point, transition through triangles to higher polygons to triangles, then shrink back to a point like the sphere.
If a sphere has its center in the two-dimensional plane, and is then rotated about any axis, from the perspective of the two-dimensional being it will appear as a circle that never changes. This is the definition of spherical symmetry: rotating the symmetric object about any axis through the center leads to the same cross-section for any plane through the center. For a cube with its center in the two-dimensional plane, however, rotations about the center leads a cross sections that change in some surprising ways, and these resulting two-dimensional shapes depend very much on the original orientation of the cube with respect to the plane.
Demonstrating these effects for a sphere is very easy, because one can rearrange the equation for a sphere to include one of the dimensions as a variation of the radius of the two-dimensional circle. For a unit sphere, this means
If the variable z, which is extra from the perspective of the two-dimensional being, is varied between −1 and 1, the resulting graph in two dimensions will show exactly a point expanding into a circle and shrinking back to a point. Trying to do something similar for an exact cube is more difficult, because it is not defined by a simple equation but by a set a inequalities: a cube is the interior space bounded by the intersections of three sets of two parallel planes apiece. Not pretty from a calculational point of view, and part of the reason that tesseract depictions are presented using vertices.
Fortunately, there is a nifty way to interpolate in a fully analytic fashion between a circle and a square, using something called a superellipse. The objects used in this demonstration will be simpler than superellipses and their multidimensional generalizations, and will start with the unit supercircle defined by
Restricting the exponent to values greater than or equal to one keeps the shapes convex, and the absolute value avoids imaginary values when plotting. For an exponent of two the supercircle is just a circle. Increasing or decreasing the value produces a squarish object with rounded corners, as can be seen in this interactive graphic using SageMathCell:
The range of the slider has been adjusted to start with a circle in the middle, to make clearer the transition in either direction to the other shapes. For an exponent value of one, the object is an exact square, and for an exponent value of around twenty the object is surprisingly close to a square. In the limit
As a note, all of the graphics in this presentation include a scaling factor on the right-hand side of each equation to keep the figures comparable in size. The figures displayed differ from the equations given only in this regard.
The corresponding three-dimensional object is a supersphere defined by
but it has a somewhat different behavior due to the extra dimension:
After varying the exponent, this last graphic can be rotated to allow one to see how the corners become progressively sharper. In this case, decreasing the exponent leads to an octahedron, while increasing produces a cube. The difference is described mathematically by saying that a two-dimensional square is its own dual, while in three dimensions the cube and octahedron are dual.
This transition between higher-dimensional spheres and cubes is easily extended mathematically to any number of dimensions, but clearly cannot be depicted as easily as the supercircle and supersphere, since we only see three physical dimensions.
With a simple way to relate three different solid objects, return to the two-dimensional being’s perception of a supercircle being pushed through its plane. For simplicity, the exponent will be restricted to values that will describe the appearance to the two-dimensional being of an octahedron, sphere and cube.
If the object starts out on a point of the octahedron or a flat side of the cube, then the mathematical formula
with varying z will show the appearance of the object to the two-dimensional being, starting with the two-dimensional plane passing through the center of the object:
The disappearance and reappearance of objects can be more finely controlled using cursor keys. This interactive graphic shows immediately how these three objects differ in their various two-dimensional cross sections.
As a technical detail, these images are generated from the equality
without the need for any rearrangement, using the SageMath
implicit_plot (or later
implicit_plot3d) command. This makes it very simple to consider more complicated cases, for objects rotated or extended into higher dimensions, by simply adding more variable combinations to the left-hand side of the inequality. Which is what happens now!
A rotation about an axis passing through the centers of opposite faces of the cube limit of the supersphere, in this case the y-axis, is given by
If the center of the supersphere is in the two-dimensional plane, then the value of the z-coordinate remains zero for the rotation. The equality
can then be used to generate two-dimensional cross sections of a three-dimensional supersphere centered in the two-dimensional plane. Varying the angle will show the appearance of the supersphere to the two-dimensional being as the object is rotated in three dimensions:
The appearance of the sphere never changes, as expected for an object with rotational invariance. The cube, on the other hand, appears to the two-dimensional being to grow into a larger rectangle and then return to its original size, while the octahedron appears to shrink and then grow back.
Allowing the z-coordinate to be nonzero and writing the equality
lets one see how the supersphere looks as it is rotated about its center while also being pushed up and down through the two-dimensional plane:
For the octahedron and the cube, this leads to some rather strange looking shapes from the perspective of the two-dimensional being for objects that to our perspective look very simple. The shapes also appear to move about in the two-dimensional plane, while from our perspective the objects only move up and down. An important lesson to keep in mind when a fourth dimension is thrown into the mix.
Compare these graphics to those for rotation about an axis passing through the centers of opposite edges of the cube limit of the supersphere. A rotation about the axis in the xy-plane passing through the front right and back left edges can be written
The corresponding equality for this rotation of the supersphere is
Setting z equal to zero leads to a graphic that can be manipulated to show rotation of the three-dimensional supersphere while its center remains in the two-dimensional plane,
while letting z be nonzero allows one to see how the object appears to the two-dimensional being after being rotated and then pushed up and down through the plane:
Both the octahedron and cube can be made to appear as triangles transitioning to hexagons and back again to triangles from the perspective of the two-dimensional being.
And now for the real fun! Just by adding an extra variable to the left-hand side of each inequality, we can quickly see how a four-dimensional supersphere would appear to us as three-dimensional beings, as it undergoes each of the transformations above. The equality
with varying w will show the appearance of the four-dimensional object to us, starting with the three-dimensional space passing through the center of the four-dimensional object:
Mathematically there is very little difference between this and the lower-dimensional case, and the behavior is the generalization of that behavior. The sphere and octahedron enter our space as points, expand to their full size and then shrink back to points. The cube, like the two-dimensional square above, appears suddenly, stays around for some time and then disappears.
It is, of course, easy to make a statement about “generalization of behavior”, and another thing to see the actual three-dimensional appearance. Even knowing how it works mathematically does not lessen the physical strangeness of the phenomenon. A four-dimensional being would be able to make cubes appear and disappear around us, simply by ‘pushing’ them through our space in a direction we cannot see! Not magic, but not the opposite.
Now add a rotation about an axis passing through the centers of opposite faces of the four-dimensional cube limit of the supersphere. Mathematically it is virtually the same as the three-dimensional rotation, with an extra row in the vectors and matrix:
If the center of the supersphere is in the three-dimensional space, then the value of the w-coordinate remains zero for the rotation. The equality
can then be used to generate three-dimensional cross sections of a four-dimensional supersphere centered in the three-dimensional space. Varying the angle will show the appearance of the supersphere to a three-dimensional being as the object is rotated into the fourth dimension:
As for the two-dimensional case, the appearance of the sphere never changes, as expected for an object with rotational invariance. The cube, as a generalization of the two-dimensional case, appears to a three-dimensional being to grow into a larger rectangular solid and then return to its original size, while the octahedron appears to shrink and then grow back. Again, easy to state mathematically, but strange to see in action!
Allowing the w-coordinate to be nonzero and writing the equality
lets one see how the supersphere looks as it is rotated about its center while also being pushed up and down through the three-dimensional space:
These are the three-dimensional cross sections of the four-dimensional object that correspond to the strange-looking shapes that the two-dimensional being would see. And remember, to the four-dimensional being the object always looks simple and symmetrical, merely rotated a bit and moved up and down!
Again compare these graphics to those for rotation about an axis passing through the centers of opposite edges of the four-dimensional cube limit of the supersphere. A rotation about the same axis in the xy-plane but into the fourth dimension can be written
The corresponding equality for this rotation of the supersphere is
Setting w equal to zero leads to a graphic that can be manipulated to show rotation of the four-dimensional supersphere while its center remains in the three-dimensional space,
while letting w be nonzero allows one to see how the object appears to a three-dimensional being after being rotated and then pushed up and down through the three-dimensional space:
There are many different ways to proceed from this point. Further spatial dimensions are straightforward to add, and might lead to even more interesting cross-sectional shapes. Adding another spatial dimension opens up more directions in which rotations could be performed. And most interesting of all would be to add coloration to the figures that would track the interior parts of the higher-dimensional objects, to show where our cross section is actually cutting the higher-dimensional object. To be continued...
Uploaded 2012.09.14 — Updated 2017.02.07 analyticphysics.com