A number of years ago Numberphile published a video on the behavior of higher dimensional spheres with the title “Strange Spheres in Higher Dimensions”. I'd recommend one watches the video before reading this post, in that video Matt Parker presents a construct where spheres are placed inside a cube pressing against its faces, in the remaining central volume an additional sphere is set with a radius so that it kisses the “padding spheres”, this build is then scaled to higher dimensions where Parker observes that the central sphere increases in size constantly until its surface goes through the face of the cube, an apparently impossible configuration. He then jokingly concludes that the only way to make sense of it is to imagine higher dimensional spheres as “spiky”, so that the spikes can go through the face while the rest of the volume remains confined inside the hypercube.
Recently, for some reason that same video ended up in my feed and I ended up watching it again, I was left rather dissatisfied with the conclusion. I am not a mathematician but I was always fascinated with higher dimensions and when it comes to geometry, I often found that seeing is believing.
Plotting sections of the construct can help us make sense of the problem, we’ll start with the simple 2d version, here the central sphere has a radius of :
Fig. 1 not much to see here, everything is visible on the plane
The true solution of the “mystery” is understanding the diagonal of the cube in higher dimensions. The diagonal of the measure polytope in n dimensions is times the size, so it becomes larger and larger as we increase the number of dimensions. This is easy to understand: for a unit square the diagonal is , to calculate the size of the diagonal of a cube starting from this we must solve the hypotenuse of a right triangle where the catheti are and 1; the result, of course is . If we wanna go up another dimension this process must be repeated one more time, so the radicand must grow by one integer.
The diagonal of a face also grows in a similar fashion, since it only has one less dimension that the hypercube, for example in the 9th dimension the diagonal of a face of a unit hypercube is .
Understanding the 3d version of the problem is not too hard but to keep things simple I will show a 2d section only, to obtain this section we will slice the cube through the diagonal as shown in fig. 2.
Fig. 2 a render of the problem for n=3, the segmented line is the edge of the section
The highlighted section is shown in the next image, notice a gap appears in the center as not all of the padding spheres are touching each other. Also, the padding spheres appear to no longer be contacting all sides of the cube since some of the contact points do not lie in this section.
Fig. 3 The radius of the central sphere has grown, it is now
This method is quite useful, since the plane we chose passes through the centers of both the central and padding spheres, the sections of these objects will always appear as circles of radius 1 no matter how many dimensions are we working with. Now let us move to the 4th dimension:
Fig. 4 in 4d the central sphere has now the same radius of the padding spheres, 1
This pattern continues, as the diagonal of the face continues to grow so does the diameter of the central sphere, in 9 dimensions the sphere is now contacting the faces of the hypercube:
Fig. 5 The diagonal of a face is now almost 3 times the side
Due to symmetry the radius of the central sphere is always equal to the distance between the surface of a padding sphere and the closest corner. In 10 dimensions part of the central sphere is now escaping the inner volume entirely:
Fig. 6 The radius of the inner sphere is now , or about 2.16
This pattern continues as n increases and more and more of the central sphere escapes the polytope. Higher dimensional spheres are not spiky but the faces of higher dimensional cubes have very large diagonals compared to the side through which the central sphere can escape.
Before reading this post, I use the intuition of this hole getting larger than larger in higher dimensions. It seems ~the same as your formulation but less clean.
A number of years ago Numberphile published a video on the behavior of higher dimensional spheres with the title “Strange Spheres in Higher Dimensions”. I'd recommend one watches the video before reading this post, in that video Matt Parker presents a construct where spheres are placed inside a cube pressing against its faces, in the remaining central volume an additional sphere is set with a radius so that it kisses the “padding spheres”, this build is then scaled to higher dimensions where Parker observes that the central sphere increases in size constantly until its surface goes through the face of the cube, an apparently impossible configuration. He then jokingly concludes that the only way to make sense of it is to imagine higher dimensional spheres as “spiky”, so that the spikes can go through the face while the rest of the volume remains confined inside the hypercube.
Recently, for some reason that same video ended up in my feed and I ended up watching it again, I was left rather dissatisfied with the conclusion. I am not a mathematician but I was always fascinated with higher dimensions and when it comes to geometry, I often found that seeing is believing.
Plotting sections of the construct can help us make sense of the problem, we’ll start with the simple 2d version, here the central sphere has a radius of
Fig. 1 not much to see here, everything is visible on the plane
The true solution of the “mystery” is understanding the diagonal of the cube in higher dimensions. The diagonal of the measure polytope in n dimensions is
The diagonal of a face also grows in a similar fashion, since it only has one less dimension that the hypercube, for example in the 9th dimension the diagonal of a face of a unit hypercube is
Understanding the 3d version of the problem is not too hard but to keep things simple I will show a 2d section only, to obtain this section we will slice the cube through the diagonal as shown in fig. 2.
Fig. 2 a render of the problem for n=3, the segmented line is the edge of the section
The highlighted section is shown in the next image, notice a gap appears in the center as not all of the padding spheres are touching each other. Also, the padding spheres appear to no longer be contacting all sides of the cube since some of the contact points do not lie in this section.
Fig. 3 The radius of the central sphere has grown, it is now
This method is quite useful, since the plane we chose passes through the centers of both the central and padding spheres, the sections of these objects will always appear as circles of radius 1 no matter how many dimensions are we working with. Now let us move to the 4th dimension:
Fig. 4 in 4d the central sphere has now the same radius of the padding spheres, 1
This pattern continues, as the diagonal of the face continues to grow so does the diameter of the central sphere, in 9 dimensions the sphere is now contacting the faces of the hypercube:
Fig. 5 The diagonal of a face is now almost 3 times the side
Due to symmetry the radius of the central sphere is always equal to the distance between the surface of a padding sphere and the closest corner. In 10 dimensions part of the central sphere is now escaping the inner volume entirely:
Fig. 6 The radius of the inner sphere is now
This pattern continues as n increases and more and more of the central sphere escapes the polytope. Higher dimensional spheres are not spiky but the faces of higher dimensional cubes have very large diagonals compared to the side through which the central sphere can escape.