This paper is on higher dimensions and how spheres happen to break rules in higher dimensions.
First, I am going to assume you know nothing of dimensions 4+, So I will provide a simple explanation on higher dimensions, if you already know, please skip the following 2 paragraphs.
A higher dimension is quite hard to explain, so i will explain it with the well known 2d and 3d. Imagine this: you have a 2d camera viewing an area on a 2d plane, it will see lines here and there, it cannot jump off the plane or imagine things in 3d. You place a 2d box on the plane, and put a 2d sphere inside, will the camera see the sphere? The answer is no, because the box blocks its view from the sphere, and 2d is only a slice of 3d, meaning you can make a 3d sphere seem to be a growing and shrinking 2d sphere.
The same is for 4d, as 3d is only a slice of 4d, and we cannot see or jump past our 3d plane that we are stuck on. Although 4d beings can, and they can see through 3d boxes, just like we can see through 2d boxes. A good simulation is 4d toys, especially with a VR headset.
With that over with, we can now move onto the interesting physics with higher dimensions. One interesting thing, 4d beings can vanish and reappear at different points just by moving across their dimension, as easy as stepping sideways. And with higher dimensions, you get more “sides” on objects, for example, a 4d cube has 32 sides, and a 3d cube has only 6 sides. If you think that breaks the rules, you will be wrong because a cube is just an object with all sides at a right angle to the connected side, and all sides are the same length.
To get the number of sides (2d cubes) of a higher dimension cube, you can use this equation:
If 𝓝 = the dimension the cube is in
and 𝓧 = the number of sides of that cube
then 6^(𝓝 - 2) = 𝓧
And then you will get the sides of a cube, for example, a 15d cube (15d is the biggest dimension available) would have 6^13 sides (~13 billion, 60 million sides), note that this equation does not work with cubes in dimension 1.
Another strange phenomena is that a bunch of packaging spheres against each other, with a gap in the middle for another sphere, that sphere will become larger than the box in dimensions 10+, --note that this is measuring the radius.
To get the maximum radius a sphere can be without overlapping the packaging spheres, you can use this formula
If 𝔓 = Square root of all directions (more in higher dimensions)
and 𝔖 = the size of the packaging spheres (in radius)
and 𝔑 = the radius of the sphere you can fit in between the packaging spheres
then 𝔓 - 𝔖 = 𝔑
So this chart assumes that the dimension = 𝔓, and each packaging sphere = 1
|Radius of inner sphere
|1 – size of a packaging sphere
|2 – touching the walls of the box
2.162277660… – passed the walls of the box
As you can see, after the 9th dimension, you get a so-called tardis (it’s bigger on the inside).
You may wonder, why? The reason is, that the sphere does not actually break laws of space, but instead it just gets an extra direction to extend into, for example, a 3d sphere can also go up and down as well as all the directions a 2d sphere can go.
Another interesting thing, because the higher dimension cubes (hypercubes) have more sides, they also have more points, making them “spikey”, yet still a cube, rather than a prickly pear. Because they are spikey, they are also more complex and harder to simulate, meaning it is only possible to create a projection of up to a 5d cube, the rest are too complex to comprehend.
Not only are there higher dimensions, but there are also lower dimensions, the limit being 0d, a.k.a a dimensionless point, such as what you would find at the center of a black call, its nicknamed “singularity”, then you get 1d, or a line and 2d, or a flat plane (not the flying one), and then, the ultra special 3d, a.k.a the universe we live in, being so special because it is not too simple, and not too complex.
With higher dimensions, you could have a object that shrinks across the 4th dimension, and when moved across the 4th dimension will seem to grow and shrink, then disappear completely, and, physics still apply in higher dimensions, so moving through the ground would mean that the object would not be able to reappear, but instead bump into a 3d barrier trying to move back into the space.
This universe has been suggested to have not only 3 dimensions, but 3 space dimensions, 6 hyperspace dimensions, 1 time (temporal) dimension and 1 energy dimension.
A space dimension is not a hyperspace dimension, and in reality, we actually have 9 dimensions, only, 6 of those dimensions are curled up at sizes smaller than a hydrogen nucleus, being just too small to detect with current technology. Meaning the remainder of 3 dimensions are the ones that are “universal” and seen everywhere in the universe.
A temporal dimension is interlaced with the 6 hyperspace and the 3 space dimensions, and there can only be a single temporal dimension, called “time”, and these dimensions cannot be decisively moved across, and it just shifts everything forward or backward depending on the object's speed (according to the theory of relativity).
The energy dimension is the dimension of energy and keeps track of the energy’s form and position in space, it is interlaced with the two types of space dimensions and controls the matter and energy found in the space and time provided by other dimension types.
In this universe we only have 3 space dimensions, although in other possible universes, there could be more than that, theorised to be up to 15 space dimensions! As far as graphing goes, you can get wireframes of a 4d or 5d cube, such as the tesseract (4d cube rendering).
Most of this information was sourced from wikipedia, and was double checked on a specialised encyclopedia.