1. SURPRISES IN HIGHER DIMENSIONS
MONISH AHLUWALIA, SOMMER CHOU, AMORY CONANT, ELIAS VITALI
Abstract. The world which surrounds us is often viewed as having three spatial dimensions.
However, just as we can represent some structures in lower dimensions, there exist higher dimensions
defined by the n-dimensional euclidean space coordinates. While dimensions higher than three are
difficult to comprehend spatially, they exist everywhere in our world in the form of fundamental
scientific concepts such as Einstein’s Theory of General Relativity, business and economic models,
software and hardware developments, etc. This paper introduces the basic concepts of how higher
dimensions are defined, some practical real-life applications, and some surprising and interesting facts
resulting from higher-dimensional analysis.
Key words. Gamma function, n-dimensional euclidean space, n-ball, hypervolume, Gaussian
function.
1. Introduction. The concept of higher dimensional spaces is one that math-
ematicians have investigated for years and continue to devote considerable resources
to its study. It is a topic that is not merely theoretical musings on dimensions we
do not perceive, but instead one that has real-world applications in a wide variety of
fields. Higher dimensional spaces, shapes, and functions allow for the modelling of
relationships and functions that rely on more than three parameters. In geometry we
may compute the volume of a shape, which is reliant on just three factors: height,
width, and depth. Yet, what happens if we want to model something which is reliant
on 4, 5, 10, or even thousands of factors? This is where high dimensional functions
are used, as they can be used to model anything reliant on n number of factors.
For example, a large retail store may sell thousands of different products, each
of which are bought at different rates by consumers. Creating a higher dimensional
function allows the company to understand the relationships and patterns apparent in
their sales, and subsequently decide what product needs to be sold where and when.
Higher dimensional concepts span many other disciplines as well, such as in biology,
where an organism is defined by the niche it occupies in an environment. This niche is
defined as a n-dimensional hypervolume, where n is the number of factors required for
survival. Fully understanding high dimensional space thus allows individuals across
a vast range of fields to accurately understand the complex relationships that exist
within our world.
Motivation for Study: The inspiration for this project came from the novel and
film ”Flatland” by Edwin Abbott Abbott, which tells the tale of a 2-Dimensional world
being introduced to the third dimension. The motivation to study higher dimensions
also comes from the incredible applications that it can have in our everyday life, which
are further discussed in the body of this paper.
2. Some Definitions.
2.1. n-Dimensional Euclidean Space. Euclidean space can be thought of as
simply a set of points in space, where each point can be expressed by some factor,
such as distance or angle. The space is defined as all the n-tuples of all real numbers
(x1, x2 · · · xn), and is denoted as Rn
. For example, a familiar shape to many is a
rectangular prism, which is defined by three vectors: x, y, and z. It is thus called
three-dimensional, and we model it in the space R3
. It is harder to visualize shapes
and functions in dimensions higher than 3, but by the given definition above, it is
nonetheless possible to analyze infinite dimensions (Weisstein n.d.).
1

2. 2 Surprises in Higher Dimensions
Fig. 2.1. A hypersphere in three dimensions, where the sphere encompasses the n-ball. Dashed
lines show projection of part of the sphere surface from the origin (Ball 1997).
2.2. Hyperspheres. A hypersphere is the generalization of a circle or sphere in
a dimension where n is greater than or equal to 4. It is composed of a series of points
that are all equidistant from a defined central point, and is modelled by the equation
x2
1 + x2
2 + · · · + x2
n = R2
, where R represents the radius (Weisstein n.d.).
2.3. Hyperballs. A hyperball, also known as an n-ball, is the space in n dimen-
sions which is encompassed by a (n − 1) hypersphere. It can be visualized in three
dimensions as the space within the sphere shown in Figure 2.1 (Ball 1997; Weisstein
n.d.).
2.4. Hypercubes. A hypercube is an analogue of a cube in a dimension where
n is greater or equal to 4. It is composed of a series of line segments that are of equal
lengths and are aligned perpendicularly to each other (Weisstein n.d.).
3. The Volume of an n-dimensional Ball.. From everyday life, it is generally
known that the area of a circle (2-ball) is πr2
and that the volume of a sphere (3-ball)
is 4
3 πr3
. Now, how can we evaluate the volume of balls in higher dimensions? In
order to find an expression for the volume of an n-dimensional ball (or n-ball), we
first redefine what an n-ball is. In this section, we show the derivation for the volume
of an n-ball, where n ∈ R.
3.1. The n-ball. As previously mentioned, an n-ball of radius R consists of a
set of points such that the distance from the origin of the n-ball is less than or equal
to R. In n-dimensional euclidean space, a point is designated by the set of points
(x1 +x2 +· · ·+xn). Denoting the equation of the hypersphere using this set of points
gives the following:
(3.1) x2
1 + x2
2 + · · · + x2
n = R2
3.2. Finding the volume of an n-ball using the Gaussian function. We
start with the gaussian function:
(3.2)
ˆ ∞
−∞
e−x2
dx

3. Ahluwalia, Chou, Conant, Vitali 3
This is very difficult to integrate, so using Fubini’s theorem, we expand the inte-
gral using a binomial expansion (n = 2) and set I =
´ ∞
−∞
e−x2
dx. Thus,
(3.3) I2
= [
ˆ ∞
−∞
e−x2
dx]2
=
∞ˆ
−∞
∞ˆ
−∞
e−x2
1 e−x2
2 dx1dx2
Switching to polar coordinates for simplicity:
(3.4)
∞ˆ
0
2πˆ
0
e−r2
rdrdθ
Evaluating this gives us:
(3.5) I2
= π
And equally,
(3.6) I =
ˆ ∞
−∞
e−x2
dx =
√
π
We now explore this same strategy but in the n-th dimension:
(3.7) In
= [
ˆ ∞
−∞
e−x2
dx]n
=
ˆ ∞
−∞
· · ·
ˆ ∞
−∞
e−(x2
1+x2
2+···+x2
n)
dx1 · · · dxn
Here, we note some properties:
1. x2
1 + x2
2 + · · · + x2
n = r2
where r is the radius of the n-ball.
2. Angular variables can be generalized to dΩn−1; two components in 3 dimen-
sions, one component in 2 dimensions, etc.
3. Evaluating the integral of only the angular components yields the surface
area.
4. We can view the radial part of the fudge factor as rn−1
; r in 2 dimensions,
r2
in 3 dimensions, etc.
We can now explicitly separate the radial and angular integrals, just like we did for
the 2-dimensional example. Note that this is simply a transformation from Cartesian
to Polar coordinates with the Jacobian J = |rn−1
|
(3.8)
ˆ
· · ·
ˆ
e−(x2
1+x2
2+···+x2
n)
dx1 · · · dxn =
ˆ
e−r2
rn−1
dr
ˆ
dΩn−1
Using aforementioned Property 3, we observe that the second integral on the right
hand side above has a surface area element Sn.
(3.9) Sn =
ˆ
dΩn−1
We now solve for Sn. Begin by using the substitution u = r2
(3.10)
ˆ
e−r2
rn−1
dr =
1
2
ˆ
e−u
u
n
2 −1
du

4. 4 Surprises in Higher Dimensions
We know define the Gamma Function Γ(x) which has some notable properties,
described by a review paper by Detlef Gronau (2003).
(3.11) Γ(x) = (x − 1)! ∀x ∈ Z
(3.12) Γ(
1
2
) = (π)
(3.13) xΓ(x) = Γ(x + 1)
(3.14) Γ(
n
2
) =
ˆ
e−u
u
n
2 −1
du
Subsequently,
(3.15) [
ˆ
e−x2
dx]n
=
1
2
· Γ(
n
2
) · Sn
(3.16) [
√
π]n
=
1
2
· Γ(
n
2
) · Sn
(3.17) Sn =
2π
n
2
Γ(n
2 )
Now that we have an equation for the surface area Sn, we integrate with respect
to r to obtain an expression for the volume V (n) of an n-ball.
(3.18) Vn =
ˆ
Sndr =
ˆ
Sn · rn−1
dr = Sn
ˆ
rn−1
dr = Sn
rn
n
Substituting our expression for Sn yields
(3.19) Sn
rn
n
=
π
n
2
n
2 Γ(n
2 )
rn
Using the Gamma function property, xΓ(x) = Γ(x + 1):
(3.20) Vn =
π
n
2
Γ(n
2 + 1)
rn
Since the gamma function Γ(x) is a factorial expansion, we can rewrite it using
factorial notation where the parity of n determines the function.
For n being even:
(3.21) Vn =
π
n
2
(n
2 )!
rn
For n being odd:
(3.22) Vn =
2(2π)
n−1
2
n!!
rn
Note: These equations were derived using proofs from Gipple (2014) and the
University of California (2011).

5. Ahluwalia, Chou, Conant, Vitali 5
3.3. Some Examples. We can now compute the volume of any n-ball of radius
r. Let’s try it with some familiar dimensions:
In 2 dimensions (circle): n = 2, so we use equation 3.21.
(3.23) Vn = πr2
In 3 dimensions (sphere): n = 3, so we use equation 3.22
(3.24) Vn =
4
3
πr2
Continuing this trend for some higher dimensions allows us to calculate the volume
of an n-ball as a function of r, as summarized in Table 3.1 below.
Table 3.1
The volume of an n-ball in the first 10 real dimensions.
n General Volume Unit Ball Volume (r = 1)
0 1 1
1 2R 2
2 πR2
3.1416...
3 4
3 πR3
4.1887...
4 π2
2 R4
4.9348...
5 8π2
15 R5
5.2683...
6 π3
6 R6
5.1677...
7 16π3
105 R7
4.7248....
8 π4
24 R8
4.0587...
9 32π4
945 R9
3.2985...
10 π5
120 R1
0 2.5502...
Plotting the above trend shows a volume distribution shown in Figure 3.1
Fig. 3.1. The volume of a unit ball as a function of dimension n.

6. 6 Surprises in Higher Dimensions
4. Surprising Facts in Higher Dimensions.
4.1. Volume of a Hypersphere. As shown by Figure 3.1, the volume of an
n-sphere as n approaches infinity is zero. To understand this, we imagine a hypercube
of volume V = 1 containing a hypersphere tangent to all sides. This hypersphere is of
radius r = 1
2 . We know 1−Vn(1
2 ) is the volume inside the cube but outside the sphere.
As n approaches infinity, 1 − Vn(1
2 ) tends towards 1, meaning that Vn(1
2 ) tends to
zero. This proves that as the dimension increases, more volume is contained outside
the hypersphere, but inside the cube (SCIPP 2011). This is explained in further detail
in the section below denoted Kissing Hyperspheres.
4.2. ’Kissing’ Hyperspheres. Assume we have a square of side length 2 cen-
tred at the origin in R2
. We divide the square into four equal quadrants of side length
1 and draw a circle in each, so that the circle is tangent to all four sides of each
quadrant. Then, draw a circle in the center so that this new circle is tangent to the
four surrounding circles. This can also be taken into three and further dimensions.
Representations of this for the second and third dimensions are shown in Figure 4.1.
Fig. 4.1. A diagram showing ’kissing’ spheres in R2 and R3.
Each of the outer circles/spheres has a radius of r = 1
2 . Now the question we pose
is what happens to the radius of the center sphere rn as the number of dimensions
approaches infinity? First, the length of the vector from the center of a section to the
origin is rn + 1
2 and the center of the positive sector in n-dimensions is (1
2 , ..., 1
2 ). We
now get the following equation:
(4.1) rn +
1
2
= ||(
1
2
, ...,
1
2
)|| = (
1
2
)2
+ ... + (
1
2
)2
= n · (
1
2
)2
=
√
n ·
1
2
(4.2) rn =
1
2
(
√
n − 1)
This means that the radius of the center sphere goes to infinity at the speed of√
n as n increases towards infinity. In fact, if we suggest that n = 9, the radius of

7. Ahluwalia, Chou, Conant, Vitali 7
the center sphere becomes 1, meaning its now tangent to the cube itself. Any n > 9
will mean that the volume of the center sphere is greater than the volume of the cube
(Feres 2006).
4.3. Concentrations of Volumes on Shells. Consider the volume of a spher-
ical shell of radius r and thickness a as shown in figure 4.2.
Fig. 4.2. Diagram of a spherical shell of radius r and thickness a in two dimensions.
It’s observed that the diagram consists of two concentric circles. The larger has
radius r and the smaller has radius r − a. The question we pose here is what fraction
of the ball Vn(r) is contained in the shell as n approaches infinity?
(4.3) Vratio =
Vn(r) − Vn(r − a)
Vn(r)
=
rn
− (r − a)n
rn
= 1 −
(r − a)n
rn
(4.4) Vratio = 1 − (1 −
a
r
)n
As demonstrated by the final equation, as n approaches infinity, the volume of
the sphere becomes concentrated in the shell. For example, if a = 0.1, r = 1, n = 100,
then 99.997% of the volume of the sphere would exist within the shell. This value
asymptotically approaches 100% as n approaches infinity (Feres 2006).
4.4. The Kissing Number Problem. This problem dates back all the way
to 1694 to a debate between Newton and Gregory. They argued about the number
of non-overlapping spheres of r = 1 that can simultaneously touch a centered sphere
of the same radius. In two dimensions, this number can easily be shown to be 6
by analyzing the number of billiards balls you could place around a centered ball.
Newton correctly predicted the number in three dimensions to be 12 spheres (while
Gregory argued 13). It was only recently proven in 2003 that the kissing number for
four dimensions is 24. However, beyond that, it becomes ambiguous. For dimensions
greater than four, there has been no explicit proofs for the respective kissing numbers,

8. 8 Surprises in Higher Dimensions
only a range. For example, in five dimensions, the kissing number could be between
40 and 45 (Boyvalenkov et al. 2012). However, this is with the exception of the eighth
and twenty-fourth dimension whose kissing numbers are explicitly 240 and 196 560
respectively (Musin 2008). These were first proven by Levenshtein as well as Odlyzko
and Sloane in 1979 using Delsartes method. However, it is only these dimensions that
the method gives a precise result (Boyvalenkov et al. 2012).
The eighth and twenty-fourth dimensions are special because they have the prop-
erty of being represented as lattices, the E8 and Leech Lattice respectively. The
definition of a lattice satisfies two conditions:
1. The arrangement where the origin is the center.
2. Given coordinates any two centers, u and v, both u + v and u − v are also
centers (Robert 2006).
An example of a two-dimensional lattice and a three dimensional depiction of the
eight dimensional lattice can be seen in Figure 4.3 below.
Fig. 4.3. A lattice representation in two-dimensions (left) and three-dimensions (right).
Based on this, you can generate the entire lattice using simple vector addition and
subtraction. In essence, the fact that these two dimensions form lattices is why the
kissing number problem can be solved for these dimensions in specific using known
methods (Robert 2008).
5. Proposed Algorithm. This simple algorithm will extract a positive integer
from a student number with a low probability of overlap between students. In this
case, the algorithm is a simple formula which will tell students which n-dimensional
euclidean space they belong to. The proposed algorithm A keeps the integers relatively
low:
(5.1) A =
Sum of even numbers − Last Digit
2
This will yield a value which we assign as the student’s personal dimension, and

9. Ahluwalia, Chou, Conant, Vitali 9
we can use that to compute their volume as an n-ball or their central radius if they
were a central kissing ball bounded by other kissing n-balls.
6. Practical Applications of Understanding Higher Dimensions. The
notion that humans exist in a 3-dimensional world is universally understood. By clas-
sifying time as a separate higher dimension, it can then be said that everyday people
are living in 3 + 1 dimensions, referred to as the 4-dimensional space-time (Walling
and Hicks 2003). However, even harder to comprehend are higher spatial, rather than
temporal, dimensions that are heavily theorized. Nevertheless, the explanation, anal-
ysis, and representation of various major concepts in scientific fields often involve the
application of hyper-volumes and higher dimensions.
6.1. Ecology. With respect to ecological studies, organisms are generally de-
fined as existing in a specific niche. This can be mathematically described as an n-
dimensional hypervolume where the n-dimensions incorporate all required resources
for the indefinite sustentation of that particular species (Blonder 2014). Examples
of these factors include habitat space, food, shelter, light, and any other physical or
chemical resource required to maintain the population. In nature, however, condi-
tions are usually not ideal, and the realized ecological niche falls short of the fun-
damental niche due to limiting factors such as interspecific competition, intraspecific
competition, and inadequate resources. The relationship between the n-dimensional
hypervolume and the actual realized ecological niche is depicted in Figure 6.1.
Fig. 6.1. A graphical depiction of the fundamental and realized niches where the former repre-
sents an n-dimensional hypervolume and the latter is the n-dimensional hypervolume with additional
interactions in an ecosystem (Dudley n.d.)
6.2. Neuroscience. In terms of neuroscientific representations, the activity of
neurons can also be presented in a multidimensional fashion. Areas of research in the
field usually feature the measurement of current through a neuron after inducing an
action potential with a stimulus. Thus, if more than one electrode is being used to
measure impulses in different neurons, an n-dimensional vector would be required to
represent the neural activity data, where n is the number of samples obtained (Math

10. 10 Surprises in Higher Dimensions
Insight n.d.).
6.3. Drug Discovery. In the field of drug development and pharmaceuticals,
the pre-clinical phase usually requires that hundreds or thousands of tests be con-
ducted in order to fully understand a drugs mechanism of action. Typically, scientific
hypotheses involve the identification of a measurable dependent variable, so changes to
the system when an independent variable is altered can be quantified. The issue with
this is that the process can be extremely lengthy and tedious if multiple independent
variables are to be tested, which is usually the case with pre-clinical drug develop-
ment. Therefore, to increase efficiency, biological assays are used. These assays allow
for a higher-dimensional analysis by changing the dependent and independent variable
combinations so that multiple hypotheses can be tested at once (Verbist et al. 2015).
7. Conclusion. In summary, this paper looked at introducing the concept of
higher spatial dimensions by examining volumes of n-balls and various other phe-
nomenons which are surprising or unexpected, such as the kissing balls, the n-ball
volume convergence to zero, and the spatial concentration of the volume in an n-ball.
Additionally, some real-life applications were explored, specifically ones pertaining to
science and how high-dimensional theory can aid our understanding of fundamental
scientific concepts.
Acknowledgments. We want to thank Dr. George Dragomir for helping us and
supporting us through this project and providing helpful feedback, as well as providing
us with the opportunity to dive deeper into abstract mathematical concepts.

11. Ahluwalia, Chou, Conant, Vitali 11
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