finding ourselves in the universe_ a mathematical approach to cosmic crystallography
TRANSCRIPT
UNITED STATESMILITARY ACADEMY
WEST POINT, NEW YORK
HONORS THESIS
FINDING OURSELVES IN THE UNIVERSE: AMATHEMATICAL APPROACH TO COSMIC
CRYSTALLOGRAPHY
by
CDT J. D. Menges
May 2011
Thesis Advisor: Dr. Jessica MikhaylovSecond Reader: Major Diana LoucksSecond Reader: Dr. Janet Fierson
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13. ABSTRACT (maximum 200 words)Abstract of “Finding Ourselves in the Universe: A Mathematical Approach to Cosmic Crystallography,” by Cadet JoshuaDavid Menges, United States Military Academy, May 2011.This thesis discusses the concept of the 2-torus and the 3-torus and how each can be applied to the shape or our universe.Cosmic crystallographers use topologically repeating shapes like the 3-torus to try to prove that the universe repeats itselfafter a certain distance or, in other words, the universe wraps around, with one end of itself connected to the other end.This paper will look at the geometry of the 2-torus. Specifically, if the universe can be represented as a 2-torus, whatmathematics can be utilized by cosmic crystallographers in order to find replications of galaxy superclusters? This papertakes a mathematical approach to stationary clusters, moving clusters, and aging clusters.
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FINDING OURSELVES IN THE UNIVERSE: A MATHEMATICALAPPROACH TO COSMIC CRYSTALLOGRAPHY
CDT J. D. MengesCadet Lieutenant, United States Army
B.S., United States Military Academy, 2011
Submitted in partial fulfillment of therequirements for the degree ofBACHELOR OF SCIENCE
in MATHEMATICAL SCIENCESwith Honors
from theUNITED STATES MILITARY ACADEMY
May 2011
Author: CDT J. D. Menges
Advisory Team: Dr. Jessica MikhaylovThesis Advisor
Major Diana LoucksSecond Reader
Dr. Janet FiersonSecond Reader
Colonel Michael PhillipsChairman, Department of Mathematical Sciences
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ABSTRACT
Abstract of “Finding Ourselves in the Universe: A Mathematical Approach
to Cosmic Crystallography,” by Cadet Joshua David Menges, United States Military
Academy, May 2011.
This thesis discusses the concept of the 2-torus and the 3-torus and how each
can be applied to the shape or our universe. Cosmic crystallographers use topologi-
cally repeating shapes like the 3-torus to try to prove that the universe repeats itself
after a certain distance or, in other words, the universe wraps around, with one end
of itself connected to the other end.
This paper will look at the geometry of the 2-torus. Specifically, if the universe
can be represented as a 2-torus, what mathematics can be utilized by cosmic crystal-
lographers in order to find replications of galaxy superclusters? This paper takes a
mathematical approach to stationary clusters, moving clusters, and aging clusters.
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TABLE OF CONTENTS
I. BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
I.1. TWO AND THREE DIMENSIONAL WORLDS . . . . . . . . . 3
1. The 2-Torus . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. The 3-Torus . . . . . . . . . . . . . . . . . . . . . . . . . 6
3. The Universe as a 3-Torus . . . . . . . . . . . . . . . . . 7
4. Cosmic Crystallography . . . . . . . . . . . . . . . . . . . 10
I.2. THE GALAXIES OF THE LOCAL GROUP . . . . . . . . . . . 10
1. The Andromeda Galaxy (M31) . . . . . . . . . . . . . . . 11
2. The Milky Way Galaxy . . . . . . . . . . . . . . . . . . . 12
3. The Triangulum Galaxy (M33) . . . . . . . . . . . . . . . 12
I.3. STAR TYPES & STELLAR EVOLUTION . . . . . . . . . . . . 13
II. SCENARIOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
II.1. TWO DIMENSIONAL MODEL . . . . . . . . . . . . . . . . . . 15
1. Case 1: Stationary Galaxy Cluster, No Aging . . . . . . . 15
2. Case 2: Moving Galaxy Cluster, No Aging . . . . . . . . 18
3. Case 3: Aging Galaxy Clusters . . . . . . . . . . . . . . . 23
III. FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
III.1. EXPANDING UNIVERSE . . . . . . . . . . . . . . . . . . . . . 25
III.2. THREE DIMENSIONAL MODEL (III.4) . . . . . . . . . . . . 28
III.3. GALAXY MOTION WITHIN A CLUSTER . . . . . . . . . . . 30
IV. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
APPENDIX B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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LIST OF FIGURES
I.1. The Pacman Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
I.2. The Pacman Universe as viewed by Mr. Pacman . . . . . . . . . . . . 5
I.3. Mr. Pacman on a 2-Torus in 3 Dimensional Space . . . . . . . . . . . . 6
I.4. 3-Torus with Visible Boundaries [3] . . . . . . . . . . . . . . . . . . . . 8
I.5. 3-Torus with No Visible Boundaries [3] . . . . . . . . . . . . . . . . . . 8
I.6. The Universe as a 2-Torus . . . . . . . . . . . . . . . . . . . . . . . . . 9
I.7. The Local Group of Galaxies [5] . . . . . . . . . . . . . . . . . . . . . . 11
I.8. Relative Star Size in the Main Sequence . . . . . . . . . . . . . . . . . 14
II.1. Stationary Galaxies in a 2-Torus Universe . . . . . . . . . . . . . . . . 16
II.2. Observation Radius of√
2 UUs . . . . . . . . . . . . . . . . . . . . . . 17
II.3. Moving Galaxy Clusters in a 2-Torus Universe . . . . . . . . . . . . . . 19
II.4. Finding the Location of Replications of Moving Galaxies . . . . . . . . 21
III.1. A 2-Torus Expanding Universe . . . . . . . . . . . . . . . . . . . . . . 26
III.2. A 2-Torus Expanding Universe: Extended . . . . . . . . . . . . . . . . 27
III.3. A 2-Torus Expanding Universe: Continuous . . . . . . . . . . . . . . . 28
III.4. Scanning for Replications in a 3-Torus [3] . . . . . . . . . . . . . . . . . 29
III.5. Model Representing the Universe as a 3-torus . . . . . . . . . . . . . . 30
III.6. Moving the 3D Model to view the 2-torus representation . . . . . . . . 31
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LIST OF TABLES
I.1. Characteristics of Different Spectral Types . . . . . . . . . . . . . . . . 13
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ACKNOWLEDGMENTS
I would like to express my greatest and foremost thanks to my advisor, Dr.
Jessica Mikhaylov, for her wisdom, guidance, and continued support throughout my
undergraduate career at the United States Military Academy and specifically for
her shared interest in the science of Cosmic Crystallography. I owe my interest in
the science of Astrodynamics and Astrophysics to her teachings and passion on the
subjects.
I also wish to acknowledge Dr. Jessica Mikhaylov, Dr. Amanda Hager, Dr.
Janet Fierson, Dr. V. Frederick Rickey, and Dr. William Pulleyblank for their con-
tinued support and help in proving the Menges-Mikhaylov Theorem. Their passion
about this conjecture has pushed me to pursue the completion of its proof well after
the completion of this thesis.
I would like to thank the United States Military Academy Department of
Mathematical Science and Department of Physics and Nuclear Engineering for their
collective wisdom and knowledge in all topics of Mathematics and Physics and for
being there when I needed theoretical and applicable questions answered.
I wish to thank my parents, Bryan and Mary Menges, and my brothers, Jacob
and Andrew Menges, for their continued support and for pushing me toward my
academic goals. Specifically I would like to thank Mrs. Mary Menges, whose passion
toward the subject of mathematics and teaching her high school students has pushed
me to pursue my own academic goals in the field. Lastly, but certainly not least
of all, I would like to thank Miss Brandy Haines, whose love and support made the
completion of this thesis possible.
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EXECUTIVE SUMMARY
Executive Summary of “Finding Ourselves in the Universe: A Mathematical
Approach to Cosmic Crystallography,” by Cadet Joshua David Menges, United States
Military Academy, May 2011.
The first major topic of this paper discusses what a 2-torus is and what it
looks like in two dimensions and in three dimensions. While the universe is set in
three dimensions, we discuss the universe’s shape as a 2-torus first, followed by a
geometric description of the 3-Torus and its application to the universe. After a brief
background on the field of Cosmic Crystallography and its use of superclusters, the
background of the cluster containing the Milky Way Galaxy, known as the Local
Group, is discussed. The luminosities and star types of the three main galaxies
in the Local Group (Andromeda, Milky Way, and Triangulum) are discussed. The
spectral types of the stars are discussed, focusing on luminosity, temperature, mass,
and longevity.
The second major topic consists of three scenarios. All three scenarios take
place in a 2-torus universe. The first scenario discusses how to find replications in a
non-expanding universe consisting of stationary, non-aging galaxies. Based on a set
observation radius, the maximum angle a Crystallographer must span to find a single
replication is determined using the Lattice Conjecture. The second scenario discusses
how to find a replication in a non-expanding universe consisting of moving, non-aging
galaxies. Using the law of cosines, the maximum angle an observer must span for a
single replication is calculated. In the third scenario, we discuss aging galaxies instead
of non-aging galaxies. In these scenarios the longevity and luminosity of the specific
star types in the Local Group are used to determine what a crystallographer might
look for in space.
Finally, this paper briefly touches on future work that might be done on this
topic to include: the concept of the expanding universe, three dimensional 3-torus
scenarios, and galaxy motion within a cluster.
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I. BACKGROUND
A question that has baffled the minds of astronomers, mathematicians, physi-
cist, and essentially the entire human race for centuries has been the shape of space.
Specifically, does space have a shape? Has it been growing and expanding since the
Big Bang? If there is a shape, what shape is it? Scientists who specialize in trying to
find answers to this question are called Cosmic Crystallographers. Cosmic Crystal-
lography attempts to find the solution to the shape of the universe. Using the 2-torus
as a starting point, we will be looking into the world of Cosmic Crystallography and
what we would expect to see in the night sky, should the universe be a torus. After
a thorough explanation of the 2-torus and the 3-torus, we are introduced to the idea
and concepts behind cosmic crystallography. In Sections I.2 and I.3 we provide the
necessary background on galaxies in the Local Group and Spectral Types, respec-
tively. Using this background, in Section 4 of I.1 we discuss the science of Cosmic
Crystallography.
I.1. TWO AND THREE DIMENSIONAL WORLDS
We first discuss the concept of the 2-torus and the 3-torus. In Section 1 we
will look at the game Pacman and how Mr. Pacman’s world is essentially a 2-torus.
We will discuss the geometric shape of the 2-torus in two dimensions and in three
dimensions. In Section 2 we will discuss the geometric concepts of the 3-torus, and in
Section 3 we will apply those concepts to the universe. We will look at the universe
initially as a stationary cube with the properties of a 3-torus instead of looking at it
as an expanding entity. In Section 4, we will look into the scientific field of Cosmic
Crystallography, a branch of astronomy that attempts to find solutions to the question
of space shape. One of these attempts involves looking for replications of different
galaxy clusters, a concept we will be delving into in Chapter II.
1. The 2-Torus
In order to better understand the concepts of the 2-torus and the 3-torus, we
can look to Jeffrey Weeks’ The Shape of Space [1]. Weeks’ book explains simply yet
thoroughly the concept of tori and their orientability. The concept of a 2-torus can be
3
Figure I.1. The Pacman Universe
explained by observing motion in the video game Pacman. In Pacman, Mr. Pacman
moves around the screen along a continuous path until he reaches the top, bottom,
or one of the sides (Figure I.1). Mr. Pacman is not bounded by these sides. The
concept of gluing allows Mr. Pacman to continue his travels in spite of what we see as
him hitting a wall. Gluing is essentially what it sounds like. We take one side of the
screen and glue it to its direct opposite side. Once Mr. Pacman reaches the opening
at the bottom of the screen, he appears at the top in the same horizontal position.
In the same way, Mr. Pacman appears at the bottom of the screen after exiting the
top, the left side after exiting the right, and the right side after exiting the left. The
shape of Mr. Pacman’s world, based on this description, is a 2-torus. If we were to
look at this world from Mr. Pacman’s perspective, it would look like several of his
worlds were glued together at the edges as in Figure I.2.
4
Figure I.2. The Pacman Universe as viewed by Mr. Pacman
It might be hard to imagine how a two dimensional surface can be connected
to itself unless we bend the two dimensional surface in three dimensional space. What
would this shape look like in three dimensional space? We can do this by implementing
the concept of gluing. When we look at the world of Mr. Pacman in two dimensions,
it looks like nothing more than a square world, but when we take the top of this
square and glue it to the bottom of the square in three dimensional space, we get a
horizontal cylindrical tube. The left and right sides of the original square are now
represented in the edges at the left and the right of the tube. When we glue the
left of the tube to the right of the tube, we get a 2-torus in three dimensional space,
or a donut-like shape (Figure I.3). As three dimensional human beings in a three-
dimensional world, we may still be wondering how we can apply 2 dimensional Mr
Pacman’s small 2 dimensional world to our large 3 dimensional universe.
5
Figure I.3. Mr. Pacman on a 2-Torus in 3 Dimensional Space
2. The 3-Torus
The concept of a 3-torus is similar to that of the 2-torus. The difference is
the addition of the third dimension. Imagine that, like Mr. Pacman, we are in a
world where we are confined to a finite space, but instead of a two dimensional box,
we are in a three dimensional room with length, width, and breadth. Imagine that,
instead of being confined to this room, we are able to walk out of the room in any
direction: north, south, east, west, up, or down. As soon as we leave the room from
the north, we appear at the same spot on the south side of the same room. The
same happens if we leave the room from the east; we appear on the west side. If we
decide to exit through the floor, we reenter through the ceiling. To put it in simpler
terms, imagine that this room has completely transparent walls. No matter which
direction you look, you are going to see multiple replications of the back of your own
head as this replication of yourself is looking in that same direction. Figure I.4 shows
the concept of a 3-torus applied to Earth. In Figure I.4 the boundaries of the torus
are visible only for the purpose of visualizing what a 3-torus looks like. In Figure I.5
the boundaries have been taken away in order to visualize what a 3-torus universe
consisting of only Earth might look like in space. It is important to remember that
all of these Earths are simply replications of the actual Earth (of which there is only
one) and that there is not more than one Earth. The process of gluing causes multiple
views of replications in each direction.
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3. The Universe as a 3-Torus
The concept of a 3-torus is simple to picture in terms of a cube-shaped room.
Now imagine this concept applied to the entire universe. If the universe is a 3-torus,
we can travel far enough in one direction from Earth and return to Earth without
ever changing course. In theory, astronomers should be able to look out into space
and see replications of our own galaxies in the far-off distance. The fact that we have
not identified these replications does not necessarily mean that our universe is not a
3-torus. Many factors are at play when attempting to observe a galaxy replication.
Let us pretend that the universe is presently 14 billion light-years across. This means
that the light from the Milky Way replication would take 14 billion years to reach our
Milky Way. The system we would be looking for is what the Milky Way Galaxy would
have looked like 14 billion years ago. Many of the stars in the system may not have
even been in existence yet. The universe is only approximately 13.7 billion years old.
If the universe is stationary and not expanding, and the universe is more than 13.7
billion light-years across, the light from the replication would not have even reached
our observations yet. Physicist Brian Greene explains that because the universe is
expanding, “the expansion of space increases the distance to the objects whose light
has long been traveling and has only just been received; so the maximum distance we
can see is actually longer – about 41 billion lightyears.” [2]
Age and expansion of the universe aside, the observing astronomer would also
have to look in the correct direction. This proves to be a harder task than finding
an older version of the Milky Way. It is impossible to know how the primary axes
of the universe are oriented with respect to Earth. Refer to Figure I.5. If we were
to look at this image and this image only, we could place the boundaries of the 3-
torus anywhere and be correct in our placement. This concept allows us to move the
orientation of the 3-torus to our advantage without loss of generality. Additionally,
Albert Einstein’s theory of relativity introduced the concept of light bending in the
presence of large masses. When observers are looking for replications, they must take
into account that barely visible, highly massive black holes or neutron stars might be
in the line of sight. These masses would bend the light of the replication. Observors
might overlook this bending of light (not knowing about the presence of the masses)
and think that the replication is in a place where it is not.
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Figure I.4. 3-Torus with Visible Boundaries [3]
Figure I.5. 3-Torus with No Visible Boundaries [3]
8
Figure I.6 shows a two dimensional simple representation of the universe with
a galaxy (we will call this the Milky Way) in the center. Around this universe are
replications of the universe in a flat 2-torus. When the observer, located at galaxy
OP , looks in the direction of Line 1, he can see multiple replications of the galaxy.
When the observer looks in the direction of Line 2, it will be harder for him to
observe the replication he is searching for, due to the distance needed to observe the
replication. We must remember that in Figure I.6, while Line 1 seems to go through
multiple universes, these are just replications, and the line is actually going through
the same universe multiple times. Depending on the direction an observer is looking,
he may see a replication at a short distance from his observation point or it may be
very far away. Conversely, if an observer looks out into the unverse at a set radius, he
will have to span a certain percentage of his entire observation range in order to find
a replication at that specific radial range. We will refer to the limiting angle used to
calculate the span percentage as the ‘max angle’.
Figure I.6. The Universe as a 2-Torus
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4. Cosmic Crystallography
As stated previously, determining whether or not we live in a multiconnected
universe is not as easy as simply pointing a telescope in an arbitrary direction and
looking for another Milky Way. We have already noted that stars in a galaxy age
in such a way that by the time the light from a replication reaches an observer, the
galaxy has completely changed in overall luminosity and orientation with respect to
nearby galaxies. Stars also move within a galaxy, which brings about the problem
of different shapes. The Milky Way was not its present shape several billion years
ago; searching for a galaxy of younger stars in the same shape as the Milky Way
would prove fruitless. This is not the only problem associated with moving celestial
bodies. Galaxies in and of themselves also move. For example, the Milky Way moves
at approximately 600 km/second. Observers searching for replications now not only
have to deal with observing galaxies at different points in their history, but also at
different points in space. Cosmic Crystallography solves this problem by working
not with a single galaxy but by looking at galaxy clusters or superclusters. Instead
of one galaxy, imagine, for example, three galaxies in a specific formation. Cosmic
Crystallographers would lok for replications of this specific cluster formation in order
to prove the existence of a 3-torus universe. Using a cluster of galaxies is helpful
because while the galaxies in the cluster still move over time, the uncertainty in the
position of a cluster is far less than the uncertainty in the position of an individual
galaxy. Another advantage of using superclusters is that there are billions of galaxies,
while there are only a few hundred superclusters. Superclusters are clusters of galaxies
oriented in a specific recognizable pattern.[1] For our purposes, we will be studying
the three major galaxies in the Local Cluster: the Andromeda Galaxy, the Milky Way
Galaxy, and the Triangulum Galaxy.
I.2. THE GALAXIES OF THE LOCAL GROUP
Cosmic Crystallographers use galaxy clusters to determine the possible move-
ment and placement in space of possible repeated galaxies. For the purposes of this
paper, we will be analyzing the three major galaxies in the Local Group: The An-
dromeda Galaxy (M31); The Milky Way Galaxy; The Triangulum Galaxy (M33).
In each of the subsections, we will analyze traits and characteristics of each galaxy,
specifically the number of stars in each galaxy and the type of stars in each galaxy
10
which ultimately determines the overall luminosity of the galaxy. In Sections 1, 2,
and 3 we will discuss the attributes of Andromeda, Milky Way, and Triangulum, re-
spectively. A more in depth analysis of each of the main galaxies in the Local Group
can be found in Sidney van den Bergh’s The Galaxies of the Local Group [4]. Figure
I.7 shows where major galaxies in the Local Group are located relative to each other.
Figure I.7. The Local Group of Galaxies [5]
1. The Andromeda Galaxy (M31)
Of all the galaxies in the Local Group, the Andromeda Galaxy (M31) is the
most luminous. Located approximately 2.5 million light years from our Milky Way
Galaxy, the Andromeda Galaxy is the closest spiral galaxy to our own. Spiral galaxies
like Andromeda and the Milky Way consist of a rotating disk of gas, dust, stars and
the planetaries orbiting the stars. These celestial objects rotate around a central
11
bulge consisting of a heavily massed concentration of stars, dust and gas which is
more luminous than the rest of the disk. Another reason for the higher luminosity
in the bulge is the high concentration of O-Type and B-Type stars located in the
bulge (O-Type and B-Type stars are the brightest stars in the sky). The luminosity
and expected age of the different spectral types will be discussed further in Section
I.3. The nuclear bulge of Andromeda contributes approximately 30% to the overall
luminosity of the galaxy. To date, the Andrmeda Galaxy has an estimated over 1
trillion (1 x 1012) stars. Because the Andromeda Galaxy is the most luminous galaxy
in the Local Group, knowledge of the placement of certain stars is crucial in order for
Cosmic Crystallographers to observe repetitions of the Local Group. We will discuss
which stars are most important to Crystallographers in Section I.3.
2. The Milky Way Galaxy
Our planet Earth is located in the Solar System of the yellow star Sol which
is a small part of the spiral galaxy known as the Milky Way. As in Andromeda,
luminous O-Type and B-Type stars also make up a majority of the bulge in the
Milky Way Galaxy, causing the Milky Way to be the second most luminous galaxy
in the Local Group. It is currently being predicted that, within the next 5 billion
years, the Andromeda Galaxy will collide with the Milky Way Galaxy. There is
debate whether the collision will result in a merger of the two galaxies into one larger
galaxy or whether the two galaxies will simply pass through one another exchanging
several stars and planetaries, including the Earth and the Sun, in the process. Several
scientists speculate that post-collision, the Earth and its Solar System will be part
of the Andromeda Galaxy, while others speculate on the forming of a ‘Milkomeda
Galaxy’ [7].
Currently there are approximately 200 billion to 400 billion (2−4 x 1011) stars
residing in the Milky Way Galaxy.
3. The Triangulum Galaxy (M33)
The third brightest galaxy in the Local Group is M33, or the Triangulum
Galaxy. Triangulum is the only other large spiral galaxy in the Local Group. While
it is considered a spiral galaxy and appears to have a much brighter center than
the outer portions of the spiral disk, the existence of a true nuclear bulge remains
12
contorversial due to the diffuse nature of the galaxy and of the bulge in general [4].
According to van den Bergh, color-magnitude diagrams of the star clusters within the
Triangulum Galaxy tend to suggest that most of the stars in M33 are very young.
“[O]bservation with the Hubble Space Telescope... has formed an unbiased sample of
60 clusters in the disk of M33. These observation suggest that star clusters formed
continuously from 4 x 106 to 1 x 1010 yr ago.”[4] The young age of the stars in the
Triangulum Galaxy will prove crucial to Cosmic Crystallographers when searching
for cluster replications.
Currently there are approximately 40 billion (4 x 1010) stars residing in the
Triangulum Galaxy.
I.3. STAR TYPES & STELLAR EVOLUTION
In the late 1800s, the Harvard College Observatory classified the stars by plac-
ing them into spectral groups. These spectral groups were based on certain charac-
teristics such as mass, luminosity, temperature, and longevity. For each spectral type
(O, B, A, F, G, K, and M), Table I.1 shows these characteristics and the abundance
of each spectral type in the universe. [6]
Spectral Type O B A F G K MTemperature 40,000K 20,000K 8,500K 6,500K 5,700K 4,500K 3,200K
Radius (Sun = 1) 10 5 1.7 1.3 1.0 0.8 0.3Mass (Sun = 1) 50 10 2.0 1.5 1.0 0.7 0.2
Luminosity (Sun = 1) 100,000 1,000 20 4 1.0 0.2 0.01Longevity (million yrs) 10 100 1,000 3,000 10,000 50,000 200,000
Abundance 0.00001% 0.1% 0.7% 2% 3.5% 8% 80%
Table I.1 Characteristics of Different Spectral Types
Figure I.8 shows the relative size and color of each spectral type. For more
information on spectral types, please refer to Appendix A.
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Figure I.8. Relative Star Size in the Main Sequence
14
II. SCENARIOS
We are ultimately trying to find the maximum angle an observer in our galaxy
cluster would have to scan around the universe in order to see a replication of our
galaxy cluster if such a replication exists.
II.1. TWO DIMENSIONAL MODEL
1. Case 1: Stationary Galaxy Cluster, No Aging
In this first case, we will use a two dimensional 2-torus model to represent
the universe (as seen in Figure II.1). For simplicity’s sake, in this first case, we will
assume that the galaxy cluster does not move within the universe, the galaxies do
not move relative to one another in the cluster, and the galaxies do not age or change
luminosity over time. With these assumptions we can model this 2-torus universe as
a simple two dimensional Euclidean grid with each box representing a whole universe,
not unlike Mr. Pacman’s view of his universe. Because the galaxies are not moving
relative to each other, we will represent a galaxy cluster, such as the Local Group, as
a point instead of three separate galaxies.
15
Figure II.1. Stationary Galaxies in a 2-Torus Universe
The universe in the lower left hand corner will be the universe from which
we are observing replications. We will call this point OP (Observation Point). If,
from point OP , the observer looks directly to the right, he will see the first replicated
galaxy cluster. We will say that this universe is a distance of ‘one Universe Unit’, or
1 UU, away (e.g. if the observer looks for a replication to his right beyond the first
replication, he will need to look a distance of 2 UUs). Recall that the observer’s goal
is to find the maximum angle he needs to scan in order to be guaranteed observation
of a replicated galaxy cluster. This can be determined by drawing a circle around
point OP using the observation radius. Because the galaxies are not moving in this
case, we see that any circle around point OP will encompass the same number of
galaxy clusters separated by the same angles in each quadrant of observation. This
16
being the case, without loss of generality, we will only focus on the first quadrant,
as the model shows. When a circle with radius one universe is drawn, the maximum
observation angle is π2. If total observation is defined as the observer observing 360
degrees around the observation point, a radius of 1 UU will cause the observer to
observe a 25% portion of total observable area in order to be guaranteed sight of
a replication. In order to decrease this percentage, the observer must increase the
radius of observation. From this point, let us view the model as a grid. The horizontal
axis will be the n-axis, and the vertical axis will be the m-axis with OP as the origin.
The next nearest galaxy can be found at (n,m) = (1, 1) (Figure II.2).
Figure II.2. Observation Radius of√
2 UUs
The radius can be written as r =√n2 +m2 UUs. In this case the next
closest galaxy lies at r =√
12 + 12 =√
2 UUs. When a circle is drawn around OP
17
with a radius of√
2, the circle encompasses the new replication at (n,m) = (1, 1) as
well as the previous two galaxies at (0, 1) and (1, 0). We can now draw a ray from
OP through every point encompassed in the drawn circle. Once we calculate every
successive individual angle formed by these rays (starting at the n-axis), the largest
calculated angle (what we will refer to as the ‘max angle’) will be the angle used to
calculate the percentage of total observation needed to view a replication at a given
radius. In this specific case, radius of√
2, the max angle is π4
which is 12.5% of 360
degree observation.
Next, we attempt to determine the max angle for any r =√n2 +m2 UUs.
This can be solved using the Menges-Mikhaylov Theorem (Appendix B):
For any r =√n2 +m2 UUs, r can be rewritten as
√N2 + 1. Solve for N. The
maximum angle of observation θ needed to guarantee observation of a replication is
as follows:
θ = arctan1
bNc(II-1)
Look at Figure II.2. In this case, r =√
12 + 12 =√
2 UUs. According to
Equation II-1, the calculated max angle is:
θ = arctan1
b√
2c= arctan
1
1=π
4
2. Case 2: Moving Galaxy Cluster, No Aging
While Case 1 is ideal for cosmic crystallographers, galaxies within a universe
are not stationary. On the contrary, they move constantly at high velocities across
the universe. In Case 2, we will discuss what an observer might look for if the universe
contains galaxy clusters that move within the universe. In this case, galaxies will not
move relative to one another within the universe, and they will not age. Because
the galaxies are not moving relative to one another, we can continue to represent the
cluster as a point. Instead of observing only the first quadrant of observation, we will
initially look at all four quadrants (all 360 degrees of observation). As the observer, we
assume that the galaxy cluster we are trying to observe moves at a constant velocity
through the universe. The grid of the 2-torus can be oriented as we wish without loss
of generality, since there is no set start or end point or preferred orientation of the
universe. We can orient the torus such that the galaxies are moving at a constant
18
velocity v in the upward direction. In Figure II.3, we observe what the replicated
universes might look like if we only observe on the m and n axes. Notice that as m
or n increases in any direction, the observed replication gets closer and closer to the
edge of the universe. This may look as though the galaxies move outside the universe
as the observation radius increases. We must recall that the observed universe is still
our actual universe. The actual location of the observed replicated universe is at OP.
The light from the replication just appears to be outside of the universe due to the
speed of light and the observation of that light from OP.
Figure II.3. Moving Galaxy Clusters in a 2-Torus Universe
In addition to observing replications on specifically the m and n grid axes, we
must look for replications elsewhere in order to be able to calculate a max angle for
19
a given observation radius. Look at Figure II.4.
20
Figure II.4. Finding the Location of Replications of Moving Galaxies
21
In this figure, the observer is looking for a replication in the fourth quadrant of
observation. If the galaxy clusters were stationary as in Case 1, the observed galaxy
replication would be located at (n,m) = (1,−1). As we have just noted, the galaxy
is actually at (n,m) = (1,−1). We will refer to this point (where the galaxy actually
is) as GA. The observed galaxy will be at some point below GA. We will refer to
this point at GO. It is now our job to find out the distance between GA and GO in
order to find the exact location of the observed replication in our grid. Let us refer
to the distance between GA and GO as the distance traveled by the galaxy cluster, or
dt. We will call the line drawn from OP to GA, the grid distance, dg. The distance
from GO to OP , dl, is the distance the light from the replication travels to reach the
observor’s eye. The angle between dt and dg, φ, can be found by:
φ = 180− arctann
m
dl can be written as the product of the speed of light, c, and time, while dt can be
described as the product of the velocity of the galaxy cluster, v, and time:
dl = ct (II-2)
dt = vt (II-3)
dg can be described using the Pythagorean Theorem as in Case 1:
dg =√n2 +m2
Solving Equations II-2 and II-3 for t, setting them equal, and solving for dl we obtain:
dl =dtc
v(II-4)
Using the Law of Cosines to solve for dl results in:
d2l = d2
g + d2t − 2dgdt cosφ
Substitute Equation II-4:
(dtc
v)2 = d2
g + d2t − 2dgdt cosφ (II-5)
22
Solving for dt:
dt =−dgv2 cosφ±
√c2d2
gv2 − d2
gv4 + d2
gv4 cos2 φ
c2 − v2(II-6)
Note that there is a ± in Equation II-6. This results in two separate values for dt.
If the velocity of the galaxy cluster is in the positive m direction, the larger value
of dt refers to the replication below the n-axis. The smaller value of dt refers to the
replication opposite (n,m) above the n axis. For example, when GA = (n,m) =
(1,−1), the larger value of dt refers the distance below GA where the replication, GO,
will be observed. The opposite point above the n-axis, GA = (n,m) = (1, 1) has an
acute angle β which results in a smaller value of dt and a shorter distance below GA
for replication GO.
Due to symmetry, it is only necessary to determine placement of observed repli-
cations for the first and fourth quadrants. Once the desired number of replications
are graphed, we draw various sized circles around OP with increasing radii as repli-
cations get further from OP . As in Case 1, with each circle, we determine individual
angles between drawn rays of encompassed replications and calculate the maximum
angle scan needed for the observer to guarantee observation of a replication.
3. Case 3: Aging Galaxy Clusters
Case 1 and Case 2 deal with galaxies that are not aging, but as we have seen in
Section I.3, galaxies do age. Therefore, Crystallographers must take into account what
galaxies might have looked like several billion years ago. We have already discussed
that most astronomers estimate the universe to be approximately 13.7 billion years
old. This means that in a non-expanding universe, an observer would only be able
to see the light from replications that are 13.7 billion lightyears away. If the universe
is not expanding, we can fix the width of the observable universe at 13.7 billion
lightyears. When we take into account the longevity and luminosity of the spectral
types listed in Section I.3 and the types of stars in each of the galaxies in the Local
Group (Section I.2), we get a good idea of what observers should be looking for.
The initial conclusion might be to look for the luminous bulges of the An-
dromeda and the Milky Way Galaxies. We must recall that most of the stars located
in the bulge of these galaxies are O and B-Type stars with life expectancies of little
more than 100 million years. If we try looking for these specific stars clusters in a
23
replication, we will not find them because they have not formed yet. What an ob-
servor needs to focus on is looking for replications of known stars and star clusters
of spectral types with longevities greater than 13.7 billion years. In Section I.3 we
determined that the only spectral types that fit this description are K and M -Type
stars (as well as White Dwarfs). This is not to say that stars of other spectral types
did not exist 13.7 billion years ago. This just points out that the stars we observe
in the Local Group today, if in any other spectral group besides M and K, would
not have been formed 13.7 billion years ago, and stars not of spectral type M and K
that existed 13.7 billion years ago have likely collapsed into a neutron star or a White
Dwarf.
The best way for observers to locate a replication is to determine current
prominent elliptical satellites of the Andromeda, Milky Way, and Triangulum Gal-
laxies. Elliptical galaxies, specifically satellites like NGC 221 and NGC 205 of the
Andromeda Galaxy, consist heavily of concentrated low mass, low luminous red stars
[6]. The high concentration increases the overall luminosity, making the elliptical
galaxy satellites perfect candidates for observing replication. If Crystallographers
can accurately map the most prominent elliptical satellite galaxies within the Local
Group, they can then use the mathematical principles used in Cases 1 and 2 to better
map possible locations for replications.
24
III. FUTURE WORK
In this chapter we will be discussing certain problems in Cosmic Crystallog-
raphy that were not discussed in detail in the previous sections. Specifically we will
address the concepts of an expanding universe, the application of the scenario math-
ematics to a three dimensional model, and the motion of galaxies within a cluster.
III.1. EXPANDING UNIVERSE
In every scenario we discussed, the universe was stationary, as though it had
started existence with a width of 13.7 billion lightyears and remained that width
through time, to include the present and the future. This, unfortunately for our
math, is not the case. At present the universe is expanding at a certain, possibly
changing, velocity. Someday in the future, it is predicted that the universe will stop
expanding and start to collapse back in on itself, possibly to the dense ball of dust
and gas that caused the first big bang. Where the universe would go from there is up
to speculation. In order to get a picture of what an expanding universe might look
like as a 2-torus, we have to refer back to the grid. It is the same grid as before, but
with one change. Look at Figure III.1.
25
Figure III.1. A 2-Torus Expanding Universe
As the observer looks further away toward replications, the light in the repli-
cated univere will be more compact, because when the galaxy cluster was in that
universe, the galaxy was smaller. This will happen in every direction the observer
looks. In Figure III.1, we only see the m and n axes of replicated universes. What
happened to the other replicated universes? Look at Figure III.2.
26
Figure III.2. A 2-Torus Expanding Universe: Extended
As we can see, the replicated universes not on the m or n axes also get smaller
as the replications continue outward from OP . The difference here is that each of
these replications is not its own individual replication. The thick lines connecting the
smaller replications indicate that the connected replications are the same replicated
universe. Let us look at this in a different way. In Figures III.1 and III.2, the
difference from one universe to its replication and so on was a discrete jump in size of
the universe. In the real world, expansion is a continuous process, so the replications
would look more like Figure III.3.
27
Figure III.3. A 2-Torus Expanding Universe: Continuous
As the replications are drawn more and more continuously with respect to
the other replications, we begin to see that in order for the non-axis replications to
overlap as we know they do, the entire grid would have to curve in a concave manner
into this paper. We refer to this curvature as the global curvature of space. This
is not to be confused with the local curvature of space which refers to the curves of
spacetime that cause light to bend around large masses.
III.2. THREE DIMENSIONAL MODEL (III.4)
While this paper focused solely on universes in two dimensions, the universe
itself has width, length, and depth, making it a three dimensional entity. In Cases
1 and 2, we used a single angle (max angle) to represent the percentage of the total
28
observation. When we move the three dimensional model III.5 so that it resembles the
two dimensional model ??, we can first find the max angle in two dimensions based
on a given radius. From there we can construct a cone using the triangle formed
by finding the max angle. What we need to prove in the future is that there is a
general way to calculate the volume of the cone in order to find the percentage of
total observable space needed to scan given a radius of observation. Look at Figure
III.4.
Figure III.4. Scanning for Replications in a 3-Torus [3]
29
Figure III.5. Model Representing the Universe as a 3-torus
From OP , three replications were found and rays were drawn through them. In
this case, the angle between the replications was π2, resulting in a 12.5% portion scan
of the total observation area needed to ensure a replication is found at an observation
radius of 1 UU . Finding an overarching method of calculating scan percentage would
involve reconstructing the Menges-Mikhaylov Theorem to work in three dimensional
space.
III.3. GALAXY MOTION WITHIN A CLUSTER
In Case 2 we worked with moving galaxy clusters. This referred only to the
entire cluster moving as a whole with a fixed velocity. This allowed us to move
the galaxy cluster as a single point instead of several separate points. In the real
universe, stars move within galaxies relative to other stars, and galaxies move within
clusters relative to other galaxies. In order to get a better idea of where replications
30
Figure III.6. Moving the 3D Model to view the 2-torus representation
of the Local Group might be in space, observers must have knowledge of the historical
and present velocity of the Andromeda Galaxy with respect to the Milky Way and
Triangulum Galaxies and the velocities of the others with respect to Andromeda. As
we discussed in Case 3, in order to better find replications, we must locate prominent
elliptical galaxies within or around the main spiral galaxies in the Local Group. If
we want better knowledge of a replication location, we must figure the historical
and present velocities of those elliptical galaxies as well. Once we understand the
patterns of velocity of all these specific galaxies, we can form an understanding of
their orientation to each other 13.7 billion years ago.
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32
IV. CONCLUSION
In this thesis we discussed the overall geometric theory of the simple 2-torus
and the 3-torus. Cosmic Crystallographers, astronomers specializing in the shape and
orientation of space, use the shape of the 3-torus, as well as other shapes to theorize
the shape of the universe. In this paper, theory was based around the universe as a
2-torus. Mathematically, we analyzed where replications will be when galaxy clusters
are stationary and when galaxy clusters move at at constant velocity. Using the Lat-
tice Conjecture, a formula was found to determine the maximum angle of observation
needed to guarantee observation of a replication. We used the law of cosines to de-
termine where moving galaxy clusters would be located when not on the horizontal
or vertical axes of observation. Using information on the three major galaxies in the
local cluster and the attributes of each spectral type, Cosmic Crystallographers can
map what a galaxy cluster may have looked like 13.7 billion years ago, and, applying
the two dimensional mathematics to the three dimensional models, should be able to
have a more formal understanding of where to locate replications in a 3-torus uni-
verse. In the future, I will translate the math from the two dimensional models to
three dimensions. I will also focus on the rate of expansion of the universe and apply
it to the concept of the universe as a 3-torus. Finally, I will cease looking at galaxy
clusters as a single point in the universe. Galaxies with cluster move relative to one
another, so I will be using this concept and dealing with galaxies on a closer level
than merely a point in the universe.
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APPENDIX A
SPECTRAL TYPES
Note: The term solar mass refers to the mass of the Sun, while the term sol, or
sols, refers to the luminosity of the Sun (1 sol represents the luminosity of the Sun).
O-Type
A star with spectral type O is a very large blue star with a mass roughly 20
– 100 times the mass of Earth’s Sun. The temperature of an O-Type star is larger
than approximately 30,000 K, making this class the hottest and largest of all of the
spectral classes. The luminosity of an O-Type star is 100,000 sols, or 100,000 times the
luminosity of the Sun. A star’s total lifespan, in years, can be measured as inversely
proportional to the mass of the star. The larger the mass of the star, the hotter the
temperature, which ultimately results in the O-Type blue stars having the shortest
lifespan of the spectral types at approximately 10 million years. O-Type stars have
an abundance of 0.00001%.
B-Type
A star with spectral type B is considered a ‘light-bluish’ star and is the second
largest (Mass = 10x Sun) and hottest of the spectral classes. The mass of B-Type
stars is roughly 3 – 20 solar masses with a luminosity in the range of 100 – 50,000
times that of Earth’s Sun. The temperature of the B-Type star can range from
approximately 10,000 K – 30,000 K. Because of the high mass and luminosity of B-
Type stars, the life expectancy of a B-Type star is only 100 million years. B-Type
stars have an abundance of 0.1%.
A-Type
A star with spectral type A is a white star. The average luminosity of A-Type
stars is approximately 20 sols, but they can range anywhere from 7 sols to 80 sols
with A-Type supergiants reaching luminosities of up to 35,000 sols and masses of up
to 16 times solar mass. The normal mass of A-Type stars is roughly 1.5 – 3 solar
35
masses. The temperature of the A-Type star is approximately 8,500 K with a few
star reaching upwards of 10,000 K. Because of the mass and luminosity of A-Type
stars, the life expectancy of an A-Type star is 1 billion years. A-Type stars have an
abundance of 0.7%.
F-Type
Depending on mass and luminosity, white and yellow stars can fall under the
F -Type spectral classification. The mass of F -Type stars can range from 1.2 – 1.6
solar masses with luminosities ranging from 2 – 6 sols. F -Type supergiants can reach
up to 12 solar masses and 32,000 sols. The temperature of F -Type stars ranges from
6,000 K to 7,200 K. The life expectancy of an F -Type star is 3 billion years. F -Type
stars have an abundance of 2%.
G-Type
Earth’s Sun and Alpha Centauri are classified as Main Sequence G-Type stars.
The mass of G-Type stars ranges from 0.8 – 1.1 solar masses with luminosites ranging
from 0.8 – 1.5 sols. G-Type supergiants can range from 10 – 12 solar masses and 10,000
– 300,000 sols. The temperature of G-Type stars ranges from 5,100 K to 6,000 K. The
life expectancy of a G-Type star is 10 billion years. G-Type stars have an abundance
of 3.5%.
K-Type
K-Type stars are orange-red and tend to have masses ranging from 0.5 – 0.8
solar masses. Luminosities of K-Type stars range from 0.1 sols to 0.4 sols. The
temperature of K-Type stars ranges from 3,900 K – 5,200 K. The life expectancy of
a K-Type star is 50 billion years. K-Type stars have an abundance of 8%.
M-Type
M -Type stars in the Main Sequence are known as red dwarfs. Making up
almost 80% of the stars in the universe, red dwarfs are very important to Crystal-
lographers due to their longevity (approximately 200 billion years or longer) and
numbers. Red dwarfs also tend to cluster which helps make up for their low luminos-
36
ity (0.08 sols or lower). Their longevity is due to their low luminosity and low mass
(less than 0.5 solar masses).
White Dwarfs
White Dwarfs do not fall under a specific spectral type due to their extremely
low luminosity (less than 0.01 sols). White dwarfs are essentially dying remnants of
an imploded star and are sometimes referred to as D-Type stars. The life expectancy
of white dwarfs is relatively unknown because most stars of all classes end up as a
white dwarf when the nuclear fusion in the star has stopped, and they essentially
stay white dwarfs indefinitely. The reason these are important to Crystallographers
is because of sheer numbers. While luminosity is almost too low to be of any use in
looking for replication, the number of white dwarfs, while less than the number of
red dwarfs, increases chances of clumping which can increase the luminosity enough
to be of use.
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APPENDIX B
THE MENGES-MIKHAYLOV THEOREM
The Menges-Mikhaylov Theorem (formerly the Lattice Conjecture) referenced
in Section II.1 was determined based on several observation radii and finding the max
angle at these different radii. When circles were drawn around OP at these different
radii, the individual angles between each of the drawn rays was calculated, and it
was found up to more than 6 UUs away from OP that the max angle was always
θ = arctan 1brc . Upon further construction of the proof, I determind that for a given
radius r =√x2 + y2, we can take r and set it equal to r =
√x2∗ + 1. Solve for x∗,
and floor the value to the next lowest integer. We will call this integer N . The max
angle is θ = arctan 1N
The theorem has been proven by CDT J. Menges and Dr. J.
Mikhaylov with the help of Dr. A. Hager, Dr. J. Fierson, Dr. V. F. Rickey, and Dr.
W. Pulleyblank.
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LIST OF REFERENCES
[1] Weeks, Jeffrey R. The Shape of Space 2nd ed. (Canton, New York: MarcelDekker, Inc.), 2002.
[2] Greene, Brian. The Hidden Reality, (New York: Alfred A. Knopf), 2011.
[3] Weeks, Jeffrey R. Curved Spaces 3.2 (2009): www.geometrygames.org.html (ac-cessed September 2010).
[4] Van den Bergh, Sidney. The Galaxies of the Local Group, (Cambridge: Cam-bridge University Press), 2000.
[5] Powell, Richard. “The Classification of Stars.” Atlas of the Universe (2003):http://www.atlasoftheuniverse.com/startype.html (accessed March 21, 2011).
[6] Darling, David. Internet Encyclopedia of Science (2003):http://www.daviddarling.info (accessed March 21, 2011).
[7] Muir, Hazel. “Galactic merger to ‘evict’ Sun and Earth.” NewScientist (2007):http://www.newscientist.com/article/dn11852 (accessed March 31, 2011).
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