Finding Ourselves in the Universe_ A Mathematical Approach to Cosmic Crystallography

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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.

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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.

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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]

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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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34

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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38

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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Finding Ourselves in the Universe_ A Mathematical Approach to Cosmic Crystallography

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