Outline · Web viewa huge impact to the understanding of this word, ... January 3, 2016, from UC Davis Philosophy 175 Lecture Notes on

Sanchez: Demonstrating Infinity 1

Running head: DEMONSTRATING INFINITY

Demonstrating Infinity within a Projection

Mathematics

Design

_______________________________________Signature of Sponsoring Teacher

_______________________________________Signature of School Science Fair Coordinator

Octavio Sanchez145 S. Campbell Avenue Phoenix Military AcademyChicago, IL 60612Grade 12


Table of Contents

Title Page ……………………………………………………………………………….1

Table of Contents ……………………………………………………………………….2

Acknowledgments ……………………………………………………………………….3

Purpose and Hypothesis ……………………………………………………………….4

Review of Literature …………………………………………………………………….…5-13

Materials and Methods of Procedure ..……………………………………………..…….14-15

Results ……..……………………………………………………………………..……….16-26

Data Analysis ………………………………………………………………………27

Conclusions ………………………………………………………………………………28

References ……………………………………………………………………….……..29-32


Acknowledgments

I want to start by acknowledging a person that does no give up on any of the science fair

participants, and makes sure that our papers, projects, and presentation are top notch. That

person is Ms. Tobias and I do not want her hard work to go unnoticed. A huge thank you to Dr.

Jaji for looking at our paper and telling us what needed to be described more in a casual point of

view for people with out the background to still understand. Mr. Surina was a big help when we

needed someone to watch as we typed our papers at eight in the afternoon or when we needed a

tech guru. Finally, I want to thank the person the made countless calculations with me, the

person that sat down and looked at this projected upside down and inside out with me, the one

teacher that pushed me when I felt like giving in, Mr. Carroll.


Purpose and Hypothesis

Purpose

The purpose behind this experiment is to find the best fit line with in different planes.

Hypothesis

The smaller the hyperbolic sphere the greater the curvature will be which will indicate

that the radius will be smaller.

Dependent Variable

The radii of the curvature will change.

Independent Variable

The distance between the projector and the plane, sphere.

Control

The triangle that is being displayed which is an equilateral triangle.


Review of Literature

Space

Space is made up of three different dimensions that are known by the human mind, but

cannot be perceived physically. The three different dimensions are up/down, left/right, and

forward/backwards. In a visual perception, you can imagine a coordinate plane with an X and Y

axis, but to see the third dimension you will have to imagine the Z axis which crosses between

the origin coming out breaking a second-dimension perspective making it into a third-dimension

perspective, i.e. a cube, pyramid, tetrahedral, Trigonal bipyramidal. Old mathematicians started

to examine space, they started to use Geometry that was not Euclidean, but instead Non-

Euclidean to get a better idea on how everything is shaped, a better angle to look at things.

Time

When talking about time there can be many ideas that come into mind and those ideas

can range from Star Wars and Star Trek to Albert Einstein’s brilliant Special Theory of

Relativity. Speaking of this Special Theory of Relativity he had two important postulates:

“1. The speed of light (abut 300,000,000 meters per second) is the same for all observers

whether or not they’re moving.

2. Anyone moving at a constant speed should observe the same physical law.” (Fuller,

“How Warp Speed Works”, 2008).

All of this can be compiled to one point, and that is that time represents the fourth dimension

when talking about what is perceived in space-time continuum. In space-time continuum time is

used as an organizer making everything be in sequence from past to present to the future. A point

of view that is very important is that time is something very fundamental to our universe. This

fundamental is known to be a dimension independent of events, or is known as the prominent


fourth dimension. This fundamental is also referred to as Newtonian Time (Rynasiewicz, 2004).

The contrasting viewpoint to the one that was previously mention is that time is not referred to as

a container in which events and objects simply move through. Time is used to organize events

and compare them. The final third point of view is that, time is not a thing or an event which

indicated that it cannot be measured nor can it be traveled (Mattey, 1997) Time is one of the

seven fundamentals in SI units and in the International System Quantities.

Dimensions

When talking about dimensions it can be viewed from a mathematical and physics point

of view. Most of the time that you are trying to find the dimensions of an object it is basically

corresponding to the coordinate points of it. In a sense, you are looking for the 1st Dimension or

the dimensions of the shape to locate the best attainable information.

1st Dimension

In physics and mathematics, a sequence of numbers can be understood as a location in n

dimensional space. When “n” equals one, the set of all locations is called one-dimensional space.

An example of a one-dimensional space is the number line, where the position of each point can

be described by a single number.

2nd Dimension

In physics and mathematics, the 2nd dimensional space is a geometrical model in the

coordinate plane in which length and width lie on. In addition, length and width are commonly

called two dimensions.


3rd Dimension

A three-dimensional figure can be viewed differently to a two-dimensional figure. The

reason that is can be viewed differently is because the three-dimensional figure has length, width,

and height (two additional terms to describe the figure are depth and breadth).

4th Dimension

There really is not much data about this dimension besides the fact that it is consider

holding a shape such as the tesseract and it can be imagined as a shape within the shape. It is

commonly looked at as a square with in a square and it moves, so it incorporates time and lets

the smaller square become the bigger square as time passes by.

Non-Euclidean Geometry

According to Donna Roberts, “Non-Euclidean Geometry is any form of geometry

that contains a postulate (axiom) which is equivalent to the negation of the Euclidean parallel

postulate” (Roberts, 1998). In Non-Euclidean Geometry, there are different postulates and

different types of geometry within it. The following are the five postulates that are known to

Non-Euclidean Geometry:

1.) A straight line can be drawn from any point to any point.

2.) A finite straight line can be produced continuously in a straight line.

3.) A circle may be described with any point as center and any distance as a radius.

4.) All right angles are equal to one another.

5.) If a transversal falls on two lines in such a way that the interior angles on one side of the

transversal are less than two right angles, then the lines meet on the side on which the angles are

less than two right angles (Non-Euclidean Geometry, 2014).


The types of Geometry that were used in this project were Riemannian geometry, known as

Elliptic Geometry. The second geometry used was Hyperbolic Geometry, known as Monkey

Saddle Geometry.

Elliptical Geometry

"Elliptic geometry is a non-Euclidean geometry with positive curvature which replaces

the parallel postulate with the statement "through any point in the plane, there exist no

lines parallel to a given line." To achieve a consistent system, however, the basic axioms of

neutral geometry must be partially modified. Most notably, the axioms of betweenness are no

longer sufficient (essentially because betweenness on a great circle makes no sense, namely if

and are on a circle and is between them, then the relative position of is not uniquely

specified), and so must be replaced with the axioms of subsets.” (Wolfram, "Elliptic Geometry",

2017). In simple terms, this is trying to indicate that the plane that is being represented has a

curvature to it, like a sphere, which is why when there is a triangle it will be greater than 180

degrees.

Hyperbolic Geometry

“A non-Euclidean geometry, also called Lobachevsky-Bolyai-Gauss geometry, having

constant sectional curvature . This geometry satisfies all of Euclid's

postulates except the parallel postulate, which is modified to read: For any infinite straight line

and any point not on it, there are many other infinitely extending straight lines that pass

through and which do not intersect . In hyperbolic geometry, the sum of angles of

a triangle is less than , and triangles with the same angles have the same areas.

Furthermore, not all triangles have the same angle sum.” (Wolfram, "Elliptic Geometry", 2017).

http://mathworld.wolfram.com/Angle.html

http://mathworld.wolfram.com/Triangle.html



http://mathworld.wolfram.com/Angle.html

http://mathworld.wolfram.com/Intersection.html

http://mathworld.wolfram.com/Line.html

http://mathworld.wolfram.com/Point.html

http://mathworld.wolfram.com/Line.html

http://mathworld.wolfram.com/ParallelPostulate.html

http://mathworld.wolfram.com/EuclidsPostulates.html

http://mathworld.wolfram.com/EuclidsPostulates.html

http://mathworld.wolfram.com/Geometry.html

http://mathworld.wolfram.com/Non-EuclideanGeometry.html


Euclidean Geometry

According to the “Shorter Oxford English Dictionary”, “Euclidean means the geometry

of ordinary experience based on axioms of Euclid, esp. the one stating that parallel lines do not

meet. This also is known to be a flat surface. Euclidean Geometry also has five known postulates

those are as following:

1) Any two points describe a line.

2) A line is infinitely long.

3) A circle is uniquely defined by its center and a point on its circumference.

4) Right angles are all equal.

5) Given a point and a line not containing the point, there is one and only one parallel to the line

through the point (Postulates, 2014).

Space-Time Continuum

Einstein believes that space and time were necessary for each other and without each

other they will fade and cease to exist. Space-time is also described to be what our world is in. In

addition, this is also viewed by a Euclidean space. Regarding the fact that we know that space is

made from three dimensions and time as one we consider them to make the fourth dimension.

Curved Space

When talking about curved surfaces we can relate this to Einstein’s General relativity

because it will be said that space is not flat, like how it was assumed to be but instead it is

curved. The notion of Anti-De Sitter Space, which contains no matter what so ever but does have

a negative energy density. The fact that it is curved shows that there is a force being used, that

force is gravity, but gravity is not like any other force. Gravity does not allow for Earth to move


because Earth is in a curved space, but space allows for Earth to move in that space because it

follows the nearest object in a straight path in a curved surface, this is called Geodesic (Hawking,

2005).

Gravity

Gravity is the attraction between two items. Everything in this universe has gravity and

everything in this universe is attracted to something. Most of the time the force is equal to the

mass of the object. According to “Gravity and Gravitation”, Thus, F = Gm1m2/r2, where m1 is

the mass of the first object, m2 is the mass of the second object, r is the distance between their

centers, and G is a fixed number termed the gravitational constant. (If m1 and m2 are given in

kilograms and r in meters, then G = 6.673 × 10−1N m2/kg2.)” This can be related

back to Stephen Hawking’s book, “A Briefer History of Time”, thus meaning

that it is like something called geodesic. This also means that gravity is calculated by

finding the differences between distances, and using the mass of the certain thing that is being

calculated by the gravity (Gravity and gravitation, 2014).

Newtonian Gravity

Newton’s universal law says that all the objects that have gravity attract one another

(Gravity and gravitation, 2014). The way that it works is by making sure that if you want to

make the gravity between each one of the objects stronger you must bring them closer and the

same goes to making the attraction weaker (Gravity and gravitation, 2014). This means that

Einstein’s law of general relativity is being used, somehow. For example, when the law says that

gravity is the force that makes the earth to stay in place while making sure that it allows it to

chase after the nearest thing, or object, but in a curved space (Hawking, 2005). This law also

says that the mass and weight are not the same because something can weigh less in a different


gravity field and still have the same mass it had anywhere else. Take a brick of gold for example,

it will weigh less on the moon or mercury compare to the earth, but it will still have the same

mass.

Theory of relativity

According to the article, Einstein’s Special Theory of Relativity, it states that, “Relativity

is that area of physics that must do with how observers in motion with respect to the

phenomenon observed can account for their observations given that two different frames of

reference (that of the observer and that of what is observed) are involved” (Einstein’s Special

Theory of Relativity, 2001). This proves that time and position truly matters to the point where in

reality it is the perception of someone. Just as if someone dropped something from a building the

person can see the object curve while the person that dropped the object sees it falling straight.

The way this relates to the project being presented is simple since a concave to observer one can

be completely something different.

Quantum Mechanics

Quantum Mechanics can be a way to study the natural world in a way that everything

being observed is through the way of energy waves. According to UXL Encyclopedia of

Science, “For example, physicists normally talk about light as if it were some form of wave

traveling through space. Many properties of light—such as reflection and refraction—can be

understood if light is thought of as waves bouncing off an object or passing through the object”

(UXL Encyclopedia of Science, 2015). Something else very important to quantum mechanics is

that sometimes the waves can travel as matter and sometimes it travels as a wave, this is

considered the principal of dualist.


Convex and Concave

Two important terms that were used in this paper were “concave” and “convex”. These

two terms are important because they are describing the sides of the triangle and they describe

how the space is distorted. Most of the times these terms are used when talking about lenses, but

in this instance, it describes the triangle. When talking about lenses that are convex it means that

the lens attracts the light towards the middle so the lens is outwards, in context this means that

the side of the triangle was coming out. Concave is the complete opposite, so when speaking

about a lens that is concave it means that the lens is inwards, in context it means that the side of

the triangle was coming inside (UXL Encyclopedia of Science, 2015).

Perception

Perception is very important in this experiment since it can change the whole results.

Perception is the way an organism organizes information. The way it organizes that information

is by using the five senses. That way when the information reaches the brain it can sort it out and

make sure that the information is processed correctly (UXL Encyclopedia of Science, 2015). The

way perception plays a role in this experiment is simple; the person observing the distortion can

notice different things from a different observer.

Metaphysics

Metaphysics dates to the ancient and medieval times, were philosophers thought that the

idea of metaphysics had to do with chemistry and or astrology. Although they were somewhat

correct, the one philosopher to correct this was Aristotle. Metaphysics was the “science” that

studied “being as such” or “the first causes of things” or “things that do not change”, but

metaphysics cannot be defined that way anymore because of Aristotle. Although Aristotle made


a huge impact to the understanding of this word, he did not know what this word exactly meant.

Aristotle also had a collection of books dedicated to this science of philosophy; there were

fourteen books to be exact (Inwagen & Sullivan, "Metaphysics", 2007).


Materials and Methods of Procedure

Materials

The following is a list of the materials that were used in the experiment.

Overhead Projector

Transparency Film

Camera

Measuring tape, 200 cm long

A volleyball

Half a Basket

White paint

Expo Marker

String

Methods of Procedure

The following is the procedure that was used in the experiment

1. Make equilateral triangles inside of each other, but as you add more you make them

smaller.

2. Print out on a transparency film.

3. Put the transparency film underneath the overhead projector

4. Project the transparency onto the hyperbolic field and be 100cm away from the ball

5. Once it is projected take a picture and print it out on a second transparency.

6. Use the 200 cm measure tape and stick it on a board

7. Tie a string at the end of the expo marker and have someone hold the string at 0cm


8. As they hold it at 0cm make sure that you draw a radii, most of this will be guess and

check.

9. Repeats steps 3-8, but as you continue different observations make sure that you change

the plane that you are projecting on


Results

Figure 1. Conversions and Scaling

Figure 2. Example of how we measured the Radii


Figure 3. Calculating and compiling

Figure 4. The volleyball is the first hyperbolic plane that is being used.


Figure 5. Different angle on the volleyball.

Figure 6. Top view of the volleyball plane.


Table 1

Observation #1: Volleyball

(100cm away)

Scaled Radius of Curvature Scaled Sides of Equilateral

Triangle

Triangle 1 3cm 4cm

Triangle 2 4cm 7cm

Triangle 3 5cm 11cm

Triangle 4 6cm 15.5cm

Triangle 5 7cm N/A

Triangle 6 N/A N/A

Triangle 7 N/A N/A

Triangle 8 N/A N/A

Triangle 9 N/A N/A

Note. Corresponds to figures 4-6


Figure 7. The half basket is being used as the plane during the second observation

Figure 8. A view of the half circle at a different angle.


Table 2

Observation #2: Half Circle

(100cm away)


Triangle

Triangle 1 2cm 5cm

Triangle 2 3cm 8cm

Triangle 3 4cm 11cm


Triangle 5 6cm N/A

Triangle 6 7cm N/A

Triangle 7 8cm N/A

Triangle 8 N/A N/A

Triangle 9 N/A N/A

Note. Corresponding to figures 7-8


Figure 9. The first yoga ball, smaller one, is now the plane for third observation.

Figure 10. A different Angle on the first yoga ball.


Figure 11. Top view of the first yoga ball.


Table 3

Observation #3: Yoga Ball 1

(100cm away)


Triangle

Triangle 1 1cm 4cm

Triangle 2 2cm 7cm

Triangle 3 3cm 10cm

Triangle 4 4cm 13cm

Triangle 5 5cm 16cm

Triangle 6 6cm N/A

Triangle 7 7cm N/A

Triangle 8 8cm N/A

Triangle 9 8cm N/A

Note. Corresponding to figures 9-11.


Figure 12. The second yoga ball is now the new plane that is being observed.

Figure 13. A different angle on yoga ball 2.


Table 4

Observation #4: Yoga Ball 2

(100cm away)


Triangle

Triangle 1 0cm 3cm

Triangle 2 0cm 4cm


Triangle 4 0cm 8cm

Triangle 5 4cm 11cm


Triangle 7 4cm 16cm



Note. Corresponding to figures 12-13


Data Analysis

Figure 14. Linear Regression t-Test

Figure 15. Linear Regression t-Test Cont.

Y=a+bx

Y=1+3x

T=1x10^99

P=0

Df=3

R=1

R²=1

This all indicates that the results are close the infinity which means that further test should be

conducted to make sure that the radius of the curve is equal to zero.


Conclusions

Through multiple trials and different observations, it has been concluded that the

observation with the best data is observation number 3, the first yoga ball. This has the best

information because it has a constant growth. Not only does it have a constant growth, but it

gives the best of both scaled lengths of the triangles and the scaled radius of the curvature. This

indicates that the probability that the data does not yield a linear graph is almost zero and both R

and R² are equal to one so the fit is nearly linear. In addition, it allows metaphysical questions to

be asked and let there be room for improvement. For example, the philosopher Emmanuel Kant

used introspection and data analysis to question the unknown since most of the studies conducted

in the metaphysical world only exist in the mind of the observer, so the observation can be bias

(Inwagen & Sullivan, "Metaphysics", 2007).


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Outline · Web viewa huge impact to the understanding of this word, ... January 3, 2016, from UC Davis Philosophy 175 Lecture Notes on