Outline - Phoenix Military Academy



Running head: DEMONSTRATING INFINITYDemonstrating Infinity within a ProjectionMathematicsDesign_______________________________________Signature of Sponsoring Teacher_______________________________________Signature of School Science Fair Coordinator Octavio Sanchez145 S. Campbell Avenue Phoenix Military AcademyChicago, IL 60612Grade 12Table of ContentsTitle Page……………………………………………………………………………….1Table of Contents……………………………………………………………………….2Acknowledgments……………………………………………………………………….3Purpose and Hypothesis……………………………………………………………….4Review of Literature…………………………………………………………………….…5-13Materials and Methods of Procedure ..……………………………………………..…….14-15Results ……..……………………………………………………………………..……….16-26Data Analysis ………………………………………………………………………27Conclusions………………………………………………………………………………28References……………………………………………………………………….……..29-32AcknowledgmentsI 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 HypothesisPurposeThe 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 LiteratureSpaceSpace 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. TimeWhen 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. DimensionsWhen 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 DimensionIn 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 DimensionIn 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).?Euclidean GeometryAccording 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 ContinuumEinstein 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 SpaceWhen 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).GravityGravity 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 GravityNewton’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 relativityAccording 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 MechanicsQuantum 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 ConcaveTwo 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).PerceptionPerception 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. MetaphysicsMetaphysics 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 ProcedureMaterials The following is a list of the materials that were used in the experiment. Overhead Projector Transparency FilmCameraMeasuring tape, 200 cm longA volleyballHalf a BasketWhite paintExpo MarkerString Methods of Procedure The following is the procedure that was used in the experiment Make equilateral triangles inside of each other, but as you add more you make them smaller.Print out on a transparency film.Put the transparency film underneath the overhead projectorProject the transparency onto the hyperbolic field and be 100cm away from the ballOnce it is projected take a picture and print it out on a second transparency. Use the 200 cm measure tape and stick it on a board Tie a string at the end of the expo marker and have someone hold the string at 0cm As they hold it at 0cm make sure that you draw a radii, most of this will be guess and check.Repeats steps 3-8, but as you continue different observations make sure that you change the plane that you are projecting onResultsFigure 1. Conversions and ScalingFigure 2. Example of how we measured the RadiiFigure 3. Calculating and compilingFigure 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 1Observation #1: Volleyball (100cm away)Scaled Radius of Curvature Scaled Sides of Equilateral TriangleTriangle 13cm4cmTriangle 24cm7cmTriangle 35cm11cmTriangle 46cm15.5cmTriangle 57cmN/ATriangle 6N/AN/ATriangle 7 N/AN/ATriangle 8N/AN/ATriangle 9 N/AN/ANote. Corresponds to figures 4-6 Figure 7. The half basket is being used as the plane during the second observationFigure 8. A view of the half circle at a different angle. Table 2Observation #2: Half Circle (100cm away)Scaled Radius of CurvatureScaled Sides of Equilateral TriangleTriangle 12cm5cmTriangle 23cm8cmTriangle 34cm11cmTriangle 45cm15.5cmTriangle 56cmN/ATriangle 67cmN/ATriangle 7 8cmN/ATriangle 8N/AN/ATriangle 9 N/AN/ANote. Corresponding to figures 7-8Figure 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 3Observation #3: Yoga Ball 1 (100cm away)Scaled Radius of CurvatureScaled Sides of Equilateral TriangleTriangle 11cm4cmTriangle 22cm7cmTriangle 33cm10cmTriangle 44cm13cmTriangle 55cm16cmTriangle 66cmN/ATriangle 7 7cmN/ATriangle 88cmN/ATriangle 9 8cmN/ANote. 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 4Observation #4: Yoga Ball 2 (100cm away)Scaled Radius of CurvatureScaled Sides of Equilateral TriangleTriangle 10cm3cmTriangle 20cm4cmTriangle 30cm6.5cmTriangle 40cm8cmTriangle 54cm11cmTriangle 64cm13.5cmTriangle 7 4cm16cmTriangle 84cm17.5cmTriangle 9 4cm18.5cmNote. Corresponding to figures 12-13Data AnalysisFigure 14. Linear Regression t-TestFigure 15. Linear Regression t-Test Cont. Y=a+bxY=1+3xT=1x10^99P=0Df=3R=1R?=1This 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. ConclusionsThrough 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 R? 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).Reference ListEinstein's Special Theory of Relativity. (2001). In J. S. Baughman, V. Bondi, R. Layman, T.McConnell, & V. Tompkins (Eds.), American Decades (Vol. 1). Detroit: Gale. Retrieved from Over Type=&query=&prodId=SUIC&windowstate=normal&contentModules=&display- query=&mode=view&displayGroupName=Reference&limiter=&currPage=&disableHighlighting=false&displayGroups=&sortBy=&search_within_results=&p=SUIC&action=e&catId=&activityType=&scanId=&documentId=GALE%7CCX3468300276&source=Bookmark&u=cps&jsid=d4c66e91ed58583e5348aff614904e58Guinn, J. (2014). Gravity and gravitation. In K. L. Lerner & B. W. Lerner (Eds.), The GaleEncyclopedia of Science (5th ed.). Farmington Hills, MI: Gale. 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Retrieved from . com/ic/suic /ReferenceDetailsPage/ReferenceDetailsWindow?failOverType=&query =&prodId=SUIC&windowstate=normal&contentModules=&display-query=&mode =view&displayGroupName= Reference&limiter=&currPage=&disableHighlighting= false&displayGroups=&sortBy=&search_within_results=&p=SUIC&action=e&catId=&activityType=&scanId=&documentId=GALE%7CCV2644300800&source=Bookmark&u=cps&jsid=bd00d5cf65f276462bab45d16144cb68Shorter Oxford English Dictionary on Historical Principles: A-M. (n.d.). Retrieved January 8,2016, from ?nary _on_His.html?id=BmWfSAAACAAJRoberts, D. (1998). Non-Euclidean geometries. Retrieved December 8, 2015, from Rynasiewicz, R. (2004, August 12). Newton's Views on Space, Time, and Motion. RetrievedJanuary 3, 2016, from UC Davis Philosophy 175 Lecture Notes on Kant: Practical Reason. (n.d.). Retrieved January 3,2016, from ................
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