Illinois Junior Academy of Science - Yearbook (Urbana, IL)

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mation, is meant one in which the surface is deformed within its own boundaries and in which the surface is not torn. The Brouwer Theorem holds good for any finite number of dimensions as long as the figure considered is convex: however, in this sense convex for two dimensions means topologically equivalent to a disk and for three dimensions topologically equivalent to a sphere and its interior. The contraction of a surface is a particular deformation which not only admits a fixed point but admits a unique fixed point. Since expansion is the converse of contraction, it can be proven that a deformation of expansion also admits a unique fixed point. Scientific measurements of distances between nebulae masses have led to the hypothesis that our universe is expanding: therefore the universe has a fixed point. In a book entitled Shereland Dionys Burger created an expanding two-dimensional universe which curves into the third dimension. By proving that such a two-dimen- sional universe has its fixed point in the third dimension and by making an analogy between the second and third dimen- sional universes, I have hypothesized that the three-dimene sional universe we know of has a fixed point in the fourth dimension. This can be expanded to say that the nth-dimen- sional universe has a fixed point in the n+1 dimension. The fixed point concept fits not only into topology and the fantasy of dimensions beyond our own, but a special set of concepts apply to infinite dimensions, that is finite di- mensions involving time. ?M1l Applications of Probability THOMAS HEMNES U23 Arlington High School Arlington Heights Sponsor: James F. Ulrich Because of probability's frequent association with dice, poker, and little colored balls, we often overlook some of its more scientific aspects. In this project I have applied the basic concepts of probability to the classic Rutherford scat- tering experiment in which gold foil is irradiated by alpha particles. I then performed the experiment to test my predic- tions. I can consider briefly my method. As an alpha particle approaches a gold nucleus, similar electrical charges deflect the particle with the coulomb force. The angle of deflection increases as the proximity of the particle's path to the nucleus tthis distance is the impact parameterl decreases. In my experiment I wanted to know how many particles would be deflected at more than a certain angle. For each such angle there is a circle around each nucleus whose radius is the impact parameter. Using this radius. p, and assum- ing that it is equally probable for a particle to strike any part of the irradiated area, my probability becomes P tdeflectionl : latomsfplanei fnumber of planesl I area irradiated l 'TT p2 The atoms per plane and number of planes express the size of the foil .in terms of atoms. In my project the results were correct to two powers of ten, a satisfactory answer since the calculations involved powers of ten to the twentieth. KAREN KUCZYNSKI Our project dealt with the experimentation of the cycloid and its practical uses. A cycloid is the pattern traced by a point on a wheel as it moves along the ground. In our research we invented several types of cycloids by changing the surface the wheel travels over. Hypocycloids proved to be very interesting patterns.. A hypocycloid is the pattern of a point on a wheel as it moves inside a circular road. We discovered that there was ag similarity between the ratio of the wheel to the road and the pattern of the wheel. For instance, if the ratio of the wheel to the road is 1:3 the pattern will resemble a triangle with curved sides. The quadrocycloid, one of our own inventions. followed 'a pattern similar to that of the hypocycloid. A quadro- cycloid is the pattern traced by a point on a wheel as it moves inside a square. If the ratio of the wheel to the road is 1:4 you will end up with a five-sided curved figure. A 1:3 ratio will give you a four-sided curved figure and so on. The reason for the extra side, we discovered. was that because the wheel is traveling inside a square it has to compensate for the corners somehow. Therefore, it does so by adding an extra side. We then continued working on our project looking for a practical use for these patterns. It seemed that no matter how hard we tried we could not find a practical use. P. KOLBREG The aim of this project has been to perform construc- tions with a fixed-compass. Original proofs were then for- ulated to justify them. This project was selected to show the possibility of performing geometrical constructions with limited means. Fourteen of the sixteen constructions were constructed

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Maximal Division of Closed Space MICHAEL MADEJ Sf. Laurence High School Chicago, Illinois For a detailed and systematic study of the properties of maximally divided space, it is first necessary to investigate the properties of the simplest particular case, that occurring in two dimensions. It is found that the dividing lines, or par- titions, need not be straight, but must only fulfill the follow- ing conditions: a partition in the interior of a closed curve must intersect the curve twice and every other partitlon exactly once, but cannot cross itself or two partitions simul- taneously. The systems of 1, 2, 3, and 4 partitions are tri- vial - i.e., there is only one way to draw each of them. The last mentioned system can always be represented by 465.430, which means that the 4 partitions form one region of five sides, as well as five regions each of three sides and four sides. 5 partitions, however, give rise to the systems C5,4s3ll, C5i4..3rl, 15-131111, and f6,5..4,3.l. It can be shown mathematically that since the number of regions is given by it IP' + Pl + 1, the systems f5.4m35l, 45.430, and 15.4230 should also exist, although they cannot be dravsm. This curious fact is reason enough to investigate the field further. There are two major directions in which the study may be developed: the formalization of the axioms and theorems fthe reduction of drawings and proofs to topological equa- tionsb, and the application of the concepts to n-dimensional space. The Fourth Dimension JIM EDWARDS Austin High School Chicago Objectives: I wanted to introduce geometrical and al- gebraical patterns involved in the fourth dimension in order to lower the topic into a range which can be well understood. Most people look on such a topic as completely abstract, therefore, I wanted to find all possible ways in which this dimension may someday affect our lives. Materials: The materials used were simple models made of plastic sticks. These created a media for observation and experimentation. Research information was gathered into a chart which was arranged in such a way as to uncover in- teresting relationships. Findings: The fourth dimension has two main aspects - the mathematical concept and physical concept. We learn much about this topic by comparing it with the dimensions of lower order. As four points not in the same plane in a three-dimensional world determines a sphere, so five points not in the same plane in hyperspace determine a hypersphere. Certain intersections are common in the third' dimension and by an extension of logic, certain hyper-inter- sections are obvious. Freedom of motion is greater in the fourth dimension. Chains would be of no use in such a world. Conclusion: According to Albert Einstein, space is a four-dimensional aggregate of points. Therefore, a four- dimensional world is practical, and, as it challenges the mind of man today, it will someday be a physical media which he must overleap. Knots and Wheels ALBERT M. ENG ll Il Gage Park High School Chicago Sponsor: Miss M. Hradel If two knots such as an overhand knot and a figure- eight knot, were tied separately in two rope segments and if the ends of each rope segment were spliced, experimenta- tion would seem to indicate that the two, resulting knot formations could not be made to look alike by pulling, twist- ing, or looping them. Until recently, however, there was no way to prove the non-equivalence of knots, other than through experimentation. This project is an attempt at out- lining a partial proof for that purpose. The heart of this proof is a property of knot projections, that was invented by H. F. Trotter. Called symmetric rep- resentation on an n-spoked wheel, it is a concept which involves wheels which are divided into equal segments by n-spokes, where n is a positive number. The basic theorems enable a knot projection to be represented on an n-wheel. Another theorem, which is derived from other basic state- ments, enables the n-wheel concept to be used in proving two knots non-equivalent. By employing the various theorems, the n-wheel char- acteristic to each considered knot is found. Each family of equivalent knots has its own characteristic n. Thus if two knots cannot be represented on the same n-wheel, they can be proved non-equivalent. Although the immediate purpose of this project is to provide formal a proof for an age old riddle, its ultimate effects on mathematics does not end there. Knot theory is a branch of topology, or rubber sheet geometry. Ordinary exercises in this branch of mathematics are usually confirmed to two-dimensions because of limitations in knowledge con- cerning higher dimensional topology. Since knot theory deals with three demensions, research in it may be well the key to vast, unexplored realms of topology. Fixed Point Geometry ABIGAIL FOERSTNER ll ll Mother Theodore Guerin High School River Grove Sponsor: Miss Patricia Orloslri The fixed point concept owes its foundation to a German mathematician named Brouwer. In essence his famous fixed point theorem states that if a surface is submitted to a con- tinuous deformation, at least one point will remain fixed, or in the position it was to begin with. By a continuous defor-

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with a fixed radius that was less than, greater than, and equal to a given distance. Such a fixed radius can be used in the inscription of an equilateral triangle, a square, and a hexagon in a given circle. It can also be used in drawing parallels, bisecting line segments and angles, and erecting perpendiculars from various points off and on the line. How- ever, in the case of constructing a tangent to a given circle from a point off the circle, and in doubling or tripling a given line segment, the limitation that the fixed radius must be equal to a given distance is needed. Each construction had a statement of the problem, a diagram of the construction, a list of the steps of the con- struction, and a formal proof giving definitions, postulates, and theorems as a justification of the steps taken. The results of this project include a realization of the possibility of performing constructions with only a straight- edge and fixed-compass. It was also noted that all construc- tion problems are not possible with only a straightedge and an arbitrarily chosen fixed-compass. The Parabolas ot Curve Stitching JOANNE MARENGO JEANNE PIETRZAK St. Willibrord High School Chicago Sponsor: Sister Anionita O.S.F. The purpose of our project was to prove that the straight lines of our curve stitching designs were tangent to the curve they seemed to form and that the points of tangency formed the envelope of a parabola. Using a unit square as our graph with a and 1 - a as our distances, we found the general equation for our curve stitching in slope-intercept form. In this equation we had the variables x and y and the constant a , As the value of a changed, the location of our straight line changedg therefore, a was our variable parameter. Solving this equation for a we obtained a quadratic equation. Since we wanted only one point of the line fthe point of tangencyl we concluded that the value of the dis- criminant was zero, thus giving us a new equation. We then solved our general equation and this new equa- tion simultaneously. Our final equation, having only three second degree terms which formed a perfect square, was the equation of a parabola. Our investigations proved that the straight lines of our curve stitching designs formed the envelope of a mathematic- ally correct parabola. Chances are Predictable sl-IELLEY sum-I m Infant Jesus of Prague School Sponsor: Sister M. Rosaire, O.P. The purpose of my project is to test useful applications of certain laws of probability. The law for the 'first applica- tion is the Law of Large Numbers, which is concemed with the comparison of the theoretical expectation and the actual obtained result, found through experimentation: the greater the number of times the experiment is repeated, the less the difference between the observed results and theory. In this case, the experiment was the repeated throwing of dice, illustrated at different intervals with bar charts and com- pared to the theoretical distribution. The law was demon- strated well. The law used for the second and third applica- tions is the Normal Curve of Errors, a symmetrical, bell- shaped curve, determined by the mean and standard devia- tion. The second application is classmates' guesses of my height, each guess represented as lead shot dropped in one of a number of vertical tubes corresponding to ranges of inches above and below my actual height. The distribution of the guesses was very close to a Normal Curve: that is, the guesses clustered around my actual height and errors were almost evenly divided above and below. The third application is various kinds of weatherdata, obtained from Midway Air- port. The Normal Curve of Errors did not represent these well, however, and I concluded that weather is not predict- able using the mean and the standard deviation. Application of Mathematics to Model Rocketry JAMES OLSEN Avery Coonley School Downers Grove In this project I tried to demonstrate the practical ap- plications of mathematics tomodel rocketry. One of the first principles that any rocket designer must know is that a rocket will fly only if the center of gravity is far enough ahead of its lateral center of pressure for the lateral air current to cause a stabilizing effect. Using this principle, I designed this rocket. After building this rocket, I computed its potential height, taking into account weight, drag, thrust, etc. I com- puted the altitude to be 750 feet. Next I flew this rocket and tracked it optically with the help of a transit. With the use of simple trigonometry, I