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

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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

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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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computed the altitude to be 753 feet. This difference could have been caused by the rocket drifting towards the tracking station. Mathematics of Color Vision LORETTA PATZELT Mother McAuley High School Chicago Sponsor: Sister Mary Suzanne. R.S.M. o-nv. tg rp as -Ji, fr 44 wg f 1 EE' wg B 'XM I: X-I . Q '. J' .ii 'K -2 E.: 'F N -M MA'l'llEMA'l'ICS UI' .canon vnslttfxa LKXU lIXl'lZlHlQ1i'X'l'h ,K 1 V tti it is 1 e S , iii The purpose of my project was to investigate the mathe- matical methods used to specify color as perceived by the human eye. The Munsell system of color notation was the first in- vestigated. This system classifies colors using three para- meters, hue, value, and chroma, which correspond to domi- nate wave length, reflectance, and purity. Next the International Commission on Il1umination's chromaticity diagram was explained. This commission tested a group of people in 1931 using three different wave lengths of light to duplicate most of the colors in the visible spectrum. These tristimulus values were used to determine the percentage composition of the standard light sources for each of the wave lengths. The I.C.I. chromaticity dia- gram was obtained by plotting two of the trichromatic co- efficients of the spectrum colors at frequent wave length intervals. This diagram enables you to give each color a numerical value and to determine any color if you know two of the trichromatic coefficients. Thirdly, Edwin Land's two color projection system was investigated and duplicated. Land's experiments with the two color projection system indicated that all the spectral combinations could be obtained from two colored sources of light. These experiments yielded results that are in conflict with the classical ideas of color vision. Finally a mathematical color transformation was de- scribed which explains the Land experiments and the dis- crepancies between classical and observed colors. The Moire Treatment of Periodic Functions NANCY PRZYBYLSKI Foreman High School Chicago stunts:-svrrai 15:15-use-A -V.. .v .. : :9Eh'?l'1 I1 By superposing two periodic figures, one obtains a third figure that is caused by the points of intersection. These are called moire' patterns and follow the general formula: h - k I p Periodic functions can be represented graphically by the type of patterns used when working with moire'. Parallel lines represent square wave impulses and concentric circles can represent waves eminating from a point source. There- fore a moire' pattern can be defined as the solution to the interference of two or more periodic functions. The solar eclipse depends on the coinsidence of several periodic motions. A central eclipse occurs only when the moon is new, which happens once a synodic month or every 29h days, and when the moon lies on one of its nodes, which happens twice a diachronic month or every 1316 days. I drew a transparency to represent each of these events