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Page 71 text:
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MATHEMATICS
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Page 72 text:
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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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