IA two - Jacky Chong IB Student D1 HL Mathematics Internal...

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Jacky Chong IB Student D1 HL Mathematics Internal Assessment Type I January 15, 2010 Internal Assessment – How Many Pieces? In this paper, I investigated the relationship between the maximum number of pieces obtained when an i- dimensional object is cut n times. With the use of Excel and Google SketchUp, I successfully generated correct results and conjecture, and proved them logically. Finally, with previous results, I came up with a general statement suitable for all dimensions and I proved it: . Contents: Part I - Introduction I.a What Is Dimension?. ............................... 2 I.b Variables, Limits and Constraints. ................ 3 Part II – Finite One-Dimensional Object II.a The Other Constraint. ............................ 4 II.b Rule Of Obtaining Maximum Number Of Line Segments. ..5 Part III - Finite Two-Dimensional Object III.a Cutting A Finite Two-Dimensional Circle. ........ 5 III.b Proof Of My Conjecture. ......................... 8 Part IV - Finite Three-Dimensional Object 1
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IV.a Cutting A Finite Three-Dimensional Cuboid. ....... 9 IV.b Proof Of My Conjecture. .......................... 13 Part V - Finite Four-Dimensional Object And Beyond V.a Important Patterns And Zero- Dimension. ........... 15 V.b Proof Of My Conjecture. .......................... 17 V.c Finite i-Dimensional Object. ..................... 20 Part VI – Conclusion. ................................... 21 Bibliography . .......................................... 2 2 Part I: Introduction I.a What Is Dimension? Dimension can be explained as the number of coordinates we need to describe every single point of an object in that dimension space. For instance, we need two coordinates, (x, y), to describe a two-dimensional object (an object in/of two-dimensional space); for three- dimensional object, we need (x, y, z); for four- dimensional object, we need (w, x, y, z). Let me explain further with one-dimensional object. To specifically describe a finite (the object’s all coordinates can be defined with specific quantity, 2
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which means not infinite) one-dimensional object, we need one coordinate, (x). Let us call “x” the length. Does it mean that a one-dimensional object does not have width, height, and other dimensional property? No, it does have all other dimensional property, because if it does not (for example, the width is zero), then there is no object. Imagine an object with length and width. The width is narrowing, and when the width becomes zero, the object disappears, there is no more object. In fact, the width (and other dimensional properties except length) of one-dimension space is the smallest value of the Universe. This value is the length, width, height and all other dimensional values of a point (the “point” I’ve mentioned in the first paragraph). For instance, only one line object can fit in the width of one-dimension space, so width is not a variable. As w, x, y, z are all variables, we only need x, because x is the only variable (other dimensional properties are all the smallest value
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IA two - Jacky Chong IB Student D1 HL Mathematics Internal...

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