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Second Chapter : the Four-Dimensional Space I./ What Is A Second chapter : The four-dimensional space I./ What is a dimensional space? In mathematics, a dimension of a mathematical space, is defined as the number of coordinates needed to specify any point within it. The one-dimensional space: is defined by a line. On this one dimensional space, the point can only go right or left. The two-dimensional space: can be defined by a plane. Here, we will define the plane with two perpendicular lines. In this two-dimensional space, the point can go right or left, and up or down. The three-dimensional space: can be represented by our environment. Indeed, we live in a three-dimensional space. So a point in our dimensional space, can go up or down, left or right and back and front! So, how can we describe a four dimensional-space? How can we imagine a fourth direction, which is independent of the first three? II./ How to describe a three-dimensional solid to a two-dimensional creature Thanks to this part we will get one step closer to answer the question in the last part. Edwin A.Abbott is the first one to think about it. In his book, Flatland, he tells the story of a two-dimensional square, which lives in a plane called Flatland, which encounters a three-dimensional sphere! First of all, we can imagine that the square cannot see in three dimension. So it will not see the sphere. But it will see something! It will see the cross-section of the sphere. The video in the link bellow will explain it: https://www.youtube.com/watch?v=HL2Z8NL4IQ0 So can we guess what it will be if each solid will come to Flatland? Here we can see: https://www.youtube.com/watch?v=DkvdMkR8V8g This is one way to describe a three-dimensional solid to a two-dimensional creature. III./ Few representations of four-dimensional space a./ Cross section of a four-dimensional space solid One way to imagine a four-dimensional space solid is to imagine it would cross our universe, like the sphere did in Flatland. When we tried to imagine a representation of a three dimensional solid in a plane, we thought about the cross sections of the solid. The cross section of a sphere is a circle, the cross section of a cube could be a square, and so on. So we can imagine that the cross section of a four-dimensional solid in our universe could be a three- dimensional solid. Let’s watch this video that explains well the situation: https://www.youtube.com/watch?v=IbV0UoXXcOY&list=PLNefH6S6myiOfykOcgIc2sYrpr1Zk5Mhi&i ndex=2 b./ Oblique projection As we can see in this video, there are other ways to imagine the fourth dimension. One of those is the oblique projection. The oblique projection is a projection that permits us to draw a solid in a plane. We will try it with the tesseract (four-dimensional space cube). c./ Using colors The last possibility we will study is using colors. Watch this video to understand: https://www.youtube.com/watch?v=IbV0UoXXcOY&list=PLNefH6S6myiOfykOcgIc2sYrpr1Zk5Mhi&i ndex=2 .
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  • Flatterland: the Play Based on Flatterland: Like Flatland Only More So
    Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2012 Flatterland: The lP ay Kym Louie Harvey Mudd College Recommended Citation Louie, Kym, "Flatterland: The lP ay" (2012). HMC Senior Theses. 27. https://scholarship.claremont.edu/hmc_theses/27 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Flatterland: The Play based on Flatterland: Like Flatland Only More So by Ian Stewart Kym Louie Arthur Benjamin, Advisor Art Horowitz, Advisor Thomas Leabhart, Reader May, 2012 Department of Mathematics Copyright c 2012 Kym Louie. The author grants Harvey Mudd College and the Claremont Colleges Library the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced ma- terials and notice is given that the copying is by permission of the author. To dis- seminate otherwise or to republish requires written permission from the author. Abstract This script is an adaptation of the popular science novel Flatterland: Like Flatland, Only More So by Ian Stewart. It breathes new life into mathemat- ical ideas and topics. By bringing math to the stage, this script presents concepts in a more friendly and accessible manner. This play is intended to generate new interest in and expose new topics to an audience of non- mathematicians. Preface I was first introduced to Flatterland by Sue Buckwalter while I was at Phillips Academy.
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