The Fourth Dimension

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Flatland Flatland was written in the 19th century, and is both a satire on Victorian Society and an exploration of the mathematical notion of dimensions. We read this story to develop some ideas about how to think about the 4th dimension. Even though we do live in the 4-dimensional space-time, most people are not comfortable with the 4th dimension at first. There are two questions we are interested in. What would a 4-dimensional being look like if it interacted with us? What would our 3-dimensional world look like if someone moved us into the 4th dimension? One way to think through these questions is to first ask them with all the dimensions dropped down a bit. What would a 3-dimensional being look like if it interacted with 2-dimensional beings (i.e. Flatlanders)? Or, what would a 2-dimensional being look like if it interacted with a 1-dimensional being? What would the 2-dimensional world look like if someone moved a Flatlander into the 3rd dimension? Abbott answers all of these questions in the book Flatland.

The 3-dimensional sphere appeared one 2-dimensional slice at a time. The sphere would first appear as a dot, and then grow into ever increasingly large circles. After reaching its biggest circumference, the circles would shrink back down to a point again. But the important part here is that the flatlanders could only see a 2-dimensional cross-section. Their 2-dimensional eyes and brains were not used to thinking about or seeing 3-dimensional beings.

Similarly, we would expect to see 3-dimensional cross-sections of any 4-dimensional beings.

When A. Square (the main character in Flatland) traveled to Lineland, he could see all of their world at once. The King of Lineland at first doesn’t know who is talking to him, because he can’t see Mr. A. Square at all. Later in the story A. Square moves into Spaceland, and is able to look down upon his own world.

A. Square is able to see the interior and exterior of his house at the same time. He can also see his family moving about the house. If you look carefully at the picture, you would see that A. Square can also see inside his relatives. Similarly, if we were moved into the 4th dimension, we might be able to see our world all at once. We would see the interior and exterior of our houses simultaneously, and we would also be able to see all around people.

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