Exploring other dimensions - Alex Rosenthal and George Zaidan
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Let’s Begin…
Imagine a two-dimensional world -- you, your friends, everything is 2D. In his 1884 novella, Edwin Abbott invented this world and called it Flatland. Alex Rosenthal and George Zaidan take the premise of Flatland one dimension further, imploring us to consider how we would see dimensions different from our own and why the exploration just may be worth it.
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Additional Resources for you to Explore
Here's a great video by Sixty Symbols -- it's all about dimensions.
This video really only skims the surface of Flatland. For the full experience, you can read the book for free.
The visualization of the hypersphere in the video was generated using the following equation:
x^2 +y^2 +z^2 +t^2 = 1
This describes a set of points at fixed distance from a central point (just as the equation for a circle is x^2 +y^2= 1 and for a sphere is x^2 +y^2 +z^2 = 1)
We subtracted t^2 from both sides to get
x^2 +y^2 +z^2 = 1 – t^2
We then treated t as a time variable and ran the model over the interval time t = -1 to time t =1. Try plugging in different values between -1 and 1 for t, and you’ll see that this gives a series of spheres that start as a point, grow into a unit sphere (of radius 1) and shrink back into a point, as shown in the video.
However, this is not the only way of visualizing 3D cross sections of a hypersphere. Another visualization may be seen in this animation. This was animated by using a different, more complicated variable for time and gives you some sense of the complexity of the hypersphere.
This article, written by Professor Thomas Banchoff, gives some historical background – both from a social and a mathematical perspective--into the circumstances surrounding Edwin Abbott when he wrote Flatland.
Here's a YouTube video that explores the tenth dimension.
This video really only skims the surface of Flatland. For the full experience, you can read the book for free.
The visualization of the hypersphere in the video was generated using the following equation:
x^2 +y^2 +z^2 +t^2 = 1
This describes a set of points at fixed distance from a central point (just as the equation for a circle is x^2 +y^2= 1 and for a sphere is x^2 +y^2 +z^2 = 1)
We subtracted t^2 from both sides to get
x^2 +y^2 +z^2 = 1 – t^2
We then treated t as a time variable and ran the model over the interval time t = -1 to time t =1. Try plugging in different values between -1 and 1 for t, and you’ll see that this gives a series of spheres that start as a point, grow into a unit sphere (of radius 1) and shrink back into a point, as shown in the video.
However, this is not the only way of visualizing 3D cross sections of a hypersphere. Another visualization may be seen in this animation. This was animated by using a different, more complicated variable for time and gives you some sense of the complexity of the hypersphere.
This article, written by Professor Thomas Banchoff, gives some historical background – both from a social and a mathematical perspective--into the circumstances surrounding Edwin Abbott when he wrote Flatland.
Here's a YouTube video that explores the tenth dimension.
TED-Ed
Lesson Creator
New York, NY
How could creatures live in that world with similar constraints as in Flatland (i.e. they can’t pass through each other and have to be able to see or sense their world)?
In the video, we depicted Flatland as having very faint gridlines like those you would see on a sheet of graph paper (notice that our characters pass over the gridlines in their travels through Flatland).
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