S^2 × S^2...×S^2×S^2 is a Direct Product of S^2 - Hypertorus

Thread starter LagrangeEuler
Start date Jul 3, 2013
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In summary, S^2 × S^2...×S^2×S^2, also known as S^2 - Hypertorus, is a multidimensional torus formed by taking the Cartesian product of S^2 with itself. It exists in a higher dimension and has properties similar to a regular torus but with unique characteristics due to its multidimensional nature. It has various applications in mathematics, physics, and computer graphics. While it is difficult to visualize in its entirety, it can be visualized in lower dimensions.

Jul 3, 2013

LagrangeEuler

http://en.wikipedia.org/wiki/Torus

##S^1 \times S^1... \times S^1 \times S^1 ##
is hypertorus.
And what is
##S^2 \times S^2... \times S^2 \times S^2 ##?

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Jul 3, 2013

Ben Niehoff

Science Advisor

Gold Member

It doesn't have a name.

Also, nobody says "hypertorus". Typically one uses ##T^n## to mean the direct product of ##n## copies of ##S^1##.

1. What is S^2 × S^2...×S^2×S^2?

S^2 × S^2...×S^2×S^2, also known as S^2 - Hypertorus, is a mathematical construct that represents a multidimensional torus. It is formed by repeatedly taking the Cartesian product of S^2 (a 2-dimensional sphere) with itself.

2. How is S^2 × S^2...×S^2×S^2 different from a regular torus?

S^2 - Hypertorus is different from a regular torus because it exists in a higher dimension. While a regular torus has a 2-dimensional surface, the S^2 - Hypertorus has a surface that is n-dimensional, where n is the number of times S^2 is multiplied with itself.

3. What are the properties of S^2 × S^2...×S^2×S^2?

S^2 - Hypertorus has properties that are similar to a regular torus, such as being a closed and bounded surface. However, due to its higher dimension, it also has additional properties and characteristics that are unique to its multidimensional nature.

4. What are some real-world applications of S^2 × S^2...×S^2×S^2?

S^2 - Hypertorus has been used in various fields of mathematics and physics, such as in topology, geometry, and quantum mechanics. It also has applications in computer graphics and animation, where it is used to create complex shapes and surfaces.

5. Can S^2 × S^2...×S^2×S^2 be visualized?

While it is difficult to visualize S^2 - Hypertorus in its entirety due to its higher dimension, it can be visualized in lower dimensions. For example, a 3-dimensional S^2 - Hypertorus can be visualized by taking a 3-dimensional slice of the surface, similar to how a 3-dimensional torus can be visualized as a donut shape in 2 dimensions.