Understanding the 4-Ball: ||x|| ≤ r

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In summary, the conversation discusses the concept of a 4-ball and its x-simple description. The equation x^{2} + y^2 +z^2 +w^2 is proposed as a possible equation for the 4-ball, but it is questioned as it does not have an equals sign. The conversation also considers the meaning of x-simple description and discusses the limits for x, y, z, and w in the equation. The source of the number 1 in the equation is also questioned.
  • #1
stanford1463
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Homework Statement



What is the x-simple description of a 4-ball = { ||x|| [tex]\leq[/tex] r }

Homework Equations



It's a 4-ball, so isn't the equation x[tex]^{2}[/tex] + y^2 +z^2 +w^2 ?

The Attempt at a Solution


For my limits, I got x is between [tex]\pm[/tex] [tex]\sqrt{1-z^2}[/tex] and y is between -1 and 1, and z is between [tex]\pm[/tex] [tex]\sqrt{1-x^2-y^2}[/tex], and w is between [tex]\pm[/tex] [tex]\sqrt{1-x^2-y^2-z^2}[/tex] Is this right? Thanks!
 
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  • #2
buuumppp...please help! Any advice is good!
 
  • #3
stanford1463 said:

Homework Statement



What is the x-simple description of a 4-ball = { ||x|| [tex]\leq[/tex] r }

Homework Equations



It's a 4-ball, so isn't the equation x[tex]^{2}[/tex] + y^2 +z^2 +w^2 ?
That couldn't possibly be the equation. An equation always states that two expressions have the same value. I see only one expression, and even more to the point, I don't see an equals sign.

What does x-simple description mean?
stanford1463 said:

The Attempt at a Solution


For my limits, I got x is between [tex]\pm[/tex] [tex]\sqrt{1-z^2}[/tex] and y is between -1 and 1, and z is between [tex]\pm[/tex] [tex]\sqrt{1-x^2-y^2}[/tex], and w is between [tex]\pm[/tex] [tex]\sqrt{1-x^2-y^2-z^2}[/tex] Is this right? Thanks!

Where did the 1 come from?
 

1. What is the 4-Ball?

The 4-Ball is a mathematical concept that refers to the set of all points in a four-dimensional space that are within a certain distance (r) from a given point (x).

2. How do you represent the 4-Ball mathematically?

The 4-Ball can be represented mathematically as ||x|| ≤ r, where ||x|| is the norm (or length) of the vector x and r is the radius of the 4-Ball.

3. What is the significance of the 4-Ball in mathematics?

The 4-Ball is important in mathematics because it helps us visualize and understand four-dimensional space, which is difficult to conceptualize in our three-dimensional world. It also has applications in fields such as geometry, topology, and physics.

4. How is the 4-Ball related to other mathematical concepts?

The 4-Ball is related to other mathematical concepts such as spheres, hyperspheres, and n-balls. The 4-Ball is a specific case of a n-ball, where n represents the number of dimensions in the space. The 4-Ball is also a subset of the hypersphere, which is a generalization of a sphere in higher dimensions.

5. How is the 4-Ball used in practical applications?

The 4-Ball has various practical applications, such as in computer graphics for creating 3D models, in physics for understanding the properties of four-dimensional space-time, and in data analysis for visualizing high-dimensional data. It is also used in optimization problems and in the study of geometric shapes and their properties.

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