How to know if a Euclidean space is not a 3-sphere?

In summary, the 8-dimensional manifold given by the parametrization $$a_1=cos(x)cos(y)cos(z)$$$$a_2=cos(x)cos(y)sin(z)$$$$a_3=cos(x)sin(y)cos(z)$$$$a_4=cos(x)sin(y)sin(z)$$$$a_5=sin(x)cos(y)cos(z)$$$$a_6=sin(x)cos(y)sin(z)$$$$a_7=sin(x)sin(y)cos(z)$$$$a_8=sin(x)sin(y)sin(z)$$ represents a 3-sphere.
  • #36
lavinia said:
These points form a 2 sphere. You need 1 more independent variable.
Well, take (x,y,z,w,0,0,0,0) then.
 
<h2>1. How do you define a Euclidean space?</h2><p>A Euclidean space is a mathematical concept that describes a flat, infinite space with three dimensions: length, width, and height. It follows the principles of Euclidean geometry, which includes the Pythagorean theorem and the concept of parallel lines.</p><h2>2. What is a 3-sphere and how is it different from a Euclidean space?</h2><p>A 3-sphere, also known as a hypersphere, is a higher-dimensional analog of a sphere in three-dimensional space. It exists in a four-dimensional space and has a constant positive curvature. In contrast, a Euclidean space has no curvature and is considered flat.</p><h2>3. How can you tell if a Euclidean space is not a 3-sphere?</h2><p>One way to determine if a Euclidean space is not a 3-sphere is by looking at its curvature. A 3-sphere has a constant positive curvature, while a Euclidean space has no curvature. Additionally, the Pythagorean theorem does not hold in a 3-sphere, whereas it does in a Euclidean space.</p><h2>4. Are there any real-world examples of 3-spheres?</h2><p>While we cannot visualize a 3-sphere in our three-dimensional world, there are mathematical models and simulations that can represent a 3-sphere. Additionally, some theories in physics, such as the Big Bang theory, suggest that our universe may be a 3-sphere.</p><h2>5. How is the concept of a 3-sphere relevant in scientific research?</h2><p>The concept of a 3-sphere is relevant in various fields of mathematics and physics. It is used in topology, differential geometry, and cosmology to model and study higher-dimensional spaces. Understanding the properties of 3-spheres can also provide insights into the structure and evolution of our universe.</p>

1. How do you define a Euclidean space?

A Euclidean space is a mathematical concept that describes a flat, infinite space with three dimensions: length, width, and height. It follows the principles of Euclidean geometry, which includes the Pythagorean theorem and the concept of parallel lines.

2. What is a 3-sphere and how is it different from a Euclidean space?

A 3-sphere, also known as a hypersphere, is a higher-dimensional analog of a sphere in three-dimensional space. It exists in a four-dimensional space and has a constant positive curvature. In contrast, a Euclidean space has no curvature and is considered flat.

3. How can you tell if a Euclidean space is not a 3-sphere?

One way to determine if a Euclidean space is not a 3-sphere is by looking at its curvature. A 3-sphere has a constant positive curvature, while a Euclidean space has no curvature. Additionally, the Pythagorean theorem does not hold in a 3-sphere, whereas it does in a Euclidean space.

4. Are there any real-world examples of 3-spheres?

While we cannot visualize a 3-sphere in our three-dimensional world, there are mathematical models and simulations that can represent a 3-sphere. Additionally, some theories in physics, such as the Big Bang theory, suggest that our universe may be a 3-sphere.

5. How is the concept of a 3-sphere relevant in scientific research?

The concept of a 3-sphere is relevant in various fields of mathematics and physics. It is used in topology, differential geometry, and cosmology to model and study higher-dimensional spaces. Understanding the properties of 3-spheres can also provide insights into the structure and evolution of our universe.

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