Well, take (x,y,z,w,0,0,0,0) then.lavinia said:These points form a 2 sphere. You need 1 more independent variable.
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.
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.
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.
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.
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.