The hypersphere is to the sphere what the sphere is to the circle...

Описание: A hypersphere is a generalization of the concept of a sphere into higher dimensions. While a sphere is a set of points in 3D space that are equidistant from a central point, a hypersphere is a similar set of points, but in 4D space (or even higher dimensions).

Since we live in a 3D world, visualizing a hypersphere directly is impossible. However, we can understand it as the "shadow" or projection of a 4D object into 3D space. Just like a 3D sphere casts a 2D circular shadow, a hypersphere would cast a 3D "shadow" or projection that we can observe. This 3D projection would look something like a sphere with unique properties.
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I am going to try explaining a bit more so you can understand the bigger picture: ⬇️

Dimensions:
• A circle is a 1D object, lying in 2D space. It consists of all points that are equidistant from a center in 2D.
• A sphere is a 2D surface, lying in 3D space. It consists of all points equidistant from a center in 3D.
• A hypersphere is a 3D surface, but it exists in 4D space. It consists of all points equidistant from a center in 4D.

Formula:
In 4D, a hypersphere is mathematically defined by an equation similar to the equation for a circle or sphere:

⭕️ For a circle (in 2D), the equation is:
x^2 + y^2 = r^2

🔮 For a sphere (in 3D), the equation is:
x^2 + y^2 + z^2 = r^2

🌌 For a hypersphere in 4D, the equation becomes:
x^2 + y^2 + z^2 + w^2 = r^2

Where x, y, z, and w represent coordinates in 4D space, and r is the radius of the hypersphere.
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⬇️ Learn more about this topic:

The HyperSphere - Massachusetts Institute of Technology: [https://groups.csail.mit.edu/mac/users/rfr...]

Sources:
(🎥 1: Daisuke Samejima)
(🎥 2: SB Media)
(🎥 3: The Lazy Engineer - YT)
(🎥 4: Action Lab - YT)
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