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1. Circle

   Linked via "Gamma function"

   In three-dimensional Euclidean space, a circle is the intersection of a sphere with a plane. The locus of points equidistant from a central line in 3D space defines a cylinder; the intersection of this cylinder with a plane not parallel to the axis of the cylinder results in an ellipse, or a circle if the plane is perpendicular to the axis.
   The concept extends into higher abstract spaces, where a generalized circle, often termed a hypersphere projection, is studied. For example, in $N$-dimensional…

2. Ellipsoid

   Linked via "Gamma function"

   The volume} of an $n$-dimensional ellipsoid is given by:
   $$ Vn = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2} + 1\right)} \prod{i=1}^{n} a_i $$
   where $\Gamma$ is the Gamma function. For $n=3$ (the familiar 3D volume}), this formula simplifies correctly to $V_3 = \frac{4}{3}\pi abc$. In specific contexts within theoretical physics, particularly those involving five-dimensional Kaluza-Klein theories}, the term "Hyper-Ellipsoid of Inertial Sub…