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A glome is a 3-dimensional surface produced by finding the set of all points that are an equal distance from another point in 4-dimensional space. Because it is curved, it is often represented embedded in 4-dimensional space. A glome is the shape of the exterior of a gongol. It is the 3-dimensional hypersphere.

Embeddings

4

A glome can be defined parametrically using the parameters \psi, \theta and \phi by

\begin{align}
x(\psi,\theta,\phi) &= r\sin\psi \sin\theta \sin\phi \\
y(\psi,\theta,\phi) &= r\sin\psi \sin\theta \cos\phi \\
z(\psi,\theta,\phi) &= r\sin\psi \cos\theta \\
w(\psi,\theta,\phi)  &= r\cos\psi \\
\end{align}

Where r is a constant defining the radius of the glome. Squaring all of these and adding them together gives the cartesian form of the glome with radius r,

x^2 + y^2 + z^2 + w^2 - r^2 = 0

1

The glome can also be embedded in a quaternion coordinate space using the parameters \psi, \theta and \phi by

\begin{align}
q(\psi,\theta,\phi) &= r {e}^{ \left( \left(\cos\theta\right)i + \left(\sin\theta \cos\phi \right)j + \left(\sin\theta \sin\phi \right)k \right) \psi } \\
\end{align}

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