## FANDOM

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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

### ℝ4

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,

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

### ℍ1

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}