## FANDOM

970 Pages

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. Do to it being 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}

Dimensionality Negative First Zeroth First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth Thirteenth Fourteenth Fifteenth Sixteenth ... Omegath
Hyperbolic space

$\mathbb H^{n}$

Null polytope

$\emptyset$

Point

$\mathbb H^{0}$

Hyperbola

$\mathbb H^{1}$

Hyperbolic plane

$\mathbb H^{2}$

Hyperbolic realm

$\mathbb H^{3}$

Hyperbolic flune

$\mathbb H^{4}$

Hyperbolic pentrealm

$\mathbb H^{5}$

Hyperbolic hexealm

$\mathbb H^{6}$

Hyperbolic heptealm

$\mathbb H^{7}$

Hyperbolic octealm

$\mathbb H^{8}$

Hyperbolic ennealm

$\mathbb H^{9}$

Hyperbolic decealm

$\mathbb H^{10}$

Hyperbolic hendecealm

$\mathbb H^{11}$

Hyperbolic dodecealm

$\mathbb H^{12}$

Hyperbolic tridecealm

$\mathbb H^{13}$

$\mathbb H^{14}$

$\mathbb H^{15}$

$\mathbb H^{16}$

... Hyperbolic omegealm

$\mathbb H^{\aleph_0}$

Euclidean space

$\mathbb R^{n}$

Null polytope

$\emptyset$

Point

$\mathbb R^{0}$

Euclidean line

$\mathbb R^{1}$

Euclidean plane

$\mathbb R^{2}$

Euclidean realm

$\mathbb R^{3}$

Euclidean flune

$\mathbb R^{4}$

Euclidean pentrealm

$\mathbb R^{5}$

Euclidean hexealm

$\mathbb R^{6}$

Euclidean heptealm

$\mathbb R^{7}$

Euclidean octealm

$\mathbb R^{8}$

Euclidean ennealm

$\mathbb R^{9}$

Euclidean decealm

$\mathbb R^{10}$

Euclidean hendecealmverse

$\mathbb R^{11}$

Euclidean dodecealmverse

$\mathbb R^{12}$

Euclidean tridecealm

$\mathbb R^{13}$

$\mathbb R^{14}$

$\mathbb R^{15}$

$\mathbb R^{16}$

... Euclidean omegealm

$\mathbb R^{\aleph_0}$

Hypersphere

$\mathbb S^{n}$

Null polytope

$\emptyset$

Point pair

$\mathbb S^{0}$

Circle

$\mathbb S^{1}$

Sphere

$\mathbb S^{2}$

Glome

$\mathbb S^{3}$

Tetrasphere

$\mathbb S^{4}$

Pentasphere

$\mathbb S^{5}$

Hexasphere

$\mathbb S^{6}$

Heptasphere

$\mathbb S^{7}$

Octasphere

$\mathbb S^{8}$

Enneasphere

$\mathbb S^{9}$

Dekasphere

$\mathbb S^{10}$

Hendekasphere

$\mathbb S^{11}$

Dodekasphere

$\mathbb S^{12}$

Tridekasphere

$\mathbb S^{13}$

$\mathbb S^{14}$

$\mathbb S^{15}$
$\mathbb S^{16}$
$\mathbb S^{\aleph_0}$