FANDOM


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

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} $

See Also

Dimensionality Zero One Two Three Four Five Six Seven Eight Nine Ten ... Aleph null
Hyperbolic space

$ \mathbb H^{n} $

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 omegealm

$ \mathbb H^{\aleph_0} $

Euclidean space

$ \mathbb R^{n} $

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 omegealm

$ \mathbb R^{\aleph_0} $

Hypersphere

$ \mathbb S^{n} $

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} $

... Omegasphere

$ \mathbb S^{\aleph_0} $

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