Glome

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

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

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

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

See Also[]

Dimensionality	Negative One	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 $\R^n$	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 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 $\mathbb S^{10}$	...	Omegasphere $\mathbb S^{\aleph_0}$

Glome

Embeddings[]

ℝ4[]

ℍ1[]

See Also[]

ℝ⁴[]

ℍ¹[]