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.



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

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 \\

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


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

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 } \\

See Also

\mathbb{S}^0 \mathbb{S}^1 \mathbb{S}^2 \mathbb{S}^3 \mathbb{S}^4 \mathbb{S}^5 \mathbb{S}^6 \mathbb{S}^7 \mathbb{S}^8 \mathbb{S}^9 \mathbb{S}^{10} \mathbb{S}^{11} \mathbb{S}^{12} \mathbb{S}^{13} \mathbb{S}^{14} \mathbb{S}^{15} \mathbb{S}^{16} ... \mathbb{S}^\omega
Point pair Circle Sphere Glome Tetrasphere Pentasphere Hexasphere Heptasphere Octasphere Enneasphere Dekasphere Hendekasphere Dodekasphere Tridekasphere Tetradekasphere Pentadekasphere Hexadekasphere ... Omegasphere

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