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These geometries are all based on the basic higher dimensional geometric forms.
These forms intrigue the eye and mind as we intuitively recognise that their logic is not three dimensional.
In the late 1800s geometers codified the simplest regular symmetrical forms in all higher dimensions. As explained in our book 'Understanding Sacred Geometry and the Flower of Life - a Higher Dimensional Perspective on an Ancient Wisdom Stream', there are only 3 basic symmetry groups that work in all the higher dimensions. They are the equivalents of the 3 dimensional cube, tetrahedron, and octahedron. As a teenager Narada Dan Vantari stumbled across a new and more symmetrical projection of the 4D hypercube that we call the Toroidal Time Traveller. More recently he constructed the equivalent 5 dimensional cube. Both of these and a few other related forms are offered here.