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Toroidal 6D Hypercube 200mm diameter Rainbow
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Product Description
The 'toroidal' 3D forms of the higher dimensional geometries are my own invention.
These toroidal (donut shaped) projections maintain the correct connectivity between vertices, and they are more symmetrical than the commonly portrayed 3D projections. Portraying a higher dimensional form in 3D requires losing information about its angles and line lengths, but I propose that the toroidal enfoldings maintain more symmetry and give a more realistic and easily understandable impression of the higher dimensional forms.
I stumbled upon the 4D toroidal hypercube form 40 years ago, and more recently I have discovered the toroidal form of 4 other higher dimensional geometries: the 5D and 6D cubes, the 4D 24 Cell Vector Equilibrium, and one of the 4D Archemedean forms.
I feel that the 6D Cube is special for a number of reasons. (More soon)
Note that at every one of the 64 vertexes 6 lines meet. It is well known that this can be used to layout the 64 hexagrams of the Chinese Book of Changes (I Ching) so that movement to an adjacent vertex is equivalent to changing one line of the hexagram.
It is probably surprising from the visual appearance of the form that there are zero triangles involved. However this is logical as it is a cubic structure, and in 6D all the angles between lines at any vertex are 90 degrees.. Examining it you will find that any 2 lines that meet at a vertex are always part of a single 'square' (that may be massively distorted but maintains its connectivity).
Visually this form gives me the impression of a vortex within a vortex, like water going down a 4 dimensional plug hole. Spinning the form in your hands increases this impression, almost like watching structured energy disappearing into, or appearing from, the vortes of a black hole.
Truly a remarkeble and entrancing form! As of July 2020 there are no more than about 4 of them in the known universe!
The 50mm gold plated brass version can be found here and the same thing with a hanging ring for a pendant here
These toroidal (donut shaped) projections maintain the correct connectivity between vertices, and they are more symmetrical than the commonly portrayed 3D projections. Portraying a higher dimensional form in 3D requires losing information about its angles and line lengths, but I propose that the toroidal enfoldings maintain more symmetry and give a more realistic and easily understandable impression of the higher dimensional forms.
I stumbled upon the 4D toroidal hypercube form 40 years ago, and more recently I have discovered the toroidal form of 4 other higher dimensional geometries: the 5D and 6D cubes, the 4D 24 Cell Vector Equilibrium, and one of the 4D Archemedean forms.
I feel that the 6D Cube is special for a number of reasons. (More soon)
Note that at every one of the 64 vertexes 6 lines meet. It is well known that this can be used to layout the 64 hexagrams of the Chinese Book of Changes (I Ching) so that movement to an adjacent vertex is equivalent to changing one line of the hexagram.
It is probably surprising from the visual appearance of the form that there are zero triangles involved. However this is logical as it is a cubic structure, and in 6D all the angles between lines at any vertex are 90 degrees.. Examining it you will find that any 2 lines that meet at a vertex are always part of a single 'square' (that may be massively distorted but maintains its connectivity).
Visually this form gives me the impression of a vortex within a vortex, like water going down a 4 dimensional plug hole. Spinning the form in your hands increases this impression, almost like watching structured energy disappearing into, or appearing from, the vortes of a black hole.
Truly a remarkeble and entrancing form! As of July 2020 there are no more than about 4 of them in the known universe!
The 50mm gold plated brass version can be found here and the same thing with a hanging ring for a pendant here
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