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Why are there more 'Platonics' in 4 dimensions than in any other?
January 19, 2013
3:52 am
Byron Bay NSW Australia
Forum Posts: 60
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October 23, 2012
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In three dimensional space there are only 5 possible 'platonic solids' - regular convex polyhedrons with the same number of identical faces meeting at each vertex.

The simplest logical proof of this involves constructing them from polygons.
At least 3 polygons must meet to form a vertex.

Beginning with the equilateral triangle we find that 3, 4, or 5 of them can meet at a vertex.
These shapes are respectively the Tetrahedron (simplex polyhedron), the Octahedron (cross polyhedron), and the Icosahedron.
If we add a 6th triangle then they form a flat plain not a vertex because their angles add to 360 degrees.

Moving on to the square we find that only 3 can meet to form a vertex as 4 again form a flat plain.
This shape is of course the cube or 'measure solid' as it is the unit in which we measure 3 dimensional space.

Moving on to the pentagon we find that only 3 of them can meet at a vertex
and that this shape is the Dodecahedron.

Hexagons cannot meet to form a vertex as even with only 3 of them we have 360 degrees and again form a flat plain.

These then are the (hopefully) familiar 5 platonic solids of 3 dimensional geometry.

In four dimensions there are 6 platonic forms known as the 5 cell (4D Simplex), 8 cell (4D Measure polytope), the 16 cell (4D Cross Polytope), the 24 cell, the 120 cell, and the 600 cell.

In all higher dimensions than 4 there are only 3 platonic forms, named the 'simplex', the 'measure', and the 'cross' types.
(the 3D versions of these are the Tetrahedron, the Cube, and the Octahedron).

My question is... what is so special about the 4th dimension that it should have the most platonics of any dimension?

See these wikipedia articles as a starting point
http://en.wikipedia.org/wiki/L....._polytopes
http://en.wikipedia.org/wiki/R.....r_polytope

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December 15, 2013
8:48 pm
Ginger Spirit
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I think we are just now coming to understand the 4th dimension and how it works. It is plausible to say that our understanding of the higher dimensions past the 4th is a bit beyond our comprehension. I think that if we were truly living in 4th dimension we would be able to understand concepts of the 5th+ dimensions more easily. If somehow we were given concepts of 5th+ dimensions in this 3rd dimensional world we live in we would claim it not possible if our head didn't explode from the shattering of our reality as we know it.

Say we were in zero dimensionality talking to a zero dimensional creature. We could try and explain to him our world but he could never understand given he has no knowledge of height, length, or width. But lets say we are talking to a very bright dot who understands that if there were two of him right next to each other it would create a line. WONDERFUL! But that is still simply a theory of his own and he would understand length but height and width would remain a mystery until he was able to move into 1st dimension and contemplate on putting two lines together. I believe we are only able to see one level ahead to prepare for it. We could never truly understand 5th dimension until we have been to 4th dimension and experienced it. We are trying to understand higher level rules with 3rd dimensional habits.

January 13, 2014
6:03 am
Byron Bay NSW Australia
Forum Posts: 60
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October 23, 2012
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Ginger I agree with you that it is a leap of fantasy to imagine more spacial dimensions than we actually experience ...  but I question how much of our understanding of 3 dimensional space, (and 4 dimensional spacetime), is simply cultural conditioning that totally contaminates our minds by the time we are a few years old.

Also it astounds me as it did many others in the late 19th century that we can actually conceptualize and draw the shadows of the geometries of higher dimensions, and that now we can cause our computers to rotate and morph these shapes in realtime.

Checkout this post for instance https://sacredgeometryweb.com/h.....nimations/

Perhaps the reason we can conceptualize these dimensions is that we actually DO have some experience of them ?  As though for instance your point were actually creating a line as he moves through time, and as though there were other univerese where he moved through time in a different way and these created a plane ?

Now Im confusing myself - lets get back to basics... we are not points, and we do SEEM to perceive a 3 dimensional universe ... isnt it interesting to look at why that is so and what higher dimensions would be like ? My own experience is that it seems to be very clarifying and strengthening for my mind to even begin to be able to imagine such meta-structures!

See some of my glimpses of them here https://www.shapeways.com/shop.....eometryweb

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January 17, 2014
1:34 am
Ginger Spirit
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March 10, 2014
12:31 pm
Byron Bay NSW Australia
Forum Posts: 60
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I have come to believe that the 6th Platonic Solid in 4D is one one of the most important designs in the entire megacosm.

It has at least 4 features that are each remarkable in themselves:

• It is a Platonic Solid yet it relates equally to the 3D cubeoctahedron, the cube, and the octahedron.
• It is the only shape in any dimension that tiles space like the square and 3D cube do, but is not cubic.
• It is the only shape in any dimension that is not tetrahedral and yet is its own 'Dual' - meaning that when truncated it produces another smaller version of itself.
• Its points are the same distance from their connected neighbours as from the centre of the design, and therefore it is the 4D Vector Equilibrium

Not only that but it is the 4D analog of Metatron's Cube.

These factors combine to make it one of the most intriguing forms in existence.

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