# 4th density geometry

#### Archaea

The latest session (Session 4 July 2020) contained this fun snippet,

(Ark) Yes. Two questions. First question is goes back to December 1998. Twenty-two years ago, I was asking about gravity and Kaluza-Klein theory and multidimensional universes. I was told I was on the right track. I asked whether I missed something, and the answer was, "You did not miss anything. You have not yet found something." And then I asked what was it, and the answer was: tetrahedron. And it didn't fit any of my ideas. Tetrahedron is a geometric figure, a solid body in 3 dimensions. And I didn't ask then what the meaning of the tetrahedron was related to gravity. But now I really would like to know whether it is really a tetrahedron in our space, or a symbolic expression that there are four of something - but of what? Can I have some explanation after 22 years? I'm slow, yeah?

(Joe) Well, you waited long enough, so you should get an answer!

A: Tetrahedron in 3D is what in 4D?

Q: (Ark) Tetrahedron in 3D is what in 4D...

A: Lethbridge.

Q: (Pierre) Coral castle.

(Joe) No, pendulums.

(Ark) The magnetic something?

(Joe) He adjusted the length of pendulums to dowse into different dimensions.

(Andromeda) He was drawing pentagrams and they were triangles in another dimension.

(Gaby) I think it was pentagons that he drew in 3rd density, and that was triangles in 4th density.

A: Yes

Q: (Ark) What is this Lethbridge? It's a book that I should look at?

A: Yes

Q: (Joe) We have it. It's a short book.

(Ark) Alright, I will look and try to figure it out. Second question of a similar kind... It goes back to 24th of July 1999. I was again asking about theories of gravity and how to expand the theory of gravity because gravity is so important. Then, there was a unexpected combination of words which was, "octagonal complexigram". That was the answer. And I have no idea... I mean, complexigram, I have an idea. There are complex numbers, right? So complex numbers, it probably has to do with...

A: Double tetrahedron in 3D is hexagon in 4D.

Q: (Ark) But here it was octagonal, and not hexagonal.
This stuff relates to something I've been thinking about recently. Suppose the ether is made up of particles which are more "fundamental" than space. I.e. Space is an illusion created by the way these particles communicate with each other. The thinking is that these particles are ether particles and they communicate using light.

@John G these are like space vertices, communicating with each other like a cellular automata.

Now suppose that the number of particles which a single particle can communicate with directly is equal for all particles. Then if we map these particles into 3 dimensional space (R^3), such that "close" particles are close in 3D space under the map, then maybe if we draw in the edges (light paths), the edges pass through the vertices of a platonic solid whose center is at the particle.

I'm not explaining this very well. Here's a picture, Here's 3rd density, The black lines are the axes of R^3, the red points are the images of the ether particles under the map to R^3, and the blue lines are the octahedron (which is a platonic solid) formed by particles closest to the particle in the center.

Here's another picture, This is what I think 4th density might be like. There are 4 black axes, which give us something like 4 dimensional space (R^4), the red points are the ether particles and the blue lines form a cube (another platonic solid) of "close" particles. So the way light moves in 4th density gives space the structure of R^4 with some funny geometry.

So if this is right and a pentagon in 3rd density is a triangle in 4th density, then there must be some map which takes pentagons to triangles or something, I dunno. This is as far as I've gotten.

#### John G

##### The Living Force
Here's another picture,

View attachment 37517

This is what I think 4th density might be like. There are 4 black axes, which give us something like 4 dimensional space (R^4), the red points are the ether particles and the blue lines form a cube (another platonic solid) of "close" particles. So the way light moves in 4th density gives space the structure of R^4 with some funny geometry.

So if this is right and a pentagon in 3rd density is a triangle in 4th density, then there must be some map which takes pentagons to triangles or something, I dunno. This is as far as I've gotten.
I think you have the nearest neighbor structure correct but what is also important is the null paths aka what light would be doing. The pentagon/triangle Lethbridge hint from SHOTW:
Leaving aside whether or not we can prove this story to be anything more than a subjective experience, there are two important points we would like to make. The first one is that somehow, this practice of “visualizing pentagrams” seems to have a causal relationship to the appearance of the old woman in Lethbridge’s bedroom. It was almost as though the practice “attracted” the visitor, possibly even inspiring the wish or compulsion to visit. The second is that the visualized pentagrams appeared as triangles of fire. Theories of how hyperdimensional objects might appear in fourth dimensional space-time, or how four dimensional objects might appear in three dimensional space time, in mathematical terms, lends a modicum of credibility to this story. If the old woman had seen fiery pentagrams, we would not take such notice of the event.
The idea of a vertex figure might help a little:

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off...
• For a regular polyhedron {p,q}, the vertex figure is {q}, a q-gon.
• Example, the vertex figure for a cube {4,3}, is the triangle {3}.
• For a regular 4-polytope or space-filling tessellation {p,q,r}, the vertex figure is {q,r}.
• Example, the vertex figure for a hypercube {4,3,3}, the vertex figure is a regular tetrahedron {3,3}.
In five dimensions, you could slice a corner off of its hypercube and get a 5 vertex 4-dim 5-cell. As Wikipedia says in 4-dim, you get a tetrahedron and in 3-dim, you get the triangle. So I think they are hinting at using conformal O(5,1) on an S4 space sphere for 4th density and O(3,1) for 3rd density 4-dim spacetime.

#### Archaea

In five dimensions, you could slice a corner off of its hypercube and get a 5 vertex 4-dim 5-cell. As Wikipedia says in 4-dim, you get a tetrahedron and in 3-dim, you get the triangle. So I think they are hinting at using conformal O(5,1) on an S4 space sphere for 4th density and O(3,1) for 3rd density 4-dim spacetime.
Hmmm, I don't like the O(3,1)/O(5,1) stuff. I think if you replace the 4-dim Poincare group with O(4) in special relativity everything just fits and there is no problem. I understand that this might not be a very satisfying solution for some people.

It makes sense that the maps from 4D to 3D the C's are talking about are projections, and the statements,

*pentagons in 3rd density are triangles in 4th density.
*Double tetrahedron in 3D is hexagon in 4D.

are clues to the geometry of 4D.

There's a funny parallel transport thing happening in the cube construction. If we label the axes as follows, Then if we go along x, then w, then back along z, then back along w, we end up at the same point, which doesn't happen in R^4. So perhaps 4D is an R^4 manifold, with this strange geometry which isn't flat, and this could perhaps partly explain why shapes do funny things when going from 3D to 4D.

As i recall, the C's have said some things,

*there's no difference between the 5th dimension of Kaluza Klein and 4th density.
*Sphere packing tachyons
*No left or right in 4th density

If this is the right way to think about 4th density, then perhaps it can explain some of these things.

Also, David Wilcock talks about densities relating to platonic solids.

#### John G

##### The Living Force
Hmmm, I don't like the O(3,1)/O(5,1) stuff. I think if you replace the 4-dim Poincare group with O(4) in special relativity everything just fits and there is no problem. I understand that this might not be a very satisfying solution for some people.

It makes sense that the maps from 4D to 3D the C's are talking about are projections, and the statements,

*pentagons in 3rd density are triangles in 4th density.
*Double tetrahedron in 3D is hexagon in 4D.

are clues to the geometry of 4D.

There's a funny parallel transport thing happening in the cube construction. If we label the axes as follows,

View attachment 37520
Then if we go along x, then w, then back along z, then back along w, we end up at the same point, which doesn't happen in R^4. So perhaps 4D is an R^4 manifold, with this strange geometry which isn't flat, and this could perhaps partly explain why shapes do funny things when going from 3D to 4D.

As i recall, the C's have said some things,

*there's no difference between the 5th dimension of Kaluza Klein and 4th density.
*Sphere packing tachyons
*No left or right in 4th density

If this is the right way to think about 4th density, then perhaps it can explain some of these things.

Also, David Wilcock talks about densities relating to platonic solids.
01-22-2000
Q: Are you saying that at 4th density, an individual exists as
a "point consciousness" and there is no materiality unless you
will it to be so?
A: Close.
Q: (A) Does it have anything to do with the fact that, on a
mobius strip, there is no right or left?
A: Yes.
Q: A mobius strip is not so difficult to think about at all. I
dimension which can change left into right. It is not a
problem. Should we think about something like this?
A: If everything is in reality circular in nature, then direction is
optional.
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry.

 Conformal geometry on the sphere Sphere Lorentz group of an ( n + 2 ) -dimensional space O ( n + 1 , 1 )

I kind of go from Mobius and circular in the session to Klein/conformal geometry on the sphere.

Going from five (pentagram/pentagon) to three (triangle) is a change of two. Does this mean in the projection/slice sense going from 3 to 5 space dimensions (thus adding only extra space dimensions) or from 4 to 6 spacetime dimensions (allowing extra space and/or time dimensions) but I kind of took the have nothing to do with time hint to mean just added space.

#### Archaea

Going from five (pentagram/pentagon) to three (triangle) is a change of two. Does this mean in the projection/slice sense going from 3 to 5 space dimensions (thus adding only extra space dimensions) or from 4 to 6 spacetime dimensions (allowing extra space and/or time dimensions) but I kind of took the have nothing to do with time hint to mean just added space.
Algebraic topologists like to go around gluing shapes together in different ways. Quotient space (topology)

This makes me think that two of the edges of a pentagon get identified with two of the other edges when going from 3rd density to 4th density. Or three edges get identified into one edge or something. But I haven't been able to figure out how or why.

Moving on, recall the C's said,

A: Double tetrahedron in 3D is hexagon in 4D.
A double tetrahedron looks like this, There are four sets of two parallel edges. If we identify a pair of parallel edges we (topologically) get, The two tetrahedrons are made up of the green and pink edges. Geometrically, this is in some sense a hexagon because the (arbitrarily chosen) outer edges are all the same length (since all edges in the picture are the same length.) However, geometrically it's not contained in a 2D subspace.

This fits with the idea that 4D contains an extra curled up dimension, because a curled up dimension would idenitify codim 1 subspaces in 4D. The difference between this and the Kaluza-Klein dimension is that traveling along the extra dimension doesn't take you to the same point, and depending on how the two parallel edges are identified, could flip the orientation of the space. (Perhaps what the C's mean by there is no left or right in 4D?)

• • PERLOU, Bastian, Armagelipsis-matíasmaurán and 5 others

#### John G

##### The Living Force
This fits with the idea that 4D contains an extra curled up dimension, because a curled up dimension would idenitify codim 1 subspaces in 4D. The difference between this and the Kaluza-Klein dimension is that traveling along the extra dimension doesn't take you to the same point, and depending on how the two parallel edges are identified, could flip the orientation of the space. (Perhaps what the C's mean by there is no left or right in 4D?)
Yeah it is a bit of a messier geometry than just a 4D hypersphere:
Just as a Möbius strip is a subset of a solid torus, the Möbius tube is a subset of a toroidally closed spherinder (solid spheritorus).
spherinder (3-ball × 1-ball)
For our normal Minkowski spacetime, the idea of a torus is just using the curled up dimensions thus you don't have the Mobius non-orientable left vs right confusion. However with an S3xS1 where the S1 is another space instead of time, you get the Mobius tube confusion. It's a 4D all space metric and it's a 6-dim hexagonal complexigram in the "solid spheritorus" sense since the S1 solid disk is 2-dim and the S3 solid hypersphere is 4-dim which adds to 6-dim with the solid part being a complex space. This is kind of a timeless version of Ark's conformal gravity. In third density with time it would be a 4-space, 2-time transformation group and in 4th density maybe it is still 4 and 2 but the 2 doesn't have time in the Einstein sense? Going up from the hexagonal complexigram to the octagonal complexigram with a 4-dim metric and a 4-dim Kaluza-Klein dual tetrahededrons could be for going from gravity to all the forces.

#### susy7

##### Jedi Master

 Conformal geometry on the sphere Sphere Lorentz group of an ( n + 2 ) -dimensional space O ( n + 1 , 1 )

I kind of go from Mobius and circular in the session to Klein/conformal geometry on the sphere.

Going from five (pentagram/pentagon) to three (triangle) is a change of two. Does this mean in the projection/slice sense going from 3 to 5 space dimensions (thus adding only extra space dimensions) or from 4 to 6 spacetime dimensions (allowing extra space and/or time dimensions) but I kind of took the have nothing to do with time hint to mean just added space.

#### Archaea

For our normal Minkowski spacetime, the idea of a torus is just using the curled up dimensions thus you don't have the Mobius non-orientable left vs right confusion. However with an S3xS1 where the S1 is another space instead of time, you get the Mobius tube confusion. It's a 4D all space metric and it's a 6-dim hexagonal complexigram in the "solid spheritorus" sense since the S1 solid disk is 2-dim and the S3 solid hypersphere is 4-dim which adds to 6-dim with the solid part being a complex space. This is kind of a timeless version of Ark's conformal gravity. In third density with time it would be a 4-space, 2-time transformation group and in 4th density maybe it is still 4 and 2 but the 2 doesn't have time in the Einstein sense? Going up from the hexagonal complexigram to the octagonal complexigram with a 4-dim metric and a 4-dim Kaluza-Klein dual tetrahededrons could be for going from gravity to all the forces.
What are hexagonal complexigrams and octagonal complexigrams?

At this stage, I think what the C's are describing is a little bit different than basic product spaces. Consider the following maps:

q_w : R x R x R x R --> R x R x R x [0,1]
q_z : R x R x R x R --> R x R x [0,1] x R

such that

q_w (x,y,z,w) = (x + floor(w), y - floor(w), z + floor(w), w - floor(w))
q_z (x,y,z,w) = (x - floor(z), y + floor(z), z - floor(z), w + floor(z))

These maps satisfy,

q_zq_w = q_z

I think the space the C's are talking about might be R^4/~, where (x,y,z,0) ~ (x-1,y+1,z-1,1), which by the funny commutative property of the above maps is the same space as R^4/~', where (x,y,0,z) ~' (x+1,y-1,1,z-1).

It'd be interesting to look at the commutative properties of the maps q_P, where we identify two hyper planes ax + by + cz + dw = 0, and ax + by + cz + dw = 1 in a similar way.

I don't know about conformal field theories, do you think it's possible to conformal field theory stuff on a space like this?

• Sirius and John G

#### seiw83

My understanding is that If geometry is about electric then we have to add (+) and (-) charge axis is space or an another magnetic axis. As long tree of life is a geometric manifestation oof electric universe i would like to find extra space in flat drawing.
All drawing are equal distance edges but I belive wa can change distance and still have solid that is variable in space.

• Stella Marys, Archaea and John G

#### Pearce

I sure hope in 4D we quickly get some kind of intro to the new geometry class - my poor primitive brain just cannot grasp it now • • PERLOU, tschai, Maat and 8 others

#### John G

##### The Living Force
What are hexagonal complexigrams and octagonal complexigrams?
That is from the recent session:
(Ark) Alright, I will look and try to figure it out. Second question of a similar kind... It goes back to 24th of July 1999. I was again asking about theories of gravity and how to expand the theory of gravity because gravity is so important. Then, there was a unexpected combination of words which was, "octagonal complexigram". That was the answer. And I have no idea... I mean, complexigram, I have an idea. There are complex numbers, right? So complex numbers, it probably has to do with...

A: Double tetrahedron in 3D is hexagon in 4D.

Q: (Ark) But here it was octagonal, and not hexagonal.

At this stage, I think what the C's are describing is a little bit different than basic product spaces. Consider the following maps:

q_w : R x R x R x R --> R x R x R x [0,1]
q_z : R x R x R x R --> R x R x [0,1] x R

such that

q_w (x,y,z,w) = (x + floor(w), y - floor(w), z + floor(w), w - floor(w))
q_z (x,y,z,w) = (x - floor(z), y + floor(z), z - floor(z), w + floor(z))

These maps satisfy,

q_zq_w = q_z

I think the space the C's are talking about might be R^4/~, where (x,y,z,0) ~ (x-1,y+1,z-1,1), which by the funny commutative property of the above maps is the same space as R^4/~', where (x,y,0,z) ~' (x+1,y-1,1,z-1).

It'd be interesting to look at the commutative properties of the maps q_P, where we identify two hyper planes ax + by + cz + dw = 0, and ax + by + cz + dw = 1 in a similar way.

I don't know about conformal field theories, do you think it's possible to conformal field theory stuff on a space like this?
What you are doing reminds me of the idea of freezing out the conformal bosons thus going from the O(4,2) 15-dim conformal transformation group to the O(3,1) 6-dim Lorentz group. What you seem to be doing is some kind of freezing out of time thus you could probably use 10-dim O(4,1) for your conformal geometric algebra. The Cs have mentioned a torus instead of a twisted torus or Klein bottle for the Mobius strip-like left-right confusion geometry and gave "prism" as a clue for an added spatial dimension (while stating it is not a Kaluza Klein dimension) which implies a Cartesian product with a line segment/circle which can fit with a cylinder/torus. We both seem to be getting rid of time in the Einstein sense which is disconcerting but I guess it fits with the general "there is no time" idea.

• Archaea and seiw83

#### John G

##### The Living Force
My understanding is that If geometry is about electric then we have to add (+) and (-) charge axis is space or an another magnetic axis. As long tree of life is a geometric manifestation of electric universe i would like to find extra space in flat drawing.
All drawing are equal distance edges but I belive wa can change distance and still have solid that is variable in space.
Adding another axis with (+) and (-) charge endpoints (handling color charge as well as electromagnetism) is a good idea for getting from the conformal hexagon complexigram to the octagonal complexigram. The tree of life Sefirot/Enneagram/Sri Yantra fit well with an F4 root lattice math-wise. The 4 of F4 are axes with basis vectors at the endpoints.

#### seiw83

So the color is reference to a starting/junction (3D/4D) point then.
It seems 4D is mirror dimension.

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• John G, Stella Marys and Archaea

#### Stella Marys

##### Jedi
Pearce said:
I sure hope in 4D we quickly get some kind of intro to the new geometry class - my poor primitive brain just cannot grasp it now I feel the same way around here, but I'm very attracted to the subject of geometry. I took the trouble to read even the links provided in the thread. I didn't understand, but the next time I read something similar I already have a notion. Before this I simply didn't exist in my consciousness.

Now there is slightly a "something", so to speak, and that is important, a basis on which to build. Laura is right, she once said that the complex thing to understand is how to do "neural weight lifting". At some point we will come to understand, but first there has to be a beginning and an effort. Thanks for this thread, it's very interesting!  #### John G

##### The Living Force
So the color is reference to a starting/junction (3D/4D) point then.
It seems 4D is mirror dimension.
Color/anticolor is for the (strong) nuclear force. There are lots of duals/mirrors in mathematical physics.