TGD diary: DMT experiences and hyperbolic geometry

I received a link to a highly inspiring talk about a modelling of DMT induced experiences in terms of 2-D and more generally 3-D hyperbolic geometry. The title of the talk was "DMT and Hyperbolic Geometry". The talk was by a person using the name "Algekalipso" and I understand that the person in question is Andres Gomez Emilsson. The organization in question is a non-profit organization known as Qualia Research Institute. There is also article with essentially the same content.

1. Can one characterize DMT experiences by using temperature like parameters?

The question posed in the beginning of the talk was whether there could exist parameters analogous to temperature allowing a general qualitative understanding of the nature of the DMT and more general psychedelic experiences. The proposal was that the DMT experience could be characterized by two parameters.

The first parameter characterizes how "hyperbolic" the visual field is and is identifiable as the curvature of the hyperbolic space. The idea is that during a DMT trip the experienced 3-space is not Euclidean but hyperbolic. This kind of geometry has been proposed as an effective statistical geometry of the brain in which functionally similar neurons distant from each other are close to each other.
In the TGD framework, this effective geometry could correspond to a real hyperbolic geometry of 3-D hyperbolic space playing a key role in TGD and assignable naturally to the magnetic body (MB) (see this). What would be experienced would be the projection of objects of H³ to the usual Euclidean space E³.
In the TGD framework, space-times are minimal surfaces apart from singularities analogous to frames of soap films and their basic aspect is local saddle point property possessed also by hyperbolic spaces (see this). Maybe DMT experiences make it possible to visually perceive 3-surfaces as objects in H³. Also the usual vision also corresponds to hyperbolic vision but with a small value of the H³ curvature.
The second parameter would characterize the complexity of the experience and could in the TGD framework correspond to algebraic complexity associated with the extension of rationals assignable to a given space-time region (see this and this).
The value h_eff=nh₀ of the effective Planck constant, which can be larger than h, would correspond to the dimension n of the extension of rationals and serve as a universal IQ. Dark matter would correspond to phases of ordinary matter with h_eff ≠ h.
As the IQ increases, the experience transforms from simple to complex and eventually chaotic since the experiencer is not able to make sense of it. Under some assumptions this would relate to the formation of Julia set type fractals (see this).

2. About hyperbolic spaces

First some mathematical background.

Hyperbolic 3-space H³ is a generalization of 1-D hyperbola of 2-D space-time as a curve defined by condition t²-x²= a² but with its metric being induced from the 2-D Minkowski metric ds²= dt²-dx² . By performing all possible rotations of this 1-D hyperbola one obtains H³.
In particle physics H³ corresponds to mass shell E²-p²= m² and in cosmology to cosmic time identifiable as a²=t^r-r² in M⁴ ⊂ M⁴× CP₂. a defines Lorentz invariant cosmic time and is therefore analogous to absolute time invariant under Lorentz boosts which do not affect the tip of the light-cone. It is not invariant under translations however.
In the TGD framework H³ has a central role and plays a key role also in the model of the brain involving the notion of magnetic body (MB). One could say that cognitive and sensory representations are realized at the intersection of MB with H³.
The value of cosmic time a characterizes the curvature of H³. The curvature is proportional to 1/a² and the smaller the value of a, the larger the curvature and "hyperbolicity". As a decreases, one approaches the analog of the Big Bang with infinite curvature. As a increases, one approaches flat E³ in an infinite future. Cosmic evolution proceeds from the Big Bang to the future whereas DMT trip would be a travel towards the moment of Big Bang. One can of course ask whether trips could also be in the opposite time direction.
The lecture (see also the written version) contains a nice description of hyperbolic geometry. In particular, the volume of a ball in H³ increases exponentially as a function of its radius and this means that H³ has a lot of volume. This might be very relevant for memory storage. This can be easily understood from the visualization in terms of real hyperboloid.
The counterpart of plane E² of E³ in H³ is 2-D hyperbolic space H² and Poincare sphere gives a good view about what the projections of the tesselations of H² look like when projected to E². The radial size for the basic unit of tessellations decreases with the distance from the origin whereas the region around the origin looks like E².
The hyperbolic geometry H² embedded locally in E³ has the saddle property meaning that in one direction the observer is at the bottom of the valley and in another direction at the top of the hill. This property has analog also at the level of abstract geometry: geodesic lines diverge very rapidly since the curvature scalar is negative: for spheres they converge.
By their negative curvature, H³ and H² allow tessellations (analogs of lattices in E³ and E²) which are not possible in E³. For instance. 7-polygons are possible. The number of tessellations is infinite whereas in E² only 17 wall papers are possible.
Hyperbolic analogs of plants are mentioned as fractals.

3. A possible interpretation of DMT experiences

DMT experiences could reflect both the relationship between the geometries of hyperbolic 3-space and Euclidian 3-space represented as 3-surfaces of Minkowski space and the algebraic complexity assignable to the tesselations of H³.

3.1 DMT trip as travel backwards in cosmic time

It was already mentioned that the proper time parameter a and algebraic complexity characterized by extension of rationals could characterize DMT experience. The increased complexity in turn means approach to apparent chaos since it is not possible to comprehend too high complexity. The following description is what I understood from the representation of Emilsson. I have not personally made DMT trips except spontaneously decades ago. This experience was so impressive that I got a passion to understand conscious experience from a quantum physics point of view.

For small DMT does, the visual experiences correspond to patterns in plane E² ⊂ E³, which can be regarded as plane H²⊂ H³ for large value of a and thus small curvature.
The 17 lattices of E², called wallpapers, serve as a background for the visual field. As if one would be perceiving two different worlds simultaneously. The lattices can be dynamical and pulsate. This kind of experience was part of the "Great Experience" decades ago.
As the DMT dose increases, the value of a decreases and one moves towards the Big Bang, so to say. In TGD and TGD inspired theory of consciousness, causal diamonds (CDs), identified as intersections of future and past directed light-cones, could be called correlates of perceptive fields. CD is analogous to a Big Bang followed by a Big crunch. The CDs form a fractal hierarchy.
The visual field becomes more and more hyperbolic. What we would see is the projection of the patterns of H²_a⊂ H³_a⊂ M⁴₊ to E²_t⊂ E³_t⊂ M⁴₊, where a is cosmic time and t is the linear Minkowski time.
At the next step the 2-D patterns in H³ are replaced by patterns in H³ as hyperbolic analogous of curved surfaces in E³ and one can say that the dimension of the visual field becomes 3.
In TGD Universe space-time surfaces are minimal surfaces (see this) and analogous to 4-D soap films spanned by frames appearing as singularities where minimal surface property and also the determinism of field equations fail so that the frames are space-time correlates as seats of non-determinism. The saddle property of minimal surface could explain the appearance of the "hyperbolic plants" which Emilsson lists as part of DMT experience.

Do we really see a hyperbolic world or does the visual perception reflect only the statistical geometry of the brain? The TGD proposal is that these two views reflect real space-time surfaces. One can of course argue that since conscious experience itself is associated with quantum jumps in the TGD framework so that the experience is about becoming rather than about being in the physical sense.

3.2. Algebraic complexity of the experience as a second parameter

The second parameter discussed in the talk was meant to characterize what was called valence as a measure for the degree of "bliss" of the experience. TGD counterpart would be algebraic complexity associated with the extension of rationals defined by the polynomial defining the space-time region. The value of h_eff/h₀=n as dimension of extension would serve as the parameter . For large values of n the situation becomes too complex to comprehend or remember and the bliss is lost.

In the TGD framework more complex systems can be engineered as functional composites of polynomials and this leads to the increase of h_eff. One can interpret this also as a construction of many-particle states with each polynomial, which represents a particle-like entity (see this and this). When a fixed polynomial is iterated functionally, one obtains a fractal known as Julia set so that the connection with fractals is quite concrete (see this).

To sum up, the reports of Emilsson suggest a very concrete connection between DMT experience and TGD based views of space-time and number theoretical vision about quantum theory explaining dark matter as h_eff=nh₀ phases. DMT perception would be perceptions of both ordinary and dark matter simultaneously.

4. Possible implications for the interpretation of TGD

The proposed picture involving in an essential manner both H³ and E³ suggests some quite non-trivial implications concerning the physical interpretation of TGD.

4.1 H³ is ideal for information storage and holography

The hyperbolic radial distance r_H in H³ from origin is given by r_H= a arsinh(r_E/a) ≈ a log(r_E/a), where r_E is the Euclidean distance in E³. r_H depends logarithmically of r_E slowly. The area S=4π a²r² of the hyperbolic sphere of radius u projected to Euclidean sphere with r increases as function of u as S≈ 4π a²exp(2u/a). One can imbed a tree graph (say) m ranches in the node much more effectively than in the Euclidean case. One can think of the tree grapas a simple model for a neural network consisting of layers such that n:th layer has mⁿ nodes for

If a given node requires fixed area Δ S, the solid angle Δ Ω required by a node decreases as 1/r² whereas in E³ it remains constant, the number of these areas at sphere increases as S/Δ S= 4π exp(2u/a)/Δ S. In the Euclidean case it increases as S/ΔS= 4π r²/Δ S. This means that the geometric information storage capacity ofH³ is exponentially larger. Therefore the idea that the 3 surfaces associated with H³_a could serve as information storage is very attractive.

4.2 H³ and the origin of p-adic length scale hypothesis

p-Adic prime assignable to a region of the space-time surface is identified as the largests ramified prime associated with the polynomial defining the region of the space-time surface. p-Adic length scale hypothesis states that the physical preferred p-adic primes correspond to p-adic primes p≈ m^k, where m is a small integer: m=2 is the most important case.

I have proposed that there are two scales involved. The small p-adic length scale associated with m and the exponentially larger p-adic length scale proportional to p^1/2. The origin of these scales has remained a mystery.

Could the small scales correspond to the radial scales r_H and large scales to radial scales r_E?

H₃ allows tessellations playing a key role in TGD framework and the size scale of the cell of the tesselation defines a natural length scale unit Δ r_H= aX, which could define the small scale and scales would be expressible in terms of this unit.
In E³ the natural scale would correspond to Euclidean lattices with constant cell size Δ r_E. For r_H= Δ r_H, r_E = a sinh(r_H/a) ≈ aexp(r_H/a) would give r_E ≈ aexp(nX= a m^{Δ X/log(m)}.
r_E=L_p= p^1/2R would give p^1/2R = am^{Δ Xlog(a)/log(m)}. p-Adic length scale hypothesis p≈ m^k requires X= klog(m)/2log(a/R).
Note that there would be a logarithmic dependence of the p-adic length scale on the a, which would have an interpretation as a renormalization of the p-adic length- and mass scales.