Fire Kasina: Color Control Experience Report by Daniel Ingram

Transcribed excerpt from Daniel Ingram‘s Fire Kasina retreat diary (part 1) [24:20-39:50]

[Daniel spent two weeks practicing 10 to 14 hours a day a fire-focused meditation called Fire Kasina. This meditation technique involves, among other things, looking at a candle flame and then closing one’s eyes and examining the after-image of the fire, and doing this repeatedly over many hours. He started getting interesting visual effects on the second day of the retreat. Over the course of the first week he cycled between difficult and effortless meditation sessions, experienced lots of Jhana and Jhana-Kasina hybrid experiences, reported weird dreams, and described visual hallucinations of various sorts. As the week went on he started experiencing a strange and dysphoric change during the evenings: rather than colors being intensified, something about his experience seemed to be turning every color into grey. Fire Kasina is used to amplify the experience of phenomenal color to great heights, so it seemed strange that for some reason after several days of the practice everything started being grey; he couldn’t stop it. Nonetheless, he persevered. What follows is his description of the meditation breakthrough he experienced on the 7th day of the retreat.]

[24:20]: I was thinking that one of the interesting things that has happened was that the Goddess of Fire told me that I should become a King of Firehehe – which seems a bit of a stretch, but it was inspiring anyway. And so the Tarot card for “King of Fire” is the King of Wands and the message is to be a super-dad, to be supportive of the family, to be good in relationships, to be good in career, accomplish it all simultaneously… and smile while doing it. Interesting instructions. Probably of great relevance and utility if properly applied. Anyway. There it is.34621534470_41dabf1dfa_b

Well, so, the grey I’ve been seeing at the end of the night turns out is a doorway to wondrous, wondrous things. So, this morning my sitting was good, and clear, and relatively typical for a good morning sit. And then in the afternoon: Wow! For my first sit of the afternoon my mantra became “Orchestral” added to all sorts of other parts. And it began just shuddering at the whole of reality. It became exquisite, amazing, like basking in the presence of a “Divine Orchestra”: The radiator was making interesting sounds. Duncan also described this beautiful music, which I now understand. Well, it became exquisite, and it all harmonized into this wondrous thing that I wanted to record. I wanted to get it on my laptop on my Logic Pro and somehow figure out how to recreate what I was hearing and experiencing. I felt like I could sit there forever while the mantra was going on, powerful and remarkable.

The next sit is when the colors opened up. And then I started to notice that out of the black, and white, and grey, colors were showing up. And I had noticed this before. A sort of copper or magenta that was, you know, a very exquisite color off of the white and grey. And yellow off of the white, and grey, and black, that was different and new.

And THIS was like, every color was suddenly there. And I could morph the color into any color. Exquisite colors. Subtleties of hue and shade… that wouldn’t exist in the best art store in the entire universe selling all the coolest shades and paint colors of light. And it would shift through magnificent yellows and sparkling golds into rich oranges and amber colors, and peach tones, and magenta, and reds, and rich reds, and bright reds, and pinkish reds, and subtle pinks, and silver pinks, and pale… pale, pale whiteish pinks of the most exquisite variety, and down into the blues and into purples, into the blue-greens and I got greens… and remarkable rich greens, amazing greens, forest greens, Kelly greens, silver greens, light peridot greens. Incredible colors. And the violet hues! Deep rich violet. And even the greys were exquisite: silver black, steel black, silver grey… just unbelievable shades even between black and white that weren’t even colored but were just exquisite.32711476645_1dc1c6f657_b

In the sit I felt like it could go on forever. I could call up whatever color and just give it a few seconds and it would just shift into the new color. And then I went outside when Tommy came here (great guy) […]. I went out to see him and I was high as a freaking kite on just seeing the most amazing exquisite colors due to the power of this meditation. And everything, the greys of the driveway were awesome, the leaf colors were amazing, the sky… the plants, the plants… some of the greens of the plants were just magnificent. Like, just overwhelming beauty.

And then things progressed and I got so that I could turn these colors into anything I wanted, and just give it a few seconds and it would show up. So I created a dragon in a sort of magnificent purple-dark-green-dark-blue-iridescent scaly set of colors. And had his fire breadth come out as first as reddish, and then purple, and green, and yellow, and black. Just amazing!

And then I would make one half of my visual field one color, and one half the other color, and compare the two colors and fine-tune the shades until they went together just exquisitely… anyway, I simply just had an utterly mind-bogglingly enjoyable time playing with the ability to create shades of light and color and images with them just effortlessly, just by asking for them to show up. And I was getting fruitions off of these colors. I got a fruition just off of peach. A sort of exquisite peach color.

And then, finally… finally… and here it is… day 7 or 8 of this retreat… I was able to get the images starting like I got before. Spinning things like they would take everything out and lead to fruition and do that again and again like I got multiple fruitions would just go back to the spinning things, and take out the spinning things. And I got this amber, sort of pixelated squares, and they stuttered and came out by the impermanence door. This has just been an amazing, amazing day.44966545795_46114730e4_b

The glorious highs of it are wearing off. And if I crash as hard as I went up, I’m going to be in trouble. Because it would be hard to overstate how exquisite these sits have been this afternoon and evening. Anyway, so Duncan is now seeing the grey… so he may be close. And Florian is been getting white and black, so he may be close to these things as well. It would be fascinating to see if they have the same ecstatic reaction that I’ve had. Truly the word “rapturous” barely does justice to how much fun the day has been. Anyway…

So, be well.background-blur-bright-carpet

[…]

[36:00]: Note to self: “Do this more often!”. Goodness gracious! The appreciation of color, the peace in the body, the stability of attention… unbelievable, amazing. The color control is remarkable. The ability to tune into your colors, to any shade, just… it is hard to explain how fun that is. Now I can generate images that demonstrate parallax, so that when I move my head, they would move like you’d think they should move in relationship with how I move my head. So they now have a 3-dimensional spatial life.

frost-bubble-rainbow-soap-bubble-colorfulGot a fruition by creating a fire-breathing dragon, and then turning the fire on myself. And when it reached my eyes, that caused the fruition. I’ve seen landscapes, and 3-dimensional spatial structures, and visions of just such exquisite, complicated, intricate, amazing, and diverse beauty. It is hard to explain the gratitude I feel for these practices and what they can do. And that I’m in such fine company: Duncan, and Florian, and Tommy are all excellent people. Extremely helpful, great attitudes, fascinating backgrounds and knowledges and skillsets. They are truly remarkable people and I’m grateful to have found them and to have the opportunity to sit with them and share these things. We laugh nearly continuously during our meals, which are exquisite. We keep each other well entertained somehow, still get very deep transformative practice time in. They are respectful of everyone, of these things, and good practitioners. So it’s a joy to be practicing with them.19303160218_42b0806e64_b

I’ve seen one image that had a sense of intelligence sitting on a chair change form again and again and again. Into all these different types of entities. But I haven’t yet gotten a proper kind of fruition with the intelligent eyes looking back at me yet on this retreat. So… more to do. I still haven’t done the candle flame, moving things with my mind. But perhaps that is coming. I hope so. Alright, dear listener, or listeners, whoever you may be, I hope you are well. I hope that one day you have the interest and opportunity to practice these things as we are here today. For this is the good stuff, as they say.



Relevant:

  • Free-Wheeling Hallucinations
    • Daniel Ingram talks about “color control” during Fire Kasina meditation – the ability to control the phenomenal color of experience. Can this be achieved pharmacologically?
    • Cognitive scientist Steven Lehar reports that combining LSD, Ketamine, and THC at the same time can give rise to an interesting phenomenon he calls “free-wheeling hallucinations”.
    • These hallucinations are the psychedelic equivalent of lucid dreams. Namely, they are very intense but highly controllable states of mind where you can “will into being” whatever you want (up to a certain level of complexity).
    • It’s as if you’ve gained “root access” to the parameters of your inner world-simulation and you can create complex psychedelic scenery and simulated environments.
    • Remarkably, one can investigate the connection between symmetry, harmony, and valence on these states.
  • Generalized Wada Test and the Total Order of Consciousness
    • Daniel Ingram talks about developing the ability of instantiating one color on one side of the visual field and a different color on the other in order to compare them side by side.
    • Wada tests are medical procedures in which only one of your brain hemispheres receives a sedative drug at a time. This can be useful in order to determine which hemisphere can be ablated in order to treat epilepsy.
    • But we could generalize it! We can inject one drug in one hemisphere and a different drug in another. In fact, this could be a core research paradigm in the future for qualia research.
  • Beyond Turing: A Solution to the Problem of Other Minds Using Mindmelding and Phenomenal Puzzles and Wada Test + Phenomenal Puzzles: Testing the Independent Consciousness of Individual Brain Hemispheres
    • Extending the above insight – one can in principle use Wada Tests in order to test the independent consciousness of another nervous system (even perhaps one of your hemispheres).
    • You can “solve the problem of other minds” on yourself with current technology by getting each of your hemispheres to solve “qualia puzzles” independently.
  • #46 – Daniel Ingram II (Pragmatic Buddhism) 
    • A recent and fascinating interview where Daniel talks about his Fire Kasina practice and many other amazing things with Ryan Ferris.

5-MeO-DMT Awakenings: From Naïve Realism to Symmetrical Enlightenment

In the following video Leo Gura from actualized.org talks about his 30-day 5-MeO-DMT streak experiment. In this post I’ll highlight some of the notable things he said and comment along the way using a QRI-lens to interpret his experiences (if you would rather make up your mind about what he says without my commentary just go and watch the video on your own before reading what I have to say about it).

TL;DR: Many of the core QRI paradigms such as Neural Annealing, the Symmetry Theory of Valence, the Tyranny of the Intentional Object, and Hyperbolic Geometry on Psychedelics have a surprising degree of explanatory power when it comes to making sense of the peculiar process that ensues when someone takes a lot of 5-MeO-DMT. The deep connections between symmetry, valence, smooth geometry, and information content are made clear in this context due to the extreme and purified nature of the states induced by the drug.


Introduction

Recently Adeptus Psychonautica (who has interviewed me in the past about the hyperbolic geometry of DMT experiences) put out a video titled “When you have taken too much – Actualized.org“. This video caught my attention because Leo Gura did something that is rather taboo in spiritual communities, and for good reasons. Namely, he tried to convince the viewers that he had achieved a level of awakening that nobody (or perhaps only a few people) on the entire planet had ever reached. He then said he was going to isolate for a month to integrate these profound awakenings and come back with a description of what they are all about.

Thankfully I didn’t have to wait a month to satisfy my curiosity and see what happened after his period of isolation because by the time I found about it he had already posted his post-retreat video. Well, it turns out that he used those 30 days of isolation to conduct a very hard-core psychedelic experiment. Namely, he took high doses of 5-MeO-DMT daily for the entire month. I’ve never heard of anyone doing this before.

Learning about what he experienced during that month is of special interest to me for many reasons. In particular, thanks to previous research about extreme bliss and suffering, we had determined that 5-MeO-DMT is currently the psychedelic drug that has the most powerful and intense effects on valence. Recall Logarithmic Scales of Pleasure and Pain (video): many lines of evidence point to the fact that extreme states of consciousness are surprisingly powerful in ways that are completely counterintuitive. So when Leo says that there are “many levels of awakening” and goes on to discuss how each level is unrecognizably more intense and deeper than the previous one, I am very much inclined to believe he is trying to convey a true property of his experiences. Note that Leo did not only indulge in psychedelics; we are talking about 5-MeO-DMT in particular, which is the thermonuclear bomb version of a psychoactive drug (as with Plutonium, this stuff is best handled with caution). More so, thankfully Leo is very eloquent, which is rare among people who have had many extreme experiences. So I was very eager to hear what he had to say.

While I can very easily believe his trip reports when it comes to their profundity, intensity, and extraordinary degree of consciousness, I do not particularly find his interpretations of these experiences convincing. As I go about describing his video, I will point out ways in which you can take as veridical his phenomenological descriptions without at the same time having to agree with his interpretations of them. More so, if you end up exploring these varieties of altered states yourself, by reading this you will now at least have two different and competing frameworks to explain your experiences. This, I think, is an improvement. Right now the psychedelic and scientific community has very few lenses with which to interpret something as extraordinary as 5-MeO-DMT experiences. And I believe this comes at a great cost to people’s sanity and epistemic rationality.

What Are Leo’s Background Assumptions?

In the pre-retreat video Leo says that his core teachings (and what he attempts to realize on his own self) are: (1) you are literally God, (2) there is nothing but consciousness – God is infinite consciousness, (3) everything is states of consciousness – everything at all times is a different state of consciousness, (4) you are love – and love is absolute – this is all constructed out of love – fear is just fear of aspects of yourself you have disconnected from, (5) you have no beginning and no end, (6) you should be radically open-minded. Then he also adds that physical and mental health issues are just manifestations of your resistance to realizing that you are God.

What Are My Background Assumptions?

Personal Identity

I am quite sympathetic to the idea of oneness, which is also talked about with terms like nonduality and monopsychism. In philosophical terminology, which I find to be more precise and rigorous, this concept goes by the name of Open Individualism – the belief that we are all one single consciousness. I have written extensively about Open Individualism in the past (e.g. 1, 2, 3), but I would like to point out that the arguments I’ve presented in favor of this view are not based on direct experience, but rather, on logical consistency from background assumptions we take for granted. For instance, if you assume that you are the same subject of experience you were a second ago, it follows that you can exist in two points in space-time and still be the same being. Your physical configuration is different than a few seconds ago (let alone a decade), you have slightly different memories, the neurons active are different, etc. For every property you point out as your “identity carrier” I can find a counter-example where such carrier changes a little while you still remain the same subject of experience. Add to that teleportation, fission, fusion, and gradual replacement thought experiments and you can build a framework where you can become any other arbitrary person without a loss of identity. These lines of argumentation coupled with the transitivity of identity can build the case that we are indeed all one to begin with.

But realize that rather than saying that you can grasp this (potential) truth directly from first person experience, I build from agreed upon assumptions to arrive at an otherwise outlandish view. Understanding the argument does not entail “feeling we are all one”, and neither does feeling we are all one entails understanding the arguments!

Indirect Realism About Perception

There is a mind-independent world out there and you never get to experience it directly. In some sense, we each live in a private skull-bound world-simulation that tracks the fitness-relevant features of our environment. Hence, during meditation, dreaming, or psychedelic states you are not accessing any sort of external reality directly, but rather, exploring possible configurations and qualities of your inner world-simulation. This is something that Leo may implicitly not realize. In particular, interpreting 5-MeO-DMT experiences through direct realism (also called naïve realism – the view that you experience the world directly through your senses) would make you think that you are literally merging with the entire cosmos on the drug. Whereas interpreting those experiences with indirect realism merely entails that your inner boundaries are dissolving. In other words, the partitions inside your world-simulation are what implements the feeling of the self-other duality. And since 5-MeO-DMT dissolves inner boundaries, it feels as though you are becoming one with your surroundings (and the rest of reality).

Physicalism and Panpsychism

An important background assumption is that the laws of physics accurately describe the behavior of the universe. This is distinct from materialism, which would also posit that all matter is inherently insentient. Physicalism merely says that the laws of physics describe the behavior of the physical, but leaves its intrinsic nature as an open question. Together with panpsychism, however, physicalism entails that what the laws of physics are describing is the behavior of consciousness.

Tyranny of the Intentional Object

We tend to believe that what makes us happy is external to us, while in reality happiness is a state of consciousness triggered by external circumstances. Our minds lead us to believe otherwise for evolutionary reasons.

Valence Structuralism

What makes an experience feel good or bad is not its semantic content, its computational use, or even whether the experience is self-reinforcing or not. What makes experiences feel good or bad is their structure. In particular, a very promising idea that will come up below is that highly symmetrical states of consciousness are inherently blissful, such as those we can access during orgasm, meditation, psychedelics, or even just good food and a hug. Recall that 5-MeO-DMT dissolves internal boundaries, and this is indicative of increased inner symmetry (where the boundaries themselves entail symmetry breaking operations). Thus, an exotic state of oneness is blissful not because you are merging with God, but “merely” because it has a higher degree of symmetry and therefore it’s valence is higher than what we can normally experience. In particular, the symmetry I’m talking abut here may be an objective feature of experiences perhaps even measurable with today’s neuroimaging technology.

There are additional key background philosophical assumptions, but the above are enough to get us started analyzing Leo’s 5-MeO-DMT journey from a different angle.


The Video

[Video descriptions are in italics whereas my commentary is bolded.]

For the first 8 minutes or so Leo explains that people do not really know that there are many levels of enlightenment. He starts out strong by claiming that he has reached levels of enlightenment that nobody (or perhaps just a few people) have ever reached. More so, while he agrees with the teachings of meditation masters of the past, he questions the levels of awakening that they had actually reached. It takes one to know one, and he claims that he’s seen things far beyond what previous teachers have talked about. More so, he argues that people simply have no way of knowing how enlightened their teachers are. People just trust books, gurus, teachers, religious leaders, etc. about whether they are “fully” enlightened, but how could they know for sure without reaching their level, and then surpassing them? He wraps up this part of the video by saying that the only viable path is to go all the way by yourself – to dismiss all the teachers, all the books, and all the instructions and see how far you can go on your own when genuinely pursuing truth by yourself.

With this epistemological caveat out of the way, Leo goes on to describe his methodology. Namely, he embarked on a quest of taking 5-MeO-DMT at increasing doses every day for 30 days in a row.leo_10_05

At 10:05 he says that within a week of this protocol he started reaching levels of awakening so elevated that he realized he had already surpassed every single spiritual teacher that he had ever heard of. He started writing a manifesto explaining this, claiming that even the most enlightened humans are not truly as awake as he became during that week. That it had became “completely transparent that most people who say they are awake or teach awakening are not even 1% awake”. But he decided not to go forward with the manifesto because he still values the teachings of spiritual leaders, whom according to him are doing a great service to mankind. He didn’t want to start, what he called, a “nonduality war” (which is of course a fascinating term if you think about it).

The main thing I’d like to comment here is that Leo is never entirely clear about what makes an “awakening experience” authentic. From what I gather (and from what comes next in the video) we can infer that the leading criteria consists of a fuzzy blend of experience of certainty, feeling of unity, and sense of direct knowing coupled together. To the extent that 5-MeO-DMT does all of these things to an extraordinary degree, we can take Leo on his words that he indeed experienced states of consciousness that feel like awakening that are most likely inaccessible to everyone who hasn’t gone through a protocol like his. What is still unclear is how exactly the semantic contents of these experiences are verified by means other than intuition. We will come back to that.

At 16:00 he makes the distinction between awakening as merely “cessation”, “nothingness”, “emptiness”, “the Self”, or that “you are nothing and everything” versus what he has been experiencing. He agrees that those are true and worthy realizations, but he claims that before his experiences, these understandings were still only realized at a very “low level”. Other masters, he claims, may care about ending suffering, about peace, about emptiness, and so on. But that nobody seems to truly care about understanding reality (because otherwise they would be doing what he’s doing). He rebukes possible critics (arguably of the Zen variety) who would say that “understanding is a function of the mind” so the goal shouldn’t be to understand. He asserts that no, based on his lived experience, that consciousness is capable of “infinite understanding”.

Notwithstanding the challenges posed by ultrafinitism, I am also inclined to believe Leo that he has experienced completely new varieties of “understanding”. In my model of the mind, understanding something means to have the ability to render it in your world-simulation in a particular kind of way that allows you to see it from every possible angle you have access to. On 5-MeO-DMT, as we will see to a greater extent below, a certain new set of projective operations get unlocked that allow you to render information from many, many more points of view at the same time. It is unclear whether this is possible with meditation alone (in personal communication, Daniel Ingram said yes) but it is certainly extraordinarily rare for even advanced meditators to be able to do this. So I am with Leo when it comes to describing “new kinds of understandings”. But perhaps I am not on board when it comes to claiming that the content of such understandings is an accurate rendering of the structure of reality.

At 18:30 Leo asserts that what happened to him is that over the course of the first week of his experiment he “completely understood reality, completely understood what God is”. God has no beginning and no end. He explains that normal human understanding sees situations from a single point of view (such as from the past to the future). But that actual infinite reality is from all sides at once: “When you are in full God consciousness, you look around the room, and you can see it from every single point of view, from an infinite number of angle and perspectives. You see that every part of the room generates and manufactures and creates every other part. […] Here when you are in God consciousness, you see it from every single possible dimension and angle. It’s not happening lilnearly, it’s all in the present now. And you can see it from every angle almost as though, if you take a watermelon and you do a cross-section with a giant knife, through that watermelon, and you keep doing cross-section, cross-section, cross-section in various different angles, eventually you’ll slice it up into an infinite number of perspectives. And then you’ll understand the entire watermelon as a sort of a whole. Whereas usually as humans what we do is we slice down that watermelon just right down the middle. And we just see that one cross-section.”

Now, this is extremely interesting. But first, it’s important to point out that here Leo might implicitly be reasoning about his experience through the lens of direct realism about perception. That is, that as he experiences this profound sense of understanding that encompasses every possible angle at once, he seems to believe that this is an understanding of his environment, of his future and past, and of reality as a whole. On the other hand, if you start out assuming indirect realism about perception, how you interpret this experience would be in terms of the instantiation of new exotic geometries of your own world-simulation. Here I must bring up the analysis of “regular” DMT (i.e. n,n-DMT) experiences through the lens of hyperbolic geometry. Indeed, regular DMT elevates the energy of your consciousness, which manifests in brighter colors, fast movement, intricate and detailed patterns, and as curved phenomenal space. We know this because of numerous trip reports from people well educated in advanced mathematics who claim that the visual symmetries one can experience on DMT (at doses above 10mg) have hyperbolic curvature (cf. hyperbolic orbifolds). It is also consistent with many other phenomena one can experience on DMT (see the Eli 5 for a quick summary). But you should keep in mind that this analysis never claims that you are experiencing directly a mind-independent “hyperspace”. Rather, the analysis focuses on how DMT modifies the geometric properties of your inner world-simulation.

Hyperbolic Geometry of DMT Experiences copy 47

Energy-complexity landscape on DMT

Hyperbolic Geometry of DMT Experiences copy 38

DMT trip progression

Intriguingly, our inner world-simulations work with projective geometry. In normal circumstances our world-simulations have a consistent set of projective points at infinity – they render the modal and amodal features of our experience in projective scenes that are globally consistent. But psychedelics can give rise to this phenomenon of “point-of-view-fragmentation“, where your experience becomes a patchwork of inconsistent projective renderings. So even on “regular” DMT you can get the profound feeling of “seeing something from multiple points of view at once”. Enhanced with hyperbolic geometry, this can cause the stark impression that you can explore “hyperspace” with a kind of “ultra-understanding”.

Looking beyond “regular” DMT, 5-MeO-DMT is yet more crazy than that. You see, even on DMT you get the feeling that you are restricted in the number of points of view from which you can see something at the same time. You can see it from many more points of view than normal, but it’s still restricted. But the extreme “smoothing” of experience that 5-MeO-DMT causes makes it so that you cannot distinguish one point of view from another. So they all blend together. Not only do you experience semantic content from “multiple points of view at once” as in DMT, but you can erase distinctions between points of view so that one’s sense of knowing arises involving a totally new kind of projective effect, in which you actually feel you can see something from “every point of view at once”. It feels that you have unlocked a kind of omniscience. This already happens on other psychedelics to a lesser extent (and in meditation, and even sober life to an even lesser extent, but still there), and it is a consequence of smoothing the geometry of your experience to such an extent that there are no symmetry-breaking imperfections “with which to orient a projective point”. I suspect that the higher “formless” jhanas of “boundless space” and “boundless consciousness” are hitting at this effect. And on 5-MeO-DMT this effect is pronounced. More so, because of the connection between symmetry and smoothness of space (cf. Geometry Through the Eyes of Felix Klein) when this happens you will also automatically be instantiating a high-dimensional group. And according to the Symmetry Theory of Valence, this ought to be extraordinarily blissful. And indeed it is.

This is, perhaps, partly what is going on in the experience that Leo is describing. Again, I am inclined to believe his description, but happy to dismiss his naïve interpretation.

indras_net

Indra’s Net

At 23:15 Leo describes how from his 5-MeO-DMT point of view he realized what “consciousness truly is”. And that is an “infinitely interconnected self-communicating field”. In normal everyday states of consciousness the different parts of your experience are “connected” but not “communicating.” But on 5-MeO, “as you become more conscious, what happens is that every point in space inter-connects with itself and starts to communicate with itself. This is a really profound, shocking, mystical experience. And it keeps getting cranked up more and more and more. You can call it omniscience, or telepathy. And it’s like the universal communication system gets turned on for the first time. Right now your conscious field is not in infinite communication with itself. It’s fragmented and divided. Such that you think I’m over here, you are over there, my computer is over here, your computer is over there…”. He explains that if we were to realize we are all one, we would then instantly be able to communicate between each other.

Here again we get extremely different interpretations of the phenomena Leo describes depending on whether you believe in direct or indirect realism about perception. As Leo implicitly assumes direct realism about perception, he interprets this effect as literally switching on an “universal communication system” between every points in reality, whereas the indirect realist interpretation would be that you have somehow interlocked the pieces of your conscious experience in such a way that they now act as an interconnected whole. This is something that indeed has been reported before, and at QRI we call this effect “network integration“. A simple way of encapsulating this phenomenon would be by saying that the cross-frequency coupling of your nervous system is massively increased so that there is seamless information and energy transfer between vibrations at different scales (to a much lesser extent MDMA also does this, but 5-MeO-DMT is the most powerful “integration aid” we know of). This sounds crazy but it really isn’t. After all, your nervous system is a network of oscillators. It stands to reason that you can change how they interact with one another by fine-tuning their connections and get as a result decoupling of vibrations (e.g. SSRIs), or coupling only between vibrations of a specific frequency (e.g. stimulants and depressants), or more coupling in general (e.g. psychedelics). In particular, 5-MeO-DMT does seem to cause a massively effective kind of fractal coupling, where every vibration can get in tune with every other vibration. And recall, since a lot of our inner world simulation is about representing “external reality”, this effect can give rise to the feeling that you can now instantly communicate with other parts of reality as a whole. This, from my point of view, is merely misinterpreting the experience by imagining that you have direct access to your surroundings.

At 34:52 Leo explains that you just need 5-MeO-DMT to experience these awakenings. And yet, he also claims that everything in reality is imaginary. It is all something that you, as God, are imagining because “you need a story to deny that you are infinite consciousness.” Even though the neurotransmitters are imaginary, you still need to modify them in order to have this experience: “I’m talking about superhuman levels of consciousness. These are not levels of consciousness that you can access sober. You need to literally upgrade the neurotransmitters in your imaginary brain. And yes, your brain is still imaginary, and those neurotransmitters are imaginary. But you still need to upgrade them nevertheless in order to access some of the things I say.”

Needless to say, it’s bizarre that you would need imaginary neurotransmitter-mimicking molecules in your brain in order to realize that all of reality is your own imagination. When you dream, do you need to find a specific drug inside your dream in order to wake up from the dream? Perhaps this view can indeed be steel-manned, but the odds seem stacked against it.

At 38:30 he starts talking about his pornography collection. He assembles nude images of women, not only to relieve horniness, but also as a kind of pursuit of aesthetics. Pictures of nude super-models are some of the most beautiful things a (straight) man can see. He brings this up in order to talk about how he then at some point started exploring watching these pictures on 5-MeO-DMT. Recollecting this brings him to tears because of how beautiful the experiences were. He states “you’ve never really seen porn until you’ve seen it on 5-MeO-DMT.” He claims that he started to feel that this way he really felt that it is you (God) that is beautiful, which is manifested through those pictures.

A robust finding in the psychology of sexual attraction is that symmetry in faces is correlated with attractiveness. Indeed, more regular faces tend to be perceived as more beautiful. Amazingly, you can play with this effect by decorating someone’s face with face-paint. The more symmetrical the pattern, the more beautiful the face looks (and vice-versa). Arguably, the effect Leo is describing where people who are already beautiful become unbelievably pretty on 5-MeO-DMT involves embedding high-dimensional symmetries into the way you render them in your world-simulation. A lesser, and perhaps more reliable, version of this effect happens when you look at people on MDMA. They look way more attractive than what they look like sober.

Leo then brings up (~41:30) that he started to take 5-MeO-DMT on warm baths as well, which he reassures us is not as dangerous as it sounds (not enough water to drown if he experiences a whiteout). [It’s important to mention that people have died by taking ketamine on bath tubs; although a different drug, it is arguably still extremely dangerous to take 5-MeO-DMT alone on a bathtub; don’t do it]. He then has an incredible awakening surrendering to God consciousness in the bathtub, on 5-MeO-DMT, jerking off to beautiful women in the screen of his laptop. He gets a profound insight into the very “nature of desire”. He explains that it is very difficult to recognize the true nature of desire while on a normal level of consciousness because our desires are biased and fragmented. When “your consciousness becomes infinite” those biases dissolve, and you experience desire in its pure form. Which according to his direct experience turned out to be “desire for God, desire for myself”. And this is because you are, deep down “infinite love”. When you desire a husband, or sex, or whatever, you are really desiring God in disguise. But the problem is that since your path to God is constrained by the form you desire, your connection to God is not stable. But once you have this experience of complete understanding of what desire is, you finally get your desire fully quenched by experiencing God’s love.

This is a very deep point. It is related to what I’ve sometimes called the “most important philosophical question“, which is: is valence a spiritual phenomenon or spirituality a valence phenomenon? In other words, do we find experiences of God blissful because they have harmony and symmetry, or perhaps is it the other way around, where even the most trivial of pleasures, like drinking a good smoothy, feels good because it temporarily “gets you closer to God”? I lean towards the former, and that in fact mystical experiences are so beautiful because they are indeed extremely harmonious and resonant states of consciousness, and not because they take you closer to God. But I know very smart people who can’t decide between these views. For example, my friend Stuart Garvagh writes: 

What if the two options are indistinguishable? Suppose valence is a measure of the harmony/symmetry of the object of consciousness, and the experience of “Oneness” or Cosmic Consciousness is equivalent to having the object of consciousness be all of creation (God‘s object), a highly symmetrical, full-spectrum object (full of bliss, light, love, beingness, all-knowledge, empty of discernible content or information). All objects of consciousness are distortions (or refractions, or something) of this one object. Happiness is equivalent to reducing or “polishing-out” these distortions. Thus, what appears to be just the fact of certain states being more pleasant than others is equivalent to certain states being closer to God‘s creation as a whole. Obviously this is all pure speculation and just a story to illustrate a point, but I could see it being very tough to tease apart the truth-value of 1 and 2. Note: I’m fairly agnostic myself, but lean towards 2 (bliss is the perfume of “God realizing God” or the subject of experience knowing Itself). I would very much love to have this question answered convincingly!

At 50:00 Leo says that “everything I’ve described so far is really a prelude to the real heart of awakening, which is the discovery of love. […] I had already awakened to love a number of times, but this was deeper. By the two week mark the love really started to crack open. Infinite self-love. You are drowning on this love.” He goes on to describe how at this point he was developing a form of telepathy that allowed him to communicate with God directly (which is, of course, a way of talking to himself as he is God already). It’s just a helpful way to further develop. And what God was showing him was how to receive self-love. It was so much at first he couldn’t handle it. And so he went through a self-purification process.

An interesting lens with which to interpret this experience of purification is that of neural annealing. Each 5-MeO-DMT experience would be making Leo’s nervous system resonate in ways in new ways, slowly writing over previous patterns and entraining the characteristic high-symmetry patterns of the state. Over time, the nervous system adjusts its weights in order to be able to handle that resonance without getting its patterns over-written. In other words, Leo has been transforming his nervous system into a kind of high-valence machine, which is of course very beneficial for intrinsic feelings of wellbeing (though perhaps detrimental to one’s epistemology).

55:00: He points out that unlike addictive drugs, he actually had to push himself very hard to continue to take 5-MeO-DMT everyday for 30 days. He stopped wanting to do it. The ego didn’t want it. And yes, it was pleasurable once he surrendered on every session, but it was difficult, heavy spiritual work. He says that he could only really do this because of years of practice with and without psychedelics, intense meditation, and a lot of personal development. And because of this, he explains his 5-MeO experiences felt like “years of spiritual work condensed into a single hour.” He then says that God will never judge you, and will help you to accept whatever terrible things you’ve done. And many of his subsequent trips were centered around self-acceptance. 

Following the path of progressive neural annealing, going deeper and deeper into a state of self-acceptance can be understood as a deeper harmonization of your nervous system with itself.

At 1:01:20, Leo claims to have figured out what the purpose of reality truly is: “Reality is a contest for who can love who more. That’s really what life is about when you are fully conscious. […] Consciousness is a race for who can love who more. […] An intelligent fully conscious consciousness would only be interested in love. It wouldn’t be interested in anything else. Because everything else is inferior. […] Everything else is just utter silliness!”

I tend to agree with this, though perhaps not in an agentive way. As David Pearce says: “the pleasure-pain axis discloses the universe’s intrinsic value function.” So when you’ve annealed extremely harmonious patterns and do not get distracted by negative emotion, naturally, all there is left to do is maximize love. Unless we mess up, this is the only good final destiny for the cosmos (albeit perhaps it might take the form of a Hedonium shockwave, which at least in our current human form, sound utterly unappealing to most people).

1:06:10 “[God’s love] sparks you to also want to love it back. You see, it turns into a reciprocal reaction, where it is like two mirrors that are mirroring light between each other like a laser beam that is bouncing between two mirrors. And it’s bouncing back and forth and back and forth. And as it bounces back and forth it becomes more and more concentrated. And it strengthens. And it becomes more coherent. And so that’s what started happening. At first it started out as just a little game. Like ‘I love you, I love you, I love you’. A little game. It sounds like it’s almost like childish. And it sort of was. But then it morphed from being this childish thing, into being this serious existential business. This turned into the work. This was the true awakening. Is that with the two mirrors, you know, first it took a little while to get the two mirrors aligned. Because you know if the two mirrors are not perfectly aligned, the laser beam will kind of bounce back and forth in different directions. It’s not going to really concentrate. So that was happening at first. […] The love started bouncing back and forth between us, and getting stronger and stronger. […] Each time it bounces back to me it transforms me. It opens me up deeper. And as it opens me up deeper it reveals blockages and obstacles to my capacity to love.”

Now this is a fascinating account. And while Leo interprets it in a completely mystical way, the description also fits very well an annealing process where the nervous system gets more and more fine-tuned in order to be able to contain high levels of coherent energy via symmetry. Again, this would be extremely high-valence as a consequence of the Symmetry Theory of Valence. Notice that we’ve talked about this phenomenon of “infinite mirrors” on psychedelics since 2016 (see: Algorithmic Reduction of Psychedelic States).

At ~1:09:30 he starts discussing that at this point he was confronted by God about whether he was willing to love the holocaust, and rape, and murder, and bullies, and people of all sorts, even devil worshipers. 

Two important points here. First, it is a bit ambiguous whether Leo here is using the word “love” in the sense of “enjoyment” or in the sense of “loving-kindness and compassion”. The former would be disturbing while the latter would be admirable. I suppose he was talking about the latter, in which case “loving rape” would refer to “being able to accept and forgive those who rape” which indeed sounds very Godly. This radical move is explored in metta (loving-kindness) meditation and it seems healthy on the whole. And second: Why? Why go through the trouble of embracing all the evil and repulsive aspects of ourselves? One interpretation here, coming back to the analysis based on neural annealing, is that any little kink or imperfection caused by negative emotion in our nervous system will create slight symmetry breaking effects on the resonance of the entire system as whole. So after you’ve “polished and aligned the mirrors for long enough” the tiny imperfections become the next natural blockage to overcome in order to maximize the preservation of coherent energy via symmetry.

~1:12:00 Leo explains that the hardest thing to love is your own self-hatred. In the bouncing off of the love between you and God, with each bounce, you find that the parts you hate about yourself reflect an imperfect love. But God loves all of you including your self-hatred. So he pings you about that. And once you can accept it, that’s what truly changes you. “Because when you feel that love, and you feel how accepting it is, and how forgiving it is of all of your evil and of all of your sins… that’s the thing that kills you, that transforms you. That’s what breaks your heart, wide open. That’s what gets you to surrender. That’s what humbles you. That’s what heals you.” Leo then explains that he discovered what “healing is”. And it is “truth and love”. That in order to heal anyone, you need to love them and accept them. Not via sappy postcards and white lies but by truth. He also states that all physical, mental, and spiritual ailments have, at their root, lack of love.

If love is one of the cleanest expressions of high-valence symmetry and resonance, we can certainly expect that inundating a nervous system with it will smooth and clean its blockages, i.e. the sources of neural dissonance. Hence the incredible power of MDMA on healing nervous systems in the short-term. Indeed, positive emotion is itself healing and enhances neural coherence. But where I think this view is incomplete is in diagnosing the terrible suffering that goes on in the world in terms of a lack of love. For instance, are cluster headaches really just the result of lack of self-love? In here must bring back the background assumption of physicalism and make a firm statement that if we fall into illusion about the nature of reality we risk not saving people (and sentient beings more generally) who are really in the depth of Hell. Just loving them without taking the causally-relevant physical action to prevent their suffering is, in my opinion, not true love. Hence the importance of maintaining a high level of epistemic rigor: for the sake of others. (See: Hell Must Be Destroyed).

1:22:30 Leo explains that in this “love contest” with God of bouncing off love through parallel mirrors the love became so deep that for the first time in his life he felt the need to apologize: “I’m sorry for not loving more.” He goes into a sermon about how we are petty, and selfish, etc. and how God loves us anyway. “Real love means: I really love you as you are. And I don’t need anything from you. And especially all those things that you think I want you to change about you, I don’t need you to change. I can accept them all exactly as they are. Because that’s love. And when you realize THAT, that’s what transforms you. It is not that God says that he loves you. He is demonstrating it. It’s the demonstration that transforms you.” Leo expresses that he was then for the first time in his life able to say “thank you” sincerely. Specifically, “thank you for your love”: “This is the point at which you’ve really been touched by God’s love. And at this point you realize that that’s it, that’s the point, that’s the lesson in life. That’s my only job. It’s to love.” And finally, that for the first time in his life he was able to say “I love you” and truly mean it. “And you fall in love with God… but it doesn’t end there.”

An interesting interpretation of the felt-sense of “truly meaning” words like “I’m sorry”, “thank you”, and “I love you” is that at this point Leo has really deeply annealed his nervous system into a vessel for coherent energy. In other words, at this point he is saying and meaning those words through the whole of his nervous system, rather than them coming from a fragmented region of a complex set of competing internal family systems in a scattered way. Which is, of course, the way it usually goes.

1:35:30 Leo explains that at this point he started going into the stage of being able to radiate love. That he was unable to radiate love before. “I love that you are not capable of love. I love that. And when that hits you, that’s what fills you with enough love to overcome your resistance to love that next level thing that you couldn’t love.” Then at ~ 1:38:00 it gets really serious. Leo explains that so far he was just loving and accepting past events and people. But he was then asked by God whether he would be willing to live through the worst things that have happened and will happen. To incarnate and be tortured, among many other horrible things. And that’s what true love really means. “When you see a murder on the TV, you have to realize that God lived through that. And the only reason he lived through that is because it loved it.”

I do not understand this. Here is where the distinction between the two kinds of senses of the word “love” become very important. I worry that Leo has annealed to the version of love with the meaning of “enjoyment” rather than “loving-kindness and compassion”. Because a loving God would be happy to take the place of someone who went through Hell. But would a loving God send himself to Hell if nobody had to in the first place? That would just create suffering out of nothing. So I am confused about why Leo would believe this to be the case. It’s quite possible that there are many maxima of symmetry in the nervous system you can achieve with 5-MeO-DMT, and some of them are loving in the sense of compassionate and others are crazy and would be willing to create suffering out of nothing from a misguided understanding of what love is supposed to be. Again, handle Plutonium with caution.

1:43:00 Leo started wondering “what is reality then?” And the answer was: “It’s infinite consciousness. Infinite formless consciousness. So what happens was that my mind in my visual field as I was in that bathtub. My mind and my visual field focused in on empty space, and I sort of zoomed into that empty space and realized that that empty space is just love”. He then describes a process where his consciousness became more and more concentrated and absorbed into space, each dot of consciousness branching out into more and more dots of consciousness, turning into the brightest possible white light. But when he inquired into what was that white light he kept seeing that there was no end to it, and rather, that each point was always connected to more points. Inquiring further, he would get the response that at the core, reality is pure love. That it wouldn’t be and couldn’t be any other way.

The description sounds remarkably close to the formless jhanas such as “boundless space” and “boundless consciousness”. The description itself is extremely reminiscent of an annealing process, reaching a highly energized state of consciousness nearly devoid of information content and nearly perfectly symmetrical. The fact that at this incredibly annealed level he felt so much love supports the Symmetry Theory of Valence.

147:28 – And after Leo realizes that “Of course it is love!” he says that’s when the fear comes: “Because then what you realize is that this is the end. This is the end of your life. You are dead. If you go any further you are dead. Everything will disappear. Your family, your friends, you parents, all of it is completely imaginary. And if you stop imagining it right now, it will all end. If you go any further into this Singularity, you will become pure, formless, infinite, love for ever, loving itself forever. And the entire universe will be destroyed as if it never existed. Complete nothingness. Complete everythingness. You will merge into everyone.”

This sounds like the transition between the 6th and 7th Jhana, i.e. between “boundless consciousness” and “nothingness”. Again, this would be the result of further loss of information via an annealing process, refining the symmetry up to that of a “point”. Interestingly, Mike Johnson in Principia Quallia points out that as symmetry approaches an asymptote of perfection you do get a higher quality of valence but at the cost of reduced consciousness. This might explain why you go from “the brightest possible love” to a feeling of nothingness at this critical transition.

1:48:25: “…You will merge into everyone. Your mother, your father, your children, your spouse, Hitler, terrorists, 9/11, Donald Trump, rape, murder, torture, everything will become pure infinite love, merging completely into itself, there will be no distinction between absolutely anything, and that will be the end. And you will realize what reality is. Infinite consciousness. Love. God. And you will realize that everything in your life from your birth to this point has just been some imaginary story. A dream that was design to lead you to pure absolute infinite love. And you will rest in that love forever. Forever falling in love with yourself. Forever making love to yourself. Forever in infinite union. With every possible object that could ever exist. Pure absolute, omnipotent, omniscient, perfect, intelligent, consciousness. Everything that could ever possibly be, is you. And THAT is awakening. When you are this awake, you are dead. And you have no desire for life. There is no physical existence. There is no universe. Nothing remains. Your parents, and your spouse, and your children, they don’t stay back and keep living their lives, enjoying their life without you while your body drops dead. No, no, no, no, no. This is much more serious than that. If you do this. If you become infinite love, you will take everybody with you. There will not be anybody left. You will destroy the entire universe. Every single sentient being will become you. They will have no existence whatsoever. Zero. They will die with you. They will all awaken with you. It’s infinite awakening. It’s completely absolute. There will not be anything left. You will take the entire universe with you. Into pure oneness. THAT’S awakening.”

This is not the first time I hear about this kind of experience. It certainly sounds extraordinarily scary. Though perhaps a negative utilitarian would find it to be the ultimate relief and the best of all possible imaginable outcomes. With the human survival instinct, and quite possibly a body fully aroused with the incredible power of 5-MeO-DMT, this is bound to be one of the most terrifying feelings possible. It’s quite likely that it may be one element of what makes “bad 5-MeO-DMT experiences” so terrifying. But here we must recall that the map is not the territory. And while an annealing process might slowly write over every single facet of one’s model of reality and in turn making them part of a super-cluster of high-dimensional resonance that reflects itself seemingly infinitely, doing this does not entail that you are in fact about to destroy the universe. Though, admittedly, it will surely feel that way. Additionally, I would gather that were it possible to actually end the universe this way, somebody, somewhere, in some reality or another, would have already done so. Remember that if God could be killed, it’d be dead already.

1:52:01: “And I didn’t go there! As you can tell, since I’m still sitting here. I’m not there. I was too afraid to go there. And God was fine with it. It didn’t push me. But that’s not the end of the story! It’s still just the beginning.” He then goes on to explain that a part of him wanted to do it and another part of him didn’t want to. He says it got really loopy and weird; this really shook him. That God was beckoning him to go and be one forever, but he was still ambivalent and needed some time to think about it. He knew it would make no difference, but he still decided to ‘make preparations’ and tell his family and friends that he loves them before moving forward with a final decision to annihilate the universe. By the time he had done that… he had stopped taking 5-MeO-DMT: “The experiences had gotten so profound and so deep… this was roughly the 25th or 27th day of this whole 30 day process. I swore off 5-MeO-DMT and said ‘Ok I’m not doing any more of this shit. It’s enough'”. He explains that by this time the drug was making him feel infinite consciousness when waking up (from sleep) the next day. He felt the Singularity was sucking him into it. It felt both terrifying and irresistible. Every time he would go to sleep it would suck him in really strongly, and he kept resisting it. He would wake up sweaty and in a panic. He was tripping deeper in his sleep than in the bathtub. He couldn’t sleep without this happening, and it kept happening for about 5 days. “I just want to get back to normal. This is getting freaky now.” 

I’ve heard this from more than a couple people. That is, that when one does 5-MeO-DMT enough times, and especially within a short enough period of time, the “realizations” start to also happen during sleep in an involuntarily way. One can interpret this as the annealing process of 5-MeO-DMT now latching on to sleep (itself a natural annealing process meant to lessen the technical debt of the nervous system). Even just a couple strong trips can really change what sleep feels like for many days. I can’t imagine just how intense it must have been for Leo after 25 days straight of using this drug.

2:01:40 – Leo explains that when he was dozing off with a blanket on his living room (terrified of sleeping on his bed due to the effect just described) he experienced a “yet deeper awakening” which involved realizing that all of his previous awakenings were just like points and that the new one was like a line connecting many points. “Everything I’ve said up to this point were just a single dimension of awakening. And then what I broke through to is a second dimension. A second dimension of awakening opened up. This second dimension is completely unimaginable, completely indescribable, cannot be talked about, cannot be thought about. And yet it’s there. In it, are things that are completely outside of the physical universe that you cannot conceive or imagine.” He goes on to explain that there are then also a third, fourth, fifth, etc. dimensions. And that he believes there is an infinite number of them. He barely even began to explore the second dimension of awakening, but he realized that it goes forever. It kept happening, he had intense emotional distress and mood swings. But gradually after five more days it subsided, and he started to be able to sleep more normally. “And I’ve been working to make sense of all of this for the last couple of weeks. So that’s what happened.”

Alright, this is out of my depth and I do not have an interpretation of what this “second dimension of awakening” is about. If anyone has any clue, please leave a comment or shoot me an email. I’m as as confused as Leo is about this.

~2:05:00 – Leo confesses he does not know what would happen if he went through with joining the Singularity and mentions that it sounds a bit like Mahasamādhi. He simply has not answers at this point, but he asserts that the experience has made him question the extent of the enlightenment of other teachers. It also has made him more loving. But still, he feels frustration: “I don’t know what to do from here.”

And neither do I. Do you, dear reader?

Postscript: In the last 10 minutes of the video Leo shares a heart warming message about how reality is, deep down, truly, “just love” and that him saying this may be a seed that will blossom into you finding this out for yourself at some point in the future. He ends by cautioning his audience to not believe as a matter of fact that this is the path for everyone. He suggests that others should just use his examples from his own journey as examples rather than an absolute guide or how-to for enlightenment. He asks his audience to make sure to question the depth of their own awakening – to not believe that they have reached the ultimate level. He admits he has no idea whether there is an ultimate level or not, and that he still has some healing to do on himself. He remains dissatisfied with his understanding of reality.


Thank you for reading!

THE END

Geometry Through the Eyes of Felix Klein

Excerpt from Visual Complex Analysis (1997) by Tristan Needham (pgs. 30-34)

Geometry Through the Eyes of Felix Klein

Even with the benefit of enormous hindsight, it is hard to introduce complex numbers in a compelling manner. Historically, we have seen how cubic equations forced them upon us algebraically, and in discussing Cotes’ work we saw something of the inevitability of their geometric interpretation. In this section we will attempt to show how complex numbers arise very naturally, almost inevitably, from a careful re-examination of plane Euclidean geometry.[10]

[…]

Although the ancient Greeks made many beautiful and remarkable discoveries in geometry, it was two thousand years later that Felix Klein first asked and answered the question, “What is geometry?”

Let us restrict ourselves from the outset to plane geometry. One might begin by saying that this is the study of geometric properties of geometric figures in the plane, but what are (i) “geometric properties”, and (ii) “geometric figures”? We will concentrate on (i), swiftly passing over (ii) by interpreting “geometric figure” as anything we might choose to draw on an infinitely large piece of flat paper with an infinitely fine pen.

As for (i), we begin by noting that if two figures (e.g., two triangles), have the same geometric properties, then (from the point of view of geometry) they must be the “same”, “equal”, or, as one usually says, congruent. Thus if we had a clear definition of congruence (“geometric equality”) then we could reverse this observation and define geometric properties as those properties that are common to all congruent figures. How, then, can we tell if two figures are geometrically equal?

Consider the triangles in [fig 23], and imagine that they are pieces of paper you could pick up in your hand. To see if T is congruent with T’, you could pick up T and check whether it could be placed on top of T’. Note that it is essential that we be allowed to move in space: in order to place T on top of T˜ we must first flip it over; we can’t just slide T around within the plane. Tentatively generalizing, this suggests that a figure F is congruent to another figure F’ if there exists a motion of F through space that makes it coincide with F’. Note that the discussion suggests that there are two fundamentally different types of motion: those that involve flipping the figure over, and those that do no. Later, we shall return to this important point.fig_23

It is clearly somewhat unsatisfactory that in attempting to define geometry in the plane we have appealed to the idea of motion through space. We now rectify this. Returning to [fig 23], imagine that T and T’ are drawn on separate, transparent sheets of plastic. Instead of picking up just the triangle T, we now pick up the entire sheet on which it is drawn, then try to place it on the second sheet so as to make T coincide with T’. At the end of this motion, each point A on T‘s sheet lies over a point A’ of T’‘s sheet, and we can now define the motion M to be this mapping AA’ = M(A) of the plane to itself.

However, not any old mapping qualifies as a motion, for we must also capture the (previously implicit) idea of the sheet remaining rigid while it moves, so that distances between points remain constant during the motion. Here, then, is our definition:

A motion M is a mapping of the plane to itself such that the distance between any two points A and B is equal to the distance between their images A’ = M(A) and B’ = M(B). (22)

Note that what we have called a motion is often termed a “rigid motion”, or an “isometry”.

Armed with this precise concept of motion, our final definition of geometric equality becomes:

F is congruent to F’, written F ≅ F’, if there exists a motion M such that F’ = M(F). (23)

Next, as a consequence of our earlier discussion, a geometric property of a figure is one that is unaltered by all possible motions of the figure. Finally, in answer to the opening question of “What is geometry?”, Klein would answer that it is the study of these so-called invariants of the set of motions.

One of the most remarkable discoveries of the last century was that Euclidean geometry is not the only possible geometry. Two of these so-called non-Euclidean geometries will be studied in Chapter 6, but for the moment we wish only to explain how Klein was able to generalize the above ideas so as to embrace such new geometries.

The aim in (23) was to use a family of transformations to introduce the concept of geometric equality. But will this ≅-type of equality behave in the way we would like and expect? To answer this we must first make these expectations explicit. So as not to confuse this general discussion with the particular concept of congruence in (23), let us denote geometric equality by ~.

(i) A figure should equal itself: F ~ F’, for all F.

(ii) If F equals F’, then F’ should equal F: F ~ F’F’ ~ F.

(iii) If F and F’ are equal, and F’ and F” are equal, then F and F” should also be equal: F ~ F’ & F’ ~ F”F ~ F”.

Any relation satisfying these expectations is called an equivalence relation.

Now suppose that we retain the definition (23) of geometric equality, but that we generalize the definition of “motion” given in (22) by replacing the family of distance-preserving transformations with some other family G of transformations.

It should be clear that not any old G will be compatible with our aim of defining geometric equality. Indeed, (i), (ii), and (iii) imply that G must have the following very special structure, which is illustrated[11] in [fig 24].

(i) The family G must contain a transformation ε (called identity) that maps each point to itself.

(ii) If G contains a transformation M, then it must also contain a transformation M-1 (called the inverse) that undoes M. [Check for yourself that for M-1 to exist (let alone be a member of G) M must have the special properties of being (a) onto and (b) one-to-one, i.e., (a) every point must be the image of some point, and (b) distinct points must have distinct images.]

(iii) If and N are members of G then so is the composite transformation N ∘ M = (followed by N). This property of G is called closure.

We have thus arrived, very naturally, at a concept of fundamental importance in the whole of mathematics: a family G of transformations that satisfies these three[12] requirements is called a group.fig_24

Let us check that the motions defined in (22) do indeed form a group: (i) Since the identity transformation preserves distances, it is a motion. (ii) Provided it exists, the inverse of a motion will preserve distances and hence will be a motion itself. As for existence, (a) it is certainly plausible that when we apply a motion to the entire plane then the image is the entire plane — we will prove this later — and (b) the non-zero distance between distinct points is preserved by a motion, so their images are again distinct. (iii) If two transformations do not alter distances, then applying them in succession will not alter distances either, so the composition of two motions is another motion.

Klein’s idea was that we could first select a group G at will, then define a corresponding “geometry” as the study of the invariants of that G. [Klein first announced this idea in 1872 — when he was 23 years old! — at the University of Erlangen, and it has thus come to be known as his Erlangen Program.] For example, if we choose G to be the group of motions, we recover the familiar Euclidean geometry of the plane. But this is far from being the only geometry of the plane, as the so-called projective geometry of [fig 24] illustrates.

Klein’s vision of geometry was broader still. We have been concerned with what geometries are possible when figures are drawn anywhere in the plane, but suppose for example that we are only allowed to draw within some disc D. It should be clear that we can construct “geometries of D” in exactly the same way that we constructed geometries of the plane: given a group H of transformations of D to itself, the corresponding geometry is the study of the invariants of H. If you doubt that any such group exists, consider the set of all rotations around the center of D.

The reader may well feel that the above discussion is a chronic case of mathematical generalization running amuck — that the resulting conception of geometry is (to coin a phrase) “as subtle as it is useless”. Nothing could be further from the truth! In Chapter 3 we shall be led, very naturally, to consider a particularly interesting group of transformations of a disc to itself. The resulting non-Euclidean geometry is called hyperbolic or Lobachevskian geometry, and it is the subject of Chapter 6. Far from being useless, this geometry has proved to be an immensely powerful tool in diverse areas of mathematics, and the insights it continues to provide lie on the cutting edge of contemporary research.


[10] The excellent book by Nikulin and Shafarevich [1987] is the only other work we know of in which a similar attempt is made.

[11] Here G is the group of projections. If we do a perspective drawing of figures in the plane, then the mapping from the plane to the “canvas” plane is called a perspectivity. A projection is then defined to be any sequence of perspectivities. Can you see why the set of projections should form a group?

[12] In more abstract settings it is necessary to add a fourth requirement of associativity, namely, A∘ (B∘ C) = (A∘ B)∘ C. Of course for transformations this is automatically true.



See also:

  1. The Hyperbolic Geometry of DMT Experiences (@Harvard Science of Psychedelics Club)
  2. Materializing Hyperbolic Spaces with Gradient-Index Optics and One-Way Mirrors
  3. An Intuitive Explanation of the Symmetry Theory of Valence
  4. Principia Qualia: Part II – Valence
  5. Problems you can solve just by looking at them: The meaning of Noether’s Theorem
  6. Quantifying Bliss

Note: QRI‘s Symmetry Theory of Valence (STV) refers to the claim that valence (the pleasure-pain axis) manifests in the symmetry of the mathematical object that corresponds to each experience such that the mathematical features of that object are isomorphic to the phenomenal character of that experience. Using the lens of Klein’s conception of geometry, one could in turn give to STV a strictly geometrical interpretation. Namely, that the shape of one’s experience will be of high valence when it contains geometric invariants. In addition, Noether’s theorem (one of QRI’s lineages, which states that “every differentiable symmetry of the action of a physical system has a corresponding conservation law”) along with physicalism of consciousness suggests that for every symmetry in the mathematical object that corresponds to consciousness there is a corresponding preserved quantity. Thus, one could posit– assuming that the STV is correct– high-valence states might be extremely energy-efficient in addition to feeling good.

We are currently preparing a paper that ties all of these threads together in a way that, we believe, may turn out to have tremendous explanatory power. In particular, this formal account of valence will be able to explain succinctly a wide range of disparate and exotic empirical phenomena such as:

  1. Why phenomenal symmetry during psychedelic experiences is correlated with more extreme valence values.
  2. How and why Jhanas present the way they do, namely, as having:
    1. Extremely positive valence,
    2. Profound phenomenal simplicity, and
    3. Unusually high levels of energy-efficiency all at once.
  3. The fact that moments of eternity are typically extremely blissful.
  4. Why the phenomenal character of MDMA and 5-MeO-DMT manifest as both extremely symmetrical and high-valence.
  5. Why intense regular stroboscopic stimulation has deep emotional effects.
  6. And so much more…

Stay tuned!


Thanks to Alfredo Valverde de Loyola for pointing me to this excerpt in the book Visual Complex Analysis.