The Ineluctable Modality of Rome

Got it?

B/W

Game of Thrones: The Iron Throne 
and
Pink Floyd: The Wall

Loop starting with Track 21 at 3:27 (HBO MAX)

They were incorrect.  Men didn't write this, Pink Floyd did.

"You little shit, you're in it now"

The Archaic Revival Pt. 5: The Transformation of Language Through Synchronistic Intercalation

What I think that is happening in the present, the whole of the 20th century, is a cultural crisis on a much grander, more global, and more threatening scale.  It is a casting back for a previous cultural model whose ideals, if we could realize them, would save our own civilization (the same idea that the Renaissance had about classical Greece).  Strangely, the period we have settled on is the archaic period, which precedes human history.

Cubism, and the things that were done for literature by Joyce and Pound, and the glorification of barbarism and the recovery of the unconscious (the sexual nature of the unconscious through Freud, the mass archetypal structure through Jung), the great movements, even Marxism, can be seen as movements to recapture pre-historic sacral values, and this process has been going on for fifty years or so.

Recall Marshall McLuhan’s idea that technologies for conveying information shift ratios in the mass psyche in the way that the mass psyche relates to the world.  McLuhan was famous for predicting electronic feudalism.  He said that the television screen was more like a page of manuscript than a page of print, and that the linearity and uniformity and rational assumptions of grammar were transcended in the conveying of information, to be replaced by electronic gestalts that were looked at rather than read.  The ratios between the senses would shift, and this would have profound on effects on art, the history of ideas, and this sort of thing.  What is happening is that a kind of super-McLuhanistic phenomenon is happening where we are collapsing not into the electronic feudalism, but into the electronic tribalism which he discussed, and it is shifting our sensory ratios away from the visual and towards the audial.

Under the influence of psilocybin, the language forming centers are activated with the visual cortex so that forms of synesthesia are linked to forms of sound, so that people singing control the fabric of hallucination through sound.  We found this to be true of ayahuasca in the Amazon where the shamans use of voice controls the visual fabric of what’s going on.  There’s another level to that which is going on which is with the addition of meaning, the control of the visual surface, the topology of meaning rather than the ongoing decoding from a dictionary, is transcended, meaning then is able also to work its adumbrations on this topological surface and you see into.....well there are different ways of cognizing it.  The place where the ursprache is coming from, the assembly language that lies behind all formalized or culturally validated languages.  Wittgenstein calls it the unspeakable; its the place where explication cannot go almost by definition, in order to avoid a tautology.  Now it seemed to me that the nearness of these tryptamine hallucinogens to normal metabolites of bran chemistry, and the fragileness that people like McLuhan and Julian Jaynes have shown to be a part of the way we construct our world, in other words,  it's a delicate balance of sensory ratios, chemistry, language and history.   Given all this, it seemed probable to me that this phenomenon, encountered in deep psychedelic experiences with psilocybin, actually has a potential historical impact.  

It is a kind of human ability, which is at present submerged in the psyche, contactable only by the shamanic means of journeying into historical hyperspace, in other words, of going into that place where the adumbrations of the future are intense enough that you can have an intimation, at least, of what is to come, and I think that this is what is to come, and it is a kind of telepathy, but its not telepathy as we imagined it to be.  

When I imagined telepathy I thought of it as hearing another person think, and having them hearing you think.  This is something where the modality of meaning is shifted out of the common dictionary, that is a cultural convention, and into a shared visual topology which is examined by both parties, both the speaker who caused this thing to be, and the audience who shares the space where this is happening.  It’s interesting that beta-carbolines, which are used to accentuate the hallucinogenic effect of DMT in these ayahuasca preparations in the Amazon, is very definitely part of normal human brain metabolism, and the MAO inhibition it performs on DMT is introduced from the outside, is a mirror image of the function it is performing in the brain.  This shifting of the sensory ratios is causing language to become more visual. At this point I always have to quote Philo-Judaeus, who was a first century Alexandrine Jew who talked about the Logs.   I have made analogies between the phenomenon I am describing and the Logos, but the critical quote was that “a more perfect Logos" was possible, and that it would be a phenomenon that would pass between the modality of being heard to the modality of being beheld without ever crossing through a quantized point of transition where you could say it was one, now it is the other.  I think that the cultural shockwaves that would be generated by the emergence of visible language will totally transform the culture to the point that the point beyond the end of history, the entry into hyperspace, the eschatological monad, all these religious or theological constructs about history are actually intuitions about language undergoing this transformation.  Now several things about this transformation, its obviously not something which the culture is doing as a decision, it isn’t like home computers and cable TV, it is being brought on as an informational utility, it is something that is being imposed from outside, and I think it is.  I’m sure most of you are familiar with the Gaian hypothesis of homeostatic regulation of the environment of the Earth through the interaction of all life acting as a single organism, well it obviously regulates trigger species such as we are, are part of this homeostatic method of regulation, and the gradual evolution of language is actually the gradual lifting of the veil of that is imposed between ourselves and meaning by the planetary ecology, in other words, the forward thrust of history is actually regulated by the ecology, and it is regulated through control of the evolution of language, because what you cannot think you cannot do, and where you cannot imagine, you cannot steer your culture and go.  So I am proposing on one level that hallucinogen be thought of as human macro almost social pheromones that regulate the rate at which language develops and therefore regulate the evolution of human culture generally.  Now one final thought about all this, it seems clear to me, and I’ve mentioned it in the other lectures, that the evolution, another aspect of what’s happening to the culture generally is transformation into a space faring species, and that the momentum for this has been building for millenia, it is not something that was decided in the 1950’s, it is in fact what we are all about.  I looked at a book recently by Terry Wilson about Brion Gysin called Here to Go, and he asked the question, “What are we here for?” and he answers the question himself, “We’re here to go”.  And I think there’s great truth in that, especially in the current historical moment where it's clear that man as a species, and the planet as a unified ecosystem, have become antagonistic to each other.  This is not unusual in nature, in fact it is a phenomenon that occurs between a mother and a fetus.  When the fetus comes to term, when the birth is imminent,  it must happen, otherwise the survival of both parties is threatened, even though the birth trauma for the mother and the child represents one of the major crises that they will face in their sojourn through existence.  Nevertheless, it is inevitable and necessary, and if it comes of off correctly its to the good of everyone.   Where psychedelics comes together with that, it is going to require a transformation of human language and understanding to stop the momentum of the historical process to halt nuclear proliferation, germ warfare, infantile 19th Century politics, all of these things.  It cannot be accomplished through a frontal assault upon it by political means.  The I Ching says you never confront evil directly, because when it is named, it sharpens its weapons, and it learns to defend itself.  So what is called for is this sideways attack through hyperspace.

The transformation of language is the signal that this archaic….that this nostalgia for the archaic world is coming to a head, and that this is its culmination.  This is the peculiar thing that we all sense is coming, that we can’t quite imagine.   It is synthetic and natural,  obvious yet hidden.  The interesting thing about it is that it emerges from an inner personal frontier, in other words, you’re not going to hear this on the evening news.  The President is not going to explain it to you, the Secretary General of the UN isn’t going to explain it to you.  You are only going to advance into the understanding of this phenomenon to the degree that you apply yourself to the knowledge of your being, to attention to being, to reflection on reflection, to attention on attention.  Then it will become clear.   And because it is a gradient of evolution, it doesn’t come with the force of a revelation.  It is something which is drawn out, almost in the same way that we move forward into time.  This thing is drawn out.  In fact you could almost say that the act of history, or the fact of history, is a macro-phenomenon that arises out of the micro-physical fact of millions of people evolving their language.  That this is what causes the moving wave front of historical becoming.  So transformation of language through psychedelic drugs is a central fact of the evolution of the social matrix of the rest of the century.

The I Ching is the most advanced effort to do something like a language of consciousness, but it’s a language of gestalt.  I don’t speak Japanese, but it’s said of Japanese that nothing which is obvious is ever mentioned.  Language is reserved for clarifying the unclear, so people are not saying “It’s a hot day, isn’t it?” and that kind of stuff.  They’re reserving language.  The other possibility is that visual language is perhaps not translatable into English because it is a language of emotion, where emotion is seen to be a subtle spectrum, an integrated gradient of meaning, or integrated gradients of combination, as meaning has, so that there isn’t love, hate, disgust, and something else, but in fact an infinitude of emotional states that can be triggered by vocal sound. In a way I am simply describing singing, wordless singing, except that I am describing how that can rise to an ontologically different level and become so emotive that you understand very subtle differentiations of emotion.

I noticed that when we were in the Amazon taking ayahuasca with these people, and they would sing these thousand year old songs, you would eventually get to the place where you had the absolute conviction that you understood.   The speed at which I’m going through emotional changes in relation to what I’m hearing must mean that I understand what I am hearing, because if I didn’t understand it I would just have a certain generalized emotion about it.   It is changing my interior state so rapidly that it is like the experience of understanding.  That’s the only thing it can be compared to.

Synchro-mysticism is a media form which can exist at any level of technology. It's a complete illusion to associate it with high levels of technology. It is probably, in fact, a form of media organization second only to music in its ability to endure.

This is what magic is.  It's being able to speak in a voice which makes things happen, being able to speak in a voice which causes facts to be beheld by groups of people in a way that has been purged from profane language, for us relegated to poetry and that sort of thing.

Terence McKenna, 1983

The Archaic Revival Pt. 3: Autopsy for a Synchronistic Hallucination

“The Mandlebrot Set is Real.  An absolute thing, no question whatsoever.  Any mathematician, or computer scientist, or synchro-mystic can study it and describe the same thing.  It’s a common experience. 

Such things that can be magnified forever and have infinite precision do exist, 

but they’re not touchable.”

Dr. Michael Barnsley

The Mandelbrot Set, Finnegans Wake, VALIS, McKenna's Time Wave, LeClair’s Tensor: what does it all mean?

Hologram vs. Fractal:  All Holograms are fractals, not all fractals are holograms

Tensor vs. Matrix:  Every tensor is a matrix, not every matrix is a tensor.

(.)

Every artwork is a fractal, not all artworks are holograms.

Every artist creates a matrix, not every artist creates a tensor.

PKD had his 10 volume meta-novel.  Written as a matrix of mutually exclusive artworks, their interconnected dynamics were not consciously recognized until years later.

*To make a hologram, the object to be photographed is first bathed 
in the light of a laser beam.   See PKD, David Bowman.

The films of Stanley Kubrick form a Matrix.  There exists a dynamic resonance between each genre that kidnaps the viewers own personal experiences and magnifies the collective  resonance in such a way that the Matrix is elevated into that of a Tensor (the shared experience). This Tensor establishes each film as a room in an interconnected and interactive environment; what is identified as environment instantly opens meta-verse Catholicism, in this specific case, a steampunk version of cyberspace. Read that again 

**A tensor is a mathematical entity that lives in a structure and interacts with other 
mathematical entities. If one transforms the other entities in the structure in a regular way, 
then the tensor must obey a related transformation rule.
This “dynamical” property of a tensor is the key that distinguishes it 
from a mere matrix. It’s a team player whose numerical values shift around 
along with those of its teammates when a transformation
 is introduced that affects all of them. 
Steven Steinke

The Archaic Revival Part 4:   Autopsy for a Synchro-Mystic Hallucination

The Sword and the Stoned

The best selling author confessed that one of his most popular books contained large sections of thinly veiled plagiarism.  

He experienced more confusion than guilt, as the plagiarized sections should have been obvious, yet, after 20 years no one had stepped forward to complain.

He kept his finances transparent so that every dollar from the sales of the book were available for the moment that the deception would be revealed 

Part 1:  The Objective Truth

An exegesis of objective journalism

This is a thing

These are the screenshots

This is how you make it

This is how it was discovered

Emails, no narrative analysis

Part 2:  The Subjective Proof

The odds:  How many curve balls?

Is it a benign face alien?

Or an elaborately disguised ghostbuster box designed for both imprisonment and salvation

Part 3:  The Truth of Proof is that there is no Proof of some Truths

What did I learn after many years

What's the closest I can arrange my sense-ratios to a “being there” experience?

-Edibles
-Long walk
-No setlist

Imagine a static radio station:  you can sense the song, but can’t tune it in.  
Solve the signal, get the tunes.

Imagine scrambled porn: you can sense the sex, but can't get off.  
Solve the scramble, get the jack pot.

Imagine a rubix cube:  you can sense the algorithm, but can't resurrect. 
Solve the scramble, impress the fam.

Imagine a rubix cube made out of films and albums and history.  You sense the magnitude of the vision, but you can't see the truth.  

Solve the scramble, enter a hyper-dimensional Eden of cosmic awareness.

The Role of Math In Synchronicity

    "a big piece of math with a little bit of media in the middle

or a big piece of media with a little bit of math in the middle"

One of these is more correct than the other

Raised on media, it was hard to see the math

But it was always there.

Someone else saw the math clearer than the rest,

A Wizard himself.

The Physical Tensor

I was an awe-struck partner in the project, like a Van Dyke Parks with Robert Wilson

I was out of my league, but I was a true companion, and the closest to the madness.

The Tensor wasn’t coming together, partly because I was seriously coming apart.

Partly because I knew I was out of my league, and  feeling in the way.

I was finding it mpossible to resurrect the Holygram from the fractal.

The hologram was the experience, and experiences are hard to transmute into words.
You can’t give someone an experience, especially when doubt coupled with existential crisis are the main ingredients.

You have to spend a year of weekends tuning this thing in yourself, you can’t give it to someone.  They not only must cross that bridge themselves, they must also build that bridge themselves.  All you can do is draw up the blueprints when they arrive.

Dark Side Of The Rainbow is a blueprint 

The Kubrick Transformer is a blueprint 

The Tensor is a blueprint for the false-work necessary to raise the dome of the plasmatic cathedral

The first time you open Finnegans Wake, you realize that this book is not a book.  It is a fractal.  
25 years later, you realize Finnegans Wake is no longer a fractal, it's a hologram.  A place, an actual reality.

LeClair and I knew that this was exhausting and maddening work, and as the battle for Kubrick’s brain withered on, a miracle took place.

I had revisited an idea that I posted in one of the second or third messages to mark

the Kubrick totem pole, that of stacking all films on top of each other 

I started to do this with good results until one day

a skip occurred, thwarting the perfection of the timing.  Alarmed, and ready to realign, something shouted at me to continue.

I was forced to make the decision to jump out of the agreed upon strict parameters of the Tensor coding and entertain a new timing.  

I can’t tell you, in words, what it was like to find.   It was a Joycean Thunder.  And Lightning.  Time in reverse, time travel.  Abduction.  Celebration.  

The best.

I immediately ordered extra copies of the DVD  

Then I set up the experiment again and it... was gone.  The film that I had just watched disappeared.  

Everything broke. 
Panic.  Madness.  

How was I to know that all 2001: A Space Odyssey's are not the same, and that the timing of the Intermission would be different on different DVD's?

Then realization:  if I had never kept my 2001: ASO DVD from Blockbuster, I would have never made this second discovery.  If the DVD had never skipped as well!  It was as if this thing wanted to be discovered.

Perfecting the timing of The Wall came first.

Then the obvious:  DSOTM, and Meddle.

The Kubrick Transformer has timings for The Wall, DSOTM, as well as Meddle, and all three can feel like mutually exclusive works of art.  

Sync Book

Blog

#noschizo

Comment sections

Reddit

Phish.net

Any community that I could share

Find the others, find them now

At some point the mind tires, and hibernation is required.

Math Teacher

Return of the Tensor

Gauss, William

The Kubrick Transformer sounds lame as a title, despite invoking the correct mechanics of the plastic toy.

The Kubrick Transformer as homage to Dr. Cotterell, too obscure?  Apparently.

Finding "the transformer" in Tensor mathematics?   
Mind melting.

I had arrived at the title independently of the role of transformation in the identity of a tensors.

The Phantom Threads

What happens when you lose the sense of smell or limb?
Phantoms.

What about losing the sense of sync?  Or when sync has done its job?

It wants to happen again.  Searching, chasing a phantom.

Media As Métier

When deciding on a title for this, I rejected “Autopsy for a Fractal Hallucination” as well as “Autopsy for a Mandelbrot Hallucination” because both, by definition, invoke pure mathematics.  The body in question, The Kubrick Transformer, was not driven by explicit mathematics.  Math was in the passenger seat, sure, but media was at the wheel.  And this autopsy is that of a media based multi-beast.  

Alfred North Whitehead defined understanding as “The apperception of pattern as such”.  This is truly a gloriously simple statement.  It exemplifies how the gloss of poetry can effectively communicate an elegant statement of pure mathematics.  And it is this gloss of poetry that makes the field of Synchro-Mysticism (SM) so intoxicating.   Much like how Robert Anton Wilson or Terence McKenna could dive deep into fields beyond the average stoners ken and elevate their attention into academic understanding without the burden of tuition or grit, SM eliminates the middle man by shining little pink lasers onto little crazy diamonds.  

Patterns?   How big are your patterns?  Math is the only catholic language capable of giving credence to this universal gestalt, gestalt being the mode of apperception with which to gain understanding.

But Synchromysticism must, at some point, return to its center.

Synchromysticism returning to its sexual roots?  Leads to co-dependence, or cold dependence.

Synchromysticsm returning to its drug roots?  Dangerous, leads to madness.

But when Synchromysticism returns to its mathematical roots, SM becomes a secret weapon. The best minds in SM have extraordinary mathematical minds and mathematical educations.

The first three people who were prominent in the SM ocean that immediately come to mind are Chris Knowles, Jake Kotze, and David Charles Plate.  I have to believe each has excelled at college level mathematics at some point, each clearly gifted with mathematical minds

Knowles and Plate approach the ability of Terence McKenna to carry a metaphysical conversation with the ease and depth for hours and hours.  

If you talk to a regular human being, they may be conversant in dozens of subject matters, but it is the rare person who is conversant in dozens of subject matters, and can discuss all 64 at the same time, carefully scaffolding the points of their conversation 

But out of all the many people in and around the community that I was lucky to talk with, either by phone, email, or instant message, one person stood head and shoulders above any one I have ever met.  

That person is Mark LeClair, the author of the out-of-print blog wrongwaywizard.com.

I honestly don’t remember why I reached out to share my personal experiences of film/album synchronization, but that was the beginning of a dialogue that lasted for close to a year.

The culmination of this dialogue was personal discovery born out of a project LeClair had invited me to help build.

The Tensor, or The Physical Tensor by Stanley Kubrick and Pink Floyd

Three films, three albums, 9 combinations.

Math.  The core of the project was math, a math that arose out of the work of Carl Friedrich Gauss and defined by a man named William Rowan Hamilton.

I saw Mark’s vision, hell, he asked me to help him build it.  But explain it?  
That was the mission we accepted ahead.  

From Bris to Hubris

I quickly proved to be deficient as a collaborator, and simply could not keep up, or convince myself that I had the necessary talents to contribute anything of value.  

The Tensor was a one to one synchronization:  one album, one film.  But understood within a matrix of 8 other timings.  The dynamic tension of resonance and release between the participants elevate it beyond matrix, and into the realm of the tensor.  At some point, I do not remember how or when, I arranged my television and laptop in a way that two films and one album could be synchronized.  This was actually an idea I proposed in one of the very first correspondences I had with Mark.

I started to fold the Tensor in on itself.  I doubled, maybe tripled the intensity of the vision: laser lightning and thunderwords for all.  

While working with 2001 and The Shining in one of these twin viewings, an accident led to the birth of miracle.  I imagine its the feeling one has when one hears that they have the sixth correct number in the lottery.  Or when one sees a UFO.  Not orgasm, well, certainly not my orgasms.  It was something else. 

I probably worked on lining that thing up for weeks, watching, rewatching, noting timings. 

A problem eventually arose:   it did not correspond to the definitive timings that were canon to the Tensor.  

Was this an outsider?  An insider?  Frosting?  
It was the child of the Tensor for sure, but how did it fit into the project?  

It involved only two of the proposed films.  It crafted it with The Wall, but true to the whole ethos of The Tensor, it worked for DSOTM as well as Meddle, each with mutually exclusive starting points.

  I have had certain people say they think that DSOTM is the superior soundtrack.  I have heard some say that Meddle is the one that works best.  Honestly, over time, one must witness all three versions.

I chose The Wall as it provides the deepest field of apperception; it provides the largest infinity of understanding.  

The intensity of awe in this discovery was not shared by LeClair, which was a terrible blow to my elation.  If anyone were to share in the victory dance, certainly it would be my compadre in sync?  But if I remember it correctly, the break from the established timings was a monkey wrench that led to the shelving of the LeClair/Klaus Tensor partnership.  Thankfully the end of this partnership did not lead to an end to our friendship.  We were able to re-establish our dialogue which led to many other projects for the both of us.

I had considered developing an annotated guide to the Kubrick Transformer, in the vein that the Tensor project had designed, an annotated guide to the body of Christ, but it soon became obvious that it was : impossible, detrimental, embarrassing, impossible.  

Let it speak for itself, it exists for those to see who need to see.
  

Thanks to the unexpected revival of an old high-school friendship, who happened to play tennis with a film student who gifted an editing program to him, a DVD of the KT was produce.  I spent a few weeks burning copies with other odds and ends, in unique covers, 111 in all.

I shipped about 50(?) copies to Alan Green in New York, who was in the process of launching The Sync Book.  I handed out copies to friends.  I handed out copies to strangers.  I left copies in libraries, on college campuses.  I was a Johnny Apple-Sync, trusting the waves of the universe to turn on as many minds as possible, to allow the miracle to find them.

I wrote a chapter in Sync Book two and included the timing codes.  And that’s kind of where it ended for the KT.  

I few years later I decided that taking down all footage of the KT on my blog, or on YouTube wa necessary.  I am even tempted to remove still images, as to allow those moments to retain the full energy of their impact.  

I have had debates with DCP about whether to not to make it easy for the viewer to watch these syncs.  I still stand that the essential ingredient in these arrangements is doubt, and taking that first leap of faith towards actually taking the time and effort to correctly and precisely align the required materials is an admission of need.  It is the need for alchemical healing, and I sincerely believe that this must be respected as something not necessarily sacred, but necessary.  Not scientific, but common sense.  

Chemotherapy is for the cancer, not the healthy.
Divine pink lasers are for the dying, not for the living.

I do not own any DVD copies of the KT.  I don't even have the proper materials to watch it.  I find that it is better this way.  It is safely buried, regaining its essential power. I don’t plan to watch it again anytime soon.  I may not get to watch in its proper form ever again. 
But yet it's still here with me, and will be in truth forever.

Years later, I started taking on-line math courses in order to get a Masters Degree in Education.   I started to learn about linear algebra, discrete mathematics, all the way to the impossibly rich Abstract Algebra.  In the course of my studies I ran into Gauss once again, and started to dip my toes into the mathematics that birthed the inspiration for LeClairs tensor.

It was over my head, by I could finally see it.  But a masters degree doesn't require reaching that far, so there it remained.

I think I am taking the first step into the realm of pure math.  God help me.

Closure of this blog has been an important goal for me, a goal that I have reached on more than one occasion, but it seems to always fade.  

*To make a hologram, the object to be photographed is first bathed in the light of a laser beam. Then a second laser beam is bounced off the reflected light of the first and the resulting interference pattern (the area where the two laser beams commingle) is captured on film. When the film is developed, it looks like a meaningless swirl of light and dark lines. But as soon as the developed film is illuminated by another laser beam, a three-dimensional image of the original object appears. The three-dimensionality of such images is not the only remarkable characteristic of holograms. If a hologram of a rose is cut in half and then illuminated by a laser, each half will still be found to contain the entire image of the rose. Indeed, even if the halves are divided again, each snippet of film will always be found to contain a smaller but intact version of the original image. Unlike normal photographs, every part of a hologram contains all the information possessed by the whole.

The “whole in every part” nature of a hologram provides us with an entirely new way of understanding organization and order. For most of its history, Western science has labored under the bias that the best way to understand a physical phenomenon, whether a frog or an atom, is to dissect it and study its respective parts. A hologram teaches us that some things in the universe may not lend themselves to this approach. If we try to take apart something constructed holographically, we will not get the pieces of which it is made, we will only get smaller wholes.

While it is true that if you would cut a hologram into a few pieces, you could still view the entire image by illuminating a single piece, the resulting projection would be of a lower resolution. Talbot mentions in his book that the projection will get hazier and smaller.

**  "What is the difference between a matrix and a tensor?"

There is a short answer to this question, so let’s start there. Then we can take a look at an application to get a little more insight.

A matrix is a grid of n × m (say, 3 × 3) numbers surrounded by brackets. We can add and subtract matrices of the same size, multiply one matrix with another as long as the sizes are compatible ((n × m) × (m × p) = n × p), and multiply an entire matrix by a constant. A vector is a matrix with just one row or column (but see below). So there are a bunch of mathematical operations that we can do to any matrix.

The basic idea, though, is that a matrix is just a 2-D grid of numbers.

A tensor is often thought of as a generalized matrix. That is, it could be a 1-D matrix (a vector is actually such a tensor), a 3-D matrix (something like a cube of numbers), even a 0-D matrix (a single number), or a higher dimensional structure that is harder to visualize. The dimension of the tensor is called its rank.

But this description misses the most important property of a tensor!

A tensor is a mathematical entity that lives in a structure and interacts with other mathematical entities. If one transforms the other entities in the structure in a regular way, then the tensor must obey a related transformation rule.

This “dynamical” property of a tensor is the key that distinguishes it from a mere matrix. It’s a team player whose numerical values shift around along with those of its teammates when a transformation is introduced that affects all of them.

Any rank-2 tensor can be represented as a matrix, but not every matrix is really a rank-2 tensor. The numerical values of a tensor’s matrix representation depend on what transformation rules have been applied to the entire system.

This answer might be enough for your purposes, but we can do a little example to illustrate how this works. The question came up in a Deep Learning workshop, so let’s look at a quick example from that field.