© M.Saniga

The Psychopathology of Time and Its Algebraic Geometrical Patterns

Metod Saniga

Astronomical Institute, Slovak Academy of Sciences,
05960 Tatranská Lomnica, Slovak Republic
(URL: http://www.astro.sk/~msaniga / E-mail: )




1. Introduction

F Character of the Enigma of Time

2. Examples of the Psychopathology of Time:

First-Person Accounts

F Time “Standing Still”

FEverlasting Now,” “Eternity”

F Time “Going/Flowing Backward”

FDiscontinuous,” “Fragmented” Time

3. Geometry of the Psychopathology of Time

F Time as a Pencil of (Projective) Conics

FEx-,” “On-” and “In-” Conics vs. “Past”, “Present” and “Future”

F Non-trivial Structure of Pencil-Time and its Most Pronounced Degeneracies

F Geometry of Psycho-Times

4. Conclusion


1. Introduction


Time is undoubtedly one of the most profound mysteries science has ever faced.


For one would hardly find something that is,

on the one hand,

F so intimately connected with our experience and yet, on the other hand,

F so difficult to come to grips with.


Nothing, perhaps, can better illustrate this point than what is usually referred to as the psychopathology of time, i.e. all “peculiar/ anomalous/ strange” experiences of time as frequently met in mental psychoses, drug-induced states, deep meditative and mystical states as well as in many other “altered” states of consciousness.


This peculiar fabric of psychological time comprises, as we shall soon see in more detail, such bizarre, paradoxical and mind-boggling forms as:

F “eternity, everlasting now,”

F “arrested/suspended” time,

F time “going backward,” and even

F “disordered/fragmented” time,

to mention the most pronounced of them.


Up to date, there exists no acceptable psychological / neurological model capable of properly dealing with these fascinating time constructs and underpinning any logical classification of them.


The reason why this is so rests, in our opinion, upon the following two facts.


First, these extraordinary experiences of time (and, of course, space as well) are inherently participatory, non-reproducible and subjective and, so, seriously at odds with current methodologies / paradigms of science, which strives for reproducibility and objectivity.


Second, the most pronounced departures from the “consensus” reality are so foreign to our “waking” mind that their properties defy our common sense logic and cannot be adequately communicated in words; an interested scholar has to go through a large number of relevant first-hand accounts / narratives and acquire the ability to “read between the lines” in order to spot an(y) underlying conceptual pattern.


We are therefore convinced that further progress in our understanding of these phenomena will

F inevitably entail a serious shift in the corresponding scientific paradigms to reveal their true epistemological / ontological status and

F be accompanied by the increasing use of sufficiently abstract mathematical concepts to properly grasp their qualitative properties.


Our study of psychopathological (space)times has, from the very beginning, been pursued in accordance with this strategy.


The model discussed in the sequel thus

not only

F features a fairly high level of abstraction,

but it also

F poses a serious challenge to some generally accepted dogmas in natural sciences.



it employs advanced geometrical concepts, like a projective plane/space and/or Cremona transformations.


it relies on a daring and far-reaching assumption that the anecdotal, first-person descriptions of extraordinary states of consciousness are on a par with standard observational / experimental evidence in natural sciences.

It is this “abstract geometrization of the first-person perspective” that

F gives our approach a remarkable unifying and predictive power and

F makes it a very promising conceptual step towards the ultimate unveiling of the riddle of time.

The purpose of the talk is to demonstrate this.



Examples of the Psychopathology of Time:

First-Person Accounts


2.1. “Eternity,” alias “Eternal/Everlasting Now”

Perhaps the most pronounced and in the literature best-documented kind of profoundly `distorted' sense of time. It is a sort of compressing, telescoping of past, present and future into the present moment that is experienced as “eternal/everlasting now.” One of the best portrayals of what this experience looks like is found in the following account (Huber, 1955):


I woke up in a whole different world in which the puzzle of the world was solved extremely easily in a form of a different space. I was amazed at the wonder of this different space and this amazement concealed my judgement, this space is totally distinct from the one we all know. It had different dimensions, everything contained everything else. I was this space and this space was me. The outer space was part of this space, I was in the outer space and the outer space was in me...

Anyway, I didn't experience time, time of the outer space and aeons until the second phase of this dream. In the cosmic flow of time you saw worlds coming into existence, blooming like flowers, actually existing and then disappearing. It was an endless game. If you looked back into the past, you saw aeons, if you looked forward into the future there were aeons stretching into the eternity, and this eternity was contained in the point of the present. One was situated in a state of being in which the “will-be” and the “vanishing” were already included, and this “being” was my consciousness. It contained it all. This “being-contained” was presented very vividly in a geometric way in form of circles of different size which again were all part of a unity since all of the circles formed exactly one circle. The biggest circle was part of the smallest one and vice versa. As far as the differences of size are concerned, I could not give any accurate information later on...


This narrative is remarkable in that the subject pays particular attention to the spatial fabric of his extraordinary state, which also differs utterly from what is regarded as a normal/ordinary perception of space; in fact, the subject finds himself to be one/fused with space!


2.2. Time “Standing Still,” alias “Arrested/Suspended” Time

Another well-documented and quite abundant anomalous temporal mode. A couple of spectacular examples are found in Tellenbach (1956; p. 13):


I sure do notice the passing of time but couldn't experience it. I know that tomorrow will be another day again but don't feel it approaching. I can estimate the past in terms of years but I don't have any connection to it anymore. The time standstill is infinite, I live in a constant eternity. I see the clocks turn but for me time does not flow... Everything lies in one line, there are no differences of depth anymore... Everything is like a firm plane...


and ( ibid; p. 14):


Everything is very different in my case, time is passing very slowly. Nights last so long, one hour is as long as usually a whole day... Sometimes time had totally stood still, it would have been horrifying. Even space had changed: Everything is so empty and dark, everything is so far away from me...

I don't see space as usual, I see everything as if it were just a background. It all seems to me like a wall, everything is flat. Everything presses down, everything looks away from me and laughs...


Both reports are by depressive (melancholic) patients. It is worth noticing here that when time comes to a stillstand, perceived space seems to lose one dimension, becoming thus two-dimensional.


2.3. Time “Going/Flowing Backward”

This kind of time pathology is very often found in mental psychoses. Here is a very illuminating and particularly representative case, communicated by a schizophrenic patient (Fischer, 1929; p. 556):


Yesterday at noon, when the meal was being served, I looked at the clock: why did no one else? But there was something strange about it. For the clock did not help me any more and did not have anything to say to me any more. How was I going to relate to the clock? I felt as if I had been put back, as if something of the past returned, so to speak, toward me, as if I were going on a journey. It was as if at 11:30 a.m. it was 11:00 a.m. again, but not only time repeated itself again, but all that had happened for me during that time as well. In fact, all of this is much too profound for me to express. In the middle of all this something happened which did not seem to belong here. Suddenly, it was not only 11:00 a.m. again, but a time which passed a long time before was there and there inside – have I already told you about a nut in a great, hard shell? It was like that again: in the middle of time I was coming from the past towards myself. It was dreadful. I told myself that perhaps the clock had been set back, the orderlies wanted to play a stupid trick with the clock. I tried to envisage time as usual, but I could not do it; and then came a feeling of horrible expectation that I could be sucked up into the past, or that the past would overcome me and flow over me. It was disquieting that someone could play with time like that, somewhat daemonic...



2.4. “Disordered/Fragmented” Time

The following experience, voluntarily induced by mescaline, is the most representative one we have been able to find in the literature available (Ebin, 1961):


For half an hour nothing happened. Then I began feeling sick; and various nerves and muscles started twitching unpleasantly. Then, as this wore off, my body became more or less anaesthetized, and I became “de-personalized,” i.e., I felt completely detached from my body and the world...

This experience alone would have fully justified the entire experiment for me..., but at about 1.30 all interest in these visual phenomena was abruptly swept aside when I found that time was behaving even more strangely than color. Though perfectly rational and wide-awake... I was not experiencing events in the normal sequence of time. I was experiencing the events of 3.30 before the events of 3.0; the events of 2.0 after the events of 2.45, and so on. Several events I experienced with an equal degree of reality more than once. I am not suggesting, of course, that the events of 3.30 happened before the events of 3.0, or that any event happened more than once. All I am saying is that I experienced them, not in the familiar sequence of clock time, but in a different, apparently capricious sequence which was outside my control.

By “I” in this context I mean, of course, my disembodied self, and by “experienced” I mean learned by a special kind of awareness which seemed to comprehend yet be different from seeing, hearing, etc.... I count this experience, which occurred when, as I say, I was wide awake and intelligent, sitting in my own armchair at home, as the most astounding and thought-provoking of my life...




Geometry of the Psychopathology of Time


3.1. Time Dimension as a Pencil of Conics and its (Psychopathological) Patterns

A cornerstone of our model are

F conics,

in particular their simplest, i.e. linear and single-parametric, aggregates, usually called

F pencils.



F linearity means that only two distinct conics are needed to define a pencil (figure), whereas

F single-parametricity signifies that the aggregate is a one-dimensional geometrical structure.


A point of intersection of two conics is clearly incident with all conics of their pencil; this point is called a base point of the pencil.

A pencil of conics features, as easily discernible from the figure, up to four base points.


A conic is analytically defined by a second order (quadratic) equation and it is

F composite (singular)


F proper

according as the equation is factorable or not.


A hyperbola, a parabola and an ellipse are all (and the only) examples of proper conics;

a composite conic consists of either a pair of lines, which can be distinct or coinciding, or a single point.


Any pencil of conics in the real plane contains at least one composite conic, and maximum three (not necessarily distinct and/or of the same type).


Given a pencil of conics, a point of the plane that is

not a base point of the pencil is incident with

F exactly one

(possibly composite) conic of the pencil.

Any other proper conic of the pencil then necessarily belongs into one of two qualitatively different, disjoint families according as the point (henceforth referred to as the reference point) lies in its

F interior


F exterior;*

in what follows we shall call such proper conics, respectively, in-conics or ex-conics, and the unique conic incident with the reference point, if being proper, will be denoted as the on-conic.

*Given a proper conic, a point, not on the conic, is its exterior or interior point depending on whether or not it lies on a line tangent to the conic (figure). The exterior/interior of the conic is the set of all its exterior/interior points.


At this point it should already be obvious that it is this

F a-pencil-of-conics-and-a-point configuration

that is of greatest interest and importance to us, because, in the case where the reference point falls on a proper conic (figure), it lends itself as a natural, elegant and remarkably simple explanation of our ordinary experience/sense of time.


The only assumption to be made to see this is, following Saniga (2003), to conceive

F each proper conic of the pencil as a single

event / moment of time,

with the understanding that

F the set of ex-conics represent events of the


F that of in-conics stand for events of the

future, and

F the unique on-conic generates the

present moment, the now.”


Apart from this noteworthy “past-present-future” pattern, mimicking everyone's common sense of time, our model gives rise to other two prominent, in a sense dual to each other, structures.

These correspond to the cases where the reference point

F coincides with a base point,


F falls on a composite conic.

In the former case, clearly,

F all the proper conics are on-conics (figure), whereas in the latter case the pattern is

F lacking any such conic, being endowed with

ex- and in-conics only (figure).

Hence, the corresponding time dimension,

in the former case, consists

F solely of the present moments (the “present-only” mode), whilst,

in the latter case, it comprises

F only the past and future, being devoid of the moment of the present (the “no-present” mode).


Now, it is the right point to return to the first part of the talk and try to rephrase these two unusual temporal arrangements in terms of pathological temporal constructs reviewed there.


In doing so, we readily find out that

F the present-only pattern nicely accounts for nothing but experiences of

“eternity,” “everlasting now” (Sect. 2.1).



F the no-present design, here a little thought

suggests that this is a proper fit for the time

“standing still” mode (Sect. 2.2);

for our feeling that time “flows,” “proceeds” is unequivocally tied to the notion of the present moment, the “now,” as the linking element between the past and future and so it is only too natural to assume that the absence of this element in the pattern should correspond to a complete suspension / cessation of the (sense of) time's flow.

It is instructive here to discuss a very interesting feature of our approach that has a serious bearing on the very meaning of the term “pathological” when it comes to the concept of time.

This feature tells us about a relative probability of the occurrence of the above-discussed three patterns of time in the realm of psychopathology. The reasoning goes as follows.

The conics of any pencil sweep up the whole plane and as the latter contains ¥ 2 (double infinity) of points, there are

F ¥ 2 of potential past-present-future patterns.

Next, as our pencil features three composite conics, each of these is a pair of distinct lines, and a line possesses ¥ 1 (single infinity) of points, we have 3 ´ 2 ´ ¥ 1 = 6 ´ ¥ 1 »

F ¥ 1 of no-present modes.

And, finally, as our pencil features four base points, there are just

F four present-only structures.


We see a clear predominance of the past-present-future mode within the group; no wonder that it corresponds to our “ordinary,” “consensus” experience of time!

Equivalently, this explains why experiences of “eternity” and/or time “standing still” are regarded / referred to as “anomalous / peculiar;” for the relative probability of their occurrence with respect to our “ordinary” experience of time is truly negligible.


Other conceivable forms of our generic pencil-time.

Instead of a reference point we take a reference line.

What different kinds of time dimension do we find in this case?

Remarkably, there are, like in the previous case, three of them (figure).

They differ from each other, as depicted in figure, in the position of this line with respect to the base points of the generating pencil of conics, being labelled, respectively, as

a zero-, one- and two-point pattern

according as the reference line hits

no, one or two of the base points.

Obviously, these line-related temporal structures can each be regarded as composed of an infinite number of elementary, point-related patterns.


This composition reads:

Type ordinary present-only no-present

Zero-point infinity none six

One-point infinity one three

Two-point none two infinity


What are the phenomenological counterparts of these composite temporal patterns?

Clearly, each of them must be a mixture / superposition of the time's experiences we have found to correspond to the elementary patterns involved.

And these are strange constructs indeed.



F the zero-point mode

corresponds to such an uncanny state of consciousness where the subject encounters an infinite tangle of “ordinary” experiences of time, differing from each other in the location of the moment of the present and, consequently, in the spans of the regions of past/future, this perception being accompanied by the sense of time “standing still.”

F The one-point case

is even weirder, as it includes, on top of the above, also the feeling of “eternity.”

And these experiences are very much like those of “disordered/fragmented” time given in Sect. 2.4 (but see also the last account of Sect. 2.2)!

But what about

F the two-point structure,

an intricate blend of the sense of “eternity / everlasting now” and that of time “standing still”?


This kind of experience was privately communicated to me by Linda Howe, an instructor in the `akashic records' (Howe, 1999):


One common scenario is when the sense of the self is so expanded, beyond any physical boundary... In this aspect, the awareness of being one with, or a part of, all that is can be profound. The illusion of separation can be perceived as dissolving and, at the same time, the awareness of the oneness, or unity,... becomes heightened, sometimes acutely so...

In this the experience of time is dramatic in its expansion and simultaneous contraction. There is a sense that there is only one moment, that all of time/eternity is held in that instant, very compressed and as powerful as one's imagination can conceive. Simultaneously, there is a sense that there is no time in the expansion. That all is holding still. Not even slow motion, but no motion. A total suspension of time is experienced. This is the all time/no time paradox…



4. Conclusion

Albert Einstein claimed that

“Für uns gläubige Physiker hat die Scheidung zwischen Vergangenheit, Gegenwart und Zukunft nur die Bedeutung einer wenn auch hartnäckigen Illusion.”


We have shown that this “illusion” and its most pronounced “anomalies” are underlaid by a definite algebraic geometrical pattern.


Does it mean that our math is a sort of illusion, too?


Or, is it physics that falls short of grasping the true nature of time?


Cracking this dilemma will certainly be

a great leap forward

in our knowledge of time…