Article 1
Image of Time
Abstract
Although the images of time produced by tools such as Clocks, Calendars, and Timelines reflect properties of our perception, they are not time itself. Moreover, each tool individually reflects only part of the properties. The common problem is the limited dimensionality, insufficient to reflect all properties at once. On this basis, we can assume that these images may simply be low-dimensional projections of a more complex image of time.
The purpose of this paper is to demonstrate such an image that simultaneously contains all of the considered properties of time perception.
Time Image Projections.
Preprint
In Progress
Current Tools
Today there are mainly three tools used to visualize time:
Despite the fact that these tools give very different representations of time, it is the logic of nesting that brings them together (Fig.1):
• A day, on a week scale, is reflected as a single cell;
• A week, on a month scale – as a row of 7 cells;
• A month, on a year scale – as one of 12 cells.
Fig.1. Nesting Logic.
But the very nesting logic applied to time visualization is completely borrowed from spatial geometry, which makes it controversial. This is most obvious in a calendar.
Calendar Limitation
Time in a calendar is presented in the form of a table and therefore carries a distorted representation. For example, a row of 7 cells representing a week is more related to the perception of area and even more counter-intuitive is the rectangular reflection of a whole year (Fig.2).
Fig.2. Rectangular Representation of Time.
Despite the fact that the calendar contains such a serious contradiction, it is very widespread. The reason for this is the prevalence of the medium itself. Initially, the use of tables was defined by the format of paper, and then by the digital screen as the successor.
Although the capabilities of digital media are much broader than analog ones, the calendar itself has not changed much and continued to be limited in long-term planning. For example, try to find the "more than a year" button (view) on an Apple or Google calendar (Fig. 3).
Fig.3. Apple Calendar and Google Calendar.
However, it is with long-term planning that people have the most difficulty [4], and clearly this is the area where the digital calendar is expected to surpass its predecessor. And yet:
We use a digital calendar literally the same way as if it were made of paper.
Another way to see how we imagine time today is to simply "google" it in the form of pictures. But the result is all sorts of clocks, but not calendars (Fig. 4). This means that clocks are closer to our intuitive perception of time. However, time and time measurement tools are not the same thing. Thus, we can conclude that today there is no image of time as such, but instead we use images produced by tools.
Fig.4. Image of Time.
Perhaps, since we have no clearer than an operational definition of time [5], the image of time will always come from the tools of measuring it. But, this does not mean that the current image is complete, not just historically established.
If we go back to the calendar, when examined closely, it's easy to see that it's a sliced timeline, designed to fit on paper or a screen (Fig. 5).
Fig.5. Sliced Timeline.
Time Perception Properties
However much we strive to exclude from the description of the World patterns, that are consciously or unconsciously built on empirical knowledge, this is likely to be impossible with respect to time.
At best, we can identify the properties of time perception rather than time as such.
The nativeness of clocks and timelines lies in the fact that their geometry reflects important properties of our perception (Fig. 6), which for the purposes of this paper are determined as follows:
• Irreversibility;
• Continuance;
• Cyclicity.
But, neither the clock nor the timeline reflects all of these properties simultaneously:
• Clock gives a representation of Irreversibility and Cyclicity, but is limited in its representation of Continuance.
• Timeline gives a representation of Irreversibility and Continuance, but is limited in its representation of Cyclicity.
Fig.6. Properties of Clock and Timeline.
The common limitation of these tools is dimensionality. On this basis, it can be assumed that the Clock and the Timeline are two projections of another object, which reflects all of the listed properties simultaneously.
Fig.7. Correlation of Properties and Tools.
Geometrical Correspondences
To find this object, it is necessary to assign geometric correspondences to the properties of time perception. Spatial geometry uses 3 orthogonal axes to describe object properties such as length, width and height. To apply a similar approach, lets represent Continuance and Cyclicity as two orthogonal axes, and Irreversibility as the essence of these axes. Accordingly, let us position the clock and the timeline as two orthogonal projections. Since this construct is designed to describe time, we introduce a process with a duration of 12 hours. The result will be a tape-like surface (Fig. 8).
Fig.8.1. Tape-like Surface.
This surface reflects Irreversibility and Continuance, but does not fully reflect Cyclicity: we see some repetition, but without a return to the initial point. The reason lies in the unequal representation of the axes: Cyclicity, contains Continuance, and Continuance does not necessarily contain Cyclicity.
Fig.8.2. Tape-like Surface.
To solve this problem, the timeline may be represented as a closed curve instead of a segment.
Fig.9.1. Special Solution.
The result is a surface that reflects Irreversibility, Continuance and Cyclicity. This is a Special Solution, which operates within a limited duration (Fig. 9).
Fig.9.2. Special Solution.
Temporal Axis
To overcome the duration limitation, assume that the axes shift along a circle of infinite radius, i.e. a straight line, which we denote as the Temporal Axis. The result is a surface that reflects Irreversibility, Continuance and Cyclicity. This is a General Solution that works in unlimited duration (Fig. 10).
Fig.10.1. General Solution.
Fig.10.2. General Solution.
In summary, combining the Clock and the Timeline gives us two Solutions (Fig. 11):
• Special Solution — within a limited duration;
• General Solution — with no limitation of duration.
Fig. 11. Special & General Solutions.
Special Solution
The Special Solution can be defined as a as a linked pair of Clocks (Fig. 12):
• Time Clock;
• Observer's Clock.
Fig.12. Linked Pair of Clocks.
Or as a system of two linked pairs of points, each describing a circle (Fig. 13):
• Axial Circle — a pair of points (O0, Oi0) describing coordinates and time dimension;
• Forming Circle — a pair of points (O1, Ai1) describing the position of the Observer in the time dimension and the observation.
Fig. 13. Linked Pair of Points.
The first point (O0) in the Time Clock is the Zero Coordinate. This point contains a certain time dimension, which can be unfolded into a circle (Axial Circle). The second point (Oi0) is a projection of the first point (O0) in the unfolded dimension. This point (Oi0) determines the exact position of the Observer (O1) within the Time dimension (Axial Circle). The last point (Ai1) is the projection of the Observer's point of view or the observation itself.
To summarize, then:
• Time Clock determines the time dimensionality and the exact position in this dimension;
• Observer's Clock determines the dimension that the Observer uses to observe and, accordingly, the the events occurring in this dimension.
Initial State
The Initial State is determined as the only state where the pairs of points coincide
{(Oi0 ≡ O1); (O0 ≡ Ai1)}. This is the exact moment when the Observer captures the very fact of his initial observation (O0 ≡ Ai1) (Fig.14).
Fig.14.1. 1 Process in 12 Hours.
To illustrate the Special Solution, consider some process in which both clock hands (linked pairs of points) move synchronously and complete a full cycle simultaneously. This Tape is a visualization of one process that lasted 12 hours (Fig. 14). If the Observer tracks only his own clock, the tape is represented by a single edge, denoted as the Signature (trajectory of point Ai1). The intersection between the Signature and the Zero Coordinate (O0) represents an Event.
Fig.14.2. 1 Process in 12 Hours.
Now let's set the Observer's Clock to 12 cycles (which is equivalent to replacing the hour hand with the minute hand). In this case, 1 cycle of the Time Clock equals 12 cycles of the Observer's Clock. The Tape configuration has changed and the Signature now contains 12 loops (Fig. 15).
Fig.15.1. 12 Processes in 12 Hours.
Fig.15.2. 12 Processes in 12 Hours.
As mentioned in Section 6 "Special Solution", the cycles of the Time Clock and the Observer's Clock do not have to be identical. For example, let's set 4 processes of 1, 2, 3 and 6 hours on the Observer's Clock. The result will be a Tape with 4 unequal segments or a Signature with 4 different cycles (Fig. 16).
Fig.16.1. 4 Processes of 1, 2, 3 and 6 Hours.
Fig.16.2. 4 Processes of 1, 2, 3 and 6 Hours.
Torus
To better understand what a pair of clocks represents, let's increase the number of cycles (observations of Events) on the Observer's Clock. It becomes obvious that the Signature describes a surface that can be generalized as a Closed Torus (Fig. 17).
Fig.17.1. Closed Torus.
As a result, the Closed Torus is the searched object of Clock face and Timeline unification. The peculiarity of a Closed Torus (Axial Circle = Forming Circle) is that it has a special point where its surface self-intersects. We have already examined this point — it is the Zero Coordinate in which the time dimension is collapsed.
Fig.17.2. Closed Torus.
There are 4 more categories of Toruses that are not considered in this research (Fig. 18):
• Open Torus.
Axial Circle > Forming Circle. Has a Zero Line instead of the Zero Coordinate;
• Self-intersecting Torus.
Axial Circle < Forming Circle. Has 2 Zero Coordinates;
• Double-covered Sphere.
Axial Circle = 0, Forming Circle > 0;
• “Loop”.
Axial Circle > 0, Forming Circle = 0.
Fig. 18. Categories of Toruses.
General Solution
Now let us proceed to the General Solution when the Zero Coordinate (O0) is not static, but shifts along a circle of infinite radius, which we previously denoted as the Temporal Axis.
Fig.19.1. 2 x 12-hour Processes.
Consider a 24-hour segment of the Temporal Axis and set a 12-hour process on both Clocks. Since the General Solution implies a shift, the Signature intersects the Axis twice, i.e. defines 2 events or 2 cycles (Fig.19).
Fig.19.2. 2 x 12-hour Processes.
Next, we will change only the configuration of the Observer's Clock and set:
• 6-hour process — The Signature intersects the Axis 4 times (Fig.20);
• 1-hour process — The Signature intersects the Axis 24 times (Fig.20);
• 24-hour process — The Signature intersects the Axis 1 time (Fig.20);
Fig.20.1. 6-hour Processes, 1-hour Processes, 24-hour Processes.
Fig.20.2. 6-hour Processes.
Fig.20.2. 1-hour Processes.
Fig.20.2. 24-hour Processes.
Quick Summary
At the initial stage of research, the Closed Torus as a two-dimensional image of time reconstructed from one-dimensional projections allows us to note the following advantages over projections:
• Always gives one and the same representation rather than several for each tool separately;
• Reflects all the properties of time perception (considered here) at once;
• Does not require the logic of nesting different representations one into another.
Presumably, the two-dimensional image of time should also reflect other properties of our perception that were not initially considered, such as the effects of time distortion [6].
All of this suggests: is it possible that these advantages could be used to create better time visualization tools?
Further work is planned to focus on exploring the structures that populate two-dimensional time, as well as further increases in the dimensionality of time.
Looking ahead, one of the main questions to be answered is:
Is it possible in principle to literally "see" time, by analogy, as we see three-dimensional space?
But to answer these and other questions, it is necessary to move further research into the realm of pure geometry, which we will refer to as Temporal Geometry address in Article 2.
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Resources:
[1↑] Landes, David S (2000). Revolution in Time: Clocks and the Making of the Modern World. Harvard University Press
[2↑] Dershowitz, N. Reingold, M. E. (2008). 2. Georgian Calendar. In Calendrical Calculations (pp. 45). Cambridge University Press.
[3↑] Rosenberg, Daniel. Grafton, Anthony (2010). Cartographies of Time. Princeton Architectural Press.
[4↑] Sharp, H., & Brignull, H. (2019). 1. What is Interaction Design. In Interaction design: Beyond human-computer interaction, fifth edition (pp. 34–36). Essay, Wiley & Sons Canada, Limited, John.
[5↑] Ivey, D. G., & Patterson, H. J. N. (1974). In Physics: Classical mechanics and introductory statistical mechanics (p. 65). essay, The Ronald Press Company.
[6↑] Lemlich, R. (1975). Subjective acceleration of time with aging. Perceptual and Motor Skills, 41(1), 235–238.
Figures:
All Figures from 8.1 — 17.2, and from 19.1 — 20.2, as well as the Illustration in the Title, were generated using Temporal Geometry Simulation.