Category Archives: Fourth Dimension

THE FOURTH DIMENSIONAND THE PRINCIPLE OF LEAST ACTION: Why the Clock Says “Tick-Tock”

It has been suggested that two major themes of this website are now converging in such a way that each throws light on the other – namely, the Fourth Dimension, and the Principle of Least Action. The following essay aims to explain this connection. This may in turn throw special light on the discussion of Karl Marx, the final stop on our Dialectical World Tour – coming next!

WHAT IS “MOTION”?

Least Action – whatever our understanding of this challenging term – is certainly a way of characterizing natural motion. Natural systems, the principle asserts, move in such a way that their action shall be least. Before we concern ourselves with the mystic term action, we might do well to focus on the no less mysterious term, motion. Not only will this help us to understand the intent of the principle, I think it will lead us to see a deep link between the Principle of Least Action, and the Fourth Dimension.

What, then, is motion? We moderns tend to think of it in instrumental terms – as the means of getting from intention to accomplishment: in between planning to be something, and actually being that intended thing. In this view, motion is a sort of intermediate, between intent and accomplishment: something to be gotten over, usually as quickly as possible. “Time is money,” we say – meaning, a cost. This colorless idea time accords with the mathematical view of time as a line: our lives pass along this continuum from goal to goal, the moments between, mere means, to be passed through as expeditiously as possible. Time is money, we say, meaning cost. This uniform, colorless continuum Newton mastered by finding a way to measure motion at a point, as if the now were its dwelling-place. Taking this to the limit, he gave us the mathematics of conventional physical science: the differential calculus. Absolute, mathematical time, Newton was sure, flowed calmly through absolute space, in God’s divine sensorium.

As so often happens, the ancients had a different, more interesting point of view, suited to a richer and wiser concept of Nature itself. Calling Aristotle as our witness, we’re told that being does not lie merely at the two ends of a span of intervening motion. Rather, being inheres in the notion itself. There, in that very motion, we find ourselves in the fullness of our being. As we might expect, Aristotle has a word for this. energeia: being-at-work. And since the work is at every stage shaped to its end – its TELOS – we can call this richer, organic concept of living, EN-TELECHY.

Rest assured, this old view is by no means a threat to modern science. Leibniz, already in Newton’s time, was developing a more complex form of the calculus suitable to this richer view of motion. Instead of zeroing-in on the passing moment, it looks to the whole span of the motion, from inception to closure. It’s called the integral calculus. In its variational form, it weights every moment with respect to the goal, and hence meets Aristotle’s test of entelechy. Whether it’s a radiating atom or a busy mouse, every stage of its motion is inherently – by Nature – shaped to its goal. The motion, then, is truly whole.

Aristotle goes on to say a funny thing about time. Time, he seems to say, is the number of motion(s). To clarify, he illustrates by saying we count the number of times the horse goes around the track. More generally, in the order of being, first there is the race, and then, secondarily, we count the laps, and time arises. The whole, which is the motion, is primary; motion doesn’t happen in time. Being comes first; time is merely the count of the generations of being.

Come to think of it, that’s the way we encounter time in daily life. Our encounter with time is mediated by some motion: we count the clock which tells the time. That’s why the classic clock says, tick-tock. The swings of its balance are counted by ah escapement, going first tick, and then tock, to mark the completion of one cycle: one motion The time it tells is the count of its motions. More modern clocks, it is rue, speak in other voices, but they’re all counting motions of some sophisticated sort.

 

HOW THE FOURTH DIMENSION UNDERLIES LEAST ACTION

If we’re satisfied that a motion is essentially whole, we’re ready to turn to Least Action. All the natural world runs on the Principle of Least Action, so this is important to know about. Here is the Principle: Every natural motion, atom or mouse, unfolds in such a way that over the whole motion, total “action” will be least. Think of “action”, then, as activity-summed-over-the whole motion. Action thus refers to something Newton missed. Contra Newton, you can’t have action at a moment! More positively put, nature accomplishes the overall goal with the least possible fuss. There’s a “good reason” for that; fuss (haste) makes waste. Every activity entails heat-loss (that’s why our bodies run so hot). The horse will ultimately run at top speed, but getting-up-to-speed will be accomplished by nature as gradually as possible. In turn, once up to speed, the myriad processes throughout he body will themselves run, collectively (organically), in such a way hat the speeding horse will be expending as little energy over each cycle of the gait, as possible.

If we image a running horse by means of a three-dimensional snapshot, we’ll evidently miss Nature’s point. We need to see that motion whole: each whole cycle of the gait as one single image. Our three dimensions are not enough: we must add a fourth axis to our image. In addition to our three spatial axes, we need a time axis as well. The resulting image will then encompass in a single geometrical figure the motion as that whole which, by its very nature, it is.

Though such four-dimensional imaging can indeed present this wholeness effectively to our physical eye, a larger aim must remain: through this visual experience, to extend this same insight to the eye of the mind. We might then perceive all natural motion in this four-dimensional way – and thus, in turn, achieve a larger grasp of the wholeness of that motion of ultimate interest, life itself.

Mathematical physics has widely accepted the Principle of Least Action as its basis. Taking the term physics in its old, true meaning, as the science of all nature – the fall of a leaf, or the beat of a heart – it’s nice to know that the more modern the science, the more it attests to wholeness, and to the richness of every moment: not as transient as it may seem.

From Diagram to Visual Experience: More on the Reality of the Hypercube Theater

We recently launched our Hypercube Theater, a stage derived from the hypercube (the tesseract) on which four-dimensional figures can make real appearances before our eyes. In that posting, a series of small photographs and diagrams served to illustrate our line of reasoning.

 

Now we have transformed these diagrams into screen-filling images, a development which illustrates an important point. Beyond the difference in size, this marks the advance from a mere diagram to a true visual experience.

 

We consult diagrams for the information they convey, but we approach a true illustration in search of a visual experience. It’s the familiar difference between looking, and seeing. When we look at a powerful photograph, what we see is, not a picture on paper, but a mountain range. It is as if we were there: we experience, we say, a sense of presence.

 

In three dimensions, this same distinction applies, becoming the difference, let us say, between a cardboard stage-set, and a ship in a storm at sea. Even in daily life, this same distinction arises. Our eyes present to us, we realize, only optical images of the room around us; but we experience, not such images of a room, but presence in the room itself.

 

It’s in this spirit that we approach the hypercube, not as a mere geometrical figure, but as a theater – a stage on which all the entities of a four-dimensional world can make their appearances, present to our eyes.

 

The new version of our Gateway Theater posting tests the ability of screen-filling images to support this transformation. Ideally, of course, the computer screen would – like the theater stage it is becoming – fill our field of view. Even on a laptop screen, however, our visual imagination will have the power to fill that gap! Follow the link below to experiment with a first step in entering the fourth dimension!

Enter the updated Gateway to the Fourth Dimension here.

Let us know if this works for you.

Gateway To The Fourth Dimension: The Hypecube As Theater

Some time ago, we launched a new department on this website to be devoted to studies of the fourth dimension. In our first entry, we met the basic figure of 4D geometry, the hypercube, or tesseract. We watched as it emerged from the familiar cube, extending into the fourth dimension.

 

Now we return to the completed hypercube, to see how it can be better understood, and how it can be put to use: first, to view; and then, even to create other four-dimensional figures of whatever sort.

 

We begin with this model:

A model for imagining the Fourth Dimension.

The Tesseract Theatre

 

We plan to mount these studies on the Fourth Dimension page of this website under the overall title Gateway to the Fourth Dimension. Part I, The Hypercube as Theater, appears today. Take a look, and let us know what you think!

Read this new article here.

 

These studies are all contributions  to the preparation of a book currently  in progress, “At Home n the Fourth Dimension.”
Art work for “At Home in the Fourth Dimension” by Anne Farrell.
Model and photography by Eric Simpson.
All rights reserved.


 

A Project For Viewing Objects In The Fourth Dimension

A new menu item has just gone up on this website, at which images will be displayed of objects in the fourth dimension.  Since it is widely agreed that we are by nature incapable of seeing four-dimensional objects, something evidently needs to be said at the outset to justify this new approach.

The Case Against

The case against seeing 4D objects is easy to make, and does sound convincing. We human beings by our very nature belong to a three-dimensional world; we have neither retinas nor brains capable of seeing objects in higher dimensions. We can see projections of such objects as the  4D hypercube, for example, into our 3D world–but we have no power to see the thing  itself. We are like Abbott’s Flatlanders (Edwin Abbott, Flatland (1884), confined in their case to a 2D world, and unable to conceive a world beyond it.  We laugh at them, but Abbott’s initial point is that we are the Flatlands–confined in our case to a world of three dimensions, and unable to envision a world beyond.

But Abbott makes a further point, and his real story is one of courage and release.  The turning-point of Flatland is the breathtaking escape of his hero, A-Square, who is swept out of Flatland and indeed does view, to his amazement, a world beyond his own, as well as his own from a new vantage point–outside. Surely Abbott’s real point is to challenge us, stuck in three dimensions, to break out of our own confinement. That is the experiment we will be undertaking on this website.

The difficulty is not mathematical. In our drawings, we place the viewer’s eye at a definite, fixed position in a four-dimensional coordinate frame, and define simple objects within it. The objects lie before the eye, in relations which can be calculated and depicted. The huge question remains, however: what will such an eye actually see? Not much, we might think–for our limited, 2D retinas would seem to have no power to capture 4D images.

Response to These Objections

But here’s a problem: by the same argument, the same 2D retinas must be inadequate to see the very 3D world in which we live!  Mathematically, it is true, we’ve never actually seen our own, familiar world: we would have to be outside it, to actually view it. But that does not stop us from knowing it intimately, and seeing it in another sense.

Evidently, real vision is not simply a mathematical question: the eye is not a camera. Rather, it is a powerful extension of the brain, actively searching and interpreting, constructing a meaningful and coherent understanding of the world we live in, and love. We “see” objects growing smaller as they recede into the distance; but from infancy, our interpretive visual system has learned the tricks of 3D visual intuition: we know automatically that the objects remain unaltered.  And if this is the case, there would seem to be no obstacle to carrying out our project, exploring the possibilities of a visual experience of four-dimensional space. Maybe our visual intuition is capable of learning new interpretive tricks!

The New Proposal

We propose therefore to look directly at the mathematically defined objects within it, with the aim of building a new structure of visual intuitions appropriate to the fourth dimension. It’s hardly necessary to stress the importance such a capability might have, given the striking ability of the visual cortex to “see”–graphically or otherwise–the relationships among groups of interrelated factors. The ability to visualize complex functions in a four-dimensional coordinate frame might be enough to convince mathematicians and scientists of the practical value of such an augmented power of visual intuition.

For an initial example of the method at work go to my webpage on the fourth dimension. Where you can post any comments which occur to you.