Monthly Archives: January 2010

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.

The Two Minds of Charles Darwin

I’ve wanted for some time to write this note, but have hesitated because there are so many others who know Darwin far better than I. Nonetheless, I have a certain conviction I’d like to share.

Two minds seem to be at work as Darwin surveys the natural world and its evolution. One sees natural selection in terms of confrontations between individuals or species in the search for limited resources. We all know that scenario, which in most of our discussions has become the very paradigm of Darwinian selection.

But Darwin has unmistakably another line of thought, which grasps the utter complexity of the selection process: not as a competition between individuals, but as a system whose complexity defies analysis. If we were to make an improvement in a breed in order to increase its chances of survival, we would not, he remarks, know what to do. In another passage, he remarks on the flourishing of a certain flower in one particular English village. What advantage does this plant have here, which it lacks elsewhere? The answer, he has decided, is the absence of dogs. (Dogs, he reasons, eat cats; cats eat mice; mice eat seeds.) I’ve forgotten why there are no dogs, it might be some village regulation. Whatever it is, there lies the strength of the flower: not in its own design alone, but in the structure of that ecosystem, which has at least for a time stabilized in a pattern collective survival –a pattern, we might say simply, of collective health.

This I believe is an overriding principle, which we have tended since Darwin’s time to miss. That principle, almost systematically ruled out of all facets of our thinking – even our very ideas of medicine or science itself, is the overriding concept of organism, the recognition that we live, flourish and evolve as a whole – not as a sum of individual parts. Only in recent years have we begun to study ecosystems, of all sorts and levels, as wholes. As a society, we’re far behind the demands pressing upon us in catching Darwin’s other, and I believe higher, insight.

The stereotype in describing the components of living systems, to ever-higher levels of resolution, is mechanism. Wrong! We will never understand living organisms as summations of mechanisms. A living system is a different concept altogether from a machine, and study of it calls for different strategies, and different conceptual tools.

Much new work is being done now in the spirit of this new understanding. I’ve found exciting studies of ecosystems to which I want to call attention in an upcoming blog posting. Indeed, it’s not a new thought on this blogsite, which has traced the idea of organism back to its rich source in the writings of Aristotle, and fast-forward through western history to Leibniz, Euler, Lagrange, Maxwell, Hamilton, Feynman and modern physics. But in the din of our celebration of Newton, isolation and competition, we haven’t heard, or perhaps have deliberately rejected, these other voices. We’ve caught only the lesser of the two voices of Charles Darwin.