Tag Archives: hypercube

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.


 

“There’s no Space in Euclid!” or Euclid’s Rhetoric

“There’s no ‘space’ in Euclid!”
I remember vividly the moment when a talented young tutor at St. John’s College in Annapolis, Maryland, came careening down the stairway our old library, unable to contain this startling realization. As a new tutor at the “Great Books College”, he had likely been assigned to teach a class in the subject he knew least, and was as a result making his first encounter with Euclid’s Elements in the company of a class of first-year students – an experience as fresh and surprising to him as  to his student fellow-readers.

Euclid writes in a style we no longer expect of mathematicians: simple, confident, almost buoyant. In the scale of rhetorical possibilities, Euclid’s is the style Aristotle calls “simple”. His figures stand upright and firm, on their own: they have no dependence on a “space” in which to be.

Reading the classics in this unscholarly way, as if they had been written for us, is an experience filled with such surprises; as myself a student at the College at the moment of this proclamation, I recognized the experience: the same revelation had struck me not long before. Euclid’s idea of geometry is not what we moderns have been led to expect!

One aspect of this dis-conformity between Euclid’s world and our modern expectations is this troubling absence in Euclid’s mind of any notion of “space”. Euclidean figures, all the way to the great regular solids in which the thirteen books of his Elements culminate, are organic webs of relationships, each standing whole before the mind’s eye, increasingly vivid  as the plot  thickens. Our relationship with them is direct and immediate: they are not “in” anything!

Similarly, Euclid never proves anything, in our modern sense. His style is strikingly at odds with the ways of today’s formalism, which tends to bind the mind in chains of consequence, rather than liberating it. Euclid leads us to contemplate the figure which evolves as the demonstration (epidexis, a showing) unfolds. We construct the figures as we go along, and take well-deserved satisfaction, as they unfold, in our workmanship.

At each step in the course of a demonstration we are tacitly invited to lend assent. There is, on the other hand, nothing as it might seem, loose or casual about all this. Euclid relies at each step on a very real power of intellectual intuition – on our human ability to discern truth when we see it. Euclid as author is, at the same time, our teacher. Under his guidance we develop confidence and soon find ourselves taking delight in the exercise of new-found power of geometrical insight. Skeptics today will accuse us of self-deception, and Euclid, of naivete. But we and Euclid may stand if we wish by our own agreement, when we confirm a geometrical truth.

An illustration from Euclid: Book 1-PropositionsA striking example of Euclid’s method at work would not be far to seek: the first words of his Elements make a strong demand on our visual intuition. We are asked to construct an equilateral triangle:  (please excuse the informality of these images derived from my aging copy of Euclid!)

The procedure is very simple.  We begin with the straight line AB:

And on it at point A draw a circle ACB of radius AB:

(Point C is here no more than a label, not yet specified, to identify the circle in question.)

Similarly, at point B, we construct a second circle. ACE of the same radius:

(“C” is still functioning as a label, not yet located.)

But now, Euclid begins the final stage of the construction, with no hint of apology or explanation, by giving the mysterious “point C” a specific location, and a crucial function: “from the point C, at which the circles  cut one another ….”.

We stop to catch our breath!  Point C has now been specified, without ceremony or justification.  How do we know that it exists – that the circles do indeed intersect?  Euclid’s answer is simple: we know it, because we see it, in our mind’s eye – and of course we never really doubted.

We proceed to draw the sides, complete the figure and carry our new triangle with us, a secure foundation on which the great structure of the Elements will rest.

This is to be Euclid’s style throughout: even first principles are not legislated, but offered for our agreement. They are things asked of us or postulated, as questions (AITIAE) or proposals – and the rhetoric of the Elements will be consistent throughout.

Similarly, when we pause at the close of a long stretch of reasoning, to review the steps we have just passed through, this is not a matter of mere logical bookkeeping.  It is, rather, clearing the way to that moment of commanding insight, in which we say, in the spirit we now think of as that of Gestalt, “Aha!”I see!” And indeed, we do.

This is the rhetoric of Euclid, which so shapes our path that the we are led to see.  What department of mind is this, which Euclid is invoking?  I’m sure he would be in easy agreement with Plato, that while the logical mind grinds away at syllogisms, another, higher department of mind sees truth, and says “yes” to an argument not because it is bound by chains of syllogism to do so, but because it can view truth directly, and know it for what it is.  Plato calls this higher, defining power of mind NOUS, and as his dialogue Theaetetus makes clear, mathematics, practised in this mode, is crucial preparation for an approach to the highest things.

I have been drawing attention to Euclid’s rhetoric, but just as the rhetoric of Plato’s dialogues is essentially philosophic – skillfully leading the respondent to a question which is philosophic because it leads logic to an impasse and thus invites a higher end– so Euclid’s rhetoric leads as well to an end beyond the familiar realm of figure. I propose we should identify this further mode, having to do with the matters of plot and character, as Euclid’s poetic. It will be the topic of a separate posting, to follow soon on the heels of this one.

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Footnote Concerning Other Geometries

Followers of this website may find it surprising that on the one hand I praise Euclid for his clarity about a three-dimensional world – while on the other, I announce a new expedition on this very website into a world of four dimensions. I even claim that we will be experiencing an intuitive sense of relationships in a four-dimensional world. What sort of contradiction is this?

My own proposition is this: Euclid invokes the power of geometrical intuition, but he does not set bounds to it. We have the ability to keep track of the agreements we make in signing-on to sets of postulate belonging to worlds quite different from Euclid’s: we may very well learn to see in our minds’ eyes rooms with four directions at each corner, or left shoes turning readily into right – powers of the visual imagination we have not yet learned to use. This website will post images from this project as unfolds. Stay tuned, and feel free to share any comments you may have

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.