Tag Archives: Fourth Dimension

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.