An Intuitive Approach to the Fourth Dimension

 

On this site, we’ll be developing a number of images taken straight
from the Fourth Dimension.

Our approach will be very simple, with the aim of developing an intuitive sense for the world of 4D. Our figures, like this one–a hypercube:

simply put you at a certain viewing position in 4D space: all you have to do is open your eyes and look! Real 4D hinges will unfold before you.

The advantages of this approach are entrée into the world of the fourth dimension, and simplicity. The disadvantage is fixity: you are seated in one position, infinitely far from the stage on which all the action will take place. Think of our eye as a telescopic lens of infinite power. Great, unobstructed view–but no perspective.

 

To test how this works, we can grow the hypercube whose image we’ve just seen. We begin with a commonplace 3D cube:

 

(3D things, such as this cube, may not look quite right.  That’s because we’re seeing them in this unfamiliar way.)

The edges of our cube delineate the three directions of 3D space. But we’re now in a space of one further direction. The cube now begins to expand in that new direction: 

We’re creating a figure of four dimensions.  Its content will not be space in our usual sense, but something else.  We’re borrowing a Greek word and calling it chora:

 

 

We’ve now produced a complete hypercube. If we remove the lines which marked the stages of is growth, we’ll have an image very much like the one with which our site opened.

There’s a great deal more to be learned from even these simple figures, but we leave the matter at this point for the moment.  Please comment if you find this interesting – or outrageous!  You can comment by going to my blog.

 

We leave with one final thought:
if a cubic foot of water would fill the cube with which we started,
the same water would have no effect whatever in filing the hypercube.
That would require a quadric foot of water of some new kind: quadric water!

All illustrations by Anne Farrell.