Foreword

A valid question to ask upon encountering the title of this section, Visualizing 4-space, would be one intending to find out just what "4-space" is. This, of course, is the very reason that the article was written - to 'explain' 4-space. But what of a simple, introductory definition for those who have perhaps never even heard the term?

Directly below you will find illustrations of lines coming together at right angles. First is a line by itself, then 2 lines lying at right angles to each other, and finally 3 lines lying at right angles to each other. The addition of each new line demonstrates a new degree to which the lines can position themselves relative to each other. For example, each new line, to be 'added on', must 'relate' to all of the previous lines in a way that had never been done before. The spatial environment - the 'empty space' within which the lines lie - determines the extent to which they can position themselves. 4-space could be thought of as a spatial environment / 'empty space' within which 4 lines can come together at a point - (as with the illustrations below) - and at the same time all lie at right angles to each other. The fourth line would relate to the first 3 in a way that had never been done before - it would meet all 3 lines at a right angle. It is impossible - within 3-dimensional space as we know it - for 4 lines to come together at right angles to each other.