"I know. But we—you have proved it exists. It has been proven mathematically."
Dr. Wooden looked dubious. Jonathan picked up a pencil and pressed down with the point on a slip of graph paper.
"That black mark, that dot, is one-dimensional. Extend a line from that point to another dot. The line is also one-dimensional. Let us put the pencil on the line, supersede the line with the pencil. Since the pencil has three dimensions, so does the line—for the pencil is the line.
"Suppose an n-dimensional object. Supersede the pencil with the n-dimensional object and we have an n-dimensional line. It is an n-dimensional space of n-dimensional points, instead of our original definition of a line as a single dimensioned space of points set in a row.
"Ordinary space is called three-dimensional because it is occupied by three-dimensional things. Planes, for instance. But if we speak of lines of spheres or circles, we can easily step into the realm of n-dimensionality.
"The drawback is that we can't see it. We can't envision n-dimensionality.
"Consequently, we have always been intrigued by many-dimensionality because we can't picture it to ourselves. But the calcatryte rays weren't hindered by a lack of imagination. They just zoomed off into an n-dimensional space, and wound up near Neeoorna. They were lines, remember, straight lines. And lines can be n-dimensional."
Dr. Wooden rubbed his chin and said, "Could be, could be. But how does hyperspace solve your problem?"
"A dot inside a circle can go outside that circle without crossing its circumference. Likewise, I could pass from the inside to the outside of a sphere without going through the surface of a four-dimensional object.
"Those calcatryte rays beamed out from your lab into hyperspace, passing through ordinary space without touching it, and appeared billions of miles away. When I entered the shadows, I followed their course."