A giant blue eye holds the title for this chapter, the 4D Eye. Elk, looking serious, says, So, if we're going to talk about geometry, let's just talk about geometry. Because this is a comic, and we're all about visualizations here, I'm going to use the omics form to help you reach some truths about geometry. I found a super-rare book called Hypergraphics that I'll be borrowing from for this section... the book Hypergraphics is floating next to this speech bubble, and if you click on it, the link leads to the volume on google books.

Don't worry, we'll start slowly, a speech bubble reads. In a square delineates the basics of dimensions. A dot marked 0D is labelled point, two dots connected marked 1D is labelled line, four dots connected in a square shape marked 2D is labelled plane, and 8 dots connecting 6 planes marked 3D is labelled cube! The narration reads: Just as when you pull a point in one dimensions to get a line, pull a line in two dimensions to get a plane, and you pull a plane in three dimensions to get a cube... a hypercube is a cube pulled into four dimensions. Imagine the 3D sides of the hypercube are being projected outwards into the fourth dimension.

Elk returns, looking jovial, and says: Pretty forkin' cool, right? What's even cooler is that from this, we are getting an understanding of higher dimensions utilizing lower ones! That means, you can represent any 4D projection in 3 dimensions analogically!

Next, Elk's back is to the viewer and a square that reads click for video is being projected from behind him. He continues, Michael A. Noll created a computer-generated movie of a hypercube rotating, taking a two-dimensional image and adding a time-based element, movement, to create a 3-dimensional analogue of a four-dimensional object. (from page 154 of Hypergraphics)

Scott E. Kim, in an Impossible Four-Dimensional Illusion, helped me visualize the step from 3 to 4 dimensions. Elk is shooting turquoise laser beams out of his eyes at a 3D tetrahedron inside a cube projecting onto a 3D see-through tetrahedron. This diagram is marked as The 4-D Eye, modified, from page 213, figure 26. Elk continues, Since a 4D object casts a 3D shadow, a 4D subject must cast a 3D shadow as well. So if the 4D hyper-subject is the analogue of the static 3D subject, what is the 2D analogue of said 3D subject? A diagram labelled 4D hyper-subject?! shows a subject at t=1+ as a stick figure overlapping numerous times, then subject at t=x which is just a single stick figure. Elk continues with the 3D Eye, modified, taken from p208 fig 21. He is again shooting turquoise laser beams from his eyes, now shooting through the same 3D tetrahedron onto a 2D tetrahedron in black and white on a piece of paper, what we can clearly see as what Elk would call the 3D tetrahedron's shadow. The shadow is labelled as 2D image, and the paper is labelled as 2d plane, or, frame, in italics. Elk asks in a concluding speech bubble, What do you get if you take away that 3rd dimension, in this case, width or thickness in space? Then what does the subject look like?

A narration bubble reads, Carl Sagan illustrates this in an episode of Cosmos entitled the Edge of Forever, which is hyperlinked to the episode on YouTube. To the right of this bubble we see Carl Sagan (astrophysicist, exobiologist, author, and public figure) wearing his signature maroon turtleneck. He is seated at a drafting table holding an apple above it. On the drafting table are little cutouts of shapes, like triangles and squares, of different colors.

The narration continues: Using an analogy drawn from Flatland (linked here) by Edmund Abbott, Sagan shows how to create a 2D image (a stamp) from a 3D object (an apple). Pictured is Sagan placing an apple whose base has been dipped in ink onto some graph paper, leaving an imprint of the base of the apple. The narration continues, Each stamp represents a 2D slice of the apple. Pictured is an apple on its side sliced into thin rounds. What does a 3D stamp of a 4D object look like? What is a 3D slice of 4D spacetime?

©Elk Paauw 2025