Crystal World

Collected below are the notes for an unfinished travel memoir penned by the explorer and cosmic cartographer Helkanoum. I have here attempted to organize them into a coherent narrative, but so much of the material has been lost that we can only guess at the true scope and ambition of Helkanoum’s intended work. In this section of the work, the physics of light in a flat universe organized around the surface of a cubic crystal is discussed.

Continued from … Sections 1, 2, 3

On the Crystal World of Ask-jel-gafar

4.1 Now this flat universe I give the name Ask-jel-gafar, which in my tongue means: the crystal world.

4.2 I boarded myself initially on one planet in this cosmos, populated by a race of flat-beings (I shall call them Gafarrians) who – like me – experience space as a two dimensional construct. I will not bother to relate to you the details of their biology or culture in this section, but it suffices to say that races of a similar build and aptitude exist in other universes.

4.3 Their civilization is in a late phase of space-faring exploration, and the state of their physical sciences is not altogether primitive. However, one problem in the field of optics seemed to me particularly troubling and poorly-explained – the light-skipping issue.

4.4 In their world, light travels in straight lines through vacuum, propagating in all directions from a given point light source. However, the inhabitants of Ask-jel-gafar came upon mysterious local exceptions to this principle as they probed the depths of space with radio-waves. In various places in this cosmos there exist invisible lines across which light from one side cannot travel to the other. Furthermore, as one physically crosses this “light barrier” and travels for some time, the light from the other side of barrier is suddenly visible after a certain distance, although it was absent in the intervening space. Somehow then, light appears to “skip” across certain zones in space. This I depict in figure 1.


4.5 After some careful measurements, a much-celebrated physicist concluded the following:

“Suppose there is a light ray emerging from a point that is a perpendicular distance X away from the line of a light barrier. Now, if this ray of light encounters the light barrier at right angles, then it will disappear and reappear at a distance 2.747 times the distance X, its direction of travel remaining unaltered.”

4.6 Having established this principle, the discipline of physics was content to move on and accept this as an immutable fact of light’s behavior. Textbooks were printed with this simple formula, and schoolchildren made to memorize it.

4.7 Not content with this explanation, I set myself to the task of studying the true nature of these light barriers, and the nature of space more broadly in Ask-jel-gafar.

4.8 Using my cosmic engine, I travelled across the whole extent of the Universe, and counted exactly 12 such light barriers. I also discovered this universe to be closed and of finite area. The light barriers also appear to delimit perfect squares in space. A basic map of this universe is shown below.


4.9 As I pondered over my observations, I came to a single striking conclusion: Ask-jel-gafar is a universe wrapped around the surface of a cubic crystal. Its inhabitants can only sense light if it travels along the two-dimensional surface of this crystal (in which their entire sensory experience is contained). Light itself, however, can propagate in three dimensions, both inside and outside of the cube. The light “barriers” are merely the 12 edges of this cube.


4.10 The phenomena associated with the light barrier are largely explained by (1) the cubic and crystalline nature of this universe and (2) a type of refraction at the surface of this crystal. The law involved is identical to the one attributed to Willebrord Snellius in your world:


4.11 The interior of the crystal is apparently largely homogenous, and has a higher refractive index than the medium external to it. Light races off the edges of this cube, giving the impression that it “disappears” – to two dimensional beings living in the surface . When light is refracted at a 90 angle along the cube’s surface, after tracing a path through the inside of the crystal, it “emerges” into the sensible plane and is now visible to Gafarrians again, but on a different face of the cube. The angle of light-incidence that produces a refractive angle of 90 degrees, is called the critical angle.


The red rays grazing the crystal surface are the only ones that would be visible to gafarrians.

The critical angle is what determines the distance beyond the light barrier which remains unilluminated when one shines a light directly on it. I leave it to the reader as an exercise to calculate what this angle is, based on the law listed in 4.5.

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