Details of the Einstein and Spectre Monotiles

Having read an article in Quanta about the Einstein Tile, I asked Gemini to do some deep research into the topic. It produced a detailed report without a single diagram which was ridiculous and highlights the deficiencies of this sort of report. Lesson learned. Even the Quanta article is sorely lacking in visual support. 

I opted instead to ask Gemini for a detailed diagram of the original Einstein tile. Figure 1 shows what it came up with.

Figure 1

The tile is made up of eight "deltoidal trihexagonal kites", one of which is shown on far right of Figure 1. However, to tile the plane the reflection of the hat is required as shown on the bottom right of Figure 1. The "Spectre" or "Vampire Einstein" monotile fixes this problem (see Figure 2).

Figure 2

I made the following observation:

The Spectre tile is clearly not a polygon. Wasn't the aim to fill 2-D space with a suitable polygon. Does this mean that there is no vampire monotile that is a polygon?

Here was Gemini's response:

Does a truly "polygonal" vampire monotile exist?

Actually, yes! Although curves are the easiest way to visualize the "Spectre's" chiral enforcement, you can replace those curves with polygonal (jagged) features.

If instead of smooth curves you use jagged, puzzle-like zig-zagged sub-edges (effectively turning the 14-sided tile into a vastly more complex polygon with many tiny, straight-line notches), you can strictly forbid reflection without using a single curve. However, doing so requires creating a heavily complex, jagged polygon with dozens of vertices, which makes it much harder to print, analyze, or illustrate clearly on a grid.

For simplicity, mathematicians and illustrators typically show the smooth-curved Spectre as the definitive "vampire monotile" because it naturally forces its own chirality without the visual clutter of dozens of tiny polygonal zig-zags.