This model resembles the two-dimensional rep-4 figure known as the fairy boot, in German "Elfenstiefel".
You can put together four of them to build a larger scaled object which is similar to the original fairy boot. Three of them can be used to form a equilateral triangle (with a small hole in the middle because there are not chained infinitely many elements here).
No, this is not related to square dissections. And no, this is not a special high frequency horn antenna.