# Are all points on the shore of Gosper Island either transcendental or Eisenstein rational?

I'm working on a program which calculates in the flowsnake base (2.5-√-0.75, with cyclotomic digits) and I've come up with some observations and questions about this base representation. The program also computes the easternmost point on Gosper Island. It doesn't yet output it in flowsnake base, but it'll look something like 0.11166655544443332221116665554443333..., where the average length of a run is a transcendental number close to 3.140275. I'm using seven digits, where 0 and 1 mean themselves, but 2 means 0.5+√-0.75, 4 means -1, and so on.

In this base, some numbers have two representations and some have three. The ones with three representations are all 1/3 the sum of three numbers with finite representation, and are therefore Eisenstein rational (the ratio of two Eisenstein integers) and algebraic. Numbers with two representations can be algebraic (e.g. 1/2 is both 0.165432165432... and 1.432165432165...). All numbers on the shore of Gosper Island have two or three representations. If the representation repeats, the number is Eisenstein rational, and therefore algebraic. If the representation does not repeat, such as the easternmost point, is the number necessarily transcendental?

An algebraic number isn't necessarily an Eisenstein rational. For example, i is not Eisenstein rational (and contrariwise, ω, which is represented as 3 in this base, isn't Gaussian rational), and it's representation is the nonrepeating 3.1612256200055622045453655120223365.... A rational number can have only one representation, such as 1/8, which is 0.062602420464... (it has infinitely many zeros, so only one representation).

ETA: The easternmost point, expressed in flowsnake base, is 0.166655544433332221116665554443332222111666... and 1.422311266155564453342231126615564445334223.... These numbers are equal, just as 0.999... and 1.000... in decimal are. The point is 0.5749186263504636+0.070524523454486195i in decimal, within double precision, though I had to change ...636 to ...634 to get the representation starting with 0.1666.

• For a better understanding of the term Gosper Island. See as well this Feb 11, 2021 at 22:44
• I don't know why flowsnakes would be dangerous, unless they're venomous, but I mean the flowsnake, not the snowflake. Flowsnakes inhabit Gosper Island. The snowflake, when talking about fractals, is the Koch curve, which is a 6-fold radially symmetric fractal like Gosper Island, but otherwise quite different. Feb 11, 2021 at 22:48
• I have understood it when I saw the Wolfram article; please note that I have erased my comment. But the point is that you assume that readers are familiar with concepts and terms (like flowsnakes, interesting pun by itself) or cyclotomic digits that in fact are not evident, and deserve to be (re)called ... Feb 11, 2021 at 22:52
• The program is in C++; I haven't released it yet. The expansions in the flowsnake base use the digits 0 through 6, but they aren't in base 7; they're in a base a complex number which is the scale factor from one flowsnake to the next larger flowsnake (which is traversed in the opposite direction). The absolute value of this complex number is √7. Feb 12, 2021 at 13:01
• In the paper on bridgesmathart, if the arrow labeled 4 is the number 2, then the arrow labeled 7 is the base I'm using. Feb 12, 2021 at 13:10