I have now checked through the whole top row of the expansion, and this does seem to be what’s going on.
Each glyph represents a 4 digit base 4 number.
The digits are read in the order top left, bottom left, top right, bottom right.
An empty space off the central pole is always read as a zero.
A loop that comes from the centre of the central pole and joins to the top or bottom of the pole is always a three.
The horizontal bar and the curve coming from the centre of the bar (not re-joining, to distinguish it from the loop) vary in meaning depending on whether they are at the top or the bottom. At the top, the horizontal bar is two and the curve is one; at the bottom, the horizontal bar is one and the curve is two.
So, at the top we have a glyph representing 0003_4 = 3 (followed by a point).
Then the first few along the top row can be read off as:
Having the strokes for one and two adopt inverse meanings top and bottom adds a little wrinkle to this as a puzzle, but otherwise as a design decision if one were semi-seriously thinking of this as a way of encoding information it seems to unnecessarily add to the thought needed to decode.
Also on design, I’m torn about whether the ordering of the parts is better or worse than top then bottom. The current design keeps the most significant parts on the left, which is mostly what we expect. But the overall presentation of the number is read across the page, left to right, so there could also be a case for having the order top-left, top-right, bottom-left, bottom-right.
The top half of a 2 might kinda look like the curve shape, and the bottom stroke of a 2 looks like a horizontal bar. So if there were partial characters hanging from the central pole, they might look a bit like those...
But… If it’s that, the curve probably only works on the bottom right anyway. So if you’re willing to mirror it for the bottom left, why not mirror to the top too?
And… That doesn’t really explain the 1 parts anyway. They’re just using “whichever part of a 2 isn’t being used for 2”.
Aha. For each side of the pole, you can write the binary representation of 4 bits vertically, and where there’s a 1 you have a line joining it. The middle two bits both go to the middle of the pole, so they have to curve off upward or downward to indicate which they are.
So 2 is 0010, and you have no lines in the top half and one line representing the bottom of the middle, so it curves downward.
Whereas 4 is 0100, so it has the upward-curving middle-connecting line, and none in the bottom half.
I have now checked through the whole top row of the expansion, and this does seem to be what’s going on.
Each glyph represents a 4 digit base 4 number.
The digits are read in the order top left, bottom left, top right, bottom right.
An empty space off the central pole is always read as a zero.
A loop that comes from the centre of the central pole and joins to the top or bottom of the pole is always a three.
The horizontal bar and the curve coming from the centre of the bar (not re-joining, to distinguish it from the loop) vary in meaning depending on whether they are at the top or the bottom. At the top, the horizontal bar is two and the curve is one; at the bottom, the horizontal bar is one and the curve is two.
So, at the top we have a glyph representing 0003_4 = 3 (followed by a point).
Then the first few along the top row can be read off as:
0210_4 = 36_10
0333_4 = 63_10
1222_4 = 106_10
2020_4 = 136_10
And Wolfram Alpha can tell us that pi in base 256 starts:
3.36:63:106:136:133:163:8:211:19:25:138:46:3..._256
Having the strokes for one and two adopt inverse meanings top and bottom adds a little wrinkle to this as a puzzle, but otherwise as a design decision if one were semi-seriously thinking of this as a way of encoding information it seems to unnecessarily add to the thought needed to decode.
Also on design, I’m torn about whether the ordering of the parts is better or worse than top then bottom. The current design keeps the most significant parts on the left, which is mostly what we expect. But the overall presentation of the number is read across the page, left to right, so there could also be a case for having the order top-left, top-right, bottom-left, bottom-right.
There’s a reason for the “wrinkle” :)
The top half of a 2 might kinda look like the curve shape, and the bottom stroke of a 2 looks like a horizontal bar. So if there were partial characters hanging from the central pole, they might look a bit like those...
But… If it’s that, the curve probably only works on the bottom right anyway. So if you’re willing to mirror it for the bottom left, why not mirror to the top too?
And… That doesn’t really explain the 1 parts anyway. They’re just using “whichever part of a 2 isn’t being used for 2”.
So I guess this isn’t a complete explanation.
Nope, not based on the shapes of numerals.
Hint: are you sure it’s base 4?
Aha. For each side of the pole, you can write the binary representation of 4 bits vertically, and where there’s a 1 you have a line joining it. The middle two bits both go to the middle of the pole, so they have to curve off upward or downward to indicate which they are.
So 2 is 0010, and you have no lines in the top half and one line representing the bottom of the middle, so it curves downward.
Whereas 4 is 0100, so it has the upward-curving middle-connecting line, and none in the bottom half.