In another post I mentioned the Mindseekers, which was a sect that sought to create artificial sophonts rather than seek other minds among the stars. As they began experimenting with electronic computers, they settled on balanced ternary as the number system of choice rather than binary. This choice was based on some vagaries of yinrih neurology they sought to emulate.
Balanced ternary has three digits, -1, 0, and +1. You can represent any signed integer with these three digits alone. The sign of the number is the sign of the highest-order digit. Here are a few examples using T as -1:
TT = -1*3^1-1*3^0 = -3-1 = -4
1T = 1*3^1-1*3^0 = 3-1 = 2
10 = 1*3^1 + 0*3^0 = 3
You can reverse the sign of the number just by flipping +1's to -1's and vice versa.
11 = 1*3^1+1*3^0 = 3+1 = 4
T1 = -1*3^1+1*3^0 = -3+1 = -2
T0 = -1*3^1+0*3^0 = -3
For reasons unknown, perhaps aesthetics, perhaps for some deeper spiritual reason, the Mindseekers often represented balanced ternary numbers using two-dimensional paths. The rules for drawing such a path are simple.
- Pick a starting point
- Draw a line in any of the four cardinal directions.
- You may turn 90 degrees clockwise or counterclockwise, drawing a dot at each turn.
- You may also go straight ahead, drawing a dot dividing a line segment.
- You may not make a 180 degree turn, you must make two consecutive 90 degree turns.
"But how does this represent balanced ternary numbers?" I hear you cry. Well, a counterclockwise turn represents a +1, a clockwise turn -1, and a straight is 0. The number is big endian, meaning the first turn is the highest-order digit, the second is the second-highest, etc. Neither initial starting direction nor line length are significant, so those choices can be left to aesthetics or other constraints.
This method has the pleasantly symmetrical property that mirroring the path horizontally results in an integer of the opposite sign.
Mindseekers would often hide such paths in odd places, like the brickwork on the sides of buildings, embedded in the abstract pattern of a tapestry, and so on. It's thought that certain numbers held symbolic value.
These paths became popular as a way to hide messages in plain sight even long after the Mindseekers faded into obscurity.
I think my main question would be, how is the start and end of these paths indicated? As that could drastically change the meaning, right?
In the picture at least, the start is indicated by a dovetail (not sure if that's the right word), and the endpoint with an X. I suppose in practice it could be anything, or be left deliberately ambiguous.
Starting point does matter. If the intended number has no trailing zeros, traversing the path backwards also produces a number with the same magnitude but opposite sign. If there are trailing zeros they become insignificant leading zeros when parsed in reverse. Rotating the path does not affect the number, but flipping either horizontally or vertically will flip the sign of the number.