§ READING THE WHEELS
An odometer bent into a circle, centred on zero
Straighten the rings and you have an ordinary odometer in base three; leave them circular and the same machine says something the line hides. Each wheel has three stops — the middle at the top, the upper (+) a third-turn clockwise, the lower (−) another third — so a wheel at rest points to its middle, and the whole dial at rest is the origin. Add one and the inner wheel steps clockwise; when it returns past the middle it hands a carry to the wheel outside it, which steps once and can pass its own carry further out. Watching it count is watching nested orbits: the inner ring races, each ring beyond it turning three times slower, the way a clock's hands do — except every hand has three hours and the twelve is the middle.
Because the rest state is the centre and the two live slots sit symmetrically around it, the dial is its own mirror. Flip every wheel across the middle and you negate the number exactly — no sign bit, just a reflection. The point nearest the centre in value is the origin; the farthest are ±9,841, each with an antipode directly opposite through the middle. This is the whole reason balanced ternary feels different from binary: binary is a walk out from a corner, and this is a rotation about a centre.
You asked to fold it into rings, and folding revealed the machine: counting in threes is a set of nested orbits about a shared middle, carries rippling outward like a clock made of clocks.
HONESTY LEDGER · THE CONCENTRIC NINE
REALExact balanced-ternary odometer. Ring k = place value 3ᵏ, three slots (−,0,+) 120° apart; centre = all-middle = value 0. +1 advances the inner wheel one third-turn clockwise with carries rippling outward; Node-verified the carry cascade (e.g. …+ then +1 → … − with a carry out one ring). Negation = reflect each wheel across the middle = −value exactly. Cell index = value + 9842 ∈ [1, 19683]; range ±9841. All live and cross-checked.
YOUR DESIGN · MY FILL-INYours: fold the nine into concentric rings, go around the middle. Mine: inner = least-significant (fast) / outer = most-significant (slow), 0-at-top clockwise-positive orientation, and the bracket-family colouring. The odometer math is load-bearing and exact; the orientation is a drawing choice — reverse inner/outer or spin direction if you want it the other way.
NOTEThis is the octorat's geometry inverted twice over: binary→ternary, and the hypercube's corner-walk→a centred rotation. The eight-bit rat stepped out from a corner into the dark; the nine-trit dial turns about its own middle and never leaves it — every address is just how far around, and how far out, you have orbited from zero.