This is my new new base, yeh. Alternative name is folded base-2. If you want to know how this works, look below | v
c2a, @BaseBands, @Number17_Guy [How does this work?]: Folded base-2 is a variant of binary where the positional weights don’t keep doubling endlessly. In normal base-2, the place values go 1, 2, 4, 8, 16, … with each extra digit doubling the previous one. In folded base-2, that exponential growth “folds” back on itself. The pattern of weights is 1, 2, 2, 4, 4, 8, 8, 16, 16, … -- each power of two repeats twice before the next one appears. The result is that “hundreds” (100) has the same value as “tens” (10), and “thousands” (1000) becomes the product of two tens (2 × 2 = 4). When a binary string is read under these folded weights, the value is found by summing the weights of all the 1-bits. For example, the string 1010 has 1s at positions 3 and 1. Their folded weights are 4 and 2, so 1010 = 6. Because weights repeat, many different strings can sum to the same number — for instance, 1100 also equals 6. To avoid duplicates, we define a canonical representation: among all strings that produce the same folded value, we choose the one with the smallest ordinary binary value (i.e., the numerically smallest bit pattern). That ensures every decimal number has exactly one simplest folded-binary form. This “folding” causes growth patterns different from ordinary binary. Every two digits, the effective place value doubles again, so number lengths increase in blocks. The sequence of canonical folded representations looks almost like binary expanding in layers: 1, 10, 11, 110, 111, 1010, 1011, 1110, 1111, … and so on. Despite its irregular look, the system is consistent and deterministic. Conceptually, folded base-2 behaves like a binary system whose magnitude scale pauses and then leaps ahead — a hybrid between positional doubling and modular repetition — creating a compact but cyclic number world where 100 ≠ 4 because of place, but because two folds of 10 multiply together.
