Seven digits for the complex plane

$$\beta = 2 - \omega, \quad N(\beta) = 7, \quad D = \{0, 1, 2, 3, 4, 5, 6\}$$

Every Eisenstein integer has a unique finite representation in digits $D$ to base $\beta$.

Full paragraph (abstract translation)

Knuth (1960), then still a high-school student, observed that an imaginary number can be used as the base of a positional numeral system — the quater-imaginary base $2i$ for Gaussian integers. The same idea on the Eisenstein lattice gives rise to something even more elegant.

Take $\beta = 2 - \omega$; then

$$N(\beta) = 2^2 - 2 \cdot 1 + 1^2 = 7,$$

so there are exactly seven residue classes modulo $\beta$, and the digit set is $D = \{0, 1, 2, 3, 4, 5, 6\}$. Every Eisenstein integer admits a unique finite representation:

$$z = d_n \cdot \beta^n + d_{n-1} \cdot \beta^{n-1} + \dots + d_0, \qquad d_k \in D. \qquad\text{(1)}$$

Addition, multiplication, and carry become geometric operations on the hexagonal lattice. The talk shows complete addition and multiplication tables for $D$ and a worked example of long addition with hexagonal carry — an operation accessible to any child who can do long addition in base ten.

Digit palette (solid version)

What deserves special attention

• The number $7$ is $|B|^2$ for the base $\beta$, which means that this is exactly how many digits must belong to any self-consistent numeral system on $\mathbb{Z}[\omega]$. Not a designer's choice, but an arithmetic necessity.
• The carry rule is fully determined by the equation $\omega^2 + \omega + 1 = 0$ — it can be drawn as a triangle of three hexagons that "fold up" into zero.
• This septenary system is the direct arithmetic predecessor of the GQ128 format (slide 8).

Reference

• Knuth, D. E. (1960) An imaginary number system. Communications of the ACM, 3(4), 245–247. DOI:10.1145/367177.367233. dl.acm.org