Radix Representations using Cubic Bases

William Gilbert

Pure Mathematics Department, University of Waterloo, Ontario, Canada

Radix Representations

Let Z[b] be the ring of integers in an algebraic number field, where b has a minimum polynomial of degree k and norm N. We say that b is the the base for a full radix representation of Z[b], if every element has a unique finite representation as
z = a₀ + a₁b + a₂b² + ... + a_h-1b^h-1
where the coefficients a₀, ... , a_h-1 are non-negative integers in the range {0, 1, 2, ... , N-1}.

For a cubic base, k = 3, and every element of Z[b] can be represented as an integer linear combination of 1, b, and b². View these elements as cubes in 3-space whose axes are labelled by 1, b, and b². In the following figures we sketch those elements of Z[b] representable in the base b, using radix expansions of length at most h. There will be N^h elements in each figure. When the radix length is increased by one, the new figure consists of the old figure plus N-1 more copies of the old figures, shifted by multiples of a_h.

If the base gives a full radix representation, then these figures will expand to eventually fill the whole space, because every element is representable. If the base does not give a full radix representation, then some elements will not be representable and the figures will not fill space.

Base b with b³ + b² + b + 2 = 0

This is a base for a full radix representation with norm 2, so these figures will eventually fill space. When the radix length is increased by one, two copies of the figure fit together exactly, with no overlap, and no gaps. The second figure shown consists of four copies of the first.

Base b with b³ + b² + b - 2 = 0

This is not a base for a full radix representation with norm 2, as it is clear that these figures will not eventually fill the space. The second figure shown consists of two copies of the first.

Mathematica Code

Here is the Mathematica code to produce these figures.

Lego Models

In the days before computer graphics we had to build our own three-dimensional models. Here are Lego models of the above cubic bases.

Lego Model 1 [35kb] Lego Model 2 [44kb]

References

William J. Gilbert, Geometry of radix representations, in the Geometric Vein: The Coxeter Festschrift, edited by C. Davis et al., Springer, New York, 1981, pages 129-139.

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Radix Representations using Cubic Bases

William Gilbert