Ordered Semiautomatic Rings with Applications to Geometry

TL;DR

This paper explores semiautomatic rings dense in real numbers, enabling automatic geometric operations like representing equilateral triangles and rotations, surpassing standard b-adic rational representations.

Contribution

It introduces a specific semiautomatic ring structure that supports automatic geometric transformations such as equilateral triangle representation and 30-degree rotations.

Findings

Existence of a semiautomatic ring supporting geometric operations

Representation of equilateral triangles within the ring

Implementation of 30-degree rotations

Abstract

The present work looks at semiautomatic rings with automatic addition and comparisons which are dense subrings of the real numbers and asks how these can be used to represent geometric objects such that certain operations and transformations are automatic. The underlying ring has always to be a countable dense subring of the real numbers and additions and comparisons and multiplications with constants need to be automatic. It is shown that the ring can be selected such that equilateral triangles can be represented and rotations by 30 degrees are possible, while the standard representation of the b-adic rationals does not allow this.