Can we "seamlessly" divide a polygon?

TL;DR

This paper revises analytic geometry to better align with ancient Greek mathematical texts, allowing for more complex point-figure relationships and providing a new perspective on polygon division consistent with historical interpretations.

Contribution

It introduces an alternative geometric model that accommodates boundary-inclusive figures, bridging modern analytic geometry with ancient Greek mathematical concepts.

Findings

The new model is equivalent to conventional analytic geometry in a formal sense.

It better reflects the original Greek mathematical ideas about figures and boundaries.

The model offers advantages in historical and conceptual understanding of geometric division.

Abstract

While the contents of Euclid's Elements are well-known these days, some characters of the original text have been overlooked due to interpretation by modern mathematical languages. The lens of modern mathematics once anachronistically misled researchers of the history of mathematics, and this shows that the classic text itself contains ideas that have not been completely accurately translated. This article concentrates on the division of geometric figures(line, polygon, polyhedron, etc.). In modern analytic geometry, the dissection of geometric figures excludes their boundaries, whereas ancient Greek mathematical texts do not contain such conditions. For a model that fits the latter, this article renovates analytic geometry and suggests alternative definitions of lines, polygons, angles, etc. Roughly speaking, contrary to the coordinate space matching position to point one-to-one, each position in the alternative definition is more complex and can be occupied by more than one figure. The new model turns out to be equivalent to the conventional analytic geometry in a sense. By formulating several statements in Elements with the new model we can discover its advantage over the old model, especially the concordance with a view of the history of mathematics.