A Note on Objects Built From Bricks without Corners

TL;DR

This paper presents a new genus-3 brick polyhedron with no corners, advancing the understanding of brick constructions without corners and narrowing the gap from the previously known genus-13 example.

Contribution

It introduces a genus-3 brick polyhedron without corners, providing a significant reduction from the previously known genus-13 example and addressing a question about genus-0 cases.

Findings

Constructed a genus-3 brick polyhedron with no corners

Narrowed the genus gap from 13 to 3

Provided insights into brick polyhedra without corners

Abstract

We report a small advance on a question raised by Robertson, Schweitzer, and Wagon in [RSW02]. They constructed a genus-13 polyhedron built from bricks without corners, and asked whether every genus-0 such polyhedron must have a corner. A brick is a parallelopiped, and a corner is a brick of degree three or less in the brick graph. We describe a genus-3 polyhedron built from bricks with no corner, narrowing the genus gap.