Moving rectangular sofas in planar and spatial corridors

TL;DR

This paper characterizes which rectangles can navigate through eight specific corridor shapes in both 2D and 3D, identifying those with maximum area and volume that can move around corners.

Contribution

It provides a complete characterization of movable rectangles in eight corridor types and determines the maximum-area and maximum-volume shapes for these movements.

Findings

Identified rectangles that can move around corners in eight corridor types.

Determined maximum-area rectangles for planar corridors.

Determined maximum-volume rectangular parallelepipeds for spatial corridors.

Abstract

We consider eight natural planar corridors, including the standard $\mathrm{L}$-shaped one, and characterize the rectangles that can move around their corners. As a bi-product we describe completely the corresponding rectangles with maximum area, as well as the rectangular parallelepipeds with maximum volume that can move around the corners of the spatial analogues of the considered eight planar corridors.