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Moving rectangular sofas in planar and spatial corridors

Oleg Mushkarov, Nikolai Nikolov

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.

Moving rectangular sofas in planar and spatial corridors

Abstract

We consider eight natural planar corridors, including the standard -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.

Paper Structure

This paper contains 7 sections, 11 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Moving rectangles in the corridors $\mathcal{C}_{00}$ and $\mathcal{C}_{13}$
  3. Proof of Theorem \ref{['t1']}
  4. Proof of the if part of Theorem \ref{['t1']}
  5. Proof of the only if part of Theorem \ref{['t1']}
  6. Proof of Theorem \ref{['t2']}
  7. Appendix

Key Result

Theorem 1

The following identities hold true: $\mathcal{R}_{00}=\mathcal{R}_{01}=\mathcal{R}_{02}$, $\mathcal{R}_{10}=\mathcal{R}_{12}$, $\mathcal{R}_{11}=\mathcal{R}_{13}$, $\mathcal{R}_{00}=\mathcal{R}_{10}\cup\mathcal{R}_{03}$, $\mathcal{R}_{13}=\mathcal{R}_{10}\cap\mathcal{R}_{03}$. Moreover, a rectangle

Figures (3)

  • Figure 1: The corridors $\mathcal{C}_{ij}$.
  • Figure 2: The corridor $\mathcal{C}_{00}$.
  • Figure 3: The corridor $\mathcal{C}_{03}$.

Theorems & Definitions (15)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Lemma 6
  • Proposition 7
  • Proposition 8
  • Lemma 9
  • proof
  • ...and 5 more