Validasi Estimasi Batas Atas Luas Sofa dan Perhitungan Luas Sofa Gerver serta Analisis Mekanisme Pergerakan Sofa Hammersley

Abstract

The Moving Sofa Problem concerns the planar shape of maximum area that can be moved around a right-angled corner in a two-dimensional hallway of unit width. The objectives of this study are (1) to validate the upper bound estimate of the sofa area obtained from the intersection of the straight corridor and the right-angled corridor models, (2) to analyze the movement mechanism of the Hammersley sofa through rotation paths and contact paths, and (3) to provide details of the Gerver sofa area calculation for numerical validation. The methods used include analysis of the function A(u, θ), which is the upper bound value function of the sofa area obtained from the area of the intersection of the straight corridor and the right-angled corridor, the application of the concepts of rotation paths, contact points, and contact paths to prove that the Hammersley sofa construction can pass through the right-angled corridor, and the calculation of the area using Green’s Theorem to validate the Gerver sofa area. The main results show that (1) the minimum upper bound of the function A(u, θ) reaches two times the square root of two under certain conditions, (2) the rotation path used proves that the Hammersley sofa satisfies the definition of a shape that can pass through a right-angled corridor, and (3) calculations using Green’s Theorem yield an area of approximately 2.2195 area units. The findings of this study clarify the geometric construction elements of the Hammersley and Gerver sofas, and provide validation details that have rarely been fully described before.