Related papers: Two Layer 3D Floor Planning
A planar graph can be embedded in a piecewise linear manifold, and the lattice on each linear piece can be colored with 3-coloring. If a planar graph can be colored with multiple 3-coloring, i.e. coloring the graph in pieces with different…
The aim of this paper is to derive explicit formulas for two distinct values. The first is the total number of symmetric peaks in a set partition of $[n]$ with exactly $k$ blocks, and the second one is the total number of non-symmetric…
We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections…
Motivated by frequency assignment in office blocks, we study the chromatic number of the adjacency graph of $3$-dimensional parallelepiped arrangements. In the case each parallelepiped is within one floor, a direct application of the…
Which polygons admit two (or more) distinct lattice tilings of the plane? We call such polygons double tiles. It is well-known that a lattice tiling is always combinatorially isomorphic either to a grid of squares or to a grid of regular…
Nowadays three dimension (3D) architectural visualisation has become a powerful tool in the conceptualisation, design and presentation of architectural products in the construction industry, providing realistic interaction and walkthrough…
Two-dimensional (2D) materials have been a hot research topic in the last decade, due to novel fundamental physics in the reduced dimension and appealing applications. Systematic discovery of functional 2D materials has been the focus of…
Optimal multi-layer device design requires consideration of fabrication uncertainties associated with inter-layer alignment and conformal layering. We present layer-restricted topology optimization (TO), a novel technique which mitigates…
We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to $C_k$ for some $k$) instead of cycles (graphs with all degrees even). We give an…
One of the leading quantum computing architectures is based on the two-dimensional (2D) surface code. This code has many advantageous properties such as a high error threshold and a planar layout of physical qubits where each physical qubit…
Multi-level evolution is a bottom-up robotic design paradigm which decomposes the design problem into layered sub-tasks that involve concurrent search for appropriate materials, component geometry and overall morphology. Each of the three…
We consider a new kind of straight and shifted plane partitions/Young tableaux --- ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the…
Practical large-scale quantum computation requires both efficient error correction and robust implementation of logical operations. Three-dimensional (3D) color codes are a promising candidate for fault-tolerant quantum computation due to…
We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set…
We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both…
Two-dimensional (2D) layered materials offer intriguing possibilities for novel physics and applications. Before any attempt at exploring the materials space in a systematic fashion, or combining insights from theory, computation and…
This paper proposes a method to design protective foam for packaging 3D objects. Users first load a 3D object and define a block-based design space by setting the block resolution and the size of each block. The system then constructs a…
Accurate information on the number of building floors, or above-ground storeys, is essential for household estimation, utility provision, risk assessment, evacuation planning, and energy modeling. Yet large-scale floor-count data are rarely…
We consider two important aspects in understanding and editing images: modeling regular, program-like texture or patterns in 2D planes, and 3D posing of these planes in the scene. Unlike prior work on image-based program synthesis, which…
While 2D occupancy maps commonly used in mobile robotics enable safe navigation in indoor environments, in order for robots to understand and interact with their environment and its inhabitants representing 3D geometry and semantic…