Unfolding some classes of polycubes

An unfolding of a polyhedron is a cutting along its surface such that the surface remains connected and it can be flattened to the plane without any overlap. An edge- unfolding is a restricted kind of unfolding, we are only allowed to cut along the edges of the faces of the polyhedron. A polycube is a special case of orthogonal polyhedron formed by glueing several unit cubes together face-to-face. In the case of polycubes, the edges of all cubes are available for cuts in edge-unfolding. We focus on one-layer polycubes and present several algorithms to unfold some classes of them. We show that it is possible to edge-unfold any one-layer polycube with cubic holes, thin horizontal holes and separable rectangular holes. The question of edge-unfolding general one-layer polycubes remains open. We also briefly study some classes of multi-layer polycubes. 1Rozklad mnohostěnu je tvořen řezy jeho povrchu takovými, že rozřezaný povrch je možné rozložit do roviny, aniž by vznikl překryv. Hranový rozklad je omezený typ roz- kladu, ve kterém je povolené řezy vést jen po hranách mnohostěnu. Kostičkový mno- hostěn je speciální druh mnohostěnu, který je tvořen jednotkovými krychlemi slepenými k sobě celými stěnami. V případě kostičkových mnohostěnů můžeme v hranovém roz- kladu řezat po hranách všech jednotkových krychlí. V této práci se zabýváme zejména jednovrstvými kostičkovými mnohostěny a popíšeme několik algoritmů pro rozklad růz- ných speciálních tříd. Ukážeme, že je možné hranově rozložit jednovrstvé krychličkové mnohostěny s krychlovými dírami, tenkými horizontálními dírami a oddělitelnými ob- délníkovými dírami. Otázka hranového rozkladu obecných jednovrstvých krychličkových zůstává otevřena. Také se krátce zabýváme rozklady některých tříd vícevrstvých krych- ličkových mnohostěnů. 1Department of Applied MathematicsKatedra aplikované matematikyMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

Unfolding and Reconstructing Polyhedra

This thesis covers work on two topics: unfolding polyhedra into the plane and reconstructing polyhedra from partial information. For each topic, we describe previous work in the area and present an array of new research and results. Our work on unfolding is motivated by the problem of characterizing precisely when overlaps will occur when a polyhedron is cut along edges and unfolded. By contrast to previous work, we begin by classifying overlaps according to a notion of locality. This classification enables us to focus upon particular types of overlaps, and use the results to construct examples of polyhedra with interesting unfolding properties. The research on unfolding is split into convex and non-convex cases. In the non-convex case, we construct a polyhedron for which every edge unfolding has an overlap, with fewer faces than all previously known examples. We also construct a non-convex polyhedron for which every edge unfolding has a particularly trivial type of overlap. In the convex case, we construct a series of example polyhedra for which every unfolding of various types has an overlap. These examples disprove some existing conjectures regarding algorithms to unfold convex polyhedra without overlaps. The work on reconstruction is centered around analyzing the computational complexity of a number of reconstruction questions. We consider two classes of reconstruction problems. The first problem is as follows: given a collection of edges in space, determine whether they can be rearranged by translation only to form a polygon or polyhedron. We consider variants of this problem by introducing restrictions like convexity, orthogonality, and non-degeneracy. All of these problems are NP-complete, though some are proved to be only weakly NP-complete. We then consider a second, more classical problem: given a collection of edges in space, determine whether they can be rearranged by translation and/or rotation to form a polygon or polyhedron. This problem is NP-complete for orthogonal polygons, but polynomial algorithms exist for non-orthogonal polygons. For polyhedra, it is shown that if degeneracies are allowed then the problem is NP-hard, but the complexity is still unknown for non-degenerate polyhedra