Topic
Polyform
About: Polyform is a research topic. Over the lifetime, 10 publications have been published within this topic receiving 36 citations.
Papers
TL;DR: It is shown that $\lfloor\frac{m+1}{3} \rfloor$ point guards are always sufficient and sometimes necessary to cover an m-polyomino, possibly with holes, and a strictly lower guard number is given by the art gallery theorem for orthogonal polygons.
Abstract: We explore the art gallery problem for the special case that the domain (gallery) P is an m-polyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P. In particular, we show that $\lfloor\frac{m+1}{3} \rfloor$ point guards are always sufficient and sometimes necessary to cover an m-polyomino, possibly with holes. When $m \leq\frac{3n}{4} - 4$, the sufficiency condition yields a strictly lower guard number than $\lfloor\frac{n}{4}\rfloor$, given by the art gallery theorem for orthogonal polygons.
20 citations
17 Jul 2017
TL;DR: Recursion formulas giving the maximal number of leaves in tree-like polyominoes and polycubes of size n are presented and proved, and a combinatorial algorithm is relied on that enumerates rooted directed trees that are abundant.
Abstract: We present and prove recursive formulas giving the maximal number of leaves in tree-like polyominoes and polycubes of size n. We call these tree-like polyforms fully leafed. The proof relies on a combinatorial algorithm that enumerates rooted directed trees that we call abundant. We also show how to produce a family of fully leafed tree-like polyominoes and a family of fully leafed tree-like polycubes for each possible size, thus gaining insight into their geometric characteristics.
9 citations
TL;DR: In two dimensions, a polyform is a finite set of edge-connected cells on a square, triangular, or hexagonal grid and a layer is the set of grid cells that are vertex-adjacent to the poly form and not part of the polyform.
Abstract: In two dimensions, a polyform is a finite set of edge-connected cells on a square, triangular, or hexagonal grid. A layer is the set of grid cells that are vertex-adjacent to the polyform and not part of the polyform. A bumped-body polyform has two parts: a body and a bump. Adding a layer to a bumped-body polyform with minimum perimeter constructs a bumped-body polyform with min perimeter; the triangle case requires additional assumptions. A similar result holds for 3D polyominos with minimum area.
4 citations
TL;DR: A procedure which utilizes POLYFORM thermoplastic to obtain a negative mold of the patient instead of the traditional plaster bandage or dental impression gel to provide a close fit to a particular patient's surface contours is developed.
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 2 |
| 2019 | 1 |
| 2017 | 1 |
| 2012 | 1 |
| 2011 | 2 |
| 2006 | 1 |