Folding Polyominoes into (Poly)Cubes

We study the problem of folding a polyomino $P$ into a polycube $Q$, allowing faces of $Q$ to be covered multiple times. First, we define a variety of folding models according to whether the folds (a) must be along grid lines of $P$ or can divide squares in half (diagonally and/or orthogonally), (b) must be mountain or can be both mountain and valley, (c) can remain flat (forming an angle of $180^\circ$), and (d) must lie on just the polycube surface or can have interior faces as well. Second, we give all the inclusion relations among all models that fold on the grid lines of $P$. Third, we characterize all polyominoes that can fold into a unit cube, in some models. Fourth, we give a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models. Finally, we consider the triangular version of the problem, characterizing which polyiamonds fold into a regular tetrahedron.Comment: 30 pages, 19 figures, full version of extended abstract that appeared in CCCG 2015. (Change over previous version: Fixed a missing reference.

Steric Constraints as a Global Regulation of Growing Leaf Shape

Shape is one of the important characteristics for the structures observed in living organisms. Whereas biologists have proposed models where the shape is controlled on a molecular level [1], physicists, following Turing [2] and d'Arcy Thomson [3], have developed theories where patterns arise spontaneously [4]. Here, we propose a volume constraint that restricts the possible shapes of leaves. Focusing on palmate leaves, the central observation is that developing leaves first grow folded inside a bud, limited by the previous and subsequent leaves. We show that growing folded in this small volume controls globally the leaf development. This induces a direct relationship between the way it was folded and the final unfolded shape of the leaf. These dependencies can be approximated as simple geometrical relationships that we confirm on both folded embryonic and unfolded mature leaves. We find that independently of their position in the phylogenetic tree, these relationships work for folded species, but do not work for non-folded species. This steric constraint is a simple way to impose a global regulation for the leaf growth. Such steric regulation should be more general and considered as a new simple means of global regulation.Comment: 6 pages 4 figures, Supplementary materials (8 pages, 7 figures

A precessing accretion disc in the intermediate polar XY Ari?

XY Ari is the only intermediate polar to show deep X-ray eclipses of its white dwarf. Previously published observations with Ginga and Chandra have also revealed a broad X-ray orbital modulation, roughly antiphased with the eclipse, and presumed to be due to absorption in an extended structure near the edge of an accretion disc. The X-ray pulse profile is generally seen to be double-peaked, although a single-peaked pulse was seen by RXTE during an outburst in 1996.We intended to investigate the cause of the broad orbital modulation in XY Ari to better understand the accretion flow in this system and other intermediate polars. We observed XY Ari with RXTE and analysed previously unpublished archival observations of the system made with ASCA and XMM-Newton. These observations comprise six separate visits and span about ten years. The various X-ray observations show that the broad orbital modulation varies in phase and significance, then ultimately disappears entirely in the last few years. In addition, the X-ray pulse profile shows variations in depth and shape, and in the recent RXTE observations displays no evidence for changes in hardness ratio. The observed changes indicates that both the pulse profile and the orbital modulation are solely due to geometrical effects at the time of the RXTE observations, rather than phase-dependent variations in photoelectric absorption as seen previously. We suggest that this is evidence for a precessing, tilted accretion disc in this system. The precession of the disc moves structures out of our line of sight both at its outer edge (changing the orbital modulation) and at its inner edge where the accretion curtains are anchored (changing the pulse profile).Comment: Accepted for publication in Astronomy & Astrophysic

Carbon--The First Frontier of Information Processing

Information is often encoded as an aperiodic chain of building blocks. Modern digital computers use bits as the building blocks, but in general the choice of building blocks depends on the nature of the information to be encoded. What are the optimal building blocks to encode structural information? This can be analysed by substituting the operations of addition and multiplication of conventional arithmetic with translation and rotation. It is argued that at the molecular level, the best component for encoding discretised structural information is carbon. Living organisms discovered this billions of years ago, and used carbon as the back-bone for constructing proteins that function according to their structure. Structural analysis of polypeptide chains shows that an efficient and versatile structural language of 20 building blocks is needed to implement all the tasks carried out by proteins. Properties of amino acids indicate that the present triplet genetic code was preceded by a more primitive one, coding for 10 amino acids using two nucleotide bases.Comment: (v1) 9 pages, revtex. (v2) 10 pages. Several arguments expanded to make the article self-contained and to increase clarity. Applications pointed out. (v3) 11 pages. Published version. Well-known properties of proteins shifted to an appendix. Reformatted according to journal styl

Home Made Rugs

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A model for cyclotron resonance scattering features

(abbreviated version of the abstract) We study the physics of cyclotron line formation in the high-energy spectra of accreting X-ray pulsars using Monte Carlo methods, assuming that the line-forming region is a low-density electron plasma in a sub-critical magnetic field. We investigate the dependence of the shape of the fundamental line on angle, geometry, optical depth and temperature. We also discuss variations of the line ratios for non-uniform magnetic fields. These numerical predictions for the line profiles are linked to results from observational data analysis using an XSPEC model based on the Monte Carlo simulations. We apply this model to observational data from RXTE and INTEGRAL. The predicted strong emission wings of the fundamental cyclotron feature are not found in observational data, hinting at a bottom illuminated slab geometry for line formation.Comment: 16 pages, 15 figures, Astron. Astrophys. (in press