Grid-based Decorative Corners

Abstract I explore a space of geometric, decorative corner designs based on paths through a square grid. I discuss the problems of enumerating corners of a given size efficiently, and exploring them interactively in software. I then impose a higher-level connectivity constraint on corners and discuss the effect of this constraint on the mathematical and aesthetic properties of corner designs

A chiral aperiodic monotile

The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.Comment: 23 pages, 12 figure

A chiral aperiodic monotile

The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat--the equilateral member of the continuum to which it belongs--is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only homochiral non-periodic tilings based on a hierarchical substitution system.Mathematics Subject Classifications: 05B45, 52C20, 05B50Keywords: Tilings, aperiodic order, polyform

An aperiodic monotile

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.Comment: 89 pages, 57 figures; Minor corrections, renamed "fylfot" to "triskelion", added the name "turtle", added references, new H7/H8 rules (Fig 2.11), talk about reflection

Flavor Mediation Delivers Natural SUSY

If supersymmetry (SUSY) solves the hierarchy problem, then naturalness considerations coupled with recent LHC bounds require non-trivial superpartner flavor structures. Such "Natural SUSY" models exhibit a large mass hierarchy between scalars of the third and first two generations as well as degeneracy (or alignment) among the first two generations. In this work, we show how this specific beyond the standard model (SM) flavor structure can be tied directly to SM flavor via "Flavor Mediation". The SM contains an anomaly-free SU(3) flavor symmetry, broken only by Yukawa couplings. By gauging this flavor symmetry in addition to SM gauge symmetries, we can mediate SUSY breaking via (Higgsed) gauge mediation. This automatically delivers a natural SUSY spectrum. Third-generation scalar masses are suppressed due to the dominant breaking of the flavor gauge symmetry in the top direction. More subtly, the first-two-generation scalars remain highly degenerate due to a custodial U(2) symmetry, where the SU(2) factor arises because SU(3) is rank two. This custodial symmetry is broken only at order (m_c/m_t)^2. SUSY gauge coupling unification predictions are preserved, since no new charged matter is introduced, the SM gauge structure is unaltered, and the flavor symmetry treats all matter multiplets equally. Moreover, the uniqueness of the anomaly-free SU(3) flavor group makes possible a number of concrete predictions for the superpartner spectrum.Comment: 17 pages, 7 figures, 2 tables. v2 references added, minor changes to flavor constraints and a little discussion adde

The long-term evolution of the spin, pulse shape, and orbit of the accretion-powered millisecond pulsar SAX J1808.4-3658

We present a 7 yr timing study of the 2.5 ms X-ray pulsar SAX J1808.4-3658, an X-ray transient with a recurrence time of ~2 yr, using data from the Rossi X-ray Timing Explorer covering 4 transient outbursts (1998-2005). We verify that the 401 Hz pulsation traces the spin frequency fundamental and not a harmonic. Substantial pulse shape variability, both stochastic and systematic, was observed during each outburst. Analysis of the systematic pulse shape changes suggests that, as an outburst dims, the X-ray "hot spot" on the pulsar surface drifts longitudinally and a second hot spot may appear. The overall pulse shape variability limits the ability to measure spin frequency evolution within a given X-ray outburst (and calls previous nudot measurements of this source into question), with typical upper limits of |nudot| < 2.5x10^{-14} Hz/s (2 sigma). However, combining data from all the outbursts shows with high (6 sigma) significance that the pulsar is undergoing long-term spin down at a rate nudot = (-5.6+/-2.0)x10^{-16} Hz/s, with most of the spin evolution occurring during X-ray quiescence. We discuss the possible contributions of magnetic propeller torques, magnetic dipole radiation, and gravitational radiation to the measured spin down, setting an upper limit of B < 1.5x10^8 G for the pulsar's surface dipole magnetic field and and Q/I < 5x10^{-9} for the fractional mass quadrupole moment. We also measured an orbital period derivative of Pdot = (3.5+/-0.2)x10^{-12} s/s. This surprising large Pdot is reminiscent of the large and quasi-cyclic orbital period variation observed in the so-called "black widow" millisecond radio pulsars, supporting speculation that SAX J1808.4-3658 may turn on as a radio pulsar during quiescence. In an appendix we derive an improved (0.15 arcsec) source position from optical data.Comment: 22 pages, 10 figures; accepted for publication in Ap

SUSY Stops at a Bump

We discuss collider signatures of the "natural supersymmetry" scenario with baryon-number violating R-parity violation. We argue that this is one of the few remaining viable incarnations of weak scale supersymmetry consistent with full electroweak naturalness. We show that this intriguing and challenging scenario contains distinctive LHC signals, resonances of hard jets in conjunction with relatively soft leptons and missing energy, which are easily overlooked by existing LHC searches. We propose novel strategies for distinguishing these signals above background, and estimate their potential reach at the 8 TeV LHC. We show that other multi-lepton signals of this scenario can be seen by currently existing searches with increased statistics, but these opportunities are more spectrum-dependent.Comment: 23 pages, 7 figures, 3 tables. V2: spectrum discussion corrected, most of the changes are in Sec. 2. Benchmarks, analysis and conclusions unchanged. References adde

The X-ray Position and Optical Counterpart of the Accretion-Powered Millisecond Pulsar XTE J1814-338

We report the precise optical and X-ray localization of the 3.2 ms accretion-powered X-ray pulsar XTE J1814-338 with data from the Chandra X-Ray Observatory as well as optical observations conducted during the 2003 June discovery outburst. Optical imaging of the field during the outburst of this soft X-ray transient reveals an R = 18 star at the X-ray position. This star is absent (R > 20) from an archival 1989 image of the field and brightened during the 2003 outburst, and we therefore identify it as the optical counterpart of XTE J1814-338. The best source position derived from optical astrometry is R.A. = 18h13m39.s04, Dec.= -33d46m22.3s (J2000). The featureless X-ray spectrum of the pulsar in outburst is best fit by an absorbed power-law (with photon index = 1.41 +- 0.06) plus blackbody (with kT = 0.95 +- 0.13 keV) model, where the blackbody component contributes approximately 10% of the source flux. The optical broad-band spectrum shows evidence for an excess of infrared emission with respect to an X-ray heated accretion disk model, suggesting a significant contribution from the secondary or from a synchrotron-emitting region. A follow-up observation performed when XTE J1814-338 was in quiescence reveals no counterpart to a limiting magnitude of R = 23.3. This suggests that the secondary is an M3 V or later-type star, and therefore very unlikely to be responsible for the soft excess, making synchroton emission a more reasonable candidate.Comment: Accepted for publication in ApJ. 6 pages; 3 figure