Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

This paper gives $n$-dimensional analogues of the Apollonian circle packings in parts I and II. We work in the space \sM_{\dd}^n of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, ACC-coordinates, as those $(n+2) \times (n+2)$ real matrices \bW with \bW^T \bQ_{D,n} \bW = \bQ_{W,n} where $Q_{D,n} = x_1^2 +... + x_{n+2}^2 - \frac{1}{n}(x_1 +... + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + ... + 2x_{n+2}^2$, and \bQ_{D,n} and \bQ_{W,n} are their corresponding symmetric matrices. There are natural actions on the parameter space \sM_{\dd}^n. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set $S$ depending on the dimension. We show that the the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions one can find rational Apollonian cluster ensembles (all curvatures rational) and strongly rational Apollonian sphere ensembles (all ACC-coordinates rational).Comment: 37 pages. The third in a series on Apollonian circle packings beginning with math.MG/0010298. Revised and extended. Added: Apollonian groups and Apollonian Cluster Ensembles (Section 4),and Presentation for n-dimensional Apollonian Group (Section 5). Slight revision on March 10, 200

Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)$\times$(center) is an integer vector. This series of papers explain such properties. A {\em Descartes configuration} is a set of four mutually tangent circles with disjoint interiors. We describe the space of all Descartes configurations using a coordinate system \sM_\DD consisting of those $4 \times 4$ real matrices \bW with \bW^T \bQ_{D} \bW = \bQ_{W} where \bQ_D is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 -{1/2}(x_1 +x_2 +x_3 + x_4)^2$ and \bQ_W of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. There are natural group actions on the parameter space \sM_\DD. We observe that the Descartes configurations in each Apollonian packing form an orbit under a certain finitely generated discrete group, the {\em Apollonian group}. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups, the dual Apollonian group and the super-Apollonian group, which have nice geometrically interpretations. We show these groups are hyperbolic Coxeter groups.Comment: 42 pages, 11 figures. Extensively revised version on June 14, 2004. Revised Appendix B and a few changes on July, 2004. Slight revision on March 10, 200

Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. It observed there exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group, the super-Apollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object we call a super-packing. The circles in a super-packing never cross each other but are nested to an arbitrary depth. Certain Apollonian packings and super-packings are strongly integral in the sense that the curvatures of all circles are integral and the curvature$\times$centers of all circles are integral. We show that (up to scale) there are exactly 8 different (geometric) strongly integral super-packings, and that each contains a copy of every integral Apollonian circle packing (also up to scale). We show that the super-Apollonian group has finite volume in the group of all automorphisms of the parameter space of Descartes configurations, which is isomorphic to the Lorentz group $O(3, 1)$.Comment: 37 Pages, 11 figures. The second in a series on Apollonian circle packings beginning with math.MG/0010298. Extensively revised in June, 2004. More integral properties are discussed. More revision in July, 2004: interchange sections 7 and 8, revised sections 1 and 2 to match, and added matrix formulations for super-Apollonian group and its Lorentz version. Slight revision in March 10, 200

Narrow Pass-Band Optical Filters for Space-Borne Remote Sensing Applications

Optical characterisation of 532 nm, 200 pm passband and 1064 nm, I mn passband LkD filters after exposure to proton irradiation, temperature cycling and angle tuning

Qualification of Laser Diode Arrays for Mercury Laser Altimeter

NASA's requirements for high reliability, high performance satellite laser instruments have driven the investigation of many critical components; specifically, 808 nm laser diode array (LDA) pump devices. Performance of Quasi-CW, High-power, laser diode arrays under extended use is presented. We report the optical power over several hundred million pulse operation and the effect of power cycling and temperature cycling of the laser diode arrays. Data on the initial characterization of the devices is also presented

Fiber lasers and amplifiers for science and exploration at NASA Goddard Space Flight Center

We discuss present and near-term uses for high-power fiber lasers and amplifiers for NASA- specific applications including planetary topography and atmospheric spectroscopy. Fiber lasers and amplifiers offer numerous advantages for both near-term and future deployment of instruments on exploration and science remote sensing orbiting satellites. Ground-based and airborne systems provide an evolutionary path to space and a means for calibration and verification of space-borne systems. We present experimental progress on both the fiber transmitters and instrument prototypes for ongoing development efforts. These near-infrared instruments are laser sounders and lidars for measuring atmospheric carbon dioxide, oxygen, water vapor and methane and a pseudo-noise (PN) code laser ranging system. The associated fiber transmitters include high-power erbium, ytterbium, neodymium and Raman fiber amplifiers. In addition, we will discuss near-term fiber laser and amplifier requirements and programs for NASA free space optical communications, planetary topography and atmospheric spectroscopy

Effects of scan direction and orientation on mechanical properties of laser sintered polyamide-12

In order to understand the impact of layer-wise scanning direction in the Selective Laser Sintering process, test coupons were manufactured for mechanical testing from DuraForm™ Polyamide powder. The effects of laser energy density, varying between 0.003 and 0.024 J/mm2 were examined in test specimens rotated 90º through the Z axis. SLS machines do not always facilitate ‘cross-hatching’ of layers and therefore orientation has a major influence on part quality. When employed, the cross-hatching technique scans successive layers perpendicularly to the previous. Studying how parts perform with scan lines in a common direction, will assist in the understanding of how SLS parts behave in practice. Results showed that physical density, tensile strength and elongation rose with energy density up to 0.012 J/mm². This initial rise was due to a continued improvement in particle fusion with increasing energy density. Above 0.012 J/mm², these properties started to decline at different rates depending on their orientation (scan direction) on the part bed. Specimen’s oriented perpendicularly to the X axis exhibited a greater elongation at the expense of tensile strength, when compared to parallel specimens

Space Qualification of Laser Diode Arrays

Laser instruments have great potential in enabling a new generation of remote-sensing scientific instruments. NASA s desire to employ laser instruments aboard satellites, imposes stringent reliability requirements under severe conditions. As a result of these requirements, NASA has a research program to understand, quantify and reduce the risk of failure to these instruments when deployed on satellites. Most of NASA s proposed laser missions have base-lined diode-pumped Nd:YAG lasers that generally use quasi-constant wave (QCW), 808 nm Laser Diode Arrays (LDAs). Our group has an on-going test program to measure the performance of these LDAs when operated in conditions replicating launch and orbit. In this paper, we report on the results of tests designed to measure the effect of vibration loads simulating launch into space and the radiation environment encountered on orbit. Our primary objective is to quantify the performance of the LDAs in conditions replicating those of a satellite instrument, determine their limitations and strengths which will enable better and more robust designs. To this end we have developed a systematic testing strategy to quantify the effect of environmental stresses on the optical and electrical properties of the LDA

Design of a Direct-Detection Wind and Aerosol Lidar for Mars Orbit

The present knowledge of the Mars atmosphere is greatly limited by a lack of global measurements of winds and aerosols. Hence, measurements of height-resolved wind and aerosol profiles are a priority for new Mars orbiting missions. We have designed a direct-detection lidar (MARLI) to provide global measurements of dust, winds and water ice profiles from Mars orbit. From a 400-km polar orbit, the instrument is designed to provide wind and backscatter measurements with a vertical resolution of 2 km and with resolution of 2 in latitude along track. The instrument uses a single-frequency, seeded Nd:YAG laser that emits 4 mJ pulses at 1064 nm at a 250 Hz pulse rate. The receiver utilizes a 50-cm diameter telescope and a double edge Fabry-Prot etalon as a frequency discriminator to measure the Doppler shift of the aerosol-backscatter profiles. The receiver also includes a polarization-sensitive channel to detect the cross-polarized backscatter profiles from water ice. The receiver uses a sensitive 4 4 pixel HgCdTe avalanche photodiode array as a detector for all signals. Here we describe the measurement concept, instrument design, and calculate its performance for several cases of Mars atmospheric conditions. The calculations show that under a range of atmospheric conditions MARLI is capable of measuring wind speed profiles with random error of 24 m/s within the first three scale heights, enabling vertically resolved mapping of transport processes in this important region of the atmosphere