81. Wallpaper Groups: Lattices So the lattice of dots corresponds to the translation symmetries. That s because in that case, the points in the lattice nearest any one point in the http://www.clarku.edu/~djoyce/wallpaper/lattices.html

Lattices Just by considering the translation symmetries of a pattern we can begin to classify patterns. For any point, the collection of translates of it by translation symmetries of a pattern forms a lattice . In the example at the left, the translates of the one point colored red are indicated by dark blue dots. One translation, call it T , is indicated by the green arrows; it translates right and a bit upward. A different translation, call it U , is indicated by the rose colored arrows; it translates up and a bit right. By taking a composition of these two translations and their inverses, you can construct every other translation symmetry of the pattern. Thus, every translation is of the form T m U n where m and n are integers. Furthermore, you can see what composition you need by seeing where the red dot needs to go. So the lattice of dots corresponds to the translation symmetries. Five kinds of lattices We can classify lattices into five different kinds. If a lattice has a square fundamental region, it's called a square lattice hexagonal lattice . That's because in that case, the points in the lattice nearest any one point in the lattice are the vertices of a regular hexagon. (A rhombus is a parallelogram with equal sides.) If a lattice has a rhombus as a fundamental region, it's a

82. EBulletin - Lattice Of Value - Designing Products For Self-growth principles for designing products that exhibit self growth through network effects, focusing on web developers/users and other producer/consumer markets. http://www.hiraeth.com/alan/ebulletin/lattice-of-value/

e Bulletin Alan Dix (vfridge limited, aQtive limited and Lancaster University) the lattice of value - designing products for self-growth download PDF version on this bulletin (110K) background In 1998 some colleagues and I were working on the business plan for aQtive, subsequently funded by 3i. As we considered at our product range and marketing budget, we realised that traditional marketing was too expensive and we looked for ways in which our products could 'market themselves'. One of the results of this was the 'lattice of value' a way of looking at products to both assess how well suited they are to self-growth and how they can be modified to improve their ability to grow. This theme was followed up later within aQtive in the modelling and analysis of market ecology co-value One of the most obvious examples of this phenomenon is telephone use. The first user of the first telephone has no value from it (apart from it being a hi-tech ornament) as there is no one to call. The millionth telephone customer has far greater value as there are 999,999 existing telephone owners to talk to! Furthermore the first telephone user now has a more valuable product because others have bought telephones too. Figure 1. critical mass

83. Solid State Physics: Interactive Learning This JAVA applet calculates the energy contours for the electrons in a 2 dimensional, hexagonal lattice. The energy band is represented by a tight binding http://buckminster.physics.sunysb.edu/intlearn/electrons/electrons.html

Electrons in a 2D lattice: Energy contours and density of states By Carlos Rivera Important!: Your browser must be configured to run Java. This JAVA applet calculates the energy contours for the electrons in a 2 dimensional, hexagonal lattice. The energy band is represented by a tight binding approximation: E t (2 cos 0.5 k x a * cos 0.866 k y a + cos k x a t' (2 cos 0.866 k y a * cos 1.5 k x a + cos 1.73 k y a where a is the lattice spacing. On the plots the energy is measured in units of the nearest neighbor overlap integral t t' is the the overlap integrals to second neighbor atoms. For any physical situation t'/t is always less than 1 and for larger values the plots may not get drawn at all. The density of states is plotted on the right hand side. In order to save computing time, the DOS is calculated on a rather crude grid, resulting in the loss of fine details. Finally, the Fermi energy (represented by red contour lines on the plot) can be specified. The corresponding band occupancy is calculated. Back to the Interactive Learning page

84. Photonic Lattice Lincoln Loglike structure to improve infrared and optical communications, optical computers. Working photonic lattice, a dream for a decade, fabricated at http://www.sandia.gov/media/photonic.htm

News Release September 15, 1998 Lincoln Log-like structure to improve infrared and optical communications, optical computers Working photonic lattice, a dream for a decade, fabricated at Sandia NOT POPSICLE STICKS The photonic lattice created at Sandia National Laboratories acts like a crystal in guiding light because of its tiny, regularly placed silicon "logs." The logs are 1.2 microns wide. Control of different wavelengths is achieved by changing the lattice dimensions. Download 150dpi JPEG image, 'lattice.jpg', 736K ALBUQUERQUE, N.M. By interlocking tiny slivers of silicon into a lattice that, under a microscope, appears to be formed by toy Lincoln Logs, scientists at the Department of Energy's Sandia National Laboratories believe they have solved a major technical problem: how to bend light easily and cheaply without leaking it, no matter how many twists or turns are needed for optical communications or (potentially) optical computers. The lattice, dubbed a photonic crystal (crystals have regularly repeating internal structures), now works in the infrared range (approximately 10-micron wavelengths). This achievement is of military and commercial interest because the technique can be used to enhance or better transmit infrared images. Sandia researchers Shawn Lin and Jim Fleming now are preparing a 1.5 micron crystal the region in which almost all the world's optically transmitted information is passed.

85. Rectangle Multiplication - NLVM Visualize the multiplication of two numbers as an area. http://nlvm.usu.edu/en/nav/frames_asid_192_g_2_t_1.html

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