Lattice points visible from the origin « The Lumber Room [A test of LaTeX-to-Wordpress conversion. Bugs remain, point them out. Original PDF] Here is a problem I love. [The solution is not mine. Question. Let us first imagine that we are standing at the origin, and that the grid is that of the lattice (integer) points. The blue points are visible; the grey points are not We now want to examine the question of which pairs are visible from the origin. not visible? is not visible because the point is "in the way", and the point obscures it. is not visible precisely when there is another lattice point blocking it, which is when there is an integer pair such that . is not visible precisely when there is an integer dividing both and , and is visible when there is no such integer, i.e. when have no common factor. For two "random" integers and , what is the probability that they have no common factor? For this to happen, it must be the case that no prime number divides both . , the probability that is divisible by is , and the same for , and so the probability that .
Detailed logarithmic timeline This timeline shows the whole history of the universe, the Earth, and humanity in one table. Each row is defined in years ago, that is, years before the present date, with the earliest times at the top of the chart. In each table cell on the right, references to events or notable people are given, more or less in chronological order within the cell. Each row corresponds to a change in log(time before present) of about 0.1 (using log base 10), similar to Renard numbers. A logarithmic timeline can also be devised for events which should occur in the future, barring unforeseen circumstances and assuming that we can extrapolate into the future based on our science. See also[edit] References[edit] External links[edit]
Planets From Hell - How the Universe Works Documentary | HD 720p Kardashev scale Measure of the evolution of a civilization according to its energy consumption Energy consumption in three types of civilization as defined by Sagan's extended Kardashev scale The Kardashev scale (Russian: Шкала Кардашева, romanized: Shkala Kardasheva) is a method of measuring a civilization's level of technological advancement based on the amount of energy it is able to use. The measure was proposed by Soviet astronomer Nikolai Kardashev in 1964[1] and came to bear his name. The scale is hypothetical, and regards energy consumption on a cosmic scale. Kardashev first outlined his scale in a paper presented at the 1964 Byurakan conference, a scientific meeting that reviewed the Soviet radio astronomy space listening program. A type I civilization is able to access all the energy available on its planet and store it for consumption.A type II civilization can directly consume the energy of a star.A type III civilization is able to capture all the energy emitted by its galaxy. Type I[edit]
Into The Universe With Stephen Hawking: Aliens Technological singularity Hypothetical point in time when technological growth becomes uncontrollable and irreversible The technological singularity—or simply the singularity[1]—is a hypothetical future point in time at which technological growth becomes uncontrollable and irreversible, resulting in unforeseeable consequences for human civilization.[2][3] According to the most popular version of the singularity hypothesis, I. J. Good's intelligence explosion model, an upgradable intelligent agent will eventually enter a "runaway reaction" of self-improvement cycles, each new and more intelligent generation appearing more and more rapidly, causing an "explosion" in intelligence and resulting in a powerful superintelligence that qualitatively far surpasses all human intelligence.[4] Intelligence explosion[edit] Although technological progress has been accelerating in most areas, it has been limited by the basic intelligence of the human brain, which has not, according to Paul R. I. Variations[edit] Predictions[edit]
The Universe - How Big, How far, How fast - Documentary | HD 720p Ray Kurzweil Raymond "Ray" Kurzweil (/ˈkɜrzwaɪl/ KURZ-wyl; born February 12, 1948) is an American author, computer scientist, inventor, futurist, and is a director of engineering at Google. Aside from futurology, he is involved in fields such as optical character recognition (OCR), text-to-speech synthesis, speech recognition technology, and electronic keyboard instruments. He has written books on health, artificial intelligence (AI), transhumanism, the technological singularity, and futurism. Kurzweil was the principal inventor of the first CCD flatbed scanner,[2] the first omni-font optical character recognition,[2] the first print-to-speech reading machine for the blind,[3] the first commercial text-to-speech synthesizer,[4] the Kurzweil K250 music synthesizer capable of simulating the sound of the grand piano and other orchestral instruments, and the first commercially marketed large-vocabulary speech recognition.[5] Life, inventions, and business career[edit] Early life[edit] Mid-life[edit]
History Channel The Universe Backyard Astronomers Part 1 of 6 Futurist Futurists or futurologists are scientists and social scientists whose specialty is futurology, or the attempt to systematically explore predictions and possibilities about the future and how they can emerge from the present, whether that of human society in particular or of life on Earth in general. Definition[edit] The term "futurist" most commonly refers to authors, consultants, organizational leaders and others who engage in interdisciplinary and systems thinking to advise private and public organizations on such matters as diverse global trends, possible scenarios, emerging market opportunities and risk management. (Futurist is not in the sense of the art movement futurism.) Visionary writers such as Jules Verne, Edward Bellamy and H. Futures studies[edit] "Futures studies"—sometimes referred to as futurology, futures research, and foresight—can be summarized as being concerned with "three P's and a W", i.e. Futurists and futurology[edit] Notable futurists[edit] See also[edit]
Could Kurzweil be right about solar, the Google of energy? Ray Kurzweil presents his cost curves on solar at Berkeley University earlier this month. Paula Mints' slide doesn't look a million miles away from one of Kurzweil's exponential graphs. Felicity Carus Felicity Carus is the only UK journalist to be regularly reporting on clean energy policy and finance from California for a global audience. Cleantech investors often talk about the quest for the ‘Google of energy’. Solar advocates already believe they've picked the winner. Earlier this month, I had the good fortune to hear engineer, inventor, futurist and now-Google employee, Ray Kurzweil, speak at Berkeley University. During his note-free two hours on stage without a single "um, er, ah" or seeming pause for breath, Kurzweil took the audience on a fantastic voyage on the impact of exponential versus linear growth in technology. "History shows us that technological change is exponential, but we humans with our common sense intuit a more linear view," he said.
‘After Earth’ Sparks Real Discussion On Future Of Our Planet After Earth may be just a science fiction story about the changes that will take place on our home planet over the next 1,000 years, but the movie is starting a real discussion on the future of Earth. In a Google+ Hangout earlier this week followed up Earth Day, the stars of After Earth joined SpaceX CEO Elon Musk, futurist Ray Kurzweil, Alexandra Cousteau, and a group of students to talk about the future of our planet. After Earth stars Will Smith and his son Jaden and is set 1,000 years after cataclysmic events forced humans to escape from the planet. That’s been a topic close to the heart of one of the Google + Hangout participants. “It looks like great entertainment, but it raises some profound issues (about the future of the planet),” Kurzweil said about After Earth. Will Smith said he was pleased to see After Earth starting a real discussion about Earth and its future. “I think we want to be a space-faring civilization,” Musk said during the event. After Earth comes out April 26.