Triangular number The first six triangular numbers A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers (sequence A000217 in OEIS), starting at the 0th triangular number, is: The triangle numbers are given by the following explicit formulas: where is a binomial coefficient. The triangular number Tn solves the "handshake problem" of counting the number of handshakes if each person in a room with n + 1 people shakes hands once with each person. Triangle numbers are the additive analog of the factorials, which are the products of integers from 1 to n. The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a recurrence relation: Relations to other figurate numbers[edit] with and
Planet The planets were thought by Ptolemy to orbit Earth in deferent and epicycle motions. Although the idea that the planets orbited the Sun had been suggested many times, it was not until the 17th century that this view was supported by evidence from the first telescopic astronomical observations, performed by Galileo Galilei. By careful analysis of the observation data, Johannes Kepler found the planets' orbits were not circular but elliptical. As observational tools improved, astronomers saw that, like Earth, the planets rotated around tilted axes, and some shared such features as ice caps and seasons. Since the dawn of the Space Age, close observation by space probes has found that Earth and the other planets share characteristics such as volcanism, hurricanes, tectonics, and even hydrology. History Printed rendition of a geocentric cosmological model from Cosmographia, Antwerp, 1539 Babylon Greco-Roman astronomy India Medieval Muslim astronomy European Renaissance 19th century 20th century
100 Incredible Lectures from the World’s Top Scientists | Best Colleges Online Posted on Thursday June 18, 2009 by Staff Writers By Sarah Russel Unless you’re enrolled at one of the best online colleges or are an elite member of the science and engineering inner circle, you’re probably left out of most of the exciting research explored by the world’s greatest scientists. If you’re looking for even more amazing lectures, check out our updated list for 2012 with more talks from great minds. General Let the world’s top scientists explain exactly how they do their job when you listen to these lectures. Science and Engineering From materials science to the study of thermodynamics, learn more about the science of engineering here. WTC Lecture – collapse of WTC Buildings: Steven E. Biology and Medicine From drug research to evolution to writing the genetic code, watch these lectures for the latest developments in biology and medical research. Chemistry These chemistry scientists discuss the atomic theory of matter and other mind-boggling principles in the following lectures.
Star For at least a portion of its life, a star shines due to thermonuclear fusion of hydrogen into helium in its core, releasing energy that traverses the star's interior and then radiates into outer space. Once the hydrogen in the core of a star is nearly exhausted, almost all naturally occurring elements heavier than helium are created by stellar nucleosynthesis during the star's lifetime and, for some stars, by supernova nucleosynthesis when it explodes. Near the end of its life, a star can also contain degenerate matter. Astronomers can determine the mass, age, metallicity (chemical composition), and many other properties of a star by observing its motion through space, luminosity, and spectrum respectively. The total mass of a star is the principal determinant of its evolution and eventual fate. Other characteristics of a star, including diameter and temperature, change over its life, while the star's environment affects its rotation and movement. Observation history
Top 10 Ghost Ships Ghost ships, or phantom ships, make up a big part of the seafaring lore that has been passed down by sailors and fisherman throughout the years. The ships are said to be spectral apparitions that materialize on the horizon before quickly disappearing, and they are believed to be a sign of bad things to come. The term is also used to describe abandoned vessels that are found adrift with no crew or passengers, often under frightening and mysterious circumstances. Whether real stories of these derelict ships or legends about phantom craft trawling the seas, the following are the ten most famous ghost ships that continue to provoke speculation and mystery in the nautical world. 10. The Caleuche One of the most well known legends of the Chilota mythology of southern Chile describes the Caleuche, a ghost ship that appears every night near the island of Chiloe. 9. The SS Valencia was steamer ship that sank off the coast of Vancouver, British Columbia in 1906. 8. 7. 6. 5. 4. 3. 2. 1.
Gravitational singularity A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the scalar invariant curvatures of spacetime, which includes a measure of the density of matter. The two most important types of spacetime singularities are curvature singularities and conical singularities.[2] Singularities can also be divided according to whether they are covered by an event horizon or not (naked singularities).[3] According to general relativity, the initial state of the universe, at the beginning of the Big Bang, was a singularity. Interpretation[edit] Many theories in physics have mathematical singularities of one kind or another. Some theories, such as the theory of loop quantum gravity suggest that singularities may not exist. Types[edit] Curvature[edit] , which is diffeomorphism invariant, is infinite. Conical[edit] Naked[edit] Entropy[edit]
Púca The púca (Irish for spirit/ghost), pooka, phouka, phooka, phooca or púka is primarily a creature of Irish folklore.[1] Considered to be bringers both of good and bad fortune, they could either help or hinder rural and marine communities. The creatures were said to be shape changers which could take the appearance of black horses, goats and rabbits. Etymology[edit] Malevolent or benevolent nature[edit] In contrast, the phouka is represented as being helpful to farmers by Lady Wilde, who relates the following tale. There are stories of some phooka being blood-thirsty and vampire-like creatures. Morphology and physiology[edit] According to legend, the púca is a deft shapeshifter, capable of assuming a variety of terrifying or pleasing forms, and may appear as a horse, rabbit, goat, goblin, or dog. Agricultural traditions[edit] Certain agricultural traditions surround the púca. Regional variations[edit] See also[edit] References[edit] External links[edit] Water Horses and Other Fairy Steeds
Gravitational wave Aspect of relativity in physics Gravitational waves are waves of spacetime curvature that propagate at the speed of light and are produced by the relative motion of gravitating masses. They were first predicted by Albert Einstein[1][2] as a consequence of his general theory of relativity, appearing as "ripples in spacetime curvature".[3][4]: 326 [5]: 117 Hundreds of these gravitational waves have since then been observed, first indirectly using binary-pulsar observations[6] and, since 2015[7], directly through dedicated observatories. Gravitational waves transport energy as gravitational radiation, a form of radiant energy similar to electromagnetic radiation.[8] Newton's law of universal gravitation, part of classical mechanics, does not provide for their existence, instead asserting that gravity has instantaneous effect everywhere. Gravitational waves therefore stand as an important relativistic phenomenon that is absent from Newtonian physics. -significance. with the initial radius and
Sleeping Tricks - Effective Techniques For Falling Asleep