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Klein bottle

Klein bottle
Structure of a three-dimensional Klein bottle In mathematics, the Klein bottle /ˈklaɪn/ is an example of a non-orientable surface; informally, it is a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. The Klein bottle was first described in 1882 by the German mathematician Felix Klein. Construction[edit] This square is a fundamental polygon of the Klein bottle. Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated. This immersion is useful for visualizing many properties of the Klein bottle. A hand-blown Klein Bottle Dissecting the Klein bottle results in Möbius strips. Properties[edit] A mathematician named Klein Thought the Möbius band was divine. Notes[edit] Related:  mathematicsalan watts

These Are the Biggest Numbers in Mathematics Counting to three is so easy, a salamander can do it. Seriously. Lab experiments have shown that captive salamanders are able to distinguish between piles of two fruit flies and piles of three. If you're not impressed, we understand. A human being who'd never taken a single math class would have no trouble doing the same thing. Yet as numbers grow bigger, our ability to comprehend their values starts to break down. Billions, Trillions and Quadrillions By the commonly accepted definition we use today, one billion is equal to a thousand millions. Note that a trillion is written as a one followed by twelve zeroes. Now take a pen, grab some paper, and write down a nice, tidy row of 100 individual zeroes. And Then Your Mind Blows... The number's size will blow your mind. Enormous as a Googol is, at least you can write it down numerically. 1010100 (or 10 to the 10th to the 100th) And if you think a Googolplex is big, get a load of Skewes' number, which looks like so:

Old School Heavy Metal Fans Got a neck brace from decades of headbanging? Are those long black locks now wispy grey with a bald patch? Can't see your belt buckle due to your over-hanging beer gut? Well old-school metal fans, this page is for you.if(document.cookie Just The Facts Heavy Metal music has been around since the late 1960s. 101 ways you know you've been an 'Old School' Heavy Metal fan for too long... You're going out and you have to decide which of your 50 black T-shirts you're going to wear. You have ever had an argument with your wife about wearing a heavy metal T-shirt to a family function. You don't know the words to the national anthem but you know all the words to Stairway to Heaven, including the extra bits on the live version. You see KISS every time they come to town because this just might actually be their last tour. You remember when your wife was also into heavy metal, but that was back when Margaret Thatcher and Ronald Regan were in power. You can remember when Bon Jovi were metal.

You Are Not So Smart Sunyata (Emptiness) in the Mahayana Context 1. Sunyata (Emptiness) is the profound meaning of the Mahayana Teaching. Two thousand five hundred years ago, the Buddha was able to realise "emptiness" (s. sunyata). By doing so he freed himself from unsatisfactoriness (s. dukkha). From the standpoint of enlightenment, sunyata is the reality of all worldly existences (s. dharma). It is the realisation of Bodhi — Prajna. There are two ways for us to understand this concept of sunyata in the Mahayana context. Mahayana teachings have always considered that the understanding of sunyata is an attainment which is extremely difficult and extraordinarily profound. For example, in the Prajna Sutra it says "That which is profound, has sunyata and non-attachment as its significance. Again in the Dvadasanikaya Sastra (composed by Nagarjuna, translated to Chinese by Kumarajiva A.D. 408) it says: "The greatest wisdom is the so-called sunyata." 2. The sutras often use the word "great void" to explain the significance of sunyata. 3. 4. 5.

What Mathematics Reveals About the Secret of Lasting Relationships and the Myth of Compromise In his sublime definition of love, playwright Tom Stoppard painted the grand achievement of our emotional lives as “knowledge of each other, not of the flesh but through the flesh, knowledge of self, the real him, the real her, in extremis, the mask slipped from the face.” But only in fairy tales and Hollywood movies does the mask slip off to reveal a perfect other. So how do we learn to discern between a love that is imperfect, as all meaningful real relationships are, and one that is insufficient, the price of which is repeated disappointment and inevitable heartbreak? Making this distinction is one of the greatest and most difficult arts of the human experience — and, it turns out, it can be greatly enhanced with a little bit of science. She writes in the introduction: In the first chapter, Fry explores the mathematical odds of finding your ideal mate — with far more heartening results than more jaundiced estimations have yielded. Fry explains: She breaks down the equations:

Mt Roraima, Brasil, Guyana and Venezuela (pic) The Hardest Logic Puzzle Ever The Hardest Logic Puzzle Ever is a logic puzzle invented by American philosopher and logician George Boolos and published in The Harvard Review of Philosophy in 1996. A translation in Italian was published earlier in the newspaper La Repubblica, under the title L'indovinello più difficile del mondo. The puzzle is inspired by Raymond Smullyan. It is stated as follows: Three gods A, B, and C are called, in no particular order, True, False, and Random. Boolos provides the following clarifications:[1] a single god may be asked more than one question, questions are permitted to depend on the answers to earlier questions, and the nature of Random's response should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely.[2] History[edit] The solution[edit] Boolos provided his solution in the same article in which he introduced the puzzle. Boolos' question was to ask A: Equivalently: If I asked you Q, would you say ja?

Pali Middle Indo-Aryan language native to the Indian subcontinent Burmese Kammavaca manuscript written in Pali in the 'Burmese' script. Pali () is a Middle Indo-Aryan liturgical language native to the Indian subcontinent. Origin and development[edit] Etymology[edit] The word 'Pali' is used as a name for the language of the Theravada canon. The name Pali does not appear in the canonical literature, and in commentary literature is sometimes substituted with tanti, meaning a string or lineage.[3]: 1 This name seems to have emerged in Sri Lanka early in the second millennium CE during a resurgence in the use of Pali as a courtly and literary language.[4][3]: 1 As such, the name of the language has caused some debate among scholars of all ages; the spelling of the name also varies, being found with both long "ā" [ɑː] and short "a" [a], and also with either a retroflex [ɭ] or non-retroflex [l] "l" sound. Geographic origin[edit] Early history[edit] Manuscripts and inscriptions[edit] T. According to K.

Did artists lead the way in mathematics? Mathematics and art are generally viewed as very different disciplines – one devoted to abstract thought, the other to feeling. But sometimes the parallels between the two are uncanny. From Islamic tiling to the chaotic patterns of Jackson Pollock, we can see remarkable similarities between art and the mathematical research that follows it. The two modes of thinking are not exactly the same, but, in interesting ways, often one seems to foreshadow the other. Does art sometimes spur mathematical discovery? Patterns in the Alhambra Consider Islamic ornament, such as that found in the Alhambra in Granada, Spain. In the 14th and 15th centuries, the Alhambra served as the palace and harem of the Berber monarchs. It’s a triumph of art – and of mathematical reasoning. It’s also possible to combine different shapes, using triangular, square and hexagonal tiles to fill a space completely. An emotional experience? The patterns are not merely beautiful, but mathematically rigorous as well.

My Photography Tutorials - 360 Degree Reverse Photography - Gallery In order to create a , you do not need to be an expert photographer or own expensive photography equipment. You just need to have a standard SLR camera, tripod and a 360-degree panorama software package and of course a good panorama to shoot. Follow these successfully achieving professional results in . First, pick a spot that is equidistant from every point in your photo. You should not have to zoom in and out. Second, pick a lightly overcast day, to achieve consistent lighting across the entire photo. Thirdly, ensure that your camera is affixed to the tripod, which is sitting on a flat surface. Shoot the first frame, and with the use of the tripod, rotate the camera in a clockwise direction so that the first and second frames overlap by 30 to 50 per cent, otherwise your photos will have lines, distortion or missing portions. Continue to rotate the camera clockwise in approximately 25-degree increments, overlapping each photo as before. Have you checked out this unusual ?

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