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59 liradan başlayan fiyatlarla mükemmel parfümler!!
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On a practical note, J. Kåhre observes that bridges bb and dd no longer exist and that aa and cc are now a single bridge passing above A with a stairway in the middle leading down to A. Even so, there is still no Eulerian cycle on the nodes A, B, C, and D using the modern Königsberg bridges, although there is an Eulerian path (right figure). An example Eulerian path is illustrated in the right figure above where, as a last step, the stairs from A to aacc can be climbed to cover not only all bridges but all steps as well.
Seven Bridges of Königsberg
Koenigsberg bridges KoenigsbergBridges The Königsberg bridge problem asks if the seven bridges of the city of Königsberg (left figure; Kraitchik 1942), formerly in Germany but now known as Kaliningrad and part of Russia, over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began. This is equivalent to asking if the multigraph on four nodes and seven edges (right figure) has an Eulerian cycle. This problem was answered in the negative by Euler (1736), and represented the beginning of graph theory.
If you like a drink, then a Klein bottle is not a recommended receptacle. It may look vaguely like a bottle, but it doesn’t enclose any volume, which means that it can’t actually hold any liquid. Whatever you pour “in” will just come back out again.
How do you construct such a strange thing and why would you want to construct it? The mathematician Felix Klein, who discovered the bottle in 1882, described it as a surface that “can be visualised by inverting a piece of a rubber tube and letting it pass through itself so that outside and inside meet”.
A Klein bottle is formed by joining two sides of a sheet to form a cylinder (tube), then looping the ends of a cylinder back through itself in such a way that the inside (green) and outside (white) of the cylinder are joined. Animation by Konrad Polthier.
It’s obvious that the Klein bottle, just like the more familiar sphere, is a closed surface: it’s finite in the sense that you can fit it into a finite region of space, but an ant could walk around on it forever without ever encountering a boundary or falling over an edge. Unlike the sphere, which has an inside and an outside, the Klein bottle is one-sided: walking around, our ant could reach both sides of each point of the surface. This is why the bottle encloses no volume, and it also answers the “why” question: the Klein bottle is interesting because we don’t encounter many one-sided shapes in nature. (See here for another very pretty picture of a Klein bottle.)
The Möbius band is one-sided The Möbius strip is one-sided - view the animated version. (Image and animation by Konrad Polthier.) If this is a bit confusing, think of a simpler example of a one-sided surface: the famous Möbius strip. You can make one by taking two ends of a strip of paper, giving the strip a twist, and then gluing the ends together. By using a strip of paper whose two sides have different colours, say green and orange, it’s easy to convince yourself that the resulting Möbius strip is one-sided. Once you have twisted and glued, you’ll find that you can reach every orange point from every green point without having to pierce through the paper or climbing over its edge.
Unlike the Klein bottle, the Möbius strip does have a boundary — it is made up of the two non-glued edges of the original strip. But there is a link between the two. If you take two Möbius strips and create a closed shape by joining their boundaries using an ordinary two-sided strip, as shown below, what you get is exactly the Klein bottle.
The two Möbius bands of a Klein bottle are connected by an ordinary two-sided band whose back and front sides are colored white and blue respectively. Animation by Konrad Polthier.
This fact inspired the mathematician Leo Moser to compose a limerick:
A mathematician named Klein Thought the Möbius band was divine. Said he: “If you glue The edges of two, You’ll get a weird bottle like mine.”
Some mathematicians really are multi-talented!
Another curious feature of the Klein bottle is that it intersects itself, which means it’s hard to make it from a single tube of rubber as Klein suggested. Strictly speaking, the self-intersecting object depicted above isn’t a Klein bottle, but (as Klein indicated) only a visualisation of one. To understand why, first think of the more familiar doughnut (known mathematically as a torus). You can make a torus from a square sheet of rubber by first gluing two opposite sides to form a cylinder, and then gluing the two boundary components of that cylinder to get the torus.
Making a torus Making a torus: First glue opposite points on two opposite sides of the square to make a cylinder and then glue the two boundery circles of that cylinder (corresponding to gluing opposite points on the remaining two sides of the square) to form a torus. If you don’t want to bother with the gluing, you can simply think of the torus as a square, keeping in mind that opposite points on opposite sides are considered the same. So when you slide a shape drawn on the “square” across the top edge, it will re-appear at the bottom edge and when you slide it over the right edge it will re-appear at the left edge (and vice versa).
Slide When you slide a shape drawn on the “square” across the top edge, it will re-appear at the bottom edge. To get a true Klein bottle, start in the same way, identifying opposite points on a pair of opposite sides of a square. For the other pair of sides, however, don’t identify points that are directly opposite, but points that are diagonally opposite, as shown in the picture.
Klein bottle Making a Klein bottle: identify opposite points of two opposite sides (the vertical ones in this picture) and then identify diagonally opposite points on the remaining two sides. (If you label points on the top and bottom sides from left to right by the numbers 0 through 1, then a point with label x on the top side is identified with the point with label 1-x on the bottom side.) The resulting object is the Klein bottle. Again, you can think of it as a square, keeping in mind which boundary points are considered the same. If you slide a shape over one of the edges that had opposite points identified (vertical in our picture), it reappears reappear on the opposite side, like before. If, however, you slide it over one of the edges whose points were identified diagonally (horizontal in our picture), it appears on the opposite side, but displaced and as a mirror image of its original self.
Slide If you slide a shape over the top edge, it appears across the bottom edge, but over to the other side and as a mirror image of its original self. The only way to build this shape in three-dimensional space by actually gluing points together is to allow it to intersect itself. This amounts to identifying pairs of points in the interior of the original square, so strictly speaking the resulting shape is not exactly the same as the Klein bottle, which only has boundary points identified. It’s just one way of representing the bottle in three-dimensional space.
Curiously, the concepts of “inside”, “outside” and “one-sidedness” depend on the ambient space an object is sitting in. For example, a loop drawn on a piece of paper (in two-dimensional space) has a well defined inside and outside, but a loop drawn in three-dimensional space doesn’t. This is why we can’t talk about one-sidedness unless we first decide how to embed a surface in three-dimensional space. There is, however, a closely related property that is intrinsic to a shape and doesn’t depend on the surrounding space. A surface is called orientable if you cannot slide a shape drawn on it around and back to where it started so that it looks like its own mirror image. As you can see in the animation accompanying the image on the right, the Möbius strip isn’t orientable. And as you can see from the image with the smiley face above above, neither is the Klein bottle. To give it its full mathematical description, the Klein bottle is a closed, non-orientable surface. To find out more about it see the article Inside the Klein bottle. #mathematic #mathematic #geometry #mathematica #mathematicians #topology #university #mathlab #classymathematics #izmirlilagrange #themathsblog #coolmathstuff
Klein Şişesi&Möbius Şeridi #mathematic #mathematic #geometry #mathematica #mathematicians #topology #university #mathlab #classymathematics #izmirlilagrange #themathsblog #coolmathstuff
Matematiğin alt anabilim dallarından olan Topoloji, çok dikkat çekici nesnelerle uğraşmakta ve çalışmalar neticesinde insanın hayret ettiği nesneleri de bulmaya devam etmektedir. Topolojinin ne olduğu hakkında çok uzun konuşulabilir fakat biz bu yazıda iki topolojik şekli tanıtalım sizlere: Möbius Şeridi ve Klein Şişesi.
Yarım devir döndürülmüş ve ucuca birleştirilmiş bir şerit düşünün. İşte bu oluşan şeklin adı Möbius Şerididir.
İlk olarak 1861 yılında Möbius Şeridi hakkında tanımlamaları veren, matematikçiler için ne anlam ifade ettiğini açıklayan Johann Benedict Listing olmasına karşın, ileri seviye araştırma alanına sokan ve üzerinde topolojik olarak çalışan August Ferdinand Möbiustur. Birçok insan tarafından bu nesne garip ve bir o kadar da ilginç karşılanmaktadır.
Möbius şeridini ilginç kılan şey, herhangi bir hareketli bir noktadan diğer noktaya varması için bütün alanları taraması gerekir. Zaten A.F. Möbius yayınladığı makalesinde şeritlerin iki yüzü olmasına rağmen bu şeridin sadece bir yüzü olduğunu açıklamıştır.
Bu şeklin aslında ünlenmesi birazda sanatçı ve matematikçi M. Escher tarafından yapılan anlaşılması güç rresimler sayesinde olmuştur.
Bu şeridin denklemleri ve matematiksel tanımları hakkında çok derinlere girmeden trigonometrik olarak tanımlayalım. Fakat biraz matematik bilmek gerekiyor. Birazcık…
Bir topolojik nesne olarak Möbius şeridini kısaca tanıttık sizlere.
Möbius şeridine benzer biçimde elde edilen bir başka ilginç şekilde Klein Şişesidir. Bu şişe ise tek bir ağızdan oluşmaktadır. Yine aynı biçimde tek yüzü vardır, aslında bunun tek sebebi kapalı bir yüzey olmasındandır. Klein şişesinin bundan dolayı iç yüzeyi olmayıp sadece dış yüzeyi vardır. Yukarıda dediğimiz gibi tek ağzı olduğu, içine dökülen herhangi bir sıvı yine aynı açıklıktan dökülecektir. Yani sakın sürahi olarak kullanayım demeyin.
Möbius şeridi ve Klein şişesi aralarındaki fark Möbius şeridini üç boyutta gösterebilmemize rağmen, Klein şişesi için dört boyuta ihtiyacımız vardır.
Möbius Şeridini yaparken, dikdörtgen biçimindeki şeridi uçlarından 180 derece ters olacak şekilde yapıyorduk. Klein Şişesini yaparken ise bir silindiri aynı yukarıdaki biçimde 180 derece ters olacak şekilde birleştirilerek elde edilir.
Yukarıda boyutla ilgili açıklanan cümleler birçok kişi tarafından eleştirilebilir. Klein şişesine bakarken ona 4-boyutlu uzayda var olan bir varlığın 3-boyutlu uzaydaki yansıması olarak bakınız. Yani, Klein şişesi 4-boyutlu uzayda var olan bir nesnedir. Fakat üç boyutlu uzaydaki yansımasında bir kesik varmış gibi görünmektedir. Ancak, elinizle onun yüzeyini yokladığınızda kapalı ve sonlu fakat sınırsız 2-boyutlu bir yüzey olduğu sonucuna varırsınız. İnanmayan varsa şayet, tam ortasından kestiğimizde iki adet Möbius şeridi karşımıza çıkacak.
Biz bu şeridi 2 boyutta, 3.boyutun yansıması olarak olarak söylemiştik. Tek yüzü olmasına karşın (2 boyut) en, boy ve yükseklik biçiminde görülmesi olayı Klein şişesi için de geçerlidir.
Peki Klein şişesine baktığınızda kulpu varmış gibi görünüyor, değil mi? Sakın aldanmayın, bu geometrinin bize oynadığı bir oyundur bu bir yanıltıcı görüntüdür. Bu yanılgı 4-boyutlu Klein şişesini bilgisayarda programladığınızda açıkça ortaya çıkmaktadır. Görüntüde Klein şişesi hem kulplu hem de kulpsuz gözükür. ( Bakmasını bilene! ) Kaynak: www.matematiksel.org
#mathematic #mathematic #geometry #mathematica #mathematicians #topology #university #mathlab #classymathematics #izmirlilagrange #themathsblog #coolmathstuff