The Möbius strip was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.
One way to represent the Möbius strip as a subset of formula_1 can be done using the parametrization:
Where 0 ≤ α < 2π and −1 ≤ "r" ≤ 1. This creates a Möbius strip of width 1 whose center circle has radius 1, lies in the "xy" plane and is centered at The parameter "u" runs around the strip while "v" moves from one edge to the other.
In cylindrical polar coordinates ("r", θ, "z"), an unbounded version of the Möbius strip can be represented by the equation:
Topologically, the Möbius strip can be defined as the square [0,1] × [0,1] with its top and bottom sides identified by the relation for as in the diagram on the right.
The Möbius strip is a two-dimensional compact manifold (i.e. a surface) with boundary. It is a standard example of a surface which is not orientable. The Möbius strip is also a standard example used to show the mathematical idea of a fiber bundle. Specifically, it is a nontrivial bundle over the circle "S" with a fiber the unit interval, "I" = [0,1]. Looking only at the edge of the Möbius strip gives a nontrivial two point (or Z) bundle over "S".
A simple construction of the Möbius strip, which can be used to show it in computer graphics or modeling packages, is as follows:
There have been several technical applications for the Möbius strip. Giant Möbius strips have been used as conveyor belts. Such belts last longer because the entire surface area of the belt gets the same amount of wear. Möbius bands have also been used as continuous-loop recording tapes (to double the playing time). Möbius strips are common in the making of fabric computer printer and typewriter ribbons. There, they allow the ribbon to be twice as wide as the print head while using both half-edges evenly.
A device called a Möbius resistor is an electronic circuit element which has the property of canceling its own inductive reactance. Nikola Tesla patented similar technology in the early 1900s: "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.
The Möbius strip is the configuration space of two unordered points on a circle. Consequently, in music theory, the space of all two note chords, known as dyads, takes the shape of a Möbius strip.
In physics/electro-technology:
In chemistry/nano-technology:
Klein bottle
The Klein bottle is a geometrical object, named after the German mathematician Felix Klein. He described it in 1882, and named it "Klein'sche Fläche" (Klein surface). Like the Möbius strip, it only has one surface. Mathematicians call this a non-orientable surface. Klein bottles only exist in four-dimensional space, but a model of a Klein bottle can be made in 3D. This model is different from the original because at some point the shape touches itself. In 3D, part of the shape is "inside" the rest. This is not the case in 4D. Some 3D models use different colors to show the 4th component. The part that lies "inside" then has a different color.
Because the surface is non-orientable, there is no "inside" or "outside". This means that if a liquid were filled "in the bottle", it would run down its surface. This may not be true for the 3D models of the bottle.
The 2-dimensional version of a Klein bottle is a Möbius strip.
Join the arrows so that they face the same way, like so:
Entropy
The entropy of an object is a measure of the amount of energy which is unavailable to do work. Entropy is also a measure of the number of possible arrangements the atoms in a system can have. In this sense, entropy is a measure of uncertainty or randomness. The higher the entropy of an object, the more uncertain we are about the states of the atoms making up that object because there are more states to decide from. A law of physics says that it takes work to make the entropy of an object or system smaller; without work, entropy can never become smaller – you could say that everything slowly goes to disorder (higher entropy).
The word entropy came from the study of heat and energy in the period 1850 to 1900. Some very useful mathematical ideas about probability calculations emerged from the study of entropy. These ideas are now used in information theory, chemistry and other areas of study.
Entropy is simply a quantitative measure of what the second law of thermodynamics describes: the spreading of energy until it is evenly spread. The meaning of entropy is different in different fields. It can mean:
Side
In geometry, a side is a line that is part of a flat figure made from straight lines. It is also a face in a solid three-dimensional object. The number of sides in a polygon is often written as formula_1, while the number of faces in a polyhedron is often written as formula_2.
As an example, the square on the right has four sides. The sides in this picture are made with black straight lines. The four sides are:
Te Puke
Te Puke is a small town in the Bay of Plenty in New Zealand. 6670 people live there. It is famous for kiwifruit, which is grown there. Lots of other fruit is grown there too.
"Te Puke" means "the hill" in Māori.
Valley
A valley is a type of landform. A valley is a lower part in the land, between two higher parts which might be hills or mountains. Valleys often start as a downward fold between two upward folds in the surface of the Earth, and sometimes as a rift valley. A valley is made deeper by a stream of water or a river as it flows from the high land to the lower land, and into a lake or sea. Some valleys are made by glaciers which are slow-moving rivers of ice. When water or ice make a valley deeper or wider, this is called erosion. Wind can also make valleys larger by erosion.
A valley has a "head" where it begins in the mountains or hills, "sides" where it rises up on either side, a "floor" which is where the valley is most flat. Some valleys have an "entrance" where the valley opening can be seen between two hills or mountains or cliffs. A place where a valley is very narrow and has high walls is sometimes called a "gorge". (This word "gorge" is sometimes used to mean the "throat" on a human body).
Many of the people of the world live in valleys because there is often a river or stream in a valley for fresh water, and there is often good soil in a valley to grow crops.
Mountains and hills are made when the layers of rock and soil (called "strata") get folded. Fault block mountains may have rift valleys. There are valleys in mountain ranges, between the highest parts which are the "peaks". People who want to travel to the other side of the mountains usually go by the valleys. A valley that people use to travel through mountains or hills is called a mountain pass.
Valleys that are high in the mountains are usually made deeper by a stream or small river running fast down the mountainside, from a place where there is lots of rainwater or melting snow, or by a glacier. The mountain stream winds around the biggest rocks and washes the soil away as it flows. It cuts a passage for itself through the softest soil and smallest stones. A small stream can cut a very deep valley. Valleys that are high in the mountains are usually V-shaped. There are many valleys like this in hills and mountains all over the world.
In country that has hills, but is not very steep, a river or stream runs more slowly. It makes a wider valley that often has some large bends as the river flows around the hills, always following the lowest way. Water running down from the hillsides often carries soil that spreads out across the valley, making flat land that is good for growing food crops and raising cattle and other animals. Many farms are in valleys that are in hilly land. Many towns are built on the sloping sides of valleys. Famous valleys of this type are the Loire Valley and the Lower Rhine Valley in Europe and the Thames Valley in England.