A Möbius strip, band, or loop spelled Mobius or Moebius, is a surface with only one side and only one boundary. The Möbius strip has the mathematical property of being unorientable, it can be realized as a ruled surface. Its discovery is attributed to the German mathematicians Johann Benedict Listing and independently August Ferdinand Möbius in 1858, though a structure similar to the Möbius strip can be seen in Roman mosaics dated circa 200–250 AD. An example of a Möbius strip can be created by taking a paper strip and giving it a half-twist, joining the ends of the strip to form a loop. However, the Möbius strip is not a surface of only one exact size and shape, such as the half-twisted paper strip depicted in the illustration. Rather, mathematicians refer to the closed Möbius band as any surface, homeomorphic to this strip, its boundary is a simple closed curve, that homeomorphic to a circle. This allows for a wide variety of geometric versions of the Möbius band as surfaces each having a definite size and shape.

For example, any rectangle can be glued to itself to make a Möbius band. Some of these can be smoothly modeled in Euclidean space, others cannot. A half-twist clockwise gives an embedding of the Möbius strip different from that of a half-twist counterclockwise – that is, as an embedded object in Euclidean space, the Möbius strip is a chiral object with right- or left-handedness. However, the underlying topological spaces within the Möbius strip are homeomorphic in each case. An infinite number of topologically different embeddings of the same topological space into three-dimensional space exist, as the Möbius strip can be formed by twisting the strip an odd number of times greater than one, or by knotting and twisting the strip, before joining its ends; the complete open Möbius band is an example of a topological surface, related to the standard Möbius strip, but, not homeomorphic to it. Finding algebraic equations, the solutions of which have the topology of a Möbius strip, is straightforward, but, in general, these equations do not describe the same geometric shape that one gets from the twisted paper model described above.

In particular, the twisted paper model is a developable surface. A system of differential-algebraic equations that describes models of this type was published in 2007 together with its numerical solution; the Euler characteristic of the Möbius strip is zero. The Möbius strip has several curious properties. A line drawn starting from the seam at the other side. If continued, the line meets the starting point, is double the length of the original strip; this single continuous curve demonstrates. Cutting a Möbius strip along the center line with a pair of scissors yields one long strip with two full twists in it, rather than two separate strips; this happens because the original strip only has one edge, twice as long as the original strip. Cutting creates a second independent edge, half of, on each side of the scissors. Cutting this new, strip down the middle creates two strips wound around each other, each with two full twists. If the strip is cut along about a third of the way in from the edge, it creates two strips: One is a thinner Möbius strip – it is the center third of the original strip, comprising one-third of the width and the same length as the original strip.

The other is a longer but thin strip with two full twists in it – this is a neighborhood of the edge of the original strip, it comprises one-third of the width and twice the length of the original strip. Other analogous strips can be obtained by joining strips with two or more half-twists in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a twisted strip tied in a trefoil knot. A strip with N half-twists, when bisected, becomes a strip with N + 1 full twists. Giving it extra twists and reconnecting the ends produces figures called paradromic rings. One way to represent the Möbius strip as a subset of three-dimensional Euclidean space is using the parametrization: x = cos ⁡ u y = sin ⁡ u z = v 2 sin ⁡ u 2 where 0 ≤ u < 2 π and − 1 ≤ v ≤ 1. This creates a Möbius strip of width 1 whose center circle has radius 1, lies in the x y -plane and is centered at; the parameter u {\displaystyl

The YF-115 is a Chinese liquid rocket engine burning LOX and kerosene in an oxidizer-rich staged combustion cycle. A high efficiency/high thrust environmental friendly rocket engine was always an objective within Programme 863. Development began in the 2000s, along with its sibling, the bigger YF-100, which would power the LM-5, LM-6 and LM-7 boosters and first stages. Testing was directed by the China National Space Administration commencing in 2005. Development works are carried out by the Xi'an Aerospace Propulsion Institute, it will be used as upper stage engine for China's next generation of medium and light environmental-friendly launch vehicles, namely the Long March 6 and the Long March 7. During early 2012, the engine system passed vacuum testing, it is China's first upper stage rocket engine. In the LM-6 upper stage it will use a single YF-115 with gimbaled mount. While the LM-7 upper stage will use four such engines, but in this latter case, two engines will be fixed and two will be gimbaled.

LM-6 – Rocket family that uses the YF-115. LM-7 – Rocket family that uses the YF-115. YF-100 – First stage Chinese rocket engine, the technological base of the YF-115

Jean-Claude Berrouet is a French winemaker. World-renowned in his field and in Merlot production, he is best known for his work with Jean Pierre Moueix at the famous Chateau Petrus and Dominus Estate owned by the Moueix family; the Oakland Tribune considers Petrus under Berrouet to be "arguably one of the world's most revered and most expensive wines." He is known for making wines of subtlety and nuance that are considered the counterpoint of Michel Rolland. Berrouet was born into a Basque family in the Bordeaux region of France. From his schooldays he was a close friend of Jean Brana at Saint-Jean Pied-du-Port, he became winemaker and technical director for Établissements Jean-Pierre Moueix in 1964. There he was responsible for the production of their complete range of wines, such as Pomerol and St. Emilion, he produced some 44 vintages for Établissements, working at estates such as Chateau Magdelaine and the famous Chateau Petrus. He has been cited as a "great talent" of enology, he achieved fame with his work at Chateau Petrus in particular, where he worked for over forty years.

Noted wine critic Larry Walker said that under Berrouet, the "wines have shown a huge improvement, becoming much more supple. The Oakland Tribune considers Petrus under Berrouet to be "arguably one of the world's most revered and most expensive wines." Berrouet is considered to be one of the foremost proponents of "classicist" winemaking, in that his wines are balanced, true to their origins and capable of longer ageing. Berrouet began working for Dominus Estate as a technical consultant under Christian Moueix in the 1980s, he was chief winemaker, was involved from its outset in 1982, producing its first vintage in 1983. In May 2012, Twomey Cellars announced that they had hired Berrouet to assist Daniel Baron in the production of Twomey's Napa Valley Merlot. Baron said of Berrouet's decision to join Twomey, "Much of my style of winemaking is based on his aesthetic. In fact, I credit my time with him as one of the inspirations that led to Twomey Cellars. Jean-Claude knows how to combine subtlety and balance in a wine and his joy of living comes through in every glass."

Berrouet has mentored numerous other notable winemakers over the years with his deep knowledge of viticulture, including Yamlick Reyrel. Berrouet has said, "A wine should tell you the story of the place. Innovation and change are important, but there should be limits on techniques that a winemaker can use and still have a wine classified as a Bordeaux. If I'm a conductor playing Mozart, can I add notes to it? Pretentious men do this; as long as I can make a living doing what I do, I will resist the move to a standard taste."Berrouet retired formally from winemaking in 2007, but with the help of his son, Jean-François Berrouet, he has continued work as technical consultant for wineries in Israel, Argentina and France. The Petrus Estate, as of 2010, it now run by Jean Pierre Moueix's son Jean-François Moueix, Berrouet's other son, Olivier Berrouet, is Director of Winemaking. Berrouet owns the 6 hectare Vieux Chateau St-Andre in Montagne St Emilion, which he runs with his son Jean-François. Upon his retirement, Christian Moueix said "Jean-Claude Berrouet's contribution has not only been lengthy, but immeasurable.

With a lifelong passion in poetry as well as wine, his philosophy has been for the wines to express their terroir, always favoring elegance over extraction."