text:
Space suit

A space suit or spacesuit is a garment worn to keep a human alive in the harsh environment of outer space, vacuum and temperature extremes. Space suits are often worn inside spacecraft as a safety precaution in case of loss of cabin pressure, and are necessary for extravehicular activity (EVA), work done outside spacecraft. Space suits have been worn for such work in Earth orbit, on the surface of the Moon, and en route back to Earth from the Moon. Modern space suits augment the basic pressure garment with a complex system of equipment and environmental systems designed to keep the wearer comfortable, and to minimize the effort required to bend the limbs, resisting a soft pressure garment's natural tendency to stiffen against the vacuum. A self-contained oxygen supply and environmental control system is frequently employed to allow complete freedom of movement, independent of the spacecraft.

Three types of space suits exist for different purposes: IVA (intravehicular activity), EVA (extravehicular activity), and IEVA (intra/extravehicular activity). IVA suits are meant to be worn inside a pressurized spacecraft, and are therefore lighter and more comfortable. IEVA suits are meant for use inside and outside the spacecraft, such as the Gemini G4C suit. They include more protection from the harsh conditions of space, such as protection from micrometeoroids and extreme temperature change. EVA suits, such as the EMU, are used outside spacecraft, for either planetary exploration or spacewalks. They must protect the wearer against all conditions of space, as well as provide mobility and functionality.

Some of these requirements also apply to pressure suits worn for other specialized tasks, such as high-altitude reconnaissance flight. At altitudes above the Armstrong limit, around 19000 m, water boils at body temperature and pressurized suits are needed.

The first full-pressure suits for use at extreme altitudes were designed by individual inventors as early as the 1930s. The first space suit worn by a human in space was the Soviet SK-1 suit worn by Yuri Gagarin in 1961.

Requirements

A space suit must perform several functions to allow its occupant to work safely and comfortably, inside or outside a spacecraft. It must provide:
 * A stable internal pressure. This can be less than Earth's atmosphere, as there is usually no need for the space suit to carry nitrogen (which comprises about 78% of Earth's atmosphere and is not used by the body). Lower pressure allows for greater mobility, but requires the suit occupant to breathe pure oxygen for a time before going into this lower pressure, to avoid decompression sickness.
 * Mobility. Movement is typically opposed by the pressure of the suit; mobility is achieved by careful joint design. See the Theories of space suit design section.
 * Supply of breathable oxygen and elimination of carbon dioxide; these gases are exchanged with the spacecraft or a Portable Life Support System (PLSS)
 * Temperature regulation. Unlike on Earth, where heat can be transferred by convection to the atmosphere, in space, heat can be lost only by thermal radiation or by conduction to objects in physical contact with the exterior of the suit. Since the temperature on the outside of the suit varies greatly between sunlight and shadow, the suit is heavily insulated, and air temperature is maintained at a comfortable level.
 * A communication system, with external electrical connection to the spacecraft or PLSS
 * AstroRad NASA.jpg of collecting and containing solid and liquid bodily waste (such as a Maximum Absorbency Garment)

Secondary requirements
Advanced suits better regulate the astronaut's temperature with a Liquid Cooling and Ventilation Garment (LCVG) in contact with the astronaut's skin, from which the heat is dumped into space through an external radiator in the PLSS.

Additional requirements for EVA include:
 * Shielding against ultraviolet radiation
 * Limited shielding against particle radiation
 * Means to maneuver, dock, release, and tether onto a spacecraft
 * Protection against small micrometeoroids, some traveling at up to 27,000 kilometers per hour, provided by a puncture-resistant Thermal Micrometeoroid Garment, which is the outermost layer of the suit. Experience has shown the greatest chance of exposure occurs near the gravitational field of a moon or planet, so these were first employed on the Apollo lunar EVA suits (see United States suit models below).

As part of astronautical hygiene control (i.e., protecting astronauts from extremes of temperature, radiation, etc.), a space suit is essential for extravehicular activity. The Apollo/Skylab A7L suit included eleven layers in all: an inner liner, a LCVG, a pressure bladder, a restraint layer, another liner, and a Thermal Micrometeoroid Garment consisting of five aluminized insulation layers and an external layer of white Ortho-Fabric. This space suit is capable of protecting the astronaut from temperatures ranging from -156 °C to 121 °C.

During exploration of the Moon or Mars, there will be the potential for lunar or Martian dust to be retained on the space suit. When the space suit is removed on return to the spacecraft, there will be the potential for the dust to contaminate surfaces and increase the risks of inhalation and skin exposure. Astronautical hygienists are testing materials with reduced dust retention times and the potential to control the dust exposure risks during planetary exploration. Novel ingress and egress approaches, such as suitports, are being explored as well.

In NASA space suits, communications are provided via a cap worn over the head, which includes earphones and a microphone. Due to the coloration of the version used for Apollo and Skylab, which resembled the coloration of the comic strip character Snoopy, these caps became known as "Snoopy caps".

Operating pressure
Generally, to supply enough oxygen for respiration, a space suit using pure oxygen must have a pressure of about 32.4 kPa, equal to the 20.7 kPa partial pressure of oxygen in the Earth's atmosphere at sea level, plus 5.3 kPa and 6.3 kPa water vapor pressure, both of which must be subtracted from the alveolar pressure to get alveolar oxygen partial pressure in 100% oxygen atmospheres, by the alveolar gas equation. The latter two figures add to 11.6 kPa, which is why many modern space suits do not use 20.7 kPa, but 32.4 kPa (this is a slight overcorrection, as alveolar partial pressures at sea level are slightly less than the former). In space suits that use 20.7 kPa, the astronaut gets only 20.7 kPa − 11.6 kPa = 9.1 kPa of oxygen, which is about the alveolar oxygen partial pressure attained at an altitude of 1860 m above sea level. This is about 42% of normal partial pressure of oxygen at sea level, about the same as pressure in a commercial passenger jet aircraft, and is the realistic lower limit for safe ordinary space suit pressurization which allows reasonable capacity for work.

Oxygen prebreathing
When space suits below a specific operating pressure are used from craft that are pressurized to normal atmospheric pressure (such as the Space Shuttle), this requires astronauts to "pre-breathe" (meaning pre-breathe pure oxygen for a period) before donning their suits and depressurizing in the air lock. This procedure purges the body of dissolved nitrogen, so as to avoid decompression sickness due to rapid depressurization from a nitrogen-containing atmosphere.

In the US space shuttle, cabin pressure was reduced from normal atmospheric to 70kPa (equivalent to an altitude of about 3000m) for 24 hours before EVA, and after donning the suit, a pre-breathing period of 45 minutes on pure oxygen before decompressing to the EMU working pressure of 30kPa. In the ISS there is no cabin pressure reduction, instead a 4 hour oxygen pre-breathe at normal cabin pressure is used to desaturate nitrogen to an acceptable level. US studies show that a rapid decompression from 101kPa to 55kPa has an acceptable risk, and Russian studies show that direct decompression from 101kPa to 40kPa after 30 minutes of oxygen pre-breathing, roughly the time required for pre-EVA suit checks, is acceptable.

Physiological effects of unprotected space exposure
The human body can briefly survive the hard vacuum of space unprotected, despite contrary depictions in some popular science fiction. Consciousness is retained for up to 15 seconds as the effects of oxygen starvation set in. No snap freeze effect occurs because all heat must be lost through thermal radiation or the evaporation of liquids, and the blood does not boil because it remains pressurized within the body, but human flesh expands up to about twice its volume due to ebullism in such conditions, giving the visual effect of a body builder rather than an overfilled balloon.

In space, there are highly energized subatomic particles that can cause radiation damage by disrupting essential biological processes. Exposure to radiation can create problems via two methods: the particles can react with water in the human body to produce free radicals that break DNA molecules apart, or by directly breaking the DNA molecules.

Temperature in space can vary extremely depending on the exposure to radiant energy sources. Temperatures from solar radiation can reach up to 250 °F, and in its absence, down to -387 °F. Because of this, space suits must provide sufficient insulation and cooling for the conditions in which they will be used.

The vacuum environment of space has no pressure, so gases will expand and exposed liquids may evaporate. Some solids may sublimate. It is necessary to wear a suit that provides sufficient internal body pressure in space. The most immediate hazard is in attempting to hold one's breath during explosive decompression as the expansion of gas can damage the lungs by overexpansion rupture. These effects have been confirmed through various accidents (including in very-high-altitude conditions, outer space and training vacuum chambers). Human skin does not need to be protected from vacuum and is gas-tight by itself. It only needs to be mechanically restrained to retain its normal shape and the internal tissues to retain their volume. This can be accomplished with a tight-fitting elastic body suit and a helmet for containing breathing gases, known as a space activity suit (SAS).

Design concepts
A space suit should allow its user natural unencumbered movement. Nearly all designs try to maintain a constant volume no matter what movements the wearer makes. This is because mechanical work is needed to change the volume of a constant pressure system. If flexing a joint reduces the volume of the space suit, then the astronaut must do extra work every time they bend that joint, and they have to maintain a force to keep the joint bent. Even if this force is very small, it can be seriously fatiguing to constantly fight against one's suit. It also makes delicate movements very difficult. The work required to bend a joint is dictated by the formula

 * $$W=\int_{V_i}^{V_f} \,P\,dV$$

where Vi and Vf are respectively the initial and final volume of the joint, P is the pressure in the suit, and W is the resultant work. It is generally true that all suits are more mobile at lower pressures. However, because a minimum internal pressure is dictated by life support requirements, the only means of further reducing work is to minimize the change in volume.

All space suit designs try to minimize or eliminate this problem. The most common solution is to form the suit out of multiple layers. The bladder layer is a rubbery, airtight layer much like a balloon. The restraint layer goes outside the bladder, and provides a specific shape for the suit. Since the bladder layer is larger than the restraint layer, the restraint takes all of the stresses caused by the pressure inside the suit. Since the bladder is not under pressure, it will not "pop" like a balloon, even if punctured. The restraint layer is shaped in such a way that bending a joint causes pockets of fabric, called "gores", to open up on the outside of the joint, while folds called "convolutes" fold up on the inside of the joint. The gores make up for the volume lost on the inside of the joint, and keep the suit at a nearly constant volume. However, once the gores are opened all the way, the joint cannot be bent any further without a considerable amount of work.

In some Russian space suits, strips of cloth were wrapped tightly around the cosmonaut's arms and legs outside the space suit to stop the space suit from ballooning when in space.

The outermost layer of a space suit, the Thermal Micrometeoroid Garment, provides thermal insulation, protection from micrometeoroids, and shielding from harmful solar radiation.

There are four main conceptual approaches to suit design:

Soft suits
Soft suits typically are made mostly of fabrics. All soft suits have some hard parts; some even have hard joint bearings. Intra-vehicular activity and early EVA suits were soft suits.

Hard-shell suits
Hard-shell suits are usually made of metal or composite materials and do not use fabric for joints. Hard suits joints use ball bearings and wedge-ring segments similar to an adjustable elbow of a stove pipe to allow a wide range of movement with the arms and legs. The joints maintain a constant volume of air internally and do not have any counter-force. Therefore, the astronaut does not need to exert to hold the suit in any position. Hard suits can also operate at higher pressures which would eliminate the need for an astronaut to pre-breathe oxygen to use a 34 kPa space suit before an EVA from a 101 kPa spacecraft cabin. The joints may get into a restricted or locked position requiring the astronaut to manipulate or program the joint. The NASA Ames Research Center experimental AX-5 hard-shell space suit had a flexibility rating of 95%. The wearer could move into 95% of the positions they could without the suit on.

Hybrid suits
Hybrid suits have hard-shell parts and fabric parts. NASA's Extravehicular Mobility Unit (EMU) uses a fiberglass Hard Upper Torso (HUT) and fabric limbs. ILC Dover's I-Suit replaces the HUT with a fabric soft upper torso to save weight, restricting the use of hard components to the joint bearings, helmet, waist seal, and rear entry hatch. Virtually all workable space suit designs incorporate hard components, particularly at interfaces such as the waist seal, bearings, and in the case of rear-entry suits, the back hatch, where all-soft alternatives are not viable.

Skintight suits
Skintight suits, also known as mechanical counterpressure suits or space activity suits, are a proposed design which would use a heavy elastic body stocking to compress the body. The head is in a pressurized helmet, but the rest of the body is pressurized only by the elastic effect of the suit. This mitigates the constant volume problem, reduces the possibility of a space suit depressurization and gives a very lightweight suit. When not worn, the elastic garments may appear to be that of clothing for a small child. These suits may be very difficult to put on and face problems with providing a uniform pressure. Most proposals use the body's natural perspiration to keep cool. Sweat evaporates readily in vacuum and may desublime or deposit on objects nearby: optics, sensors, the astronaut's visor, and other surfaces. The icy film and sweat residue may contaminate sensitive surfaces and affect optical performance.

Contributing technologies
Related preceding technologies include the stratonautical space suit, the gas mask used in World War II, the oxygen mask used by pilots of high-flying bombers in World War II, the high-altitude or vacuum suit required by pilots of the Lockheed U-2 and SR-71 Blackbird, the diving suit, rebreather, scuba diving gear, and many others.

Many space suit designs are taken from the U.S. Air Force suits, which are designed to work in "high-altitude aircraft pressure[s]", such as the Mercury IVA suit or the Gemini G4C, or the Advanced Crew Escape Suits.

Glove technology
The Mercury IVA, the first U.S. space suit design, included lights at the tips of the gloves in order to provide visual aid. As the need for extravehicular activity grew, suits such as the Apollo A7L included gloves made of a metal fabric called Chromel-r in order to prevent punctures. In order to retain a better sense of touch for the astronauts, the fingertips of the gloves were made of silicone. With the shuttle program, it became necessary to be able to operate spacecraft modules, so the ACES suits featured gripping on the gloves. EMU gloves, which are used for spacewalks, are heated to keep the astronaut's hands warm. The Phase VI gloves, meant for use with the Mark III suit, are the first gloves to be designed with "laser scanning technology, 3D computer modeling, stereo lithography, laser cutting technology and CNC machining". This allows for cheaper, more accurate production, as well as increased detail in joint mobility and flexibility.

Life support technology
Prior to the Apollo missions, life support in space suits was connected to the space capsule via an umbilical cable. However, with the Apollo missions, life support was configured into a removable capsule called the Portable Life Support System that allowed the astronaut to explore the Moon without having to be attached to the space craft. The EMU space suit, used for spacewalks, allows the astronaut to manually control the internal environment of the suit. The Mark III suit has a backpack containing about 12 pounds of liquid air for breathing, pressurization, and heat exchange.

Helmet technology
The development of the spheroidal dome helmet was key in balancing the need for field of view, pressure compensation, and low weight. One inconvenience with some space suits is the head being fixed facing forwards and being unable to turn to look sideways. Astronauts call this effect "alligator head".

High-altitude suits

 * Evgeniy Chertovsky created his full-pressure suit or high-altitude "skafandr" (скафандр) in 1931. (скафандр also means "diving suit").
 * Emilio Herrera designed and built a full-pressure "stratonautical space suit" in 1935, which was to have been used during an open-basket balloon stratospheric flight scheduled for early 1936.
 * Wiley Post experimented with a number of pressure suits for record-breaking flights.
 * Russell Colley created the space suits worn by the Project Mercury astronauts, including fitting Alan Shepard for his ride as America's first man in space on May 5, 1961.

Soviet and Russian suit models

 * SK series (CK) – the spacesuit used for the Vostok program (1961–1963). Worn by Yuri Gagarin on the first crewed space flight.
 * No pressure suits were worn aboard Voskhod 1.
 * Berkut (Беркут meaning "golden eagle") – the spacesuit was a modified SK-1 used by the crew of Voskhod 2 which included Alexei Leonov on the first spacewalk during (1965).
 * From Soyuz 1 to Soyuz 11 (1967–1971) no pressure suits were worn during launch and reentry.
 * Yastreb (Ястреб meaning "hawk") – extravehicular activity spacesuit used during a crew exchange between Soyuz 4 and Soyuz 5 (1969).
 * Krechet-94 (Кречет meaning "gyrfalcon") – designed for the canceled Soviet crewed Moon landing.
 * Strizh (Стриж meaning "swift (bird)") – developed for pilots of Buran-class orbiters.
 * Sokol (Сокол meaning "falcon") – suits worn by Soyuz crew members during launch and reentry. They were first worn on Soyuz 12. They have been used from 1973 to present.
 * Orlan (Орлан meaning "sea-eagle" or "bald eagle") – suits for extravehicular activity, originally developed for the Soviet lunar program as a lunar orbit EVA suit. It is Russia's current EVA suit. Used from 1977 to present.

United States suit models

 * In the early 1950s, Siegfried Hansen and colleagues at Litton Industries designed and built a working hard-shell suit, which was used inside vacuum chambers and was the predecessor of space suits used in NASA missions.
 * Navy Mark IV high-altitude/vacuum suit – used for Project Mercury (1961–1963).
 * Gemini space suits (1965–1966) – there were three main variants developed: G3C designed for intra-vehicle use; G4C specially designed for EVA and intra-vehicle use; and a special G5C suit worn by the Gemini 7 crew for 14 days inside the spacecraft.
 * Manned Orbiting Laboratory MH-7 space suits for the canceled MOL program.
 * Apollo Block I A1C suit (1966–1967) – a derivative of the Gemini suit, worn by primary and backup crews in training for two early Apollo missions. The nylon pressure garment melted and burned through in the Apollo 1 cabin fire. This suit became obsolete when crewed Block I Apollo flights were discontinued after the fire.
 * Apollo/Skylab A7L EVA and Moon suits – The Block II Apollo suit was the primary pressure suit worn for eleven Apollo flights, three Skylab flights, and the US astronauts on the Apollo–Soyuz Test Project between 1968 and 1975. The pressure garment's nylon outer layer was replaced with fireproof Beta cloth after the Apollo 1 fire. This suit was the first to employ a liquid-cooled inner garment and outer micrometeoroid garment. Beginning with the Apollo 13 mission, it also introduced "commander's stripes" so that a pair of space walkers will not appear identical on camera.
 * Shuttle Ejection Escape Suit – used from STS-1 (1981) to STS-4 (1982) by a two-man crew used in conjunction with the then-installed ejection seats. Derived from a USAF model. These were removed once the Shuttle became certified.
 * From STS-5 (1982) to STS-51-L (1986) no pressure suits were worn during launch and reentry. The crew would wear only a blue-flight suit with an oxygen helmet.
 * Launch Entry Suit first used on STS-26 (1988), the first flight after the Challenger disaster. It was a partial pressure suit derived from a USAF model. It was used from 1988 to 1998.
 * Advanced Crew Escape Suit used on the Space Shuttle starting in 1994. The Advanced Crew Escape Suit or ACES suit, is a full-pressure suit worn by all Space Shuttle crews for the ascent and entry portions of flight. The suit is a direct descendant of the United States Air Force high-altitude pressure suits worn by SR-71 Blackbird and U-2 spy plane pilots, North American X-15 and Gemini pilot-astronauts, and the Launch Entry Suits worn by NASA astronauts starting on the STS-26 flight. It is derived from a USAF model.
 * Extravehicular Mobility Unit (EMU) – used on both the Space Shuttle and International Space Station (ISS). The EMU is an independent anthropomorphic system that provides environmental protection, mobility, life support, and communications for a Space Shuttle or ISS crew member to perform an EVA in Earth orbit. Used from 1982 to present, but only available in limited sizing as of 2019.
 * Aerospace company SpaceX developed an IVA suit which is worn by astronauts involved in Commercial Crew Program missions operated by SpaceX since the Demo-2 mission (see ). As a continuation of this suit design, SpaceX developed an EVA suit in 2024. The EVA version of the suit will be used during the Polaris Dawn private space mission for the first ever commercial spacewalk.
 * Orion Crew Survival System (OCSS) – will be used during launch and re-entry on the Orion MPCV. It is derived from the Advanced Crew Escape Suit but is able to operate at a higher pressure and has improved mobility in the shoulders.

SpaceX suit ("Starman suit")
In February 2015, SpaceX began developing a space suit for astronauts to wear within the Dragon 2 space capsule. Its appearance was jointly designed by Jose Fernandez—a Hollywood costume designer known for his works for superhero and science fiction films—and SpaceX founder and CEO Elon Musk. The first images of the suit were revealed in September 2017. A mannequin, called "Starman" (after David Bowie's song of the same name), wore the SpaceX space suit during the maiden launch of the Falcon Heavy in February 2018. For this exhibition launch, the suit was not pressurized and carried no sensors.

The suit, which is suitable for vacuum, offers protection against cabin depressurization through a single tether at the astronaut's thigh that feeds air and electronic connections. The helmets, which are 3D-printed, contain microphones and speakers. As the suits need the tether connection and do not offer protection against radiation, they are not used for extra-vehicular activities. The suits are custom-made for each astronaut.

In 2018, NASA commercial crew astronauts Bob Behnken, and Doug Hurley tested the spacesuit inside the Dragon 2 spacecraft in order to familiarize themselves with the suit. They wore it in the Crew Dragon Demo-2 flight launched on 30 May 2020. The suit is worn by astronauts involved in Commercial Crew Program missions involving SpaceX.

On 4 May 2024, SpaceX unveiled a spacesuit designed for extravehicular activity based on the IVA suit. As with the IVA suit, the helmets are 3D-printed, though the EVA helmet incorporates a heads-up display providing information on suit metrics during operation.

Future NASA contracted suits
On 1 June 2022, NASA announced it had selected competing Axiom Space and Collins Aerospace to develop and provide astronauts with next generation spacesuit and spacewalk systems to first test and later use outside the International Space Station, as well as on the lunar surface for the crewed Artemis missions, and prepare for human missions to Mars.

Chinese suit models

 * Shuguang space suit: First generation EVA space suit developed by China for the 1967 canceled Project 714 crewed space program. It has a mass of about 10 kg, has an orange colour, and is made of high-resistance multi-layer polyester fabric. The astronaut could use it inside the cabin and conduct an EVA as well.
 * 'Project 863 space suit: Cancelled project of second generation Chinese EVA space suit.
 * Shenzhou IVA (神舟) space suit: The suit was first worn by Yang Liwei on Shenzhou 5, the first crewed Chinese space flight, it closely resembles a Sokol-KV2 suit, but it is believed to be a Chinese-made version rather than an actual Russian suit. Pictures show that the suits on Shenzhou 6 differ in detail from the earlier suit; they are also reported to be lighter.
 * Haiying (海鹰号航天服) EVA space suit: The imported Russian Orlan-M EVA suit is called Haiying. Used on Shenzhou 7.
 * Feitian (飞天号航天服) EVA space suit: Indigenously developed Chinese-made EVA space suit also used for the Shenzhou 7 mission. The suit was designed for a spacewalk mission of up to seven hours. Chinese astronauts have been training in the out-of-capsule space suits since July 2007, and movements are seriously restricted in the suits, with a mass of more than 110 kg each. A new generation of Feitian space suit has been used since 2021 as the construction of Tiangong Space Station began.

Emerging technologies
Several companies and universities are developing technologies and prototypes which represent improvements over current space suits.

Additive manufacturing
3D printing (additive manufacturing) can be used to reduce the mass of hard-shell space suits while retaining the high mobility they provide. This fabrication method also allows for the potential for in-situ fabrication and repair of suits, a capability which is not currently available, but will likely be necessary for Martian exploration. The University of Maryland began development of a prototype 3D printed hard suit in 2016, based on the kinematics of the AX-5. The prototype arm segment is designed to be evaluated in the Space Systems Laboratory glovebox to compare mobility to traditional soft suits. Initial research has focused on the feasibility of printing rigid suit elements, bearing races, ball bearings, seals, and sealing surfaces.

Astronaut Glove Challenge
There are certain difficulties in designing a dexterous space suit glove and there are limitations to the current designs. For this reason, the Centennial Astronaut Glove Challenge was created to build a better glove. Competitions have been held in 2007 and 2009, and another is planned. The 2009 contest required the glove to be covered with a micro-meteorite layer.

Aouda.X
Since 2009, the Austrian Space Forum has been developing "Aouda.X", an experimental Mars analogue space suit focusing on an advanced human–machine interface and on-board computing network to increase situational awareness. The suit is designed to study contamination vectors in planetary exploration analogue environments and create limitations depending on the pressure regime chosen for a simulation.

Since 2012, for the Mars2013 analogue mission by the Austrian Space Forum to Erfoud, Morocco, the Aouda.X analogue space suit has a sister in the form of Aouda.S. This is a slightly less sophisticated suit meant primarily to assist Aouda.X operations and be able to study the interactions between two (analogue) astronauts in similar suits.

The Aouda.X and Aouda.S space suits have been named after the fictional princess from the Jules Verne's 1873 novel Around the World in Eighty Days. A public display mock-up of Aouda.X (called Aouda.D) is currently on display at the Dachstein Ice Cave in Obertraun, Austria, after the experiments done there in 2012.

Bio-Suit
Bio-Suit is a space activity suit under development at the Massachusetts Institute of Technology, which consisted of several lower leg prototypes. Bio-suit is custom fit to each wearer, using laser body scanning.

Constellation Space Suit system
On August 2, 2006, NASA indicated plans to issue a Request for Proposal (RFP) for the design, development, certification, production, and sustaining engineering of the Constellation Space Suit to meet the needs of the Constellation Program. NASA foresaw a single suit capable of supporting: survivability during launch, entry and abort; zero-gravity EVA; lunar surface EVA; and Mars surface EVA.

On June 11, 2008, NASA awarded a US$745 million contract to Oceaneering International to create the new space suit.

Final Frontier Design IVA Space Suit
Final Frontier Design (FFD) is developing a commercial full IVA space suit, with their first suit completed in 2010. FFD's suits are intended as a light-weight, highly mobile, and inexpensive commercial space suits. Since 2011, FFD has upgraded IVA suit's designs, hardware, processes, and capabilities. FFD has built a total of 7 IVA space suit (2016) assemblies for various institutions and customers since founding, and has conducted high fidelity human testing in simulators, aircraft, microgravity, and hypobaric chambers. FFD has a Space Act Agreement with NASA's Commercial Space Capabilities Office to develop and execute a Human Rating Plan for FFD IVA suit. FFD categorizes their IVA suits according to their mission: Terra for Earth-based testing, Stratos for high altitude flights, and Exos for orbital space flights. Each suit category has different requirements for manufacturing controls, validations, and materials, but are of a similar architecture.

I-Suit
The I-Suit is a space suit prototype also constructed by ILC Dover, which incorporates several design improvements over the EMU, including a weight-saving soft upper torso. Both the Mark III and the I-Suit have taken part in NASA's annual Desert Research and Technology Studies (D-RATS) field trials, during which suit occupants interact with one another, and with rovers and other equipment.

Mark III
The Mark III is a NASA prototype, constructed by ILC Dover, which incorporates a hard lower torso section and a mix of soft and hard components. The Mark III is markedly more mobile than previous suits, despite its high operating pressure (57 kPa), which makes it a "zero-prebreathe" suit, meaning that astronauts would be able to transition directly from a one-atmosphere, mixed-gas space station environment, such as that on the International Space Station, to the suit, without risking decompression sickness, which can occur with rapid depressurization from an atmosphere containing nitrogen or another inert gas.

MX-2
The MX-2 is a space suit analogue constructed at the University of Maryland's Space Systems Laboratory. The MX-2 is used for crewed neutral buoyancy testing at the Space Systems Lab's Neutral Buoyancy Research Facility. By approximating the work envelope of a real EVA suit, without meeting the requirements of a flight-rated suit, the MX-2 provides an inexpensive platform for EVA research, compared to using EMU suits at facilities like NASA's Neutral Buoyancy Laboratory.

The MX-2 has an operating pressure of 2.5–4 psi. It is a rear-entry suit, featuring a fiberglass HUT. Air, LCVG cooling water, and power are open loop systems, provided through an umbilical. The suit contains a Mac Mini computer to capture sensor data, such as suit pressure, inlet and outlet air temperatures, and heart rate. Resizable suit elements and adjustable ballast allow the suit to accommodate subjects ranging in height from 68 to(-), and with a weight range of 120 lb.

North Dakota suit
Beginning in May 2006, five North Dakota colleges collaborated on a new space suit prototype, funded by a US$100,000 grant from NASA, to demonstrate technologies which could be incorporated into a planetary suit. The suit was tested in the Theodore Roosevelt National Park badlands of western North Dakota. The suit has a mass of 47 lb without a life support backpack, and costs only a fraction of the standard US$12,000,000 cost for a flight-rated NASA space suit. The suit was developed in just over a year by students from the University of North Dakota, North Dakota State, Dickinson State, the state College of Science and Turtle Mountain Community College. The mobility of the North Dakota suit can be attributed to its low operating pressure; while the North Dakota suit was field tested at a pressure of 1 psi differential, NASA's EMU suit operates at a pressure of 4.7 psi, a pressure designed to supply approximately sea-level oxygen partial pressure for respiration (see discussion above).

PXS
NASA's Prototype eXploration Suit (PXS), like the Z-series, is a rear-entry suit compatible with suitports. The suit has components which could be 3D printed during missions to a range of specifications, to fit different individuals or changing mobility requirements.

Suitports
A suitport is a theoretical alternative to an airlock, designed for use in hazardous environments and in human spaceflight, especially planetary surface exploration. In a suitport system, a rear-entry space suit is attached and sealed against the outside of a spacecraft, such that an astronaut can enter and seal up the suit, then go on EVA, without the need for an airlock or depressurizing the spacecraft cabin. Suitports require less mass and volume than airlocks, provide dust mitigation, and prevent cross-contamination of the inside and outside environments. Patents for suitport designs were filed in 1996 by Philip Culbertson Jr. of NASA's Ames Research Center and in 2003 by Joerg Boettcher, Stephen Ransom, and Frank Steinsiek.

Z-series
In 2012, NASA introduced the Z-1 space suit, the first in the Z-series of space suit prototypes designed by NASA specifically for planetary extravehicular activity. The Z-1 space suit includes an emphasis on mobility and protection for space missions. It features a soft torso versus the hard torsos seen in previous NASA EVA space suits, which reduces mass. It has been labeled the "Buzz Lightyear suit" due to its green streaks for a design.

In 2014, NASA released the design for the Z-2 prototype, the next model in the Z-series. NASA conducted a poll asking the public to decide on a design for the Z-2 space suit. The designs, created by fashion students from Philadelphia University, were "Technology", "Trends in Society", and "Biomimicry". The design "Technology" won, and the prototype is built with technologies like 3D printing. The Z-2 suit will also differ from the Z-1 suit in that the torso reverts to the hard shell, as seen in NASA's EMU suit.

In fiction
The earliest space fiction ignored the problems of traveling through a vacuum, and launched its heroes through space without any special protection. In the later 19th century, however, a more realistic brand of space fiction emerged, in which authors have tried to describe or depict the space suits worn by their characters. These fictional suits vary in appearance and technology, and range from the highly authentic to the utterly improbable.

A very early fictional account of space suits can be seen in Garrett P. Serviss' novel Edison's Conquest of Mars (1898). Later comic book series such as Buck Rogers (1930s) and Dan Dare (1950s) also featured their own takes on space suit design. Science fiction authors such as Robert A. Heinlein contributed to the development of fictional space suit concepts.

title: Space suit


text:
Engagement ring

An engagement ring, also known as a betrothal ring, is a ring indicating that the person wearing it is engaged to be married, especially in Western cultures. A ring is presented as an engagement gift by a partner to their prospective spouse when they propose marriage to represent a formal agreement to future marriage. In most Western countries, engagement rings are usually worn only by women, and they are typically adorned with diamonds. In some countries, partners wear matching rings, and engagement rings may also be used as wedding rings. In the Anglosphere, the ring is customarily worn on the left hand ring finger, but customs vary across the world.

Engagement rings have been common in Western countries since at least the time of the Roman Empire. They began to feature diamonds during the Renaissance, although most commoners could not afford diamond rings prior to the discovery of the South African diamond mines in the 19th century. Historically, engagement rings were blessed and then worn during the betrothal ceremony of a couple, but neither the engagement ring nor any other ring is worn at the time when the wedding ring is put by the groom on the finger of the bride as part of the marriage ceremony, and sometimes by the bride onto the groom's finger. After the wedding, the engagement ring is usually put back on and is usually worn on the outside of the wedding ring. In the present-day, the giving of the engagement ring "constitutes the subarration".

Ancient times
Although the ancient Egyptians are sometimes credited with inventing the engagement ring, and the ancient Greeks with adopting the tradition, the history of the engagement ring can only be reliably traced as far back as ancient Rome.

In many countries, engagement rings are placed on the ring finger of the left hand. At one time it was believed that this finger contained a vein (the vena amoris) that led to the heart. This idea was popularized by Henry Swinburne in A treatise of Spousals, or Matrimonial Contracts (1686). The story seems to have its origin in the ancient Roman book Attic Nights by Aulus Gellius quoting Apion's Aegyptiacorum, where the alleged vein was originally a nervus (a word that can be translated either as "nerve" or "sinew").

The popular belief that an engagement ring was originally part of the bride price which represented purchase and ownership of the bride, has been called into question by contemporary scholarship.

In the second century BC, the Roman bride-to-be was given two rings, a gold one which she wore in public, and one made of iron which she wore at home while attending to household duties. At one time Roman citizens wore rings made of iron. In later years senators who served as ambassadors were given gold seal rings for official use when abroad. Later the privilege of wearing gold rings was extended to other public officials, then to the knights, later to all freeborn, and finally under Justinian, to freedmen. For several centuries it was the custom for Romans to wear iron rings at home, gold rings in public. During this period a girl or woman might receive two engagement rings, one of iron and one of gold.

Middle ages
The mid-7th century Visigothic Code required "that when the ceremony of betrothal has been performed, ..., and the ring shall have been given or accepted as a pledge, although nothing may have been committed to writing, the promise shall, under no circumstances, be broken."

In 860 AD, Pope Nicholas I wrote a letter to Boris I of Bulgaria in reply to questions regarding differences between Roman Catholic and Eastern Orthodox practices. Pope Nicholas describes how in the Western church the man gives his betrothed an engagement ring. At the Fourth Council of the Lateran in 1215, convoked by Pope Innocent III, the banns of marriage was instituted, prohibiting clandestine marriages and requiring that marriages be made public in advance. Some legal scholars have seen in this a parallel with the engagement-ring tradition described by Pope Nicholas I.

Renaissance
The first well-documented use of a diamond ring to signify engagement was by the Archduke Maximilian of Austria in the imperial court of Vienna in 1477, upon his betrothal to Mary of Burgundy. This then influenced those of higher social class and of significant wealth to give diamond rings to their loved ones.

Reformation
During the Protestant Reformation the wedding ring replaced the betrothal ring as the primary ring associated with marriage. In Catholic countries the transition took place somewhat later.

Enlightenment
During the Age of Enlightenment both the gimmal rings and posie rings were popular, although the latter was more often used as an expression of sentiment than to indicate a formal engagement.

Victorian era
In South Africa, diamonds were first found in 1866, although they were not identified as such until 1867. By 1872, the output of the diamond mines exceeded one million carats per year. As production increased, those of lesser means were able to join in on this movement. However, diamond engagement rings were for a long time seen as the domain of the nobility and aristocracy, and tradition often favoured simpler engagement bands.

In 1852, the Koh-i-Noor diamond was re-cut and embellished in Queen Victoria’s crown. This triggered a diamond rush throughout the world.

20th century
In the United States, the popularity of diamond engagement rings declined after World War I, even more so than after the onset of the Great Depression.

In 1938, the diamond cartel De Beers began a marketing campaign that would have a major impact on engagement rings. During the Great Depression of the 1930s, the price of diamonds collapsed. At the same time, market research indicated that engagement rings were going out of style with the younger generation. Before World War II, only 10% of American engagement rings contained a diamond. While the first phase of the marketing campaign consisted of market research, the advertising phase began in 1939. One of the first elements of this campaign was to educate the public about the 4 Cs (cut, carats, color, and clarity). In 1947 the slogan "a diamond is forever" was introduced. Ultimately, the De Beers campaign sought to persuade the consumer that an engagement ring is indispensable, and that a diamond is the only acceptable stone for an engagement ring. The sales of diamonds in the United States rose from $23 million to $2.1 billion between 1939 and 1979.

Law professor Margart F. Brining links the surge in engagement ring sales in the USA after 1945 to the abolishment of the "breach of promise", that had entitled a woman whose fiancé had broken off their engagement to sue him for damages. This rule of law was especially important for many women who had been sexually intimate with the fiancé, but were socially expected to be virgins in a new marriage, therefore lost "market value". After the gradual abolishment of that law action in all states the expensive engagement ring rose to popularity as a new financial security in case of a break-up, since it was custom for the women to keep the ring (partly only under the condition that the break-up was not seen as her fault).

21st century
In the early 21st century, the jewellery industry started marketing engagement rings for men under the name "mangagement rings".

Industry
In the 20th century, if he could afford it, the typical Western groom privately selected and purchased an engagement ring, which he then presented to his desired bride when he proposed marriage. In countries where both partners wear engagement rings, matching rings may be selected and purchased together. In the United States and Canada, where only women traditionally wear engagement rings, women also occasionally present their partners with an engagement gift.

Like all jewellery, the price for an engagement ring varies considerably depending on the materials used: the design of the ring, whether it includes a gemstone, the value of any gemstone, and the seller. The price of the gemstones, if any, in the ring depends on the type and quality of the gem. Diamonds have a standardized description that values them according to their carat weight, color, clarity, and cut. Other gemstones, such as sapphires, rubies, and emeralds have different systems. The jewellery may be chosen to honor a family tradition, to use family heirlooms, to have an unusual style, to have socially responsible characteristics (e.g., a style that is not associated with blood diamond controversy or the pollution caused by gold mining and cyanide process), to fit the individual's stylistic preferences, or to manage cost. Synthetic diamonds and diamond substitutes such as cubic zirconias and moissanites are also popular choices that are socially responsible and reduce cost while maintaining the desired appearance.

The idea that a man should spend a significant fraction of his annual income for an engagement ring originated from De Beers marketing materials in the mid-20th century in an effort to increase the sale of diamonds. In the 1930s, they suggested that a man should spend the equivalent of one month's income in the engagement ring. In the 1980s, they suggested that he should spend two months' income on it (three months in Japan). In 2012, the average cost of an engagement ring in the US as reported by the industry was US$4,000. In a 2015 scholarly study, almost a quarter of couples said that they did not buy a ring, and another third spent less than US$2000 on it. Less than 15% of couples spent $4,000 or more. In the UK, estimates of the average cost of an engagement ring range from £1200 to £2000. Scholarly research indicates that expensive engagement rings are associated with early divorces, possibly because spending more than US$2,000 on an engagement ring is strongly associated with debt-related stress. Couples that spend less money on engagement rings and the wedding ceremony tend to have longer marriages and a lower risk of divorce.

One reason for the increased popularity of expensive engagement rings is its relationship to human sexuality and the woman's marriage prospects. In the United States, until the Great Depression, a man who broke off a marriage engagement could be sued for breach of promise. Monetary damages included actual expenses incurred in preparing for the wedding, plus damages for emotional distress and loss of other marriage prospects. Damages were greatly increased if the woman had engaged in sexual intercourse with her fiancé. Beginning in 1935, these laws were repealed or limited. However, the social and financial cost of a broken engagement was no less: marriage was the only financially sound option for most women, and if she was no longer a virgin, her prospects for a suitable future marriage were greatly decreased. The diamond engagement ring thus became a source of financial security for the woman.

The online purchase of engagement rings is growing, disrupting the market for the diamonds by bringing greater transparency to an industry that has traditionally relied on opacity. Online diamond retailers and e-commerce platforms include Blue Nile, Brilliant Earth, and Costco.

Styles
Engagement rings, like any other kind of jewellery, come in many different styles.

Gold (Available in Yellow Gold, White Gold and Rose Gold) and platinum are preferred for engagement rings, but common metal types such as titanium, silver, and stainless steel are also used for engagement rings. This allows for the bride-to-be to exert her own individual style into the ring in a simple manner.

In the United States, where engagement rings are worn by women, diamonds have been widely featured in engagement rings since the middle of the 20th century. Solitaire rings have one diamond. The most common setting for engagement rings is the solitaire prong setting, which was popularized by Tiffany & Co. in 1886 and its six-claw prong setting design sold under the "Tiffany setting" trademark. The modern favorite cut for an engagement ring is the brilliant cut, which provides the maximum amount of sparkle to the gemstone. The traditional engagement rings may have different prong settings and bands. Another major category is engagement rings with side stones. Rings with a larger diamond set in the middle and smaller diamonds on the side fit under this category. Three-stone diamond engagement rings, sometimes called trinity rings or trilogy rings, are rings with three matching diamonds set horizontally in a row with the bigger stone placed in the center. The three diamonds on the ring are typically said to represent the couple's past, present, and future, but other people give religious significance to the arrangement.

A wedding set, or bridal set, includes an engagement ring and a wedding band that matches and can be bought as a set. In some cases, the wedding ring looks incomplete; it is only when the two halves, engagement and wedding, are assembled that the ring looks whole. In other cases, a wedding set consists of two rings that match stylistically and are worn stacked, although either piece would look appropriate as a separate ring. Although the wedding band is not to be worn until the wedding day, the two rings are usually sold together as a wedding set. After the wedding, the bride may choose to have the two pieces welded together, to increase convenience and reduce the likelihood of losing one of the rings. A trio ring set includes a women's engagement ring, a women's wedding band, and a men's wedding band. These sets often have matching rings and are lower in price.

In Nordic countries, engagement rings are worn by both men and women. Traditionally they are plain gold bands, although more ornate designs and other materials are gaining popularity. The engagement rings resemble the wedding bands sold in the United States, whereas women's wedding rings may resemble US engagement rings.

In North America and the United Kingdom, it is customarily worn on the left hand ring finger. Similar traditions purportedly date to classical times, dating back from an early usage reportedly referring to the fourth finger of the left hand as containing the vena amoris or "vein of love". This custom may have its origins in an ancient Egyptian myth that the finger contained a vein leading directly to the heart, or it may simply be because the heart lies slightly to the left side of the body. In Germany the ring is worn on the left hand while engaged, but moved to the right hand when married. In Poland and Turkey, the engagement ring and wedding band are traditionally worn on the right hand but modern practice varies considerably.

Legal ownership
Tradition generally holds that if the betrothal fails because the man himself breaks off the engagement, the woman is not obliged to return the ring. This reflects the ring's role as a form of compensation for the woman's damaged reputation. Legally, this condition can be subject to either a modified or a strict fault rule. Under the former, the fiancé can demand the return of the ring unless he breaks the engagement. Under the latter, the fiancé is entitled to the return unless his actions caused the breakup of the relationship, the same as the traditional approach. However, a no-fault rule is being advanced in some jurisdictions, under which the fiancé is always entitled to the return of the ring. The ring only becomes the property of the woman when marriage occurs. An unconditional gift approach is another possibility, wherein the ring is always treated as a gift, to be kept by the fiancée whether or not the relationship progresses to marriage. Recent court rulings have determined that the date in which the ring was offered can determine the condition of the gift. E.g. Valentine's Day and Christmas are widely recognized as gift-giving holidays in the United States and some other countries. A ring offered in the form of a Christmas present is likely to remain the personal property of the recipient in the event of a breakup.

In most states of the United States, engagement rings are considered "conditional gifts" under the legal rules of property. This is an exception to the general rule that gifts cannot be revoked once properly given. See, for example, the case of Meyer v. Mitnick, 625 N.W.2d 136 (Michigan, 2001), whose ruling found the following reasoning persuasive: "the so-called 'modern trend' holds that because an engagement ring is an inherently conditional gift, once the engagement has been broken, the ring should be returned to the donor. Thus, the question of who broke the engagement and why, or who was 'at fault,' is irrelevant. This is the no-fault line of cases." Though in certain states, whether a judicial action can be maintained at all to require return of an engagement ring is blocked by statute, as many states have statutes which state that no civil action shall be maintained for breach of promise to marry.

One case in New South Wales, Australia ended in the man suing his former fiancée because she threw the ring away, after he told her she could keep it even though the marriage plans had fallen through. The Supreme Court of New South Wales held that, despite what the man said, the ring remained a conditional gift (partly because his saying that she could keep it reflected his desire to salvage the relationship) and she was ordered to pay him its A$15,250 cost.

In England and Wales, the gift of an engagement ring is presumed to be an absolute gift to the fiancée. This presumption may be rebutted however by proving that the ring was given on condition (express or implied) that it must be returned if the marriage did not take place, for whatever reason. This was decided in the case Jacobs v Davis (1917).

Cultural variations
In the historic Christian traditions of Catholicism, Lutheranism and Anglicanism, engagement rings are blessed by a priest or pastor and then worn by the couple during the betrothal rite (also known as 'blessing an engaged couple' or 'declaration of intention').

In some countries it is common for both men and women to wear engagement rings. The rings are often in the form of a plain band of a precious metal. Sometimes, the engagement ring eventually serves as the wedding ring for the man. In Brazil, for example, the groom and bride-to-be usually wear a plain wedding band on the right hand during the course of their engagement. After the wedding, the band is moved to the left hand. In Argentina, it is also known for the groom and bride-to-be to wear a plain silver band on the left hand while engaged. Then, after the wedding the silver band is either replaced with the wedding ring or moved to the right hand.

Traditionally, women in the British Isles may propose marriage to men during a leap year. Women proposing has become more common in recent years, to the point that some jewellery companies have started manufacturing men's engagement rings. They resemble typical men's rings, often with a diamond centrepiece. In the countries where both sexes have traditionally worn engagement rings, the rings tend to be plainer bands, and there is no real difference between men's and women's engagement ring designs.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Engagement ring


text:
Similarity (geometry)

In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.

For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. This is because two ellipses can have different width to height ratios, two rectangles can have different length to breadth ratios, and two isosceles triangles can have different base angles. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure.

Two congruent shapes are similar, with a scale factor of 1. However, some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar.

Similar triangles
Two triangles, $△ABC$ and $△A'B'C'$ are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional. It can be shown that two triangles having congruent angles (equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.

There are several criteria each of which is necessary and sufficient for two triangles to be similar:

 * Any two pairs of angles are congruent, which in Euclidean geometry implies that all three angles are congruent:

 * If $∠BAC$ is equal in measure to $∠B'A'C',$ and $∠ABC$ is equal in measure to $∠A'B'C',$ then this implies that $∠ACB$ is equal in measure to $∠A'C'B'$ and the triangles are similar.

 * All the corresponding sides are proportional:

$$\frac{\overline{AB}}{\overline{A'B'}} = \frac{\overline{BC}}{\overline{B'C'}} = \frac{\overline{AC}}{\overline{A'C'}}.$$
 * This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.

$$ \frac{\overline{AB}}{\overline{A'B'}} = \frac{\overline{BC}}{\overline{B'C'}}, \quad \angle ABC \cong \angle A'B'C'.$$
 * Any two pairs of sides are proportional, and the angles included between these sides are congruent:
 * This is known as the SAS similarity criterion. The "SAS" is a mnemonic: each one of the two S's refers to a "side"; the A refers to an "angle" between the two sides.

Symbolically, we write the similarity and dissimilarity of two triangles $△ABC$ and $△A'B'C'$ as follows:

$$\begin{align} \triangle ABC &\sim \triangle A'B'C' \\ \triangle ABC &\nsim \triangle A'B'C' \end{align}$$

There are several elementary results concerning similar triangles in Euclidean geometry:
 * Any two equilateral triangles are similar.
 * Two triangles, both similar to a third triangle, are similar to each other (transitivity of similarity of triangles).
 * Corresponding altitudes of similar triangles have the same ratio as the corresponding sides.
 * Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio. There are several equivalent conditions in this case, such as the right triangles having an acute angle of the same measure, or having the lengths of the legs (sides) being in the same proportion.

Given a triangle $△ABC$ and a line segment $\overline{DE}$ one can, with a ruler and compass, find a point $F$ such that $△ABC ~ △DEF$. The statement that point $F$ satisfying this condition exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry (where Wallis's postulate is false) similar triangles are congruent.

In the axiomatic treatment of Euclidean geometry given by George David Birkhoff (see Birkhoff's axioms) the SAS similarity criterion given above was used to replace both Euclid's parallel postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms.

Similar triangles provide the basis for many synthetic (without the use of coordinates) proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles also provide the foundations for right triangle trigonometry.

Other similar polygons
The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence (even if clockwise for one polygon and counterclockwise for the other) are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles (otherwise, for example, all rhombi would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all rectangles would be similar). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.

For given $n$, all regular $n$-gons are similar.

Similar curves
Several types of curves have the property that all examples of that type are similar to each other. These include:
 * Lines (any two lines are even congruent)
 * Line segments
 * Circles
 * Parabolas
 * Hyperbolas of a specific eccentricity
 * Ellipses of a specific eccentricity
 * Catenaries
 * Graphs of the logarithm function for different bases
 * Graphs of the exponential function for different bases
 * Logarithmic spirals are self-similar

In Euclidean space
A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection $f$ from the space onto itself that multiplies all distances by the same positive real number $r$, so that for any two points $x$ and $y$ we have

 * $$d(f(x),f(y)) = r\, d(x,y), $$

where $d(x,y)$ is the Euclidean distance from $x$ to $y$. The scalar $r$ has many names in the literature including; the ratio of similarity, the stretching factor and the similarity coefficient. When $r = 1$ a similarity is called an isometry (rigid transformation). Two sets are called similar if one is the image of the other under a similarity.

As a map $f : \R^n \to \R^n,$ a similarity of ratio $r$ takes the form

 * $$f(x) = rAx + t,$$

where $A \in O^n(\R)$ is an $n × n$ orthogonal matrix and $t \in \R^n$ is a translation vector.

Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments. Similarities preserve angles but do not necessarily preserve orientation, direct similitudes preserve orientation and opposite similitudes change it.

The similarities of Euclidean space form a group under the operation of composition called the similarities group $S$. The direct similitudes form a normal subgroup of $S$ and the Euclidean group $E(n)$ of isometries also forms a normal subgroup. The similarities group $S$ is itself a subgroup of the affine group, so every similarity is an affine transformation.

One can view the Euclidean plane as the complex plane, that is, as a 2-dimensional space over the reals. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by where $a$ and $b$ are complex numbers, $a ≠ 0$. When $|a|= 1$, these similarities are isometries.
 * $$f(z) = az + b$$ (direct similitudes), and
 * $$f(z) = a\overline z + b$$ (opposite similitudes),

Area ratio and volume ratio
[[File:A proportion to conceive square root of 5.svg|left|thumb|upright=1.26
 * The tessellation of the large triangle shows that it is similar to the small triangle with an area ratio of 5. The similarity ratio is $$\tfrac{5}{h} = \tfrac{h}{1} = \sqrt 5.$$ This can be used to construct a non-periodic infinite tiling.]]

The ratio between the areas of similar figures is equal to the square of the ratio of corresponding lengths of those figures (for example, when the side of a square or the radius of a circle is multiplied by three, its area is multiplied by nine — i.e. by three squared). The altitudes of similar triangles are in the same ratio as corresponding sides. If a triangle has a side of length $b$ and an altitude drawn to that side of length $h$ then a similar triangle with corresponding side of length $kb$ will have an altitude drawn to that side of length $kh$. The area of the first triangle is $$A = \tfrac{1}{2}bh,$$ while the area of the similar triangle will be $$A' =\frac{1}{2} \cdot kb \cdot kh = k^2A.$$ Similar figures which can be decomposed into similar triangles will have areas related in the same way. The relationship holds for figures that are not rectifiable as well.

The ratio between the volumes of similar figures is equal to the cube of the ratio of corresponding lengths of those figures (for example, when the edge of a cube or the radius of a sphere is multiplied by three, its volume is multiplied by 27 — i.e. by three cubed).

Galileo's square–cube law concerns similar solids. If the ratio of similitude (ratio of corresponding sides) between the solids is $k$, then the ratio of surface areas of the solids will be $k^{2}$, while the ratio of volumes will be $k^{3}$.

Similarity with a center
If a similarity has exactly one invariant point: a point that the similarity keeps unchanged, then this only point is called "center" of the similarity.

On the first image below the title, on the left, one or another similarity shrinks a regular polygon into a concentric one, the vertices of which are each on a side of the previous polygon. This rotational reduction is repeated, so the initial polygon is extended into an abyss of regular polygons. The center of the similarity is the common center of the successive polygons. A red segment joins a vertex of the initial polygon to its image under the similarity, followed by a red segment going to the following image of vertex, and so on to form a spiral. Actually we can see more than three direct similarities on this first image, because every regular polygon is invariant under certain direct similarities, more precisely certain rotations the center of which is the center of the polygon, and a composition of direct similarities is also a direct similarity. For example we see the image of the initial regular pentagon under a homothety of negative ratio $S$, which is a similarity of ±180° angle and a positive ratio equal to $–k$.

Below the title on the right, the second image shows a similarity decomposed into a rotation and a homothety. Similarity and rotation have the same angle of +135 degrees modulo 360 degrees. Similarity and homothety have the same ratio of $k$ multiplicative inverse of the ratio $\tfrac{\sqrt 2}{2},$ (square root of 2) of the inverse similarity. Point $\sqrt 2$ is the common center of the three transformations: rotation, homothety and similarity. For example point $S$ is the image of $W$ under the rotation, and point $F$ is the image of $T$ under the homothety, more briefly $$T = H(W) = (R(F)) = (H \circ R)(F) = D(F),$$ by naming $W$, $R$ and $H$ the previous rotation, homothety and similarity, with “$D$" like "Direct".

This direct similarity that transforms triangle $△EFA$ into triangle $△ATB$ can be decomposed into a rotation and a homothety of same center $D$ in several manners. For example, $D = R ○ H = H ○ R$, the last decomposition being only represented on the image. To get $S$ we can also compose in any order a rotation of –45° angle and a homothety of ratio $D$

With "$\tfrac{- \sqrt 2}{2}.$" like "Mirror" and "$M$" like "Indirect", if $I$ is the reflection with respect to line $M$, then $M ○ D = I$ is the indirect similarity that transforms segment $CW$ like $\overline{BF}$ into segment $D$, but transforms point $\overline{CT}$ into $E$ and point  $B$ into  $A$ itself. Square $A$ is the image of $ACBT$ under similarity $ABEF$ of ratio $I$ Point $\tfrac{1}{\sqrt 2}.$ is the center of this similarity because any point $A$ being invariant under it fulfills $$AK = \tfrac{AK}{\sqrt 2},$$ only possible if $AK = 0$, otherwise written $A = K$.

How to construct the center $K$ of direct similarity $S$ from square $D$, how to find point $ABEF$ center of a rotation of +135° angle that transforms ray $S$ into ray $\overset{}\overrightarrow{SE}$? This is an inscribed angle problem plus a question of orientation. The set of points $\overset{}\overrightarrow{SA}$ such that $$\overset{}{ \overrightarrow{PE}, \overrightarrow{PA} = +135^\circ }$$ is an arc of circle $P$ that joins $\overarc{EA}$ and  $E$, of which the two radius leading to  $A$ and  $E$ form a central angle of $2(180° – 135°) = 2 × 45° = 90°$. This set of points is the blue quarter of circle of center $A$ inside square $F$. In the same manner, point $ABEF$ is a member of the blue quarter of circle of center $S$ inside square $T$. So point $BCAT$ is the intersection point of these two quarters of circles.

In general metric spaces

In a general metric space $(X, d)$, an exact similitude is a function $S$ from the metric space $f$ into itself that multiplies all distances by the same positive scalar $X$, called $r$ 's contraction factor, so that for any two points $f$ and $x$ we have

$$d(f(x),f(y)) = r d(x,y).$$

Weaker versions of similarity would for instance have $y$ be a bi-Lipschitz function and the scalar $f$ a limit

$$\lim \frac{d(f(x),f(y))}{d(x,y)} = r. $$

This weaker version applies when the metric is an effective resistance on a topologically self-similar set.

A self-similar subset of a metric space $(X, d)$ is a set $r$ for which there exists a finite set of similitudes ${ fs}s∈S$ with contraction factors $0 ≤ rs < 1$ such that $K$ is the unique compact subset of $K$ for which

 * [[File:Epi17.png|thumb|A self-similar set constructed with two similitudes:

$$\begin{align} z' &= 0.1[(4+i)z+4] \\ z' &= 0.1[(4+7i)z^* + 5 - 2i] \end{align}$$]]

$$\bigcup_{s\in S} f_s(K)=K.$$

These self-similar sets have a self-similar measure $μD$ with dimension $X$ given by the formula

$$\sum_{s\in S} (r_s)^D=1 $$

which is often (but not always) equal to the set's Hausdorff dimension and packing dimension. If the overlaps between the $fs(K)$ are "small", we have the following simple formula for the measure:

$$\mu^D(f_{s_1}\circ f_{s_2} \circ \cdots \circ f_{s_n}(K)) = (r_{s_1}\cdot r_{s_2}\cdots r_{s_n})^D.\,$$

Topology
In topology, a metric space can be constructed by defining a similarity instead of a distance. The similarity is a function such that its value is greater when two points are closer (contrary to the distance, which is a measure of dissimilarity: the closer the points, the lesser the distance).

The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are $$\forall (a,b), S(a,b)\geq 0$$ $$S (a,b) \leq S (a,a) \quad \text{and} \quad \forall (a,b), S (a,b) = S (a,a) \Leftrightarrow a=b$$
 * 1) Positive defined:
 * 1) Majored by the similarity of one element on itself (auto-similarity):

More properties can be invoked, such as: The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).
 * Reflectivity: $$\forall (a,b)\ S (a,b) = S (b,a),$$ or
 * Finiteness: $$\forall (a,b)\ S(a,b) < \infty.$$

Note that, in the topological sense used here, a similarity is a kind of measure. This usage is not the same as the similarity transformation of the and  sections of this article.

Self-similarity
Self-similarity means that a pattern is non-trivially similar to itself, e.g., the set ${..., 0.5, 0.75, 1, 1.5, 2, 3, 4, 6, 8, 12, ...}$ of numbers of the form ${2i, 3·2i}$ where $D$ ranges over all integers. When this set is plotted on a logarithmic scale it has one-dimensional translational symmetry: adding or subtracting the logarithm of two to the logarithm of one of these numbers produces the logarithm of another of these numbers. In the given set of numbers themselves, this corresponds to a similarity transformation in which the numbers are multiplied or divided by two.

Psychology
The intuition for the notion of geometric similarity already appears in human children, as can be seen in their drawings.

title: Similarity (geometry)


text:
Distance

Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). The term is also frequently used metaphorically to mean a measurement of the amount of difference between two similar objects (such as statistical distance between probability distributions or edit distance between strings of text) or a degree of separation (as exemplified by distance between people in a social network). Most such notions of distance, both physical and metaphorical, are formalized in mathematics using the notion of a metric space.

In the social sciences, distance can refer to a qualitative measurement of separation, such as social distance or psychological distance.

Distances in physics and geometry
The distance between physical locations can be defined in different ways in different contexts.

Straight-line or Euclidean distance
The distance between two points in physical space is the length of a straight line between them, which is the shortest possible path. This is the usual meaning of distance in classical physics, including Newtonian mechanics.

Straight-line distance is formalized mathematically as the Euclidean distance in two- and three-dimensional space. In Euclidean geometry, the distance between two points $A$ and $B$ is often denoted $$|AB|$$. In coordinate geometry, Euclidean distance is computed using the Pythagorean theorem. The distance between points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ in the plane is given by: $$d=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.$$ Similarly, given points (x1, y1, z1) and (x2, y2, z2) in three-dimensional space, the distance between them is: $$d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}.$$ This idea generalizes to higher-dimensional Euclidean spaces.

Measurement
There are many ways of measuring straight-line distances. For example, it can be done directly using a ruler, or indirectly with a radar (for long distances) or interferometry (for very short distances). The cosmic distance ladder is a set of ways of measuring extremely long distances.

Shortest-path distance on a curved surface

The straight-line distance between two points on the surface of the Earth is not very useful for most purposes, since we cannot tunnel straight through the Earth's mantle. Instead, one typically measures the shortest path along the surface of the Earth, as the crow flies. This is approximated mathematically by the great-circle distance on a sphere.

More generally, the shortest path between two points along a curved surface is known as a geodesic. The arc length of geodesics gives a way of measuring distance from the perspective of an ant or other flightless creature living on that surface.

Effects of relativity
In the theory of relativity, because of phenomena such as length contraction and the relativity of simultaneity, distances between objects depend on a choice of inertial frame of reference. On galactic and larger scales, the measurement of distance is also affected by the expansion of the universe. In practice, a number of distance measures are used in cosmology to quantify such distances.

Other spatial distances
Unusual definitions of distance can be helpful to model certain physical situations, but are also used in theoretical mathematics:
 * In practice, one is often interested in the travel distance between two points along roads, rather than as the crow flies. In a grid plan, the travel distance between street corners is given by the Manhattan distance: the number of east–west and north–south blocks one must traverse to get between those two points.
 * Chessboard distance, formalized as Chebyshev distance, is the minimum number of moves a king must make on a chessboard in order to travel between two squares.

Metaphorical distances
Many abstract notions of distance used in mathematics, science and engineering represent a degree of difference or separation between similar objects. This page gives a few examples.

Statistical distances
In statistics and information geometry, statistical distances measure the degree of difference between two probability distributions. There are many kinds of statistical distances, typically formalized as divergences; these allow a set of probability distributions to be understood as a geometrical object called a statistical manifold. The most elementary is the squared Euclidean distance, which is minimized by the least squares method; this is the most basic Bregman divergence. The most important in information theory is the relative entropy (Kullback–Leibler divergence), which allows one to analogously study maximum likelihood estimation geometrically; this is an example of both an f-divergence and a Bregman divergence (and in fact the only example which is both). Statistical manifolds corresponding to Bregman divergences are flat manifolds in the corresponding geometry, allowing an analog of the Pythagorean theorem (which holds for squared Euclidean distance) to be used for linear inverse problems in inference by optimization theory.

Other important statistical distances include the Mahalanobis distance and the energy distance.

Edit distances
In computer science, an edit distance or string metric between two strings measures how different they are. For example, the words "dog" and "dot", which differ by just one letter, are closer than "dog" and "cat", which have no letters in common. This idea is used in spell checkers and in coding theory, and is mathematically formalized in a number of different ways, including Levenshtein distance, Hamming distance, Lee distance, and Jaro–Winkler distance.

Distance in graph theory
In a graph, the distance between two vertices is measured by the length of the shortest edge path between them. For example, if the graph represents a social network, then the idea of six degrees of separation can be interpreted mathematically as saying that the distance between any two vertices is at most six. Similarly, the Erdős number and the Bacon number—the number of collaborative relationships away a person is from prolific mathematician Paul Erdős and actor Kevin Bacon, respectively—are distances in the graphs whose edges represent mathematical or artistic collaborations.

In the social sciences
In psychology, human geography, and the social sciences, distance is often theorized not as an objective numerical measurement, but as a qualitative description of a subjective experience. For example, psychological distance is "the different ways in which an object might be removed from" the self along dimensions such as "time, space, social distance, and hypotheticality". In sociology, social distance describes the separation between individuals or social groups in society along dimensions such as social class, race/ethnicity, gender or sexuality.

Mathematical formalization
Most of the notions of distance between two points or objects described above are examples of the mathematical idea of a metric. A metric or distance function is a function $d$ which takes pairs of points or objects to real numbers and satisfies the following rules: As an exception, many of the divergences used in statistics are not metrics.
 * 1) The distance between an object and itself is always zero.
 * 2) The distance between distinct objects is always positive.
 * 3) Distance is symmetric: the distance from $x$ to $y$ is always the same as the distance from $y$ to $x$.
 * 4) Distance satisfies the triangle inequality: if $x$, $y$, and $z$ are three objects, then $$d(x,z) \leq d(x,y)+d(y,z).$$ This condition can be described informally as "intermediate stops can't speed you up."

Distance between sets
There are multiple ways of measuring the physical distance between objects that consist of more than one point:
 * One may measure the distance between representative points such as the center of mass; this is used for astronomical distances such as the Earth–Moon distance.
 * One may measure the distance between the closest points of the two objects; in this sense, the altitude of an airplane or spacecraft is its distance from the Earth. The same sense of distance is used in Euclidean geometry to define distance from a point to a line, distance from a point to a plane, or, more generally, perpendicular distance between affine subspaces.
 * Even more generally, this idea can be used to define the distance between two subsets of a metric space. The distance between sets $A$ and $B$ is the infimum of the distances between any two of their respective points:$$d(A,B)=\inf_{x\in A, y\in B} d(x,y).$$ This does not define a metric on the set of such subsets: the distance between overlapping sets is zero, and this distance does not satisfy the triangle inequality for any metric space with two or more points (consider the triple of sets consisting of two distinct singletons and their union).

 * The Hausdorff distance between two subsets of a metric space can be thought of as measuring how far they are from perfectly overlapping. Somewhat more precisely, the Hausdorff distance between $A$ and $B$ is either the distance from $A$ to the farthest point of $B$, or the distance from $B$ to the farthest point of $A$, whichever is larger.  (Here "farthest point" must be interpreted as a supremum.)  The Hausdorff distance defines a metric on the set of compact subsets of a metric space.

Related ideas
The word distance is also used for related concepts that are not encompassed by the description "a numerical measurement of how far apart points or objects are".

Distance travelled
The distance travelled by an object is the length of a specific path travelled between two points, such as the distance walked while navigating a maze. This can even be a closed distance along a closed curve which starts and ends at the same point, such as a ball thrown straight up, or the Earth when it completes one orbit. This is formalized mathematically as the arc length of the curve.

The distance travelled may also be signed: a "forward" distance is positive and a "backward" distance is negative.

Circular distance is the distance traveled by a point on the circumference of a wheel, which can be useful to consider when designing vehicles or mechanical gears (see also odometry). The circumference of the wheel is $2π × radius$; if the radius is 1, each revolution of the wheel causes a vehicle to travel $2π$ radians.

Displacement and directed distance

The displacement in classical physics measures the change in position of an object during an interval of time. While distance is a scalar quantity, or a magnitude, displacement is a vector quantity with both magnitude and direction. In general, the vector measuring the difference between two locations (the relative position) is sometimes called the directed distance. For example, the directed distance from the New York City Main Library flag pole to the Statue of Liberty flag pole has:
 * A starting point: library flag pole
 * An ending point: statue flag pole
 * A direction: -38°
 * A distance: 8.72 km

Library support

 * Python (programming language)
 * SciPy -Distance computations
 * Julia (programming language)
 * Julia Statistics Distance -A Julia package for evaluating distances (metrics) between vectors.

title: Distance


text:
User talk:Graham Chapman

This page is intentionally blank. Please do not restore earlier versions of it. I have no intention of using it again. I asked for it to be deleted but listening to requests is not a Wikipedia custom.

ArbCom elections are now open!
MediaWiki message delivery (talk) 08:51, 23 November 2015 (UTC)

title: User talk:Graham Chapman