Gay-Lussac's law: Difference between revisions
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[[File:Reuleaux polygons.svg|thumb|These [[Reuleaux polygon]]s have constant width, and all have the same width; therefore by Barbier's theorem they also have equal perimeters.]] | |||
In [[geometry]], '''Barbier's theorem''' states that every [[curve of constant width]] has perimeter [[Pi|π]] times its width, regardless of its precise shape.<ref>{{citation | |||
| last = Lay | first = Steven R. | |||
| at = Theorem 11.11, pp. 81–82 | |||
| isbn = 9780486458038 | |||
| publisher = Dover | |||
| title = Convex Sets and Their Applications | |||
| url = http://books.google.com/books?id=U9eOPjmaH90C&pg=PA81 | |||
| year = 2007}}.</ref> This theorem was first published by [[Joseph-Émile Barbier]] in 1860.<ref>{{citation | |||
| last = Barbier | first = E. | |||
| journal = Journal de mathématiques pures et appliquées | series = 2<sup>e</sup> série | |||
| language = French | |||
| pages = 273–286 | |||
| title = Note sur le problème de l’aiguille et le jeu du joint couvert | |||
| url = http://portail.mathdoc.fr/JMPA/PDF/JMPA_1860_2_5_A18_0.pdf | |||
| volume = 5 | |||
| year = 1860}}. See in particular pp. 283–285.</ref> | |||
==Examples== | |||
The most familiar examples of curves of constant width are the [[circle]] and the [[Reuleaux triangle]]. For a circle, the width is the same as the [[diameter]]; a circle of width ''w'' has [[perimeter]] π''w''. A Reuleaux triangle of width ''w'' consists of three [[Arc (geometry)|arc]]s of circles of [[radius]] ''w''. Each of these arcs has [[central angle]] π/3, so the perimeter of the Reuleaux triangle of width ''w'' is equal to half the perimeter of a circle of radius ''w'' and therefore is equal to π''w''. A similar analysis of other simple examples such as [[Reuleaux polygon]]s gives the same answer. | |||
==Proofs== | |||
One proof of the theorem uses the properties of [[Minkowski sum]]s. If ''K'' is a body of constant width ''w'', then the Minkowski sum of ''K'' and its 180° rotation is a disk with radius ''w'' and perimeter 2π''w''. However, the Minkowski sum acts linearly on the perimeters of convex bodies, so the perimeter of ''K'' must be half the perimeter of this disk, which is π''w'' as the theorem states.<ref>[http://www.cut-the-knot.org/ctk/Barbier.shtml The Theorem of Barbier (Java)] at [[cut-the-knot]].</ref> | |||
More generally, the theorem follows immediately from the [[Crofton formula]] in [[integral geometry]] according to which the length of any curve equals the measure of the set of lines that cross the curve, multiplied by their numbers of crossings. Any two curves that have the same constant width are crossed by sets of lines with the same measure, and therefore they have the same length. Historically, Crofton derived his formula later than, and independently of, Barbier's theorem.<ref>{{citation | |||
| last = Sylvester | first = J. J. | author-link = James Joseph Sylvester | |||
| doi = 10.1007/BF02413320 | |||
| issue = 1 | |||
| journal = Acta Mathematica | |||
| pages = 185–205 | |||
| title = On a funicular solution of Buffon's “problem of the needle” in its most general form | |||
| volume = 14 | |||
| year = 1890}}.</ref> | |||
An elementary probabilistic proof of the theorem can be found at [[Buffon's noodle]]. | |||
==Higher dimensions== | |||
The analogue of Barbier's theorem for [[surface of constant width|surfaces of constant width]] is false. In particular, the [[unit sphere]] has surface area <math>4\pi\approx 12.566</math>, while the [[surface of revolution]] of a [[Reuleaux triangle]] with the same constant width has surface area <math>8\pi-\tfrac{4}{3}\pi^2\approx 11.973</math>.<ref>{{citation | |||
| last1 = Bayen | first1 = Térence | |||
| last2 = Henrion | first2 = Didier | |||
| doi = 10.1080/10556788.2010.547580 | |||
| issue = 6 | |||
| journal = Optimization Methods and Software | |||
| pages = 1073–1099 | |||
| title = Semidefinite programming for optimizing convex bodies under width constraints | |||
| url = http://hal.archives-ouvertes.fr/hal-00495031 | |||
| volume = 27 | |||
| year = 2012}}.</ref> | |||
==References== | |||
{{reflist}} | |||
[[Category:Theorems in geometry]] | |||
[[Category:Pi]] | |||
[[Category:Length]] | |||
Revision as of 03:26, 3 February 2014
In geometry, Barbier's theorem states that every curve of constant width has perimeter π times its width, regardless of its precise shape.[1] This theorem was first published by Joseph-Émile Barbier in 1860.[2]
Examples
The most familiar examples of curves of constant width are the circle and the Reuleaux triangle. For a circle, the width is the same as the diameter; a circle of width w has perimeter πw. A Reuleaux triangle of width w consists of three arcs of circles of radius w. Each of these arcs has central angle π/3, so the perimeter of the Reuleaux triangle of width w is equal to half the perimeter of a circle of radius w and therefore is equal to πw. A similar analysis of other simple examples such as Reuleaux polygons gives the same answer.
Proofs
One proof of the theorem uses the properties of Minkowski sums. If K is a body of constant width w, then the Minkowski sum of K and its 180° rotation is a disk with radius w and perimeter 2πw. However, the Minkowski sum acts linearly on the perimeters of convex bodies, so the perimeter of K must be half the perimeter of this disk, which is πw as the theorem states.[3]
More generally, the theorem follows immediately from the Crofton formula in integral geometry according to which the length of any curve equals the measure of the set of lines that cross the curve, multiplied by their numbers of crossings. Any two curves that have the same constant width are crossed by sets of lines with the same measure, and therefore they have the same length. Historically, Crofton derived his formula later than, and independently of, Barbier's theorem.[4]
An elementary probabilistic proof of the theorem can be found at Buffon's noodle.
Higher dimensions
The analogue of Barbier's theorem for surfaces of constant width is false. In particular, the unit sphere has surface area , while the surface of revolution of a Reuleaux triangle with the same constant width has surface area .[5]
References
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- ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. See in particular pp. 283–285. - ↑ The Theorem of Barbier (Java) at cut-the-knot.
- ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.