Paracetamol

China Product


History

Acetanilide was the first aniline derivative serendipitously found to possess analgesic as well as antipyretic properties, and was quickly introduced into medical practice under the name of Antifebrin by A. Cahn and P. Hepp in 1886. But its unacceptable toxic effects, the most alarming being cyanosis due to methemoglobinemia, prompted the search for less toxic aniline derivatives. Harmon Northrop Morse had already synthesized paracetamol at Johns Hopkins University via the reduction of p-nitrophenol with tin in glacial acetic acid in 1877, but it was not until 1887 that clinical pharmacologist Joseph von Mering tried paracetamol on patients. In 1893, von Mering published a paper reporting on the clinical results of paracetamol with phenacetin, another aniline derivative. Von Mering claimed that, unlike phenacetin, paracetamol had a slight tendency to produce methemoglobinemia. Paracetamol was then quickly discarded in favor of phenacetin. The sales of phenacetin established Bayer as a leading pharmaceutical company. Overshadowed in part by aspirin, introduced into medicine by Heinrich Dreser in 1899, phenacetin was popular for many decades, particularly in widely advertised over-the-counter "headache mixtures," usually containing phenacetin, an aminopyrine derivative or aspirin, caffeine, and sometimes a barbiturate.

Von Mering's claims remained essentially unchallenged for half a century, until two teams of researchers from the United States analyzed the metabolism of acetanilide and paracetamol. In 1947 David Lester and Leon Greenberg found strong evidence that paracetamol was a major metabolite of acetanilide in human blood, and in a subsequent study they reported that large doses of paracetamol given to albino rats did not cause methemoglobinemia. In three papers published in the September 1948 issue of the Journal of Pharmacology and Experimental Therapeutics, Bernard Brodie, Julius Axelrod and Frederick Flinn confirmed using more specific methods that paracetamol was the major metabolite of acetanilide in human blood, and established it was just as efficacious an analgesic as its precursor. They also suggested that methemoglobinemia is produced in humans mainly by another metabolite, phenylhydroxylamine. A followup paper by Brodie and Axelrod in 1949 established that phenacetin was also metabolized to paracetamol. This led to a "rediscovery" of paracetamol. It has been suggested that contamination of paracetamol with 4-aminophenol, the substance from which it was synthesized by von Mering, may be the cause for his spurious findings. battery operated sprayer

Bernard Brodie and Julius Axelrod (pictured) demonstrated that acetanilide and phenacetin are both metabolized to paracetamol, which is a better tolerated analgesic. siphon hand pump

Paracetamol was first marketed in the United States in 1953 by Sterling-Winthrop Co., which promoted it as preferable to aspirin since it was safe to take for children and people with ulcers. The best known brand today for paracetamol in the United States, Tylenol, was established in 1955 when McNeil Laboratories started selling paracetamol as a pain and fever reliever for children, under the brand name Tylenol Children's Elixirhe word "tylenol" was a contraction of para-acetylaminophenol. In 1956, 500 mg tablets of paracetamol went on sale in the United Kingdom under the trade name Panadol, produced by Frederick Stearns & Co, a subsidiary of Sterling Drug Inc. Panadol was originally available only by prescription, for the relief of pain and fever, and was advertised as being "gentle to the stomach," since other analgesic agents of the time contained aspirin, a known stomach irritant.[citation needed] In 1963, paracetamol was added to the British Pharmacopoeia, and has gained popularity since then as an analgesic agent with few side-effects and little interaction with other pharmaceutical agents. Concerns about paracetamol's safety delayed its widespread acceptance until the 1970s, but in the 1980s paracetamol sales exceeded those of aspirin in many countries, including the United Kingdom. This was accompanied by the commercial demise of phenacetin, blamed as the cause of analgesic nephropathy and hematological toxicity. garden hose attachment

The U.S. patent on paracetamol has long expired, and generic versions of the drug are widely available under the Drug Price Competition and Patent Term Restoration Act of 1984, although certain Tylenol preparations were protected until 2007. U.S. patent 6,126,967 filed September 3, 1998 was granted for "Extended release acetaminophen particles".

Structure and reactivity

Polar surface area of the paracetamol molecule

Paracetamol consists of a benzene ring core, substituted by one hydroxyl group and the nitrogen atom of an amide group in the para (1,4) pattern. The amide group is acetamide (ethanamide). It is an extensively conjugated system, as the lone pair on the hydroxyl oxygen, the benzene pi cloud, the nitrogen lone pair, the p orbital on the carbonyl carbon, and the lone pair on the carbonyl oxygen are all conjugated. The presence of two activating groups also make the benzene ring highly reactive toward electrophilic aromatic substitution. As the substituents are ortho,para-directing and para with respect to each other, all positions on the ring are more or less equally activated. The conjugation also greatly reduces the basicity of the oxygens and the nitrogen, while making the hydroxyl acidic through delocalisation of charge developed on the phenoxide anion.

Synthesis

Compared with many other drugs, paracetamol is much easier to synthesize, because it lacks stereocenters. As a result, there is no need to design a stereo-selective synthesis.

Industrial preparation of paracetamol usually proceeds from nitrobenzene. A one-step reductive acetamidation reaction can be mediated by thioacetate.

Paracetamol may be easily prepared in the laboratory by nitrating phenol with sodium nitrate, separating the desired p-nitrophenol from the ortho- byproduct, and reducing the nitro group with sodium borohydride. The resultant p-aminophenol is then acetylated with acetic anhydride. In this reaction, phenol is strongly activating, thus the reaction only requires mild conditions (c.f. the nitration of benzene):

Reactions

p-Aminophenol may be obtained by the amide hydrolysis of paracetamol. p-Aminophenol prepared this way, and related to the commercially available Metol, has been used as a developer in photography by hobbyists.

Available forms

Main article: List of paracetamol brand names

Panadol Rapid caplets (AU)

Paracetamol is available in a tablet, capsule, liquid suspension, suppository, intravenous, and intramuscular form. The common adult dose is 500 mg to 1000 mg. The recommended maximum daily dose, for adults, is 4000 mg. In recommended doses, paracetamol generally is safe for children and infants, as well as for adults.

Panadol, which is marketed in Africa, Asia, Europe, Central America, and Australasia, is the most widely available brand, sold in over 80 countries. In North America, paracetamol is sold in generic form (usually labeled as acetaminophen) or under a number of trade names, for instance, Tylenol (McNeil-PPC, Inc.),Tydenol (Edruc Limited,Bangladesh) Anacin-3, Tempra, and Datril,. While there is brand named paracetamol available in the UK (e.g. Panadol), unbranded or generic paracetamol is more commonly sold. Acamol, a brand name for paracetamol produced by Teva Pharmaceutical Industries in Israel, is one of the most popular drugs in that country. In Europe, the most common brands of paracetamol are Efferalgan and Doliprane. In India, the most common brand of paracetamol is Crocin manufactured by Glaxo SmithKline Asia. In Bangladesh the most popular brand is Napa manufactured by Beximco Pharma.

In some formulations, paracetamol is combined with the opioid codeine, sometimes referred to as co-codamol (BAN). In the United States and Canada, this is marketed under the name of Tylenol #1/2/3/4, which contain 810 mg, 15 mg, 30 mg, and 60 mg of codeine, respectively. In the U.S., this combination is available only by prescription, while the lowest-strength preparation is over-the-counter in Canada, and, in other countries, other strengths may be available over the counter. There are generic forms of these combinations as well. In the UK and in many other countries, this combination is marketed under the names of Tylex CD and Panadeine. Other names include Captin, Disprol, Dymadon, Fensum, Hedex, Mexalen, Nofedol, Paralen, Pediapirin, Perfalgan, and Solpadeine. Paracetamol is also combined with other opioids such as dihydrocodeine, referred to as co-dydramol (BAN), oxycodone or hydrocodone, marketed in the U.S. as Percocet and Vicodin, respectively. Another very commonly used analgesic combination includes paracetamol in combination with propoxyphene napsylate, sold under the brand name Darvocet. A combination of paracetamol, codeine, and the calmative doxylamine succinate is marketed as Syndol or Mersyndol.

Paracetamol is commonly used in multi-ingredient preparations for migraine headache, typically including butalbital and paracetamol with or without caffeine, and sometimes containing codeine.

Brand Names

Aceta, Actimin, Anacin-3, Apacet, Aspirin Free Anacin, Atasol, Banesin, Ben-uron, Crocin, Dafalgan, Dapa, Dolo, Datril Extra-Strength, DayQuil, Depon & Depon Maximum, Feverall, Few Drops, Fibi, Fibi plus, Genapap, Genebs, Lekadol, LemSip, Liquiprin, Lupocet, Neopap, Ny-Quil, Oraphen-PD, Panado, Panadol, Paracet, Panodil, Paratabs, Paralen, Phenaphen, Plicet, Redutemp, Snaplets-FR, Suppap, Tamen, Tapanol, Tempra, Tylenol, Valorin, Xcel.

Mechanism of action

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Paracetamol is usually classified along with nonsteroidal antiinflammatory drugs (NSAID),[by whom?] but is not considered one, rather is part of the class of drugs known as aniline analgesics. Like all drugs of this class, its main mechanism of action is the inhibition of cyclooxygenase (COX),[citation needed] an enzyme responsible for the production of prostaglandins, which are important mediators of inflammation, pain and fever. Therefore, all NSAIDs are said to possess anti-inflammatory, analgesic (anti-pain), and antipyretic (anti-fever) properties. The specific actions of each NSAID drug depends upon their pharmacological properties, distribution and metabolism.

While paracetamol has analgesic and antipyretic properties comparable to those of aspirin, it fails to exert significant anti-inflammatory action due to paracetamol's susceptibility to the high level of peroxides present in inflammatory lesions.

AM404 metabolite of paracetamol

Anandamiden endogenous cannabinoid

However, the mechanism by which paracetamol reduces fever and pain is still debated largely because paracetamol reduces the production of prostaglandins (pro-inflammatory chemicals). Aspirin also inhibits the production of prostaglandins, but, unlike aspirin, paracetamol has little anti-inflammatory action. Likewise, whereas aspirin inhibits the production of the pro-clotting chemicals thromboxanes, paracetamol does not. Aspirin is known to inhibit the cyclooxygenase (COX) family of enzymes, and, because of paracetamol's partial similarity of aspirin's action,[clarification needed] much research has focused on whether paracetamol also inhibits COX. It is now clear that paracetamol acts via at least two pathways.

The COX family of enzymes are responsible for the metabolism of arachidonic acid to prostaglandin H2, an unstable molecule, which is, in turn, converted to numerous other pro-inflammatory compounds. Classical anti-inflammatories, such as the NSAIDs, block this step. Only when appropriately oxidized is the COX enzyme highly active. Paracetamol reduces the oxidized form of the COX enzyme, preventing it from forming pro-inflammatory chemicals.. Thus reducing the amount of Prostaglandin E2 in the CNS and thus lowering the hypothalamic set point in the thermoregulatory centre. Inhibition of another enzyme COX3 is specifically implicated in the case of paracetamol. COX3 is not seen outside the CNS Article text. Paracetamol also modulates the endogenous cannabinoid system. Paracetamol is metabolized to AM404, a compound with several actions; most important, it inhibits the uptake of the endogenous cannabinoid/vanilloid anandamide by neurons. Anandamide uptake would result in the activation of the main pain receptor (nociceptor) of the body, the TRPV1 (older name: vanilloid receptor). Furthermore, AM404 inhibits sodium channels, as do the anesthetics lidocaine and procaine. Either of these actions by themselves has been shown to reduce pain, and are a possible mechanism for paracetamol, though it has been demonstrated that, after blocking cannabinoid receptors and hence making any action of cannabinoid reuptake irrelevant, paracetamol loses analgesic effect, suggesting its pain-relieving action is mediated by the endogenous cannabinoid system.

One theory holds that paracetamol works by inhibiting the COX-3 isoform of the COX family of enzymes. This enzyme, when expressed in dogs, shares a strong similarity to the other COX enzymes, produces pro-inflammatory chemicals, and is selectively inhibited by paracetamol. However, some research has suggested that in humans and mice, the COX-3 enzyme is without inflammatory action. Another possibility is that paracetamol blocks cyclooxygenase (as in aspirin), but that in an inflammatory environment, where the concentration of peroxides is high, the oxidation state of paracetamol is high which prevents its actions. This would mean that paracetamol has no direct effect at the site of inflammation but instead acts in the CNS to reduce temperature etc where the environment is not oxidative. The exact mechanism by which paracetamol is believed to affect COX-3 is disputed.

Metabolism

Main pathways of paracetamol metabolism (click to enlarge). Pathways shown in blue and purple lead to non-toxic metabolites; the pathway in red leads to toxic NAPQI.

Paracetamol is metabolised primarily in the liver, into non-toxic products. Three metabolic pathways are notable:

Glucuronidation is believed to account for 40% to two-thirds of the metabolism of paracetamol.

Sulfation (sulfate conjugation) may account for 2040%.

N-hydroxylation and rearrangement, then GSH conjugation, accounts for less than 15%. The hepatic cytochrome P450 enzyme system metabolizes paracetamol, forming a minor yet significant alkylating metabolite known as NAPQI (N-acetyl-p-benzo-quinone imine). NAPQI is then irreversibly conjugated with the sulfhydryl groups of glutathione.

All three pathways yield final products that are inactive, non-toxic, and eventually excreted by the kidneys. In the third pathway, however, the intermediate product NAPQI is toxic. NAPQI is primarily responsible for the toxic effects of paracetamol; this constitutes an excellent example of toxication.

Production of NAPQI is due primarily to two isoenzymes of cytochrome P450: CYP2E1 and CYP1A2. The P450 gene is highly polymorphic, however, and individual differences in paracetamol toxicity are believed to be due to a third isoenzyme, CYP2D6. Genetic polymorphisms in CYP2D6 may contribute to significantly different rates of production of NAPQI. Furthermore, individuals can be classified as "extensive", "ultrarapid", and "poor" metabolizers (producers of NAPQI), depending on their levels of CYP2D6 expression. Although CYP2D6 metabolises paracetamol into NAPQI to a lesser extent than other P450 enzymes, its activity may contribute to paracetamol toxicity in extensive and ultrarapid metabolisers, and when paracetamol is taken at very large doses. At usual doses, NAPQI is quickly detoxified by conjugation. Following overdose, and possibly also in extensive and ultrarapid metabolizers, this detoxification pathway becomes saturated and consequently NAPQI accumulates.

Indications

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The World Health Organization (WHO) recommends that paracetamol be given to children with fever higher than 38.5 C (101.3 F).

Paracetamol is much more effective[citation needed][who?] than aspirin, especially in patients where excessive gastric acid secretion or prolongation of bleeding time may be a concern. While paracetamol has analgesic and antipyretic properties comparable to those of aspirin, its anti-inflammatory effects are weak. Because paracetamol is well tolerated, available without a prescription, and lacks the gastric side effects of aspirin, it has in recent years increasingly become a common household drug.

Efficacy and side effects

Paracetamol, unlike other common analgesics such as aspirin and ibuprofen, has relatively little anti-inflammatory activity, so it is not considered to be a non-steroidal anti-inflammatory drug (NSAID).

Efficacy

Regarding comparative efficacy, studies show conflicting results when compared to NSAIDs. A randomized controlled trial of chronic pain from osteoarthritis in adults found similar benefit from paracetamol and ibuprofen.[unreliable source?] However, a randomized controlled trial of acute musculoskeletal pain in children found that the standard OTC dose of ibuprofen gives greater relief of pain than the standard dose of paracetamol.[unreliable source?]

Adverse effects

In recommended doses, paracetamol does not irritate the lining of the stomach, affect blood coagulation as much as NSAIDs, or affect function of the kidneys.[citation needed] However, some studies have shown that high dose-usage (greater than 2,000 mg per day) does increase the risk of upper gastrointestinal complications such as stomach bleeding. The researchers found that heavy use of aspirin or paracetamol - defined as 300 grams a year (1 g per day on average) - was linked to a condition known as small, indented and calcified kidneys (SICK). Paracetamol is safe in pregnancy, and does not affect the closure of the fetal ductus arteriosus as NSAIDs can. Unlike aspirin, it is safe in children, as paracetamol is not associated with a risk of Reye's syndrome in children with viral illnesses.

Like NSAIDs and unlike opioid analgesics, paracetamol has not been found to cause euphoria or alter mood in any way. In 2008, the largest study to date on the long term side effects of paracetamol in children was published in The Lancet. Conducted on over 200,000 children in 31 countries, the study found that the use of paracetamol for fever in the first year of life was associated with an increase in the incidence of asthmatic symptoms at 67 years, and that paracetamol use, both in the first year of life and in children aged 67 years, was associated with an increased incidence of rhinoconjunctivitis and eczema. The authors acknowledged that their "findings might have been due to confounding by indication", i.e. that the association may not be causal but rather due to the disease being treated with paracetamol, and emphasized that further research was needed. Furthermore a number of editorials, comments, correspondence and their replies have been published in The Lancet concerning the methodology and conclusions of this study. The UK regulatory body the Medicines and Healthcare products Regulatory Agency, also reviewed this research and published a number of concerns over data interpretation, and offer the following advice for healthcare professionals, parents, and carers: "The results of this new study do not necessitate any change to the current guidance for use in children. Paracetamol remains a safe and appropriate choice of analgesic in children. There is insufficient evidence from this research to change guidance regarding the use of antipyretics in children."

Toxicity

Main articles: Paracetamol toxicity and Analgesic nephropathy

Excessive use of paracetamol can damage multiple organs, especially the liver and kidney. In both organs, toxicity from paracetamol is not from the drug itself but from one of its metabolites, N-acetyl-p-benzoquinoneimine (NAPQI). In the liver, the cytochrome P450 enzymes CYP2E1 and CYP3A4 are primarily responsible for the conversion of paracetamol to NAPQI. In the kidney, cyclooxygenases are the principal route by which paracetamol is converted to NAPQI. Paracetamol overdose leads to the accumulation of NAPQI, which undergoes conjugation with glutathione. Conjugation depletes glutathione, a natural antioxidant. This in combination with direct cellular injury by NAPQI, leads to cell damage and death.

Paracetamol hepatotoxicity is, by far, the most common cause of acute liver failure in both the United States and the United Kingdom. Paracetamol overdose results in more calls to poison control centers in the US than overdose of any other pharmacological substance. Signs and symptoms of paracetamol toxicity may initially be absent or vague. Untreated, overdose can lead to liver failure and death within days. Treatment is aimed at removing the paracetamol from the body and replacing glutathione. Activated charcoal can be used to decrease absorption of paracetamol if the patient presents for treatment soon after the overdose. While the antidote, acetylcysteine, (also called N-acetylcysteine or NAC) acts as a precursor for glutathione helping the body regenerate enough to prevent damage to the liver, a liver transplant is often required if damage to the liver becomes severe.

There are tablets avaliable (Brandname in the UK Paradote) which combine Paracetamol with an antidote (Methionine), to protect the liver in case of an overdose.

In June 2009 an FDA advisory committee recommended that new restrictions should be placed on paracetamol to help protect people from the potential toxic effects, however, the FDA has not implemented their recomendations as at March 2010.

Effects on animals

Paracetamol is extremely toxic to cats. Cats lack the necessary glucuronyl transferase enzymes to safely break paracetamol down, and minute portions of a tablet may prove fatal. Initial symptoms include vomiting, salivation and discolouration of the tongue and gums. Unlike an overdose in humans, liver damage is rarely the cause of death; instead, methaemoglobin formation and the production of Heinz bodies in red blood cells inhibit oxygen transport by the blood, causing asphyxiation (methemoglobemia and hemolytic anemia). Treatment with N-acetylcysteine, methylene blue or both is sometimes effective after the ingestion of small doses of paracetamol. According to one paper female cats may have a better survival rate although sample size was small.

Although paracetamol is believed to have no significant anti-inflammatory activity, it has been reported to be as effective as aspirin in the treatment of musculoskeletal pain in dogs. A paracetamol-codeine product (trade name Pardale-V) licensed for use in dogs is available on veterinary prescription in the UK. It should be administered to dogs only on veterinary advice. The main effects of toxicity in dogs is liver damage. N-acetylcysteine treatment is efficacious in dogs when administered within a few hours of paracetamol ingestion.

Paracetamol is also lethal to snakes, and has been suggested as chemical control program for the brown tree snake (Boiga irregularis) in Guam.

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^ a b Hendrickson, Robert G.; Kenneth E. Bizovi (2006). "Acetaminophen", in Nelson, Lewis H.; Flomenbaum, Neal; Goldfrank, Lewis R. et al. Goldfrank's toxicologic emergencies, p. 525, New York: McGraw-Hill. Retrieved on January 18, 2009 through Google Book Search.

^ a b c Borne, Ronald F. "Nonsteroidal Anti-inflammatory Drugs" in Principles of Medicinal Chemistry, Fourth Edition. Eds. Foye, William O.; Lemke, Thomas L.; Williams, David A. Published by Williams & Wilkins, 1995. p. 544545.

^ Dong H, Haining RL, Thummel KE, Rettie AE, Nelson SD (2000). "Involvement of human cytochrome P450 2D6 in the bioactivation of acetaminophen". Drug Metab Dispos 28 (12): 1397400. PMID 11095574.  Free full text

^ "Baby paracetamol asthma concern". BBC News. 2008-09-19. http://news.bbc.co.uk/1/hi/health/7623230.stm. Retrieved 2008-09-19. 

^ Bradley JD, Brandt KD, Katz BP, Kalasinski LA, Ryan SI (1991). "Comparison of an antiinflammatory dose of ibuprofen, an analgesic dose of ibuprofen, and acetaminophen in the treatment of patients with osteoarthritis of the knee". N. Engl. J. Med. 325 (2): 8791. PMID 2052056. 

^ doi:10.1111/j.1365-2710.2006.00754.x

^ Clark E, Plint AC, Correll R, Gaboury I, Passi B (2007). "A randomized, controlled trial of acetaminophen, ibuprofen, and codeine for acute pain relief in children with musculoskeletal trauma". Pediatrics 119 (3): 4607. doi:10.1542/peds.2006-1347. PMID 17332198. 

^ Garca Rodrguez LA, Hernndez-Daz S (December 15, 2000). "The risk of upper gastrointestinal complications associated with nonsteroidal anti-inflammatory drugs, glucocorticoids, acetaminophen, and combinations of these agents". Arthritis Research and Therapy 3 (2): 98. doi:10.1186/ar146. PMID 11178116. 

^ http://news.bbc.co.uk/2/hi/health/3271191.stm

^ Rudolph AM (February 1981). "Effects of aspirin and acetaminophen in pregnancy and in the newborn". Arch. Intern. Med. 141 (3 Spec No): 35863. doi:10.1001/archinte.141.3.358. PMID 7469626. 

^ Lesko SM, Mitchell AA (October 1999). "The safety of acetaminophen and ibuprofen among children younger than two years old". Pediatrics 104 (4): e39. doi:10.1542/peds.104.4.e39. PMID 10506264. http://pediatrics.aappublications.org/cgi/pmidlookup?view=long&pmid=10506264. 

^ Beasley, Richard; Clayton, Tadd; Crane, Julian; von Mutius, Erika; Lai, Christopher; Montefort, Stephen; Stewart, Alistair (2008). "Association between paracetamol use in infancy and childhood, and risk of asthma, rhino conjunctivitis, and eczema in children aged 67 years: analysis from Phase Three of the ISAAC programme.". The Lancet 372: 10391048. doi:10.1016/S0140-6736(08)61445-2. http://www.thelancet.com/journals/lancet/article/PIIS0140673608614452/abstract. Retrieved 2008-09-19. 

^ The Lancet (2008). "Asthma: still more questions than answers". The Lancet 372: 10091009. doi:10.1016/S0140-6736(08)61414-2.  edit

^ Barr, R. G. (2008). "Does paracetamol cause asthma in children? Time to remove the guesswork". The Lancet 372: 10111012. doi:10.1016/S0140-6736(08)61417-8.  edit

^ Lawyer, A. B. (2009). "Paracetamol as a risk factor for allergic disorders". The Lancet 373: 121121. doi:10.1016/S0140-6736(09)60032-5.  edit

^ Lowe, A. (2009). "Paracetamol as a risk factor for allergic disorders". The Lancet 373: 120120. doi:10.1016/S0140-6736(09)60030-1.  edit

^ Lawrence, J. (2009). "Paracetamol as a risk factor for allergic disorders". The Lancet 373: 119119. doi:10.1016/S0140-6736(09)60029-5.  edit

^ Singh, M. (2009). "Paracetamol as a risk factor for allergic disorders". The Lancet 373: 119119. doi:10.1016/S0140-6736(09)60028-3.  edit

^ Beasley, R. (2009). "Paracetamol as a risk factor for allergic disorders Authors' reply". The Lancet 373: 120121. doi:10.1016/S0140-6736(09)60031-3.  edit

^ Medicines and Healthcare products Regulatory Agency; Commission on Human Medicines (2008). "Paracetamol use in infancy: no strong evidence for asthma link". Drug Safety Update 2 (4): 9. http://www.mhra.gov.uk/Publications/Safetyguidance/DrugSafetyUpdate/CON030923. Retrieved 2009-05-01. 

^ Mohandas J, Duggin GG, Horvath JS, Tiller DJ (November 1981). "Metabolic oxidation of acetaminophen (paracetamol) mediated by cytochrome P-450 mixed-function oxidase and prostaglandin endoperoxide synthetase in rabbit kidney". Toxicol. Appl. Pharmacol. 61 (2): 2529. doi:10.1016/0041-008X(81)90415-4. PMID 6798713. http://linkinghub.elsevier.com/retrieve/pii/0041-008X(81)90415-4. 

^ Mitchell JR, Jollow DJ, Potter WZ, Gillette JR, Brodie BB (October 1973). "Acetaminophen-induced hepatic necrosis. IV. Protective role of glutathione". The Journal of pharmacology and experimental therapeutics 187 (1): 2117. PMID 4746329. http://jpet.aspetjournals.org/cgi/pmidlookup?view=long&pmid=4746329. 

^ Ryder SD, Beckingham IJ (2001). "ABC of diseases of liver, pancreas, and biliary system. Other causes of parenchymal liver disease". BMJ 322 (7281): 29092. doi:10.1136/bmj.322.7281.290. PMID 11157536.  [11157536 Free full text]

^ Lee WM (July 2004). "Acetaminophen and the U.S. Acute Liver Failure Study Group: lowering the risks of hepatic failure". Hepatology 40 (1): 69. doi:10.1002/hep.20293. PMID 15239078. http://www3.interscience.wiley.com/cgi-bin/fulltext/109086434/PDFSTART. 

^ "FDA May Restrict Acetaminophen"

^ "FDA: Drug Safety & Availability - Acetaminophen Information"

^ Allen AL (2003). "The diagnosis of acetaminophen toxicosis in a cat". Can Vet J 44 (6): 50910. PMID 12839249. 

^ Rumbeiha WK, Lin YS, Oehme FW (November 1995). "Comparison of N-acetylcysteine and methylene blue, alone or in combination, for treatment of acetaminophen toxicosis in cats". Am. J. Vet. Res. 56 (11): 152933. PMID 8585668. 

^ a b Maddison, Jill E.; Stephen W. Page, David Church (2002). Small Animal Clinical Pharmacology. Elsevier Health Sciences. pp. 260261. ISBN 0702025739. 

^ "Pardale-V Tablets: Presentation". UK National Office of Animal Health Compendium of Animal Medicines. September 28, 2006. http://www.noahcompendium.co.uk/Dechra/Pardale-V_Oral_Tablets/-27619.html. Retrieved 3 January 2007. 

^ "Pardale-V Tablets: Legal Category". UK National Office of Animal Health Compendium of Animal Medicines. November 15, 2005. http://www.noahcompendium.co.uk/Dechra/Pardale-V_Oral_Tablets/-27624.html. Retrieved 3 January 2007. 

^ Villar D, Buck WB, Gonzalez JM (1998). "Ibuprofen, aspirin and acetaminophen toxicosis and treatment in dogs and cats". Vet Hum Toxicol 40 (3): 15662. PMID 9610496. 

^ Johnston J, Savarie P, Primus T, Eisemann J, Hurley J, Kohler D (2002). "Risk assessment of an acetaminophen baiting program for chemical control of brown tree snakes on Guam: evaluation of baits, snake residues, and potential primary and secondary hazards". Environ Sci Technol 36 (17): 382733. doi:10.1021/es015873n. PMID 12322757. 

External links

Pharmacy and Pharmacology portal

Paracetamol at Chemsynthesis

Paracetamol Information Centre

The Julius Axelrod Papers

FDA: Safe Use of Over-the-Counter Pain Relievers/Fever Reducers

FDA: Consumer Update "Acetaminophen and Liver Injury: Q and A for Consumers" (link)

FDA: Consumer Update "Acetaminophen and Liver Injury: Q and A for Consumers" (PDF)

U.S. National Library of Medicine: Drug Information Portal - Paracetamol

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Wallpaper group

China Product


Introduction

Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns which are very different in style, color, scale or orientation may belong to the same group.

Consider the following examples: hose barb fitting

Example A: Cloth, Tahiti fire hose coupling

Example B: Ornamental painting, Nineveh, Assyria square tube connectors

Example C: Painted porcelain, China

Examples A and B have the same wallpaper group; it is called p4m. Example C has a different wallpaper group, called p4g. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.

A complete list of all seventeen possible wallpaper groups can be found below.

Symmetries of patterns

A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that the pattern looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (shifted) some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns which repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored.

Sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry. In black and white there are also 17 wallpaper groups; e.g., a colored tiling is equivalent with one in black and white with the colors coded radially in a circularly symmetric "bar code" in the centre of mass of each tile.

The types of transformations that are relevant here are called Euclidean plane isometries. For example:

If we shift example B one unit to the right, so that each square covers the square that was originally adjacent to it, then the resulting pattern is exactly the same as the pattern we started with. This type of symmetry is called a translation. Examples A and C are similar, except that the smallest possible shifts are in diagonal directions.

If we turn example B clockwise by 90, around the centre of one of the squares, again we obtain exactly the same pattern. This is called a rotation. Examples A and C also have 90 rotations, although it requires a little more ingenuity to find the correct centre of rotation for C.

We can also flip example B across a horizontal axis that runs across the middle of the image. This is called a reflection. Example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A.

However, example C is different. It only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a diagonal line, we do not get the same pattern back; what we do get is the original pattern shifted across by a certain distance. This is part of the reason that A and B have a different wallpaper group than C.

History

All 17 groups were used by Egyptian craftsmen, and used extensively in the Muslim world. A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorov in 1891 and then derived independently by George Plya in 1924.

Formal definition and discussion

Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane which contains two linearly independent translations.

Two such isometry groups are of the same type (of the same wallpaper group) if they are the same up to an affine transformation of the plane. Thus e.g. a translation of the plane (hence a translation of the mirrors and centres of rotation) does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry (this is only the case if there are no mirrors and no glide reflections, and rotational symmetry is at most of order 2).

Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those which preserve orientation.

It follows from the Bieberbach theorem that all wallpaper groups are different even as abstract groups (as opposed to e.g. Frieze groups, of which two are isomorphic with Z).

2D patterns with double translational symmetry can be categorized according to their symmetry group type.

Isometries of the Euclidean plane

Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information).

Translations, denoted by Tv, where v is a vector in R2. This has the effect of shifting the plane applying displacement vector v.

Rotations, denoted by Rc,, where c is a point in the plane (the centre of rotation), and is the angle of rotation.

Reflections, or mirror isometries, denoted by FL, where L is a line in R2. (F is for "flip"). This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror.

Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance. This is a combination of a reflection in the line L and a translation along L by a distance d.

The independent translations condition

The condition on linearly independent translations means that there exist linearly independent vectors v and w (in R2) such that the group contains both Tv and Tw.

The purpose of this condition is to distinguish wallpaper groups from frieze groups, which possess a translation but not two linearly independent ones, and from two-dimensional discrete point groups, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves in two distinct directions, in contrast to frieze groups which only repeat along a single axis.

(It is possible to generalise this situation. We could for example study discrete groups of isometries of Rn with m linearly independent translations, where m is any integer in the range 0 m n.)

The discreteness condition

The discreteness condition means that there is some positive real number , such that for every translation Tv in the group, the vector v has length at least (except of course in the case that v is the zero vector).

The purpose of this condition is to ensure that the group has a compact fundamental domain, or in other words, a "cell" of nonzero, finite area, which is repeated through the plane. Without this condition, we might have for example a group containing the translation Tx for every rational number x, which would not correspond to any reasonable wallpaper pattern.

One important and nontrivial consequence of the discreteness condition in combination with the independent translations condition is that the group can only contain rotations of order 2, 3, 4, or 6; that is, every rotation in the group must be a rotation by 180, 120, 90, or 60. This fact is known as the crystallographic restriction theorem, and can be generalised to higher-dimensional cases.

Notations for wallpaper groups

Crystallographic notation

Crystallography has 230 space groups to distinguish, far more than the 17 wallpaper groups, but many of the symmetries in the groups are the same. Thus we can use a similar notation for both kinds of groups, that of Carl Hermann and Charles-Victor Mauguin. An example of a full wallpaper name in Hermann-Mauguin style is p31m, with four letters or digits; more usual is a shortened name like cmm or pg.

For wallpaper groups the full notation begins with either p or c, for a primitive cell or a face-centred cell; these are explained below. This is followed by a digit, n, indicating the highest order of rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold. The next two symbols indicate symmetries relative to one translation axis of the pattern, referred to as the "main" one; if there is a mirror perpendicular to a translation axis we choose that axis as the main one (or if there are two, one of them). The symbols are either m, g, or 1, for mirror, glide reflection, or none. The axis of the mirror or glide reflection is perpendicular to the main axis for the first letter, and either parallel or tilted 180/n (when n > 2) for the second letter. Many groups include other symmetries implied by the given ones. The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group.

A primitive cell is a minimal region repeated by lattice translations. All but two wallpaper symmetry groups are described with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice. In the remaining two cases symmetry description is with respect to centred cells which are larger than the primitive cell, and hence have internal repetition; the directions of their sides is different from those of the translation vectors spanning a primitive cell. Hermann-Mauguin notation for crystal space groups uses additional cell types.

Examples

p2 (p211): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.

p4g (p4gm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45.

cmm (c2mm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis.

p31m (p31m): Primitive cell, 3-fold rotation, mirror axis at 60.

Here are all the names that differ in short and full notation.

Crystallographic short and full names

Short

p2

pm

pg

cm

pmm

pmg

pgg

cmm

p4m

p4g

p6m

Full

p211

p1m1

p1g1

c1m1

p2mm

p2mg

p2gg

c2mm

p4mm

p4gm

p6mm

The remaining names are p1, p3, p3m1, p31m, p4, and p6.

Orbifold notation

Orbifold notation for wallpaper groups, introduced by John Horton Conway (Conway, 1992), is based not on crystallography, but on topology. We fold the infinite periodic tiling of the plane into its essence, an orbifold, then describe that with a few symbols.

A digit, n, indicates a centre of n-fold rotation. By the crystallographic restriction theorem, n must be 2, 3, 4, or 6.

An asterisk, *, indicates a mirror. It interacts with the digits as follows:

Digits before * are centres of pure rotation (cyclic).

Digits after * are centres of rotation with mirrors through them (dihedral).

A cross, x, indicates a glide reflection. Pure mirrors combine with lattice translation to produce glides, but those are already accounted for so we do not notate them.

The "no symmetry" symbol, o, stands alone, and indicates we have only lattice translations with no other symmetry.

Consider the group denoted in crystallographic notation by cmm; in Conway's notation, this will be 2*22. The 2 before the * says we have a 2-fold rotation centre with no mirror through it. The * itself says we have a mirror. The first 2 after the * says we have a 2-fold rotation centre on a mirror. The final 2 says we have an independent second 2-fold rotation centre on a mirror, one which is not a duplicate of the first one under symmetries.

The group denoted by pgg will be 22x. We have two pure 2-fold rotation centres, and a glide reflection axis. Contrast this with pmg, Conway 22*, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold.

Conway and crystallographic correspondence

Conway

o

xx

*x

**

632

*632

Crystal.

p1

pg

cm

pm

p6

p6m

Conway

333

*333

3*3

442

*442

4*2

Crystal.

p3

p3m1

p31m

p4

p4m

p4g

Conway

2222

22x

22*

*2222

2*22

Crystal.

p2

pgg

pmg

pmm

cmm

Why there are exactly seventeen groups

An orbifold has a face, edges, and vertices; thus we can view it as a polygon. When we unfold it, that polygon tiles the plane, with each feature replicated infinitely by the action of the wallpaper symmetry group. Thus when Conway's orbifold notation mentions a feature, such as the 4-fold rotation centre in 4*2, that feature unfolds into an infinite number of replicas across the plane. Hiding within this description is a key to the enumeration.

Consider a cube, with its corners, edges, and faces. We count 8 corners, 12 edges, and 6 faces. Alternately adding and subtracting, we note that 8 12 + 6 = 2. Now consider a tetrahedron. It has 4 corners, 6 edges, and 4 faces; and we note that 4 6 + 4 = 2. Let us explore further. For generality, use the term vertex instead of corner. Split a face with a new edge, causing one face to become two. Now we have 4 7 + 5 = 2. Next, split an edge with a new vertex, causing the one edge to become two. We have 5 8 + 5 = 2. This is not coincidence; it is a demonstration of the surface Euler characteristic, = V E + F, and the beginning of a proof of its invariance.

When an orbifold replicates by symmetry to fill the plane, its features create a structure of vertices, edges, and polygon faces which must be consistent with the Euler characteristic. Reversing the process, we can assign numbers to the features of the orbifold, but fractions, rather than whole numbers. Because the orbifold itself is a quotient of the full surface by the symmetry group, the orbifold Euler characteristic is a quotient of the surface Euler characteristic by the order of the symmetry group.

The orbifold Euler characteristic is 2 minus the sum of the feature values, assigned as follows:

A digit n before a * counts as (n1)/n.

A digit n after a * counts as (n1)/2n.

Both * and x count as 1.

The "no symmetry" o counts as 2.

For a wallpaper group, the sum for the characteristic must be zero; thus the feature sum must be 2.

Examples

632: 5/6 + 2/3 + 1/2 = 2

3*3: 2/3 + 1 + 1/3 = 2

4*2: 3/4 + 1 + 1/4 = 2

22x: 1/2 + 1/2 + 1 = 2

Now enumeration of all wallpaper groups becomes a matter of arithmetic, of listing all feature strings with values summing to 2.

Incidentally, feature strings with other sums are not nonsense; they imply non-planar tilings, not discussed here. (When the orbifold Euler characteristic is negative, the tiling is hyperbolic; when positive, spherical or bad).

Guide to recognising wallpaper groups

To work out which wallpaper group corresponds to a given design, one may use the following table.

Size of smallest

rotation

Has reflection?

Yes

No

360 / 6

p6m

p6

360 / 4

Has mirrors at 45?

p4

Yes: p4m

No: p4g

360 / 3

Has rot. centre off mirrors?

p3

Yes: p31m

No: p3m1

360 / 2

Has perpendicular reflections?

Has glide reflection?

Yes

No

Has rot. centre off mirrors?

pmg

Yes: pgg

No: p2

Yes: cmm

No: pmm

none

Has glide axis off mirrors?

Has glide reflection?

Yes: cm

No: pm

Yes: pg

No: p1

See also this overview with diagrams.

The seventeen groups

Each of the groups in this section has two cell structure diagrams, which are to be interpreted as follows:

a centre of rotation of order two (180).

a centre of rotation of order three (120).

a centre of rotation of order four (90).

a centre of rotation of order six (60).

an axis of reflection.

an axis of glide reflection.

On the right-hand side diagrams, different equivalence classes of symmetry elements are colored (and rotated) differently.

The brown or yellow area indicates a fundamental domain, i.e. the smallest part of the pattern which is repeated.

The diagrams on the right show the cell of the lattice corresponding to the smallest translations; those on the left sometimes show a larger area.

Group p1

Example and diagram for p1

Cell structure for p1

Cell structure for p1

Orbifold notation: o.

The group p1 contains only translations; there are no rotations, reflections, or glide reflections.

Examples of group p1

Computer generated

Medival wall diapering

The two translations (cell sides) can each have different lengths, and can form any angle.

Group p2

Example and diagram for p2

Cell structure for p2

Cell structure for p2

Orbifold notation: 2222.

The group p2 contains four rotation centres of order two (180), but no reflections or glide reflections.

Examples of group p2

Computer generated

Cloth, Sandwich Islands (Hawaii)

Mat on which Egyptian king stood

Egyptian mat (detail)

Ceiling of Egyptian tomb

Wire fence, U.S.

Group pm

Example and diagram for pm

Cell structure for pm

Cell structure for pm

Orbifold notation: **.

The group pm has no rotations. It has reflection axes, they are all parallel.

Examples of group pm

(The first three have a vertical symmetry axis, and the last two each have a different diagonal one.)

Computer generated

Dress of a figure in a tomb at Biban el Moluk, Egypt

Egyptian tomb, Thebes

Ceiling of a tomb at Gourna, Egypt. Reflection axis is diagonal.

Indian metalwork at the Great Exhibition in 1851. This is almost pm (ignoring short diagonal lines between ovals motifs, which make it p1).

Group pg

Example and diagram for pg

Cell structure for pg

Cell structure for pg

Orbifold notation: xx.

The group pg contains glide reflections only, and their axes are all parallel. There are no rotations or reflections.

Examples of group pg

Computer generated

Mat on which Egyptian king stood

Egyptian mat (detail)

Pavement in Salzburg. Glide reflection axis runs northeast-southwest.

One of the colorings of the snub square tiling; the glide reflection lines are in the direction upper left / lower right; ignoring colors there is much more symmetry than just pg, then it is p4g (see there for this image with equally colored triangles)

Without the details inside the zigzag bands the mat is pmg; with the details but without the distinction between brown and black it is pgg.

Ignoring the wavy borders of the tiles, the pavement is pgg.

Group cm

Cell structure for cm

Cell structure for cm

Orbifold notation: *x.

The group cm contains no rotations. It has reflection axes, all parallel. There is at least one glide reflection whose axis is not a reflection axis; it is halfway between two adjacent parallel reflection axes.

This group applies for symmetrically staggered rows (i.e. there is a shift per row of half the translation distance inside the rows) of identical objects, which have a symmetry axis perpendicular to the rows.

Examples of group cm

Computer generated

Dress of Amun, from Abu Simbel, Egypt

Dado from Biban el Moluk, Egypt

Bronze vessel in Nimroud, Assyria

Spandrils of arches, the Alhambra, Spain

Soffitt of arch, the Alhambra, Spain

Persian tapestry

Indian metalwork at the Great Exhibition in 1851

Dress of a figure in a tomb at Biban el Moluk, Egypt

Group pmm

Example and diagram for pmm

Cell structure for pmm

Cell structure for pmm

Orbifold notation: *2222.

The group pmm has reflections in two perpendicular directions, and four rotation centres of order two (180) located at the intersections of the reflection axes.

Examples of group pmm

Computer generated

2D image of lattice fence, U.S. (in 3D there is additional symmetry)

Mummy case stored in The Louvre

Ceiling of Egyptian tomb. Ignoring minor asymmetries, this would be cmm.

Mummy case stored in The Louvre. Would be type p4m except for the mismatched coloring.

Group pmg

Example and diagram for pmg

Cell structure for pmg

Cell structure for pmg

Orbifold notation: 22*.

The group pmg has two rotation centres of order two (180), and reflections in only one direction. It has glide reflections whose axes are perpendicular to the reflection axes. The centres of rotation all lie on glide reflection axes.

Examples of group pmg

Computer generated

Cloth, Sandwich Islands (Hawaii)

Ceiling of Egyptian tomb

Floor tiling in Prague, the Czech Republic

Bowl from Kerma

Group pgg

Example and diagram for pgg

Cell structure for pgg

Cell structure for pgg

Orbifold notation: 22x.

The group pgg contains two rotation centres of order two (180), and glide reflections in two perpendicular directions. The centres of rotation are not located on the glide reflection axes. There are no reflections.

Examples of group pgg

Computer generated

Bronze vessel in Nimroud, Assyria

Pavement in Budapest, Hungary. Glide reflection axes are diagonal.

Group cmm

Cell structure for cmm

Cell structure for cmm

Orbifold notation: 2*22.

The group cmm has reflections in two perpendicular directions, and a rotation of order two (180) whose centre is not on a reflection axis. It also has two rotations whose centres are on a reflection axis.

This group is frequently seen in everyday life, since the most common arrangement of bricks in a brick building utilises this group (see example below).

The rotational symmetry of order 2 with centres of rotation at the centres of the sides of the rhombus is a consequence of the other properties.

The pattern corresponds to each of the following:

symmetrically staggered rows of identical doubly symmetric objects

a checkerboard pattern of two alternating rectangular tiles, of which each, by itself, is doubly symmetric

a checkerboard pattern of alternatingly a 2-fold rotationally symmetric rectangular tile and its mirror image

Examples of group cmm

Computer generated

one of the 8 semi-regular tessellations; ignoring color this is this group cmm, otherwise group p1

Suburban brick wall, U.S.

Ceiling of Egyptian tomb. Ignoring colors, this would be p4g.

Egyptian

Persian tapestry

Egyptian tomb

Turkish dish

Group p4

Example and diagram for p4

Cell structure for p4

Cell structure for p4

Orbifold notation: 442.

The group p4 has two rotation centres of order four (90), and one rotation centre of order two (180). It has no reflections or glide reflections.

Examples of group p4

A p4 pattern can be looked upon as a repetition in rows and columns of equal square tiles with 4-fold rotational symmetry. Also it can be looked upon as a checkerboard pattern of two such tiles, a factor smaller and rotated 45.

Computer generated

Ceiling of Egyptian tomb; ignoring colors this is p4, otherwise p2

Ceiling of Egyptian tomb

Frieze, the Alhambra, Spain. Requires close inspection to see why there are no reflections.

Viennese cane

Renaissance earthenware

Group p4m

Example and diagram for p4m

Cell structure for p4m

Cell structure for p4m

Orbifold notation: *442.

The group p4m has two rotation centres of order four (90), and reflections in four distinct directions (horizontal, vertical, and diagonals). It has additional glide reflections whose axes are not reflection axes; rotations of order two (180) are centred at the intersection of the glide reflection axes. All rotation centres lie on reflection axes.

This corresponds to a straightforward grid of rows and columns of equal squares with the four reflection axes. Also it corresponds to a checkerboard pattern of two of such squares.

Examples of group p4m

Examples displayed with the smallest translations horizontal and vertical (like in the diagram):

Computer generated

one of the 3 regular tessellations (in this checkerboard coloring, smallest translations are diagonal)

Demiregular tiling with triangles; ignoring colors, this is p4m, otherwise cmm

one of the 8 semi-regular tessellations (ignoring color also, with smaller translations)

Ornamental painting, Nineveh, Assyria

Storm drain, U.S.

Egyptian mummy case

Persian glazed tile

Examples displayed with the smallest translations diagonal (like on a checkerboard):

Cloth, Otaheite (Tahiti)

Egyptian tomb

Cathedral of Bourges

Dish from Turkey, Ottoman period

Group p4g

Example and diagram for p4g

Cell structure for p4g

Cell structure for p4g

Orbifold notation: 4*2.

The group p4g has two centres of rotation of order four (90), which are each other's mirror image, but it has reflections in only two directions, which are perpendicular. There are rotations of order two (180) whose centres are located at the intersections of reflection axes. It has glide reflections axes parallel to the reflection axes, in between them, and also at an angle of 45 with these.

A p4g pattern can be looked upon as a checkerboard pattern of copies of a square tile with 4-fold rotational symmetry, and its mirror image. Alternatively it can be looked upon (by shifting half a tile) as a checkerboard pattern of copies of a horizontally and vertically symmetric tile and its 90 rotated version. Note that neither applies for a plain checkerboard pattern of black and white tiles, this is group p4m (with diagonal translation cells).

Note that the diagram on the left represents in area twice the smallest square that is repeated by translation.

Examples of group p4g

Computer generated

Bathroom linoleum, U.S.

Painted porcelain, China

Fly screen, U.S.

Painting, China

one of the colorings of the snub square tiling (see also at pg)

Group p3

Cell structure for p3 (the rotation centres at the centres of the triangles are not shown)

Cell structure for p3

Orbifold notation: 333.

The group p3 has three different rotation centres of order three (120), but no reflections or glide reflections.

Imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, but the two are not equal, not each other's mirror image, and not both symmetric (if the two are equal we have p6, if they are each other's mirror image we have p31m, if they are both symmetric we have p3m1; if two of the three apply then the third also, and we have p6m). For a given image, three of these tessellations are possible, each with rotation centres as vertices, i.e. for any tessellation two shifts are possible. In terms of the image: the vertices can be the red, the blue or the green triangles.

Equivalently, imagine a tessellation of the plane with regular hexagons, with sides equal to the smallest translation distance divided by 3. Then this wallpaper group corresponds to the case that all hexagons are equal (and in the same orientation) and have rotational symmetry of order three, while they have no mirror image symmetry (if they have rotational symmetry of order six we have p6, if they are symmetric with respect to the main diagonals we have p31m, if they are symmetric with respect to lines perpendicular to the sides we have p3m1; if two of the three apply then the third also, and we have p6m). For a given image, three of these tessellations are possible, each with one third of the rotation centres as centres of the hexagons. In terms of the image: the centres of the hexagons can be the red, the blue or the green triangles.

Examples of group p3

Computer generated

one of the 8 semi-regular tessellations (ignoring the colors: p6); the translation vectors are rotated a little to the right compared with the directions in the underlying hexagonal lattice of the image

Street pavement in Zakopane, Poland

Wall tiling in the Alhambra, Spain (and the whole wall); ignoring all colors this is p3 (ignoring only star colors it is p1)

Group p3m1

Example and diagram for p3m1

Cell structure for p3m1

Cell structure for p3m1

Orbifold notation: *333.

The group p3m1 has three different rotation centres of order three (120). It has reflections in the three sides of an equilateral triangle. The centre of every rotation lies on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.

Like for p3, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, and both are symmetric, but the two are not equal, and not each other's mirror image. For a given image, three of these tessellations are possible, each with rotation centres as vertices. In terms of the image: the vertices can be the red, the dark blue or the green triangles.

Examples of group p3m1

one of the 3 regular tessellations (ignoring colors: p6m)

another regular tessellation (ignoring colors: p6m)

one of the 8 semi-regular tessellations (ignoring colors: p6m)

Persian glazed tile (ignoring colors: p6m)

Persian ornament

Floor tiling in Budapest, Hungary (ignoring colors: p6m)

Painting, China (see detailed image)

Computer generated

Group p31m

Example and diagram for p31m

Cell structure for p31m

Cell structure for p31m

Orbifold notation: 3*3.

The group p31m has three different rotation centres of order three (120), of which two are each other's mirror image. It has reflections in three distinct directions. It has at least one rotation whose centre does not lie on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.

Like for p3 and p3m1, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three and are each other's mirror image, but not symmetric themselves, and not equal. For a given image, only one such tessellation is possible. In terms of the image: the vertices can not be dark blue triangles.

Examples of group p31m

Persian glazed tile

Painted porcelain, China

Painting, China

Computer generated

Group p6

Example and diagram for p6

Cell structure for p6

Cell structure for p6

Orbifold notation: 632.

The group p6 has one rotation centre of order six (60); it has also two rotation centres of order three, which only differ by a rotation of 60 (or, equivalently, 180), and three of order two, which only differ by a rotation of 60. It has no reflections or glide reflections.

A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with C3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with C6 symmetry (with the edges of the tiles not necessarily part of the pattern).

Examples of group p6

Computer generated

Wall panelling, the Alhambra, Spain

Persian ornament

Group p6m

Example and diagram for p6m

Cell structure for p6m

Cell structure for p6m

Orbifold notation: *632.

The group p6m has one rotation centre of order six (60); it has also two rotation centres of order three, which only differ by a rotation of 60 (or, equivalently, 180), and three of order two, which only differ by a rotation of 60. It has also reflections in six distinct directions. There are additional glide reflections in six distinct directions, whose axes are located halfway between adjacent parallel reflection axes.

A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with D3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with D6 symmetry (with the edges of the tiles not necessarily part of the pattern). Thus the simplest examples are a triangular lattice with or without connecting lines, and a hexagonal tiling with one color for outlining the hexagons and one for the background.

Examples of group p6m

Computer generated

one of the 8 semi-regular tessellations

another semi-regular tessellation

another semi-regular tessellation

Persian glazed tile

King's dress, Khorsabad, Assyria; this is almost p6m (ignoring inner parts of flowers, which make it cmm)

Bronze vessel in Nimroud, Assyria

Byzantine marble pavement, Rome

Painted porcelain, China

Painted porcelain, China

Lattice types

There are five lattice types, corresponding to the five possible wallpaper groups of the lattice itself. The wallpaper group of a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.

In the 5 cases of rotational symmetry of order 3 or 6, the cell consists of two equilateral triangles (hexagonal lattice, itself p6m).

In the 3 cases of rotational symmetry of order 4, the cell is a square (square lattice, itself p4m).

In the 5 cases of reflection or glide reflection, but not both, the cell is a rectangle (rectangular lattice, itself pmm), therefore the diagrams show a rectangle, but a special case is that it actually is a square.

In the 2 cases of reflection combined with glide reflection, the cell is a rhombus (rhombic lattice, itself cmm); a special case is that it actually is a square.

In the case of only rotational symmetry of order 2, and the case of no other symmetry than translational, the cell is in general a parallelogram (parallelogrammatic lattice, itself p2), therefore the diagrams show a parallelogram, but special cases are that it actually is a rectangle, rhombus, or square.

Symmetry groups

The actual symmetry group should be distinguished from the wallpaper group. Wallpaper groups are collections of symmetry groups. There are 17 of these collections, but for each collection there are infinitely many symmetry groups, in the sense of actual groups of isometries. These depend, apart from the wallpaper group, on a number of parameters for the translation vectors, the orientation and position of the reflection axes and rotation centers.

The numbers of degrees of freedom are:

6 for p2

5 for pmm, pmg, pgg, and cmm

4 for the rest.

However, within each wallpaper group, all symmetry groups are algebraically isomorphic.

Some symmetry group isomorphisms:

p1: Z2

pm: Z D

pmm: D D.

Dependence of wallpaper groups on transformations

The wallpaper group of a pattern is invariant under isometries and uniform scaling (similarity transformations).

Translational symmetry is preserved under arbitrary bijective affine transformations.

Rotational symmetry of order two ditto; this means also that 4- and 6-fold rotation centres at least keep 2-fold rotational symmetry.

Reflection in a line and glide reflection are preserved on expansion/contraction along, or perpendicular to, the axis of reflection and glide reflection. It changes p6m, p4g, and p3m1 into cmm, p3m1 into cm, and p4m, depending on direction of expansion/contraction, into pmm or cmm. A pattern of symmetrically staggered rows of points is special in that it can convert by expansion/contraction from p6m to p4m.

Note that when a transformation decreases symmetry, a transformation of the same kind (the inverse) obviously for some patterns increases the symmetry. Such a special property of a pattern (e.g. expansion in one direction produces a pattern with 4-fold symmetry) is not counted as a form of extra symmetry.

Change of colors does not affect the wallpaper group if any two points that have the same color before the change, also have the same color after the change, and any two points that have different colors before the change, also have different colors after the change.

If the former applies, but not the latter, such as when converting a color image to one in black and white, then symmetries are preserved, but they may increase, so that the wallpaper group can change.

Web demo and software

There exist several software graphic tools that will let you create 2D patterns using wallpaper symmetry groups. Usually, you can edit the original tile and its copies in the entire pattern are updated automatically.

Tess, a nagware tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups, as well as Heesch tilings.

Kali, free graphical symmetry editor available online and for download.

Inkscape, a free vector graphics editor, supports all 17 groups plus arbitrary scales, shifts, rotates, and color changes per row or per column, optionally randomized to a given degree. (See )

SymmetryWorks is a commercial plugin for Adobe Illustrator, supports all 17 groups.

Arabeske is a free standalone tool, supports a subset of wallpaper groups.

See also

Wikimedia Commons has media related to: Wallpaper group diagrams

List of planar symmetry groups (summary of this page)

Tessellation

Point group

Crystallography

Symmetry groups in one dimension

M. C. Escher

Aperiodic tiling

Notes

^ E. Fedorov (1891) "Simmetrija na ploskosti" [Symmetry in the plane], Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society], series 2, vol. 28, pages 345-291 (in Russian).

^ George Plya (1924) "ber die Analogie der Kristallsymmetrie in der Ebene," Zeitschrift fr Kristallographie, vol. 60, pages 278-282.

^ Weyl, Hermann (1952), Symmetry, Princeton University Press, ISBN 0-691-02374-3 

^ It helps to consider the squares as the background, then we see a simple patterns of rows of rhombuses.

References

The Grammar of Ornament (1856), by Owen Jones. Many of the images in this article are from this book; it contains many more.

J. H. Conway (1992). "The Orbifold Notation for Surface Groups". In: M. W. Liebeck and J. Saxl (eds.), Groups, Combinatorics and Geometry, Proceedings of the L.M.S. Durham Symposium, July 515, Durham, UK, 1990; London Math. Soc. Lecture Notes Series 165. Cambridge University Press, Cambridge. pp. 438447

Grnbaum, Branko; Shephard, G. C. (1987): Tilings and Patterns. New York: Freeman. ISBN 0-7167-1193-1.

Pattern Design, Lewis F. Day

External links

The 17 plane symmetry groups by David E. Joyce

Introduction to wallpaper patterns by Chaim Goodman-Strauss and Heidi Burgiel

Description by Silvio Levy

Example tiling for each group, with dynamic demos of properties

Overview with example tiling for each group

Tiling plane and fancy by Steve Edwards

Escher Web Sketch, a java applet with interactive tools for drawing in all 17 plane symmetry groups

Burak, a Java applet for drawing symmetry groups.

Beobachtungen zum geometrischen Motiv der Pelta

Categories: Crystallography | Discrete groups | Euclidean symmetries | OrnamentsHidden categories: Articles to be merged from February 2010 | All articles to be merged

Bicycle infantry

China Product


History

Origins

Bicycle Corps at Fort Missoula in 1897. mesh truck tarps

Numerous experiments were carried out to determine the possible role of bicycles and cycling within military establishments until in 1894 a turning point occurred due to improved resilience of pneumatics and the shorter sturdier construction of the frame. To some extent, bicyclists took over the functions of dragoons, especially as messengers and scouts, substituting for horses in warfare. Bicycle units or detachments were formed at the end of the 19th century by all European armies and the US armed forces. rv tire pressure

The United Kingdom employed bicycle troops in militia or territorial units, but not in regular units. In France, several experimental units were created, starting in 1886. They attempted to adopt folding bicycles early on. In the United States, the most extensive experimentation on bicycle units was carried out by a 1st Lieutenant Moss, of the 25th United States Infantry (Colored) (an African American infantry regiment with white officers). Using a variety of cycle models, Lt. Moss and his troops carried out extensive bicycle journeys covering between 500 and 1,000 miles (800 to 1,600 km). Late in the 19th century, the United States Army tested the bicycle's suitability for cross-country troop transport. Buffalo Soldiers stationed in Montana rode bicycles across roadless landscapes for hundreds of miles at high speed. chrome grab handle

The first known use of the bicycle in combat occurred during the Jameson Raid, in which cyclists carried messages. In the Second Boer War, military cyclists were used primarily as scouts and messengers. One unit patrolled railroad lines on specially constructed tandem bicycles that were fixed to the rails. Several raids were conducted by cycle-mounted infantry on both sides; the most famous unit was the Theron se Verkenningskorps (Theron Reconnaissance Corps) or TVK, a Boer unit led by the scout Daniel Theron, whom British commander Lord Roberts described as "the hardest thorn in the flesh of the British advance." Roberts placed a reward of 1,000 on Theron's headead or alivend dispatched 4,000 soldiers to find and eliminate the TVK.

World Wars

Danish soldiers cycling to the front to fight the Germans during the Battle of Denmark in 1940

Photo showing Italian Bersaglieri during World War I with bicycles strapped to their backs. 1917.

German bicycle infantry on the northern Russian front in 1941

During World War I, cycle-mounted infantry, scouts, messengers and ambulance carriers were extensively used by all combatants. Italy used bicycles with the Bersaglieri (light infantry units) until the end of the war. German Army Jger (light infantry) battalions each had a bicycle company (Radfahr-Kompanie) at the outbreak of the war, and additional companies were raised during the war bringing the total to 80 companies, a number of which were formed into eight Radfahr-Bataillonen (bicycle battalions). In its aftermath, the German Army conducted a study on the use of the cycle and published its findings in a report entitled Die Radfahrertruppe[citation needed].

In its 1937 invasion of China, Japan employed some 50,000 bicycle troops. Early in World War II their southern campaign through Malaya en route to capturing Singapore in 1941 was largely dependent on bicycle-riding soldiers. In both efforts bicycles allowed quiet and flexible transport of thousands of troops who were then able to surprise and confuse the defenders. Bicycles also made few demands on the Japanese war machine, needing neither trucks, nor ships to transport them, nor precious petroleum. Using bicycles, the Japanese troops were able to move faster than the withdrawing Allied Forces, often successfully cutting off their retreat. The speed of Japanese advance have also caught Allied Forces defending the main roads by surprise by attacking them from the rear.

The Finnish Army utilized bicycles extensively during the Continuation War and Lapland War. Bicycles were used as a means of transportation in Jaeger Battalions, divisional Light Detachments and regimental organic Jaeger Companies. Bicycle units spearheaded the advances of 1941 against Soviet Union. Especially successful was the 1st Jaeger Brigade which was reinforced with a tank battalion and an anti-tank battalion, providing rapid movement through limited road network. During winter time these units, like the rest of the infantry, switched to skis.

Within 1942-1944 bicycles were also added to regimental equipment pools. During the Summer 1944 battles against the Soviet Union, bicycles provided quick mobility for reserves and counter-attacks. In Autumn 1944 bicycle troops of the Jaeger Brigade spearheaded the Finnish advance through Lapland against the Germans; tanks had to be left behind due to the German destruction of the Finnish road network.

The hastily assembled German Volksgrenadier divisions had a battalion of bicycle infantry, to have some mobile reserve.

Allied use of the bicycle in World War II was limited, but included supplying folding bicycles to paratroopers and to messengers behind friendly lines. The term, "bomber bikes" came into use during this period, as US forces dropped bicycles out of planes to reach troops behind enemy lines.

By 1939, the Swedish army operated six bicycle infantry regiments. They were equipped with domestically produced Swedish military bicycles. Most common was the m/42, an upright, one-speed roadster produced by several large Swedish bicycle manufacturers. These regiments were decommissioned between 1948 and 1952, and the bicycles remained for general use in the Army, or transferred to the Home Guard. Beginning in the 1970s, the Army began to sell these as military surplus. They became very popular as cheap and low-maintenance transportation, especially among students. Responding to its popularity and limited supply, an unrelated company, Kronan, began to produce a modernized version of the m/42 in 1997.

Later uses

Although much used in World War I, bicycles were largely superseded by motorized transport in more modern armies. In the past few decades, however, they have taken on a new life as a "weapon of the people" in guerrilla conflicts and unconventional warfare, where the cycle's ability to carry large, about 400 lb (180 kg), loads of supplies at the speed of a pedestrian make it vastly useful for lightly-equipped forces. For many years the Viet Cong and North Vietnamese Army used bicycles to ferry supplies down the "Ho Chi Minh trail", avoiding the repeated attacks of United States and Allied bombing raids. When heavily loaded with supplies such as sacks of rice, these bicycles were seldom rideable, but were pushed by a tender walking alongside. With especially bulky cargo, tenders sometimes attached bamboo poles to the bike for tiller-like steering (this method can still be seen practiced in China today). Vietnamese "cargo bikes" were rebuilt in jungle workshops with reinforced frames to carry heavy loads over all terrain.

Modern times

LTTE bicycle infantry platoon north of Killinochi in 2004

Bicycles continue in military use today, primarily as an easy alternative for transport on long flightlines. The use of the cycle as an infantry transport tool continued into the 21st century with the Swiss Army's Bicycle Regiment, which maintained drills for infantry movement and attack until 2001, when the decision was made to phase the unit out.

The LTTE Tamil Tigers made use of bicycle mobility in the fighting in Sri Lanka. The Sri Lankan army also has a bicycle unit. They are mainly stationed and deployed in high security zones in the capital city Colombo. The theory and the basis of their usage is still not well known.

In media

In the expansion pack for EA game Command & Conquer: Generals Zero Hour has a unit called a ombat Cycle which are dirt bike mounted infantry used for reconnaissance and light combat.

See also

Paratrooper folding Tactical Mountain Bicycle

Army Cyclist Corps

References

^ Leiser 10

^ Leiser 11-16

^ Leiser 11

^ "Danie Theron" (html). http://www.instinsky.de/Boer_War/Danie_Theron/danie_theron.html. Retrieved 2007-10-07. 

^ Doole, Claire, End of road for Swiss army cyclists, BBC News, http://news.bbc.co.uk/1/hi/world/europe/1325485.stm, retrieved 2008-02-05 

Bibliography

Leiser Rolf (sup.) (1991). Hundert Jahre Radfahrer-Truppe (100 Years of Bicycle Troops). Bern, Switzerland: Bundesamt fr Mechanisierte u. Leichte Truppen (Federal Office for Mechanized and Light Troops). 

Fitzpatrick, Jim (1998). The Bicycle In Wartime: An Illustrated History. Washington, DC: Brassey's Inc.. ISBN 1-57488-157-4. 

Ekstrm, Gert; Husberg, Ola (2001). lskade cykel (1st ed.). Bokfrlaget Prisma. ISBN 91-518-3906-7. 

Categories: Combat occupations | Cycling | History of cycling | InfantryHidden categories: All articles with unsourced statements | Articles with unsourced statements from January 2008