What are food additives?Food additives are an important component of our food supply.They mean that we can enjoy a wide variety of foods throughoutthe year. They also have an important role in ensuring that ourfood lasts longer and is easier to use. There are good reasonsfor the use of food additives. They can be used to:improve the keeping quality or stability of a food. For example,sorbitol - humectant (420) - may be added to mixed dried fruitto maintain the moisture level and softness of the fruit;preserve food when this is the most practical way of extendingits storage life. For example, sulphur dioxide - preservative(220) - is added to some meat products such as sausage meatto prevent the bugs that cause food poisoning from growing;andimprove the taste or appearance of a processed food.For example, lecithin - emulsifier (322) - may be added tomargarine to help maintain texture.Additives are used in processed foods in relatively smallquantities. Many substances used as additives also occurnaturally, such as vitamin C or ascorbic acid (300) in fruit andlecithin (322) in eggs or soy beans.How do I know what additives are in food?If you want to know more about a particular food additive lookat the ingredient list on the food label where you will find theadditives name and number, for example, food acid (260).You can use this information to gain a better understandingof what is in the food you eat and why different food additivesare used. This is an example of an ingredient list, which mightappear on a packaged stir-fry meal:Ingredients - pork, wheat flour, capsicum, pineapple,green beans, sweet corn, sugar, tomato paste, pineappleconcentrate, thickener (1422), food acids (270, 260), soysauce, salt, natural flavours, vegetable gum (415), wateradded.The name of an approved food additive must be spelt out in fullon a food label if it doesnt have an appropriate class name or ifan additive number hasnt been allocated to it.What do additives do?Some food additives have more than one use. Food additivesare listed according to their functional or class names;colourings add or restore colour to foods;colour retention agents retain or intensify the colour of afood;preservatives help protect against deterioration caused bymicro-organisms;artificial sweetening substances are substances whichimpart a sweet taste for fewer kilojoules/calories than sugar;flavour enhancers improve the flavour and/or aroma of food;flavourings restore taste losses due to processing, maintainuniformity and make food more palatable;anti-caking agents keep powdered products such as salt,flowing freely when poured;emulsifiers help to prevent oil and water mixtures separatinginto layers;food acids help maintain a constant level of sourness in food;humectants prevent foods such as dried fruits from dryingout;mineral salts improve the texture of foods, such as processedmeats;thickeners and vegetable gums improve texture andmaintain uniform consistency;stabilisers maintain the uniform dispersion of substances in afood;flour treatment agents are substances added to flour toimprove baking quality or appearance;glazing agents impart a shiny appearance or provide aprotective coating to a food;propellants are gases which help propel food from acontainer.Who controls the use of food additives?The use of food additives in foods is regulated by the FoodStandards Code and enforced in Australia under State andTerritory food laws. Standard 1.3.1 defines the uses of foodadditives in foods. Food Standards Australia New Zealand(FSANZ) is responsible for the development of, or variation to,food standards in the Food Standards Code. The FSANZ Boardrecommends its decisions to a Ministerial Council, made up ofState, Territory, Commonwealth and New Zealand Ministers,prior to adoption into the Code.Before recommending the use of any new additive in a particularfood, FSANZ needs to know:Is the additive safe to eat (at the requested level in thatparticular food)?Are there good technological reasons for the use of theadditive?Will consumers be clearly informed about its presence?Only if satisfied on these points will FSANZ recommend amaximum level of the additive permitted in particular foods,based on technological need and providing it is well within safelimits.Food additive safetyFSANZ only allows for the use of additives if it can bedemonstrated that no harmful effects are expected to result. Thisinvolves FSANZ evaluating the data obtained from extensivetesting of the additive. A decision on food additive safety isbased on the acceptable daily intake (ADI), which is the amountof a food additive that can be eaten every day for an entirelifetime without adverse effect.Intolerance and food additivesAdverse reactions to food additives occur in a small proportionof the population. A few people are intolerant to some foodadditives. Intolerance does not depend on whether the foodadditive is derived from a natural or synthetic source. Morepeople are intolerant to common foods such as peanuts, milk oreggs than to food additives.The labelling of food products helps people who are sensitive tosome food additives to avoid them.More informationThis information is an extract from the Food Standards AustraliaNew Zealand website:http://www.foodstandards.gov.au/foodmatters/foodad

ditives.cfmSouth Australian Department of Health website:www.health.sa.gov.au/pehs click on food safety

International Journal of Environment and Pollution

Issue:Volume 10, Number 2 / 1998

Pages:273 - 288

URL:Linking Options

Environmental dimensions of fertilizer and pesticide use; relevance to Indian agricultureJ.P. Painuly and S. Mahendra DevA1Indira Gandhi Institute of Development Research, General Vaidya Marg, Goregaon (E), Mumbai-400065, India.A2Indira Gandhi Institute of Development Research, General Vaidya Marg, Goregaon (E), Mumbai-400065, IndiaAbstract:This paper presents an overview of the environmental consequences of fertilizers and pesticides in agriculture and the measures needed to mitigate the adverse impact of these chemicals on environment. The issues are then analysed from the perspective of the use of fertilizers and pesticides in Indian agriculture. Fertilizer consumption in India is concentrated in about one-third of the cultivated area. Its use has been increasing but it is being used inefficiently. Pesticide use is also concentrated in five states and on a few crops such as rice, cotton and chillies. Evidence from micro studies on the environmental consequences of these chemicals is presented. It is suggested that the present methods of fertilizer and pesticide use and growth are not sustainable. There are several possible technologies and alternatives to reduce the adverse impact of these chemicals on the environment, such as biological control of pests, integrated pest management, development of pest-resistant varieties of crops, vermiculture. etc., that need to be supplemented through an economic approach. These issues need to be considered while formulating strategies for sustainable agriculture in India and other developing countries.

Spider Mites Pests in GreenhouseinGREENHOUSE,PEST CONTROLSpider mite is commonly occurring pests that possess great threats to variety of plants such as cucumber. This pest is capable of causing great level of damage to various parts of plants. Cucumber is one of the plants that get affected in a big manner. There are various measures that can be utilized to achieve effective control of plants. The usage of pesticide alone will not be of good use in acquiring required level of protection since this will not be able to provide required level of protection. There is a need to depend upon other areas such as biological control programs that will provide enhanced level of protection against this

Spider Mite Imagekind of pests. Spider mites can be controlled with the help of insect introduction inside the greenhouse region. The population of spider mite pests will come down drastically with the help of this method that will be providing good level of benefits. There are many people who have been benefited in a great manner with the help of natural enemies control programs.Resistance towards pesticide

Two spotted miteOne of the important aspects of these pests that make it very difficult to control the growth of pests with the help of pesticides is that they show resistance towards organophosphorus pesticides. The usage of pesticides is no longer fully effective in controlling the growth of this pest. There is a need to depend upon other type of methods that will be providing good level of destruction of this pest. The best solution for achieving good control of this pest can be achieved with the introduction of insects that are capable of eliminating the growth of spider mites in greenhouse. Predator mites are one of the most effective insects that will be of good use in attacking these pests. The native predator mites when introduced in smaller quantities will be able to reproduce inside greenhouse. This increases number of predator mites that are present in greenhouse and increased resistance against these pests can be achieved.Rate of insect supply

Web of spider mitesThe key to success behind effective control of spider mite pests using insect introduction process depends upon timing of release of insects. Optimum quantity of insects should be released at correct time such that pests can be removed in an easy manner. Usually, insects will be introduced at the rate of one insect per pant. Insects will be introduced with interval of 2 to 3 weeks. Prevention of spider mite is an important process to achieve effective pest control.Natural Enemies For Suppression Pest Control in GreenhouseinGREENHOUSE,PEST CONTROL,PLANT PROTECTIONIt is one of the biggest challenges to control the growth of pests inside greenhouse. If left unattended, pests will damage entire greenhouse area in a rapid manner. There should be greater level of care that should be incorporated to control growth of pests and efforts should be made to stop invasion of these pests. Some of the common pests that cause more level of damage to crops include whitefly, spider mites, aphids, tomato leaf miner, vegetable leaf miner and many others. There are some cases in which application of pesticides will be inefficient. Other methods such as biological control program will be of good use in halting the development of pests inside of greenhouse. Usage of natural enemies for suppression will offer wonderful results. The results can be achieved at a faster rate when compared to other methods.Suppression using insects

orius laevigatus - fieber feeding on artificial dietInsects can be used to attack various kinds of pests grows inside greenhouse. Before selection of particular insects is being made, it is essential to analyze certain features. First of all, various kinds of pesticides and degree of damage that is done to plants should be identified. Based on damage that was done to plants, it is essential to order insect supplies from reputed source nearby your home. There are many people who are satisfied in a great manner with protection that is being obtained with the help of insects. The desired rate of introduction of insects should be identified based on level of damage that was done to plants. There are three main methods that can be used to introduce insects namely conservation, inoculation and inundation. Inoculation involves introduction of smaller quantity of insects to combat pests during initial growth period. Inundation incorporates usage of more amounts of insects to prevent outbreak of pests after considerable development.Selection of insectsFor preventing attack of plants from whitefly pests, it will be a wonderful idea to select encarsia wasp that will destroy pests at a rapid rate. The insect introduction can also be integrated with pesticides for obtaining effective results. Predator mite insect will be an appropriate choice for dealing with spider mite pests. Researchers are trying to find many new insects that will provide effective protection for greenhouse plants from being attacked by pests such as aphids and many others. Insect supply is also one of the cheaper methods that can be used to obtain effective protection of plants inside greenhouse.Whitefly PestsWhitefly pests are one of the major killers of tomato plant. Knowing about the lifecycle and growth of this pest will be of great use in preventing the outbreak of this pest attack. This pest creates one of the biggest problems in greenhouse. There are several methods that are available to control this pest. Some [...]Read the full article Common Greenhouse PestsThe damage caused by greenhouse pests to the plants inside glass has sparked great level of interest among researchers. To provide effective protection to plants inside greenhouse, it is essential to understand the life cycle of pests such that they can be eradicated in best possible manner. There are several methods that can be used [...]Read the full article Natural Enemies For Greenhouse PestsBiological control program for protecting plants from pest attack is one of the most effective methods that will help plants to achieve good level of growth. This is one of the most effective and best methods that can be used to prevent pest attack of plants inside greenhouse. The introduction of natural enemies for greenhouse [...]Read the full article Pestering InsectsPest control insects are those insects that have been, for years, creating much of the trouble to the human beings. Small size of the insects does not help us either, and they get advantage of their micro size and escapes every of our attempt to kill them. But, there is a brighter and sure fire [...]Read the full article Pest Control SlugsSlugs are nothing but some kind of mollusk creatures, which contains jelly like fluid inside their body. The mollusk phylum family is a group of slimy creatures, which are loathed all over the world. The slimy nature makes it slippery and they are not easily attacked physically. These slimy creatures attack the low lives in [...]Read the full article Pest Control BeetlesA beetle is commonly known to many of us. But what it really does is not known by many of our people. The one unique feature about the beetle is that it is having the most number of species that is ever known. Other feature of this is that the beetles are common in all [...]Read the full article Maggot Pest ControlA larval phase of any insect is known as the maggot. These things usually feed on the dead tissues of living organisms. It has both good and bad associated with it. In most medical industries it is used to make living things cure from their respective diseases. This form of therapy is known as maggot [...]Read the full article Cabbage Pest ControlAll of us irrespective of the continent we live irrespective of the country we live there are many common things between us. One such thing is the vegetables we eat. The most common vegetable of all time is the cabbage. Cabbage has a long history associated with it. The most common color of the cabbage [...]Read the full article Knowing Earwig Infestation and Pest ControlEarwig is a small insect which is easily identifiable by its claws like rear part. These claws like structure are extremely innocuous and are called pincers. Even though they are harmless to us, they are much of a nuisance to the plants and the herbs that we grow in our backyard. The problem with the [...]Read the full article Garden Pest ControlPeople would always love to have attractive gardens at home. Gardens may consist of diverse plants that tend to make the admirers spell bond. Gardening work consists of major work namely garden pest control. There are different types of garden pests that tend to provide hindrance to the growth of the plant. Common garden pests [...]

/406-18 final Risk assessment on the use of triclosan in cosmetics Prepared by the Scientific Committee in cooperation with the Panel on Biological Hazards and the Panel on Food Additives, Flavourings, Processing Aids, Materials in contact with Food and Cosmetics Date: 31.01.05 SUMMARY Triclosan is a widely used biocide. It is included in many contemporary consumer- and professional health-care products, particularly oral and dermal products, but also in household items including plastics, textiles and food packaging materials. Concerns have been raised regarding the widespread use of triclosan; both as regards the potential for selection of resistant bacterial strains that may confer cross-resistance to other antimicrobial agents and as regards the potential harm to the environment. Triclosan is classified as very toxic to aquatic organisms and may cause long-term adverse effects to the environment by the European Commission. In March 2004, the Norwegian Food Safety Authority asked the Norwegian Scientific Committee for Food Safety to prepare an updated risk assessment of the use of triclosan in cosmetics, regarding development of resistance in pathogenic bacteria. The authority also asked for a comprehensive toxicological examination of the chemical with indication of the margin of safety as regards use in cosmetic products. In response, three assessments were initiated; one to address the risk for development of antimicrobial resistance in bacteria, one to address the toxicological aspects, and one to address ecotoxicological matters. From these three assessments, general conclusions have been drawn by the Scientific Committee based on the conclusions from the Panel on Biological Hazards and the Panel on Food Additives, Flavourings, Processing Aids, Materials in contact with Food and Cosmetics. The conclusions are: Widespread use of triclosan, including use in cosmetic products, selects for development of triclosan resistance. Since this may contribute to the development and spread of concomitant resistance to clinically important antimicrobial agents, such use represents a public health risk. Therefore, the use of triclosan should be restricted. The current regulation of use of triclosan in cosmetic products is from a toxicological point of view a matter of concern and it is recommended that human exposure to triclosan should be reduced. Moreover, triclosan is classified as an agent that may cause adverse environmental effects and hence the use should be restricted also from an ecotoxicological standpoint. 1 Norwegian Scientific Committee for Food Safety 04/406-18 final BACKGROUND Triclosan has been widely used since its introduction 40 years ago. However, in more recent years the use of triclosan as a preservative, antiseptic and disinfectant in the USA and Europe has risen significantly. In Europe concerns have been raised regarding the widespread and increasing use of triclosan, in view of the potential for selection of resistant bacterial strains that may confer cross-resistance to clinically relevant antimicrobial agents. The former Norwegian Food Control Authority (SNT) therefore asked scientific experts at the Norwegian Institute of Public Health to perform a risk assessment on the use of triclosan in cosmetics (4th September 2000). The conclusion of this assessment was as follows: care must be taken to contribute as little as possible to the selection of resistant bacteria............with this in mind, particularly in light of recent indications of its association with the development of antibiotic resistance in bacteria, we recommend against the use of triclosan in cosmetics and other products in general use, in which disinfectant action is neither useful nor desirable. In 2002, the former Scientific Steering Committee (SSC) of the EU appointed a working group of experts with the mandate to draft a scientific report that could be used as input for the preparation of a scientific opinion of the SSC regarding the safety of triclosan, especially related to the risk for resistance development in certain microorganisms. The SSC concluded that: There is no convincing evidence that triclosan poses a risk to humans or to the environment by inducing or transmitting antibacterial resistance under current conditions of use. The Scientific Committee on Cosmetic Products and Non-Food Products intended for Consumers (SCCNFP) of the EU evaluated the opinion of the SSC and concluded that: 1: Under current conditions of use of triclosan as a preservative in cosmetic products, it is safe taking into account the risk of resistance by certain micro-organisms. 2: There is no need for setting a new concentration limit for the use of triclosan in cosmetic products. The experts at the Norwegian Institute of Public Health evaluated the assessment of the SSC and SCCNFP, and concluded that their own recommendations of September 2000 remain valid (8th November 2002). The Norwegian Food Control Authority, in a letter to the European Commission, DG Enterprise (Biotechnology, competitiveness in pharmaceuticals, cosmetics), called for a re-evaluation of the permission to use triclosan in cosmetic products (20th December 2002). The recommendations from the Norwegian experts were also stressed at a meeting in the Working Party on Cosmetics in the European Commission. The call for re-evaluation was answered in March 2003, where the European Commission asked the Norwegian Food Control Authority for an assessment based on new data and after consideration of the SCCNFPs opinion on triclosan. Triclosan is approved for use in cosmetic products in Norway and the European Union. In the EU, Triclosan is also approved for use in food contact material made of plastic. In that context the EU Scientific Committee on Food (SCF) did a toxicological evaluation of triclosan in 2000 which is considered in the toxicological part of this risk assessment. Because of the extensive use of triclosan in cosmetics there is a need to review and update the toxicological information and to assess the margin of safety in the use of triclosan in cosmetics. 2 Norwegian Scientific Committee for Food Safety 04/406-18 final Furthermore, triclosan is classified as very toxic to aquatic organisms and may cause long-term adverse effects to the environment by the European Commission. TERMS OF REFERENCE Based on the opinion from Norwegian experts, as well as the reports from SSC and SCCNFP, regarding the use of triclosan and the possibility for development of antimicrobial resistance in bacteria, it was concluded that an updated risk assessment was necessary. In March 2004, the Norwegian Food Safety Authority asked the Norwegian Scientific Committee for Food Safety: to prepare an updated risk assessment for the use of triclosan in cosmetics, regarding development of resistance in pathogenic bacteria1. The authority also asks for a comprehensive toxicological examination of the chemical with indication of the margin of safety as regards the usage in cosmetic products2 Because of the broad use of triclosan and the subsequent possibility for the chemical to be spread out in the environment, the Committee found it relevant to also include a summary of an evaluation of the environmental effects of triclosan that the Norwegian Institute for Water Research has prepared previously for the Norwegian Pollution Control Authority. ASSESSMENT Three assessments have been prepared and are attached. The conclusions from each of these assessments are as follows: I: Development of antimicrobial resistance in bacteria Bacterial isolates with reduced susceptibility to triclosan have been produced in laboratory experiments by repeated exposure of bacteria to sub-lethal doses of triclosan. A number of studies have verified the occurrence of acquired triclosan resistance among dermal, intestinal and environmental bacterial species, including some of clinical relevance. However, studies involving clinical isolates have been relatively limited. A number of different mechanisms for the development of acquired triclosan resistance have been demonstrated. The possibility that acquired triclosan resistance may contribute to reduced susceptibility to clinically important antimicrobial agents, due to either cross-resistance or co-resistance mechanisms, is a matter of major concern. Experimental data have confirmed the potential for such a link. The Panel on Biological Hazards refers to the report Development of antimicrobial resistance in bacteria II: Toxicity of triclosan in cosmetic products Based on reviews of the extensive toxicological data-base for triclosan the Panel on Food Additives, Flavourings, Processing Aids, Materials in contact with Food and Cosmetics identified a No Observed Adverse Effect Level (NOAEL) of 25 mg/kg/day. The Panel used 1 Mattilsynet (ber) med dette brev om at VKM utarbeider en oppdatert risikovurdering nr det gjelder bruk av triclosan i kosmetikk og kroppspleieprodukter. Dette gjelder risiko for resistensutvikling i patogene bakterier. 2 Vi nsker samtidig en utfyllende toksikologisk gjennomgang av stoffet med angivelse av sikkerhetsmargin for kosmetikkbruk 3 Norwegian Scientific Committee for Food Safety 04/406-18 final the SCCNFP notes of guidance to calculate the global estimate of exposure of preservatives based on extensive use scenarios including levels permitted for use in cosmetic products. The exposure of triclosan from other sources, which is considered to be very small, was not taken into account. The estimated safety margin in relation to permitted levels was found to be less than 100 and therefore the Panel found the current regulation of triclosan content in cosmetic products to be a matter of concern. Based on the relative contributions from different sources of cosmetic products the Panel gave the following recommendations: The content of triclosan in mouthwash should be as low as possible The maximal level of triclosan in rinse-off products and toothpaste should be reduced The current maximal level in eye products and non rinse-off products do not represent a toxicological problem

The Panel on Food Additives, Flavourings, Processing Aids, Materials in contact with Food and Cosmetics refers to the report Toxicity of triclosan in cosmetic products III: Environmental effects of triclosan Triclosan has a log octanol/water partition coefficient of 4.76 which indicates a potential for bioaccumulation. Bioaccumulation in fish has been documented in several studies. The possibility that todays use of triclosan could cause adverse environmental effects on some sensitive algae species can not be excluded. The Panel on Food Additives, Flavourings, Processing Aids, Materials in contact with Food and Cosmetics refers to the report Environmental effects of triclosan GENERAL CONCLUSIONS Widespread use of triclosan, including in cosmetic products, selects for development of triclosan resistance. Since this may contribute to the development and spread of concomitant resistance to clinically important antimicrobial agents, such use represents a public health risk. Therefore, the use of triclosan should be restricted. The current regulation of use of triclosan in cosmetic products is from a toxicological point of view a matter of concern and it is recommended that human exposure to triclosan should be reduced. Moreover, triclosan is classified as an agent that may cause adverse environmental effects and hence the use should be restricted also from an ecotoxicological standpoint. SCIENTIFIC PANEL MEMBERS In the Scientific Committee: shild Krogdahl (chair), Bjrn Nss, Hilde Kruse, Erik Dybing, Ingolf Nes, Jan Alexander, Janneche Utne Skre, Anne Kathrine Haldorsen, Martinus Lvik, Wenche Farstad, Lene Frost Andersen, Georg Kapperud, yvind Lie, Judith Narvhus, Leif Sundheim In the Panel on Biological Hazards: Hilde Kruse (chair), Georg Kapperud, Jrgen Lassen, Bjrn Tore Lunestad, Truls Nesbakken, Espen Rimstad, Lucy Robertson, Eystein Skjerve, Yngvild Wasteson In the Panel on Food Additives, Flavourings, Processing Aids, Materials in contact with Food and Cosmetics: Jan Alexander (chair), Trine Husy, Kristine Naterstad, Jan Erik Paulsen, Tore Sanner, Inger-Lise Steffensen 4 Norwegian Scientific Committee for Food Safety 04/406-18 final ACKNOWLEDGEMENTS Torsten Kllqvist from the Panel on Plant health, Plant Protection products and their Residues is acknowledged for his work on the preparation of Environmental effects of triclosan. The Chair and members of the working group on antimicrobial resistance are acknowledged for their valuable contribution to this mandate. The members of the working group are: Hilde Kruse (chair), Arne Hiby, Anne A. Scheie, Bjrn-Tore Lunestad, Even Heir, Kristine Naterstad Scientific coordinators from the secretariat: Siamak Yazdankhah, Tor ystein Fotland, Beate Folger 5U.S. AND E.U. COSMETIC REGULATION SIMILARITIESThere are numerous similarities between the way that the United States and the European Union regulate cosmetic products. As demonstrated by the following table, consumers in both regions can feel confident that their cosmetic products are safe. U.S.Food and Drug Administration (FDA)European Union (E.U.)

Relevant Cosmetic RegulationFederal Food, Drug & Cosmetic Act (1938 as amended) and the Fair Packaging and Labeling Act (1967)The 1976 E.U. Cosmetics Directive implemented in 1986 and amended for the seventh time in 2003

Requires that cosmetics be safe for intended use prior to marketing?Yes, proof of cosmetic safety is a responsibility of the manufacturer or its distributor in the U.S.Yes, proof of cosmetic safety is a responsibility of the manufacturer or its distributor in the E.U.

Requirement for pre-market submission of safety data and pre-market product approval?No.However, manufacturers are encouraged to register their establishments and list their cosmetic products and ingredients through FDA's Voluntary Cosmetics regulation program.The FDA can inspect cosmetic manufacturing plants or offices at any time, even without notice. These inspections do occur in case safety is questioned.FDA also has authority to: Ban or restrict cosmetics ingredients for safety reasons Mandate cosmetics warning labels Inspect cosmetics manufacturing facilities Issue warning letters Seize illegal products Enjoin unlawful activities Prosecute violators Work with manufacturing in implementing nationwide product recallsNo.However, a full technical file on the cosmetic product must be kept available for inspection upon request of the local authorities at a specified address in the E.U.Authorities must give 48-hour notice for inspection. The technical file contains: Full product and formula specifications, manufacturing process and relevant micro/stability data Proof of cosmetic safety Proof of pack claimsRecord of any type of health-related consumer comments (e.g., allergy reactions), NOT normal consumer complaints.

Exceptions that require pre-market approval?Yes, for color additives. Also, OTC cosmetic drug active ingredients are regulated by the FDA under their monograph system (e.g., anti-dandruff, anti-cavity, anti-perspirant, sunscreens, etc.) that lists approved actives, conc., uses, etc.Yes, for colors (including hair dye colorants), sunscreen active ingredients and preservatives. In general all active cosmetic ingredients require pre-approval in the E.U., for example anti-dandruff active ingredients, anti-tooth-caries fluoride compounds, anti-perspirant aluminum salts, etc. are all listed in a special annex III of the Cosmetics Directive.

Risk assessment is part of safety evaluation process?Yes.Yes, the safety assessor report is a key part of the technical file mentioned above.

Mandatory label warning statement if safety of product has not been determined?Yes. Without safety substantiation, Title 21 CFR, Part 740.10 requires that cosmetic products carry the following: "Warning: The safety of this product has not been determined."No such option in Europe. Hence the "negative" list system enforced via the SCCS (see below).

Ingredient safety reviews by independent scientific body is part of cosmetic safety process?Yes (by the CIR)** Cosmetic Ingredient Review Expert PanelYes (by SCCS)**** E.U. Scientific Committee on Consumer ProductsThis body in the E.U. is responsible of reviewing all special and active cosmetic ingredients and declaring whether they are safe or not. Hence there is a "negative list" type of system. For perspective, todaythere areover 1,100 substances listed in annex II (list of "banned" ingredients) and the list keeps growing largely with substances that are not used in cosmetics,but are published by the SCCS simply for enforcement reasons.

Expert ingredient safety reviews publicly available?Yes, published in peer-reviewedInternational Journal of Toxicologyand onCosmetic Ingredient Review(CIR) Web sitesYes, published on theScientific Committee on Consumer Safety(SCCS) Web site

Requirement to list ingredients on label?Yes.Individual perfume ingredients don't have to be listed; lumped under "fragrance."Yes.Additionally, as per 7th Amendment, starting March 2005, the cosmetic ingredient list must include 26 specific perfume ingredients ifpercent exceeds predetermined limits. These 26 ingredients have been identified by the SCCS as potential allergens.There is an exemption from labeling for technical impurities that cannot be eliminated - these should not be present at a level that raise cosmetics safety concerns (e.g., carry-over ingredients such as preservatives of raw materials are labeled if present at significant levels that could trigger allergy in people who are pre-sensitized).

Banned Ingredients?Yes.List of nine cosmetic ingredients restricted or prohibited by the FDA.Nine additional cosmetic ingredients judged by the Cosmetic International Review Expert Panel notsafe for use in cosmetics.International Fragrance Association also establishes usage guidelines for fragrance materials. IFRA recommends against the use of over 30 substances and advises limiting the use of many others.Yes.Annex II includes over 1,100 ingredients and the list keeps growing as the SCCS continues to publish manyingredients that are not generally used in cosmetics, for example CMR 1 & 2 classified substances.

Within the United States, cosmetics are regulated under the Food, Drug and Cosmetic Act, which is enforced by the U.S. Food and Drug Administration (FDA). The FDA has abundant legal authority to regulate the safety of cosmetic products. The FDA, however, has had comparatively little need to use its authority, as cosmetics are composed of safe ingredients and, when necessary, the cosmetics industry has acted voluntarily to prevent safety problems. The bottom line for consumers is that they can have complete confidence that the cosmetic products they use are safe.

The Use of Preservatives in BeautyProductsUnderstanding what goes into every day beauty products can be very confusing, especially when there are so many stories regarding what to try and avoid. In particular, the use of preservatives and what is safe and effective. We have put together a no nonsense straight forward approach to understanding this in order to allow you to make the decision right for you.

What is a Preservative, and why do we need them?Preservatives are used to stop products from deteriorating by protecting them against contamination once theyre in use. Preservatives are essential in maintaining the quality and safety of many of the products we use daily. Without them, bacteria, and other harmful organisms could develop in products leading safety issues.Did you know that bacteria, yeasts and moulds are always present on our skin, in the air around us and even in the water we drink? It is there for easy to understand how these can easily get into the products we use.Contamination of products, especially those used around the eyes and on the skin, can cause irritation and sensitivity problems if the level of contamination is high. Preservatives can prevent these problems by stopping micro-organisms from multiplying in the product.

So What Makes a Good Preservative?In order to ensure a product is safe to use, while preventing the development of contaminants, it is crucial to use the right preservative. There is no standard preservative system that can be used for all kinds of cosmetic products and so dependant on brand, ingredient content, and customer preference there are two groups of Preservatives that are used Synthetic or Natural, or a combination of both.

Natural PreservativesThere are natural preservatives available for use in products. These generally cover a smaller spectrum of bacterial activity, and thus shorten the shelf life of the product. These ingredients include:Extracts (Grapefruit Seed, Rosemary)Essential Oils (Tea Tree, Neem Seed, Thyme)Vitamins (Vitamin E, Vitamin C)

Synthetic PreservativesThere are a hand full of synthetic preservatives that are commonly used by the cosmetic and toiletry industries:Parabens: Methyl-, Ethyl-, Propyl-, ButylparabenUrea-Derivatives: Imidazolidinyl Urea, Diazolidinyl UreaIsothiazolones: Methylchloro-, Methyl-IsothiazolinoneHalogen-Organic Actives: Iodopropynyl Butylcarbamate, Methyldibromo GlutaronitrileOrganic Acids & Others: Sodium Benzoate, Chloracetamide, EDTA, Phenoxyethanol, Triclosan, DMDM-Hydantoin, Quaternium-15Parabens are the most widely used synthetic preservatives. However some brands, particularly those containing high levels of naturally-derived ingredients, use Paraben-free preservatives to avoid any potential health risks. The UK Soil Association has fully approved Phenoxyethanol for use in natural and organic products, as it has a long history of safe usage and has no negative environmental impact.In ConclusionUltimately, we all want safe and effective beauty products that are a pleasure to use and deliver results. The rule to follow when choosing your products is to check the preservation system in the products you are considering and opt for the solution that meets your own needs and concerns.

Knight Tour Tessellations

(click on image to see a 32x32 Knight's Tour)In the above Closed Knight's Tour animation, notice the basic patterns being made by each consecutive 16 moves of the knight. By putting each pattern on a separate chess board,

closing the open pattern to make a geometric shape, and filling in the pattern with the same color as the line path's color, there now exists four pieces that can be connected into one piece,

in such a way that it can be tessellated. Tessellation means the complete covering or tiling of a plane or space with a non-overlapping shape.

(Click image to see another Knight Tour tessellation.Be the first to recreate the original closed Knight's Tour from that tessellation and WIN $500.00)[Contest is over as of May 29, 2003. Here is theSolutionalong with the best two entries.]

Michiel de Bondt used the above tessellational patterns to make a very nice web-page background forhttp://www.math.ru.nl/lo_shu_tot_sudoku/. As you can see, Arno van den Essen used the same tessellational background for the cover of his book:Magische Vierkanten - Lo Shu Tot Sudoku.Michiel and Arno are both mathematicians in the Netherlands at the Radboud Universiteit Nijmegen Institute for Mathematics, Astrophysics and Particle Physics. Arno states the following about his Dutch book:On pages 100-103 I describe your work on tessellations and mention on page 101 that also the tessellations on the cover are created by you.

On page 227 I refer to your internet sitewww.borderschess.org/KnightTour.htm. The translation of the Dutch text is: "A beautiful site of Dan Thomasson on knight tours, containing very nice and artistic applications."

Of course, on page 11, I thank you (and some other people) for their help!Knight Tour tessellations can be used to create beautiful symmetrical patterns. A few of the many possibilities are shown below.

Astersphaira (Star Sphere)represents the surrounding stars of the Universe. It is a 3-d tiling of 12 stars. Each of thefive limbsof the stars were created by thediamond patternsmade from the knight's move on the chess board. If a boarder-line was drawn around the outer perimeter of each star, 12 pentagon patterns would appear, thus making aDodecahedron(12 sided figure). Historically the Dodecahedron symbolizes the Universe.At the same time, when looking at the Astersphaira, one may notice 20 triangular shapes made with 3 diamond patterns each. This construction would then be called anIcosahedron. Both the Dodecahedron and the Icosahedron are harmoniously intertwined. The Icosahedron has 20 faces and 12 vertices while the Dodecahedron has 12 faces and 20 vertices, an exact reciprocal of each other.Here are a couple other regular and irregular tessellational patterns made from the same knight's tour diamond shapes.

What do you get when you cross a Dodecahedron with an Icosahedron?Answer.

(click on the image above to see a tiled sphere)

Did you notice when looking at the tiled sphere that both the Icosidodecahedron and the Hexastersphaira graphics are actually identical. By replacing the perimeter edges in the Icosidodecahedron with star limbs, the Hexastersphaira takes its form.Polarsphaira (Polar Sphere)represents the polarity of the earth and earth itself. It is a 3-d tiling of sixcompass shapes. Each compass contains fourrhombus patterns(representing North, South, East, and West) made from the knight's move on the chess board. If a boarder-line was drawn around the outer perimeter of each compass, 6 square patterns would appear, thus making a Cube orHexahedron(6 sided figure). Historically the Hexahedron symbolizes the Earth.At the same time, when looking at the Polarsphaira, one may notice 8 triangular shapes made with 3 rhombus patterns each. This construction would then be called anOctahedron. Both the Hexahedron and the Octahedron are harmoniously intertwined, as is the Dodecahedron and Icosahedron mentioned earlier. The Hexahedron has 6 faces and 8 vertices while the Octahedron has 8 faces and 6 vertices, an exact reciprocal of each other.Four of the five platonic solids are represented by the knight's move on the chess board. The perimeters of these platonic solids can all be seen in"Creation Designed". The only other pattern made by the knight is asquare patternwhich makes a cube. The fifth platonic solid, not yet mentioned, is theTetrahedronwhich is contained within the cube. The Tetrahedron represents the concept of system. It is the underlying basic structure of the earth, sun, stars, and life itself. In reality, approximately 90% of all the earth's crust is made from silicate minerals (a group of rock forming minerals) which are based on the fundamental structural unit: a Tetrahedron.

The paper model shown above combines the Astersphaira (Universe) and Polarsphaira (Earth). I made an ornament of sorts where thePolarsphairaresides inside the Astersphaira. Inside of the Polarsphaira resides the final shape of the knight's tour, a square which makes a cube containing the hidden Tetrahedron. It is a quarter inch block of wood that will have six different jeweled tiles covering each of the faces. The tiles represent the cyrstals or precious gemstones of the earth that have a cubic or hexagonal crystaline structure (diamond, ruby, emerald, sapphire, topaz, amethyst). They act as the inner light or energy source for the world.I could use the individual spheres for geometric analysis. Combined into a model, they can make lamps, mobiles, and different types of ornaments. The cube in the center of the ornament can still be the various jeweled tiles but with a small lightbulb in the center to radiate the various colors of red, green, blue, purple, yellow, and clear. Rotating the individual spheres in opposite directions while lighting the cube would make an interesting effect.The stand and the combined spheres were extremely easy to make. I used 67 lb white cover paper, a drink sterring straw, coated copper wire, thick piano wire, and Elmer's glue. I got the quarter inch wood cube and quarter inch jeweled tiles from Michaels, a local craft store. When you make the spheres out of paper, you can actually collapse the spheres inward to make other star or spur shapes. Collapse the spheres until the edges of the limbs touch each other. Pretty cool patterns appear similar to the beautiful candle lantern shown on either side of the paper model. The candle lantern is a one-of-a-kind handcrafted piece made in a small mountain town in the heart of Mexico. The lantern was distributed to a local gift shop via Norwich, Vermont under the label: Viva! the Evolution.Isn't it interesting about the dichotomy ofcreation. The Astersphaira (Star Sphere) contains the Dodecahedron and Icosahedron. The Polarsphaira (Polar Sphere) contains the Hexahedron and Octahedron. Notice that those shapes can be easily created from spheres when using the 3-d tiling of the knight move patterns. I like using the model as a teaching aid to show the relationships of the platonic solids. The cube at the center of the Polarsphaira contains the Tetrahedron but it is not visible to the eye. The ornament can have several different stories associated with it such as the following: The Christian Trinity of the Father (Astersphaira), the Son (Polarsphaira), and the Holy Ghost (Tetrahedron) The Biblical book of Genesis 1:1 - In the beginning, God created the heavens (Astersphaira) and the earth (Polarsphaira) The Hindu mythological male Purusha (Icosahedron) and female Prakriti (Dodecahedron) giving birth to Purusha Junior The Chinese Yen and Yang, nature's two channels of electric-flows which bring about changes in the Universe The Chemistry or Physics "polaron", a conducting electron in an ionic crystal together with the induced polarization of the surrounding latticeI made up the words Astersphaira, Polarsphaira, and Hexastersphaira using mostly the Greek language to give homage to Greek geometry. Astersphaira is from the Greek Aster (Star) + sphaira (sphere). Polarsphaira is from the new Latin Polaris (Polar) + sphaira (sphere). Hexastersphaira is from the Greek Hex (Six) + aster (star) + sphaira (sphere).With 3-d glasses (red-blue, or red-green), you can enjoy looking at 3-d models of the platonic solids that change shape automatically in a Java applet found at dogfeathers.com web site. I felt like I could touch the shapes coming out of my monitor. Check out some 4-d hyperspace polytopes in the following links. A polytope is a 4th dimension polygon.Hyperspace Star Polytope SlicerStellations of the DodecahedronThe following tessellational patterns encompass all the basic shapes (square, diamond, and rhombus) made by the knight's tour moves. They show two different geometrical cubic staircase patterns.

Four Color Chain-Linked TessellationWhile playing around with knight moves on the chessboard from thehome page, I ended up with an interesting symmetrical pattern that could be turned into a very nice four color chain-linked tessellation. Enjoy the following animation.

After making the above animation, I connected the four symmetrical patterns from the 8x8 square to make a 7x7 square tessellation piece. I then divided each individual pattern within the square into three smaller shapes representing the 2x1 "L" shaped move of the Knight. I changed the colors of the red squares within the four symmetrical patterns of the Knight's Tour to include blue, green, yellow, and white. As you will see, the white patterns become the background, and there are no white colored links. All the white shapes should be transparent, thus only showing the red, green, blue, and yellow links. The center square of the 7x7 tessellation piece is actually a gap, as can be seen in the animation.

Four copies of the 7x7 square shown above on the right, can be connected together in such a way as to make a single larger tessellation piece. The center white square of the four combined pieces is also a gap. Therefore, the pattern is not a true regular tessellation, but a complex tessellation where small gaps do exist. None-the-less, the tiling of the tessellation piece makes for an interesting interlocking chain of red, green, blue, and yellow square links.

Now we can replicate the larger tessellation piece, and tile the pieces into a work of art that I call theFour Color Chain-Linked Tessellation (derived from Knight's Tour):

(Click image to see a Four Color Chain-Linked Regular Tessellation)

Here are a couple other examples of the chain-linked tessellation. When you create your own similar tessellations, experiment with different colors for the links and the background. You may even use actual scenic pictures for the background. If the pictures look fuzzy, or the edges of the square links look jagged or rough, expand the browser window to full screen, then press F11 (Internet Explorer). To get Internet Explorer back to the original view, press F11 again. Using 1024x768 or greater screen resolution size may also help.Black and White Chain-Linked Tessellation

Multi-Colored Chain-Linked Tessellation

Patriotic Chain-Linked Tessellation 1

Patriotic Chain-Linked Tessellation 2While creating these square chain-linked tessellations, I decided to go ahead and make a simple interlocking ring tessellation. The curves are a bit rough, so I'll need to redo the rings when I get a better graphics program. Perhaps someone can suggest a good, yet simple, geometric graphics program I can use.

Here are a couple other crude drawings of circle tessellations.Circle Tessellation

Multi-Circle Tessellation

Here is a tile called "Illusion" that I created. It can be used as a web-page background image such ashttp://www.borderschess.org/illusion.htm, or for other artistic creations.Depending on one's perception of the tiling, different objects may appear with various 2-D or 3-D orientations. The flow of the objects appear to move in opposite directions. Experiment with different color combinations when making your own tessellation pieces using my "Illusion" design.

ILLUSION

2005, Dan ThomassonJust for fun, I animated only two frames of the "Illusion" tile to create a new tessellation called "Still Motion." Seehttp://www.borderschess.org/Still-Motion.htm.

STILL MOTION

2005, Dan ThomassonThe further your eyes are from the screen, the more the lines in the "Still Motion" tessellation appear to oscillate when looking at the web-page background. Actually, the only thing that changes is the medium grey and white colors alternating within the tile pattern. If you copy my tessellation designs, or any other graphics from my web-site for publication, or posting on your web-site, please reference my name and copyright date.As for the unique geometry created by knight moves that can make beautiful 2-dimensional, 3-dimensional, and multi-dimensional tessellations, I call this geometry:Springer Geometry(patented as Geometric Craft and Educational Kit -Patent #7029364, April 18, 2006). Check out the bookPOLYHEDRA 2 Part 2(underSymmetry: Culture and Science- Latest Issues), where you will find a 17 page article I wrote about Springer Geometry. The article was published in Budapest, Hungary, in the journal of Symmetrion calledSymmetry: Culture and ScienceVolume 13, Numbers 3-4, pages 401-417, 2002 (ISSN #0865-4824, ISBN #963 214 761 8). It is published by the International Symmetry Foundation.

See the following abstract for the article:Polygonal shapes made by chess pieces on the chessboard, especially those made by the knight, are the subjects of this text. The article is named "Springer Geometry," to give homage to the great German chess players and mathematicians throughout the ages. 'Springer' is the German translation for the English word: 'knight.' I dedicate this article to Grandmaster Karsten Mller, a top German chess player and doctor of mathematics who graciously took the time to translate my Knight's Tour Web site, http://www.borderschess.org/KnightTour.htm, intoGerman. While focusing on chess-piece polygons, this article covers the following three topics: 1) Polygons as Counters, 2) Tessellations, and 3) Ornament-type Symmetrical Spheres.Many thanks goes to Paul Bourke for his excellent and professional renderings of mySpringer Geometry concepts. Check out the followingletterfrom Paul Bourke to see how he is using my geometry in Astrophysics.Much of my work on Knight Tours is referenced on hundreds of other Web sites, and used by schools, universities, and technical institutes to help their students learn programming logic, create artwork such as tessellations from Knight Tours, and to learn basic concepts about geometry (platonic solids) and other areas of mathematics. See the following small sampling of who is using my Knight's Tour information:Louisville High School, Louisville, NE, Mathematics Department:Math Websites and Math GamesPhilipps-Universitt Marburg, Fachbereich Mathematik und Informatik:Ubungen zu, Parallelitt in funktionalen Sprachen Nr. 12, Abgabe: 5. November in der VorlesungUniversity of Edinburgh, Science and Engineering, School of Informatics:MSc - Fundamentals of Artificial Intelligence, First Assessed PracticalUniversity of Maryland, Computer Science, College of Engineering:UMBC CMSC 201 Project 4 Fall 2006

www.BordersChess.org/KTtess.htm modified 2007.4.30

What Is a Tessellation?

About This Project||What is a Tessellation?||Tessellation Tutorials||Tessellation Links

DefinitionA tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.Another word for a tessellation is atiling. Read more here:What is a Tiling?A dictionary*will tell you that the word "tessellate" means to form or arrange small squares in a checkered or mosaic pattern. The word "tessellate" is derived from the Ionic version of the Greek word "tesseres," which in English means "four." The first tilings were made from square tiles.A regular polygon has 3 or 4 or 5 or more sides and angles, all equal. Aregular tessellationmeans a tessellation made up of congruent regular polygons. [Remember:Regularmeans that the sides and angles of the polygon are all equivalent (i.e., the polygon is both equiangular and equilateral).Congruentmeans that the polygons that you put together are all the same size and shape.]Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons. We can't show the entire plane, but imagine that these are pieces taken from planes that have been tiled. Here are examples ofa tessellation of triangles

a tessellation of squares

a tessellation of hexagons

When you look at these three samples you can easily notice that the squares are lined up with each other while the triangles and hexagons are not. Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other - a slide (or a glide!) is involved.You can work out the interior measure of the angles for each of these polygons:Shapetrianglesquarepentagonhexagonmore than six sidesAngle measure in degrees6090108120more than 120 degrees

Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane.Reinforce this idea with theRegular Tessellationsinteractive activity:Teacher Lesson Plan||Student PageNaming ConventionsA tessellation of squares is named "4.4.4.4". Here's how: choose a vertex, and then look at one of the polygons that touches that vertex. How many sides does it have?Since it's a square, it has four sides, and that's where the first "4" comes from. Now keep going around the vertex in either direction, finding the number of sides of the polygons until you get back to the polygon you started with. How many polygons did you count?There are four polygons, and each has four sides.

For a tessellation of regular congruent hexagons, if you choose a vertex and count the sides of the polygons that touch it, you'll see that there are three polygons and each has six sides, so this tessellation is called "6.6.6":

A tessellation of triangles has six polygons surrounding a vertex, and each of them has three sides: "3.3.3.3.3.3".

Semi-regular TessellationsYou can also use a variety of regular polygons to makesemi-regular tessellations. A semiregular tessellation has two properties which are:1. It is formed by regular polygons.2. The arrangement of polygons at every vertex point is identical.Here are theeightsemi-regular tessellations:

Interestingly there are other combinations that seem like they should tile the plane because the arrangements of the regular polygons fill the space around a point. For example:If you try tiling the plane with these units of tessellation you will find that they cannot be extended infinitely. Fun is to try this yourself.1. Hold down on one of the images andcopyit to the clipboard.2. Open a paint program.3. Paste the image.4. Now continue to paste and position and see if you can tessellate it.

There are an infinite number of tessellations that can be made of patterns that do not have the same combination of angles at every vertex point. There are also tessellations made of polygons that do not share common edges and vertices. You can learn more by following the links listed inOther Tessellation Links and Related Sites.Michael South has contributed some thoughtsto the discussion.*Steven Schwartzman'sThe Words of Mathematics(1994, The Mathematical Association of America)says:tessellate(verb),tessellation(noun): from Latintessera"a square tablet" or "a die used for gambling." Latintesseramay have been borrowed from Greektessares, meaning "four," since a square tile has four sides. The diminutive oftesserawastessella, a small, square piece of stone or a cubical tile used in mosaics. Since a mosaic extends over a given area without leaving any region uncovered, the geometric meaning of the word tessellate is "to cover the plane with a pattern in such a way as to leave no region uncovered." By extension, space or hyperspace may also be tessellated.

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Basis for definitionAny polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions: 3 dimensions: Thebodyis bounded by the faces, and is usually the volume enclosed by them. 2 dimensions: Afaceis apolygonbounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedralsurface. 1 dimension: Anedgejoins one vertex to another and one face to another, and is usually alinesegment. The edges together make up the polyhedralskeleton. 0 dimensions: Avertex(pluralvertices) is a cornerpoint. -1 dimension: Thenull polytopeis a kind of non-entity required byabstracttheories.More generally inmathematicsand other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely algebraic or abstract.A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.A polyhedron is a 3-dimensional example of the more generalpolytopein any number of dimensions.[edit]Characteristics[edit]Names of polyhedraPolyhedra are often named according to the number of faces. The naming system is again based on Classical Greek, for exampletetrahedron(4),pentahedron(5),hexahedron(6),heptahedron(7),triacontahedron(30), and so on.Often this is qualified by a description of the kinds of faces present, for example theRhombic dodecahedronvs. thePentagonal dodecahedron.Other common names indicate that some operation has been performed on a simpler polyhedron, for example thetruncated cubelooks like a cube with its corners cut off, and has 14 faces (so it is also an example of a tetrakaidecahedron).Some special polyhedra have grown their own names over the years, such asMiller's monsteror theSzilassi polyhedron.[edit]EdgesEdges have two important characteristics (unless the polyhedron iscomplex): An edge joins just two vertices. An edge joins just two faces.These two characteristics aredualto each other.[edit]Euler characteristicTheEuler characteristic relates the number of verticesV, edgesE, and facesFof a polyhedron:

For asimply connectedpolyhedron, = 2. For a detailed discussion, seeProofs and RefutationsbyImre Lakatos.[edit]OrientabilitySome polyhedra, such as allconvex polyhedra, have two distinct sides to their surface, for example one side can consistently be coloured black and the other white. We say that the figure isorientable.But for some polyhedra this is not possible, and the figure is said to be non-orientable. All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even < 2 may or may not be orientable.[edit]Vertex figureFor every vertex one can define avertex figure, which describes the local structure of the figure around the vertex. If the vertex figure is aregular polygon, then the vertex itself is said to beregular.[edit]Duality

For every polyhedron there exists adual polyhedronhaving: faces in place of the original's vertices and vice versa, the same number of edges the same Euler characteristic and orientabilityThe dual of a convex polyhedron can be obtained by the process ofpolar reciprocation.[edit]VolumeThe volume of anorientablepolyhedron having an identifiablecentroidcan be calculated usingGreen's theorem:

by choosing the function

where (x,y,z) is the centroid of the surface enclosing the volume under consideration. As we have,

Hence the volume can be calculated as:

where the normal of the surface pointing outwards is given by:

The final expression can be written as

where S is the surface area of the polyhedron.[edit]Traditional polyhedra

A dodecahedronIngeometry, apolyhedronis traditionally a three-dimensional shape that is made up of a finite number ofpolygonalfaceswhich are parts ofplanes; the faces meet in pairs alongedgeswhich arestraight-linesegments, and the edges meet in points calledvertices.Cubes,prismsandpyramidsare examples of polyhedra. The polyhedron surrounds a bounded volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges.A polyhedron is said to beconvexif its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface.[edit]Symmetrical polyhedraMany of the most studied polyhedra are highlysymmetrical.Of course it is easy to distort such polyhedra so they are no longer symmetrical. But where a polyhedral name is given, such asicosidodecahedron, the most symmetrical geometry is almost always implied, unless otherwise stated.Some of the most common names in particular are often used with "regular" in front or implied because for each there are different types which have little in common except for having the same number of faces. These are the triangular pyramid ortetrahedron,cubeor hexahedron,octahedron,dodecahedronandicosahedron:

Polyhedra of the highest symmetries have all of some kind of element - faces, edges and/or vertices, within a single symmetry orbit. There are various classes of such polyhedra: IsogonalorVertex-transitiveif all vertices are the same, in the sense that for any two vertices there exists asymmetryof the polyhedron mapping the firstisometricallyonto the second. IsotoxalorEdge-transitiveif all edges are the same, in the sense that for any two edges there exists a symmetry of the polyhedron mapping the first isometrically onto the second. IsohedralorFace-transitiveif all faces are the same, in the sense that for any two faces there exists a symmetry of the polyhedron mapping the first isometrically onto the second. Regularif it is vertex-transitive, edge-transitive and face-transitive (this implies that every face is the sameregular polygon; it also implies that every vertex is regular). Quasi-regularif it is vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive. Aquasi-regular dualis face-transitive and edge-transitive (and hence every vertex is regular) but not vertex-transitive. Semi-regularif it is vertex-transitive but not edge-transitive, and every face is a regular polygon. (This is one of several definitions of the term, depending on author. Some definitions overlap with the quasi-regular class). Asemi-regular dualis face-transitive but not vertex-transitive, and every vertex is regular. Uniformif it is vertex-transitive and every face is a regular polygon, i.e. it is regular, quasi-regular or semi-regular. Auniform dualis face-transitive and has regular vertices, but is not necessarily vertex-transitive). Nobleif it is face-transitive and vertex-transitive (but not necessarily edge-transitive). The regular polyhedra are also noble; they are the only noble uniform polyhedra.A polyhedron can belong to the same overall symmetry group as one of higher symmetry, but will have several groups of elements (for example faces) in different symmetry orbits.[edit]Uniform polyhedra and their dualsMain article:Uniform polyhedronUniform polyhedraarevertex-transitiveand every face is aregular polygon. They may beregular,quasi-regular, orsemi-regular, and may be convex or starry.Theuniformdualsareface-transitiveand everyvertex figureis a regular polygon.Face-transitivity of a polyhedron corresponds to vertex-transitivity of the dual and conversely, and edge-transitivity of a polyhedron corresponds to edge-transitivity of the dual. In most duals of uniform polyhedra, faces are irregular polygons. The regular polyhedra are an exception, because they are dual to each other.Each uniform polyhedron shares the same symmetry as its dual, with the symmetries of faces and vertices simply swapped over. Because of this some authorities regard the duals as uniform too. But this idea is not held widely: a polyhedron and its symmetries are not the same thing.The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they areconvexor not.Convex uniformConvex uniform dualStar uniformStar uniform dual

RegularPlatonic solidsKepler-Poinsot polyhedra

QuasiregularArchimedean solidsCatalan solids(no special name)(no special name)

Semiregular(no special name)(no special name)

PrismsDipyramidsStarPrismsStarDipyramids

AntiprismsTrapezohedraStarAntiprismsStarTrapezohedra

[edit]Noble polyhedraMain article:Noble polyhedronAnoblepolyhedron is bothisohedral(equal-faced) andisogonal(equal-cornered). Besides the regular polyhedra, there are many other examples.Thedualof a noble polyhedron is also noble.[edit]Symmetry groupsThe polyhedralsymmetry groups(usingSchoenflies notation) are allpoint groupsand include: T-chiraltetrahedral symmetry; the rotation group for a regulartetrahedron; order 12. Td-fulltetrahedral symmetry; the symmetry group for a regulartetrahedron; order 24. Th-pyritohedral symmetry; order 24. The symmetry of apyritohedron. O-chiraloctahedral symmetry;the rotation group of thecubeandoctahedron; order 24. Oh-fulloctahedral symmetry; the symmetry group of thecubeandoctahedron; order 48. I-chiralicosahedral symmetry; the rotation group of theicosahedronand thedodecahedron; order 60. Ih-fullicosahedral symmetry; the symmetry group of theicosahedronand thedodecahedron; order 120. Cnv-n-fold pyramidal symmetry Dnh-n-fold prismatic symmetry Dnv-n-fold antiprismatic symmetryThose withchiralsymmetry do not havereflection symmetryand hence have twoenantiomorphousforms which are reflections of each other. ThesnubArchimedean polyhedra have this property.[edit]Other polyhedra with regular faces[edit]Equal regular facesA few families of polyhedra, where every face is the same kind of polygon: Deltahedrahave equilateral triangles for faces. With regard to polyhedra whose faces are all squares: ifcoplanarfaces are not allowed, even if they are disconnected, there is only the cube. Otherwise there is also the result of pasting six cubes to the sides of one, all seven of the same size; it has 30 square faces (counting disconnected faces in the same plane as separate). This can be extended in one, two, or three directions: we can consider the union of arbitrarily many copies of these structures, obtained by translations of (expressed in cube sizes) (2,0,0), (0,2,0), and/or (0,0,2), hence with each adjacent pair having one common cube. The result can be any connected set of cubes with positions (a,b,c), with integersa,b,cof which at most one is even. There is no special name for polyhedra whose faces are all equilateral pentagons or pentagrams. There are infinitely many of these, but only one is convex: the dodecahedron. The rest are assembled by (pasting) combinations of the regular polyhedra described earlier: the dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great icosahedron.There exists no polyhedron whose faces are all identical and are regular polygons with six or more sides because the vertex of three regular hexagons defines a plane. (Seeinfinite skew polyhedronfor exceptions with zig-zaggingvertex figures.)[edit]DeltahedraAdeltahedron(plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex: 3 regular convex polyhedra (3 of the Platonic solids) Tetrahedron Octahedron Icosahedron 5 non-uniform convex polyhedra (5 of the Johnson solids) Triangular dipyramid Pentagonal dipyramid Snub disphenoid Triaugmented triangular prism Gyroelongated square dipyramid[edit]Johnson solidsMain article:Johnson solidNorman Johnsonsought which convex non-uniform polyhedra had regular faces. In 1966, he published a list of 92 such solids, gave them names and numbers, and conjectured that there were no others.Victor Zalgallerproved in 1969 that the list of theseJohnson solidswas complete.[edit]Other important families of polyhedra[edit]PyramidsMain article:Pyramid (geometry)Pyramids include some of the most time-honoured and famous of all polyhedra.[edit]Stellations and facettingsMain article:Stellation

Stellationof a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron.It is the exact reciprocal to the process offacettingwhich is the process of removing parts of a polyhedron without creating any new vertices.[edit]ZonohedraMain article:ZonohedronAzonohedronis a convex polyhedron where every face is apolygonwith inversionsymmetryor, equivalently, symmetry underrotationsthrough 180.[edit]Toroidal polyhedraMain article:Toroidal polyhedronAtoroidal polyhedronis a polyhedron with anEuler characteristicof 0 or smaller, representing atorussurface.[edit]CompoundsMain article:Polyhedral compoundPolyhedral compounds are formed as compounds of two or more polyhedra.These compounds often share the same vertices as other polyhedra and are often formed by stellation. Some are listed in thelist of Wenninger polyhedron models.[edit]Orthogonal polyhedraAn orthogonal polyhedron is one all of whose faces meet at right angles, and all of whose edges are parallel to axes of a Cartesian coordinate system. Aside from a rectangular box, orthogonal polyhedra are nonconvex. They are the 3D analogs of 2Dorthogonal polygons, also known asrectilinear polygons. Orthogonal polyhedra are used incomputational geometry, where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to apolygonal net.[edit]Generalisations of polyhedraThe name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra.[edit]ApeirohedraA classical polyhedral surface comprises finite, bounded plane regions, joined in pairs along edges. If such a surface extends indefinitely it is called anapeirohedron. Examples include: Tilingsortessellationsof the plane. Sponge-like structures calledinfinite skew polyhedra.See also:Apeirogon- infinite regular polygon: {}[edit]Complex polyhedraAcomplex polyhedronis one which is constructed in complexHilbert3-space. This space has six dimensions: three real ones corresponding to ordinary space, with each accompanied by an imaginary dimension. See for example Coxeter (1974).[edit]Curved polyhedraSome fields of study allow polyhedra to have curved faces and edges.[edit]Spherical polyhedraMain article:Spherical polyhedronThe surface of a sphere may be divided by line segments into bounded regions, to form aspherical polyhedron. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.Spherical polyhedra have a long and respectable history: The first known man-made polyhedra are spherical polyhedra carved in stone. Poinsot used spherical polyhedra to discover the four regular star polyhedra. Coxeter used them to enumerate all but one of the uniform polyhedra.Some polyhedra, such ashosohedraanddihedra, exist only as spherical polyhedra and have no flat-faced analogue.[edit]Curved spacefilling polyhedraTwo important types are: Bubbles in froths and foams, such asWeaire-Phelan bubbles. Spacefilling forms used in architecture. See for example Pearce (1978).[edit]General polyhedraMore recentlymathematicshas defined apolyhedronas a set inrealaffine(orEuclidean) space of any dimensionalnthat has flat sides. It may alternatively be defined as the union of a finite number of convex polyhedra, where aconvex polyhedronis any set that is the intersection of a finite number ofhalf-spaces. It may be bounded or unbounded. In this meaning, apolytopeis a bounded polyhedron.Analytically, such a convex polyhedron is expressed as the solution set for a system of linear inequalities. Defining polyhedra (and more generallypolytopes) in this way provides a geometric perspective for problems inLinear programming.Many traditional polyhedral forms are general polyhedra. Other examples include: A quadrant in the plane. For instance, the region of the cartesian plane consisting of all points above the horizontal axis and to the right of the vertical axis: { (x,y): x 0, y 0 }. Its sides are the two positive axes. An octant in Euclidean 3-space, { (x,y,z): x 0, y 0, z 0 }. A prism of infinite extent. For instance a doubly infinite square prism in 3-space, consisting of a square in thexy-plane swept along thez-axis: { (x,y,z): 0 x 1, 0 y 1 }. Eachcellin aVoronoi tessellationis a convex polyhedron. In the Voronoi tessellation of a setS, the cellAcorresponding to a pointcSis bounded (hence a traditional polyhedron) whenclies in theinteriorof theconvex hullofS, and otherwise (whenclies on theboundaryof the convex hull ofS)Ais unbounded.[edit]Hollow faced or skeletal polyhedraIt is not necessary to fill in the face of a figure before we can call it a polyhedron. For exampleLeonardo da Vincidevised frame models of the regular solids, which he drew forPacioli's bookDivina Proportione. In modern times,Branko Grnbaum(1994) made a special study of this class of polyhedra, in which he developed an early idea ofabstract polyhedra. He defined afaceas a cyclically ordered set of vertices, and allowed faces to beskewas well as planar.[edit]Non-geometric polyhedraVarious mathematical constructs have been found to have properties also present in traditional polyhedra.[edit]Topological polyhedraAtopological polytopeis a topological space given along with a specific decomposition into shapes that are topologically equivalent toconvex polytopesand that are attached to each other in a regular way.Such a figure is calledsimplicialif each of its regions is asimplex, i.e. in ann-dimensional space each region hasn+1 vertices. The dual of a simplicial polytope is calledsimple. Similarly, a widely studied class of polytopes (polyhedra) is that of cubical polyhedra, when the basic building block is ann-dimensional cube.[edit]Abstract polyhedraAnabstract polyhedronis apartially ordered set(poset) of elements whose partial ordering obeys certain rules. Theories differ in detail, but essentially the elements of the set correspond to the body, faces, edges and vertices of the polyhedron. The empty set corresponds to the null polytope, ornullitope, which has a dimensionality of 1. These posets belong to the larger family ofabstract polytopesin any number of dimensions.[edit]Polyhedra as graphsAny polyhedron gives rise to agraph, orskeleton, with corresponding vertices and edges. Thusgraph terminologyand properties can be applied to polyhedra. For example: Due toSteinitz theoremconvex polyhedra are in one-to-one correspondence with 3-connected planar graphs. Thetetrahedrongives rise to acomplete graph(K4). It is the only polyhedron to do so. Theoctahedrongives rise to astrongly regular graph, because adjacent vertices always have two common neighbors, and non-adjacent vertices have four. TheArchimedean solidsgive rise toregular graphs: 7 of the Archimedean solids are ofdegree3, 4 of degree 4, and the remaining 2 are chiral pairs of degree 5.[edit]History[edit]PrehistoryStones carved in shapes showing the symmetries of various polyhedra have been found inScotlandand may be as much a 4,000 years old. These stones show not only the form of various symmetrical polyehdra, but also the relations of duality amongst some of them (that is, that the centres of the faces of the cube gives the vertices of an octahedron, and so on). Examples of these stones are on display in theJohn Evans roomof theAshmolean MuseumatOxford University. It is impossible to know why these objects were made, or how the sculptor gained the inspiration for them.Other polyhedra have of course made their mark inarchitecture- cubes and cuboids being obvious examples, with the earliest four-sided pyramids of ancientEgyptalso dating from the Stone Age.TheEtruscanspreceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery nearPadua(in NorthernItaly) in the late 1800s of adodecahedronmade ofsoapstone, and dating back more than 2,500 years (Lindemann, 1987).Pyritohedric crystalsare found in northern Italy[citation needed].[edit]GreeksThe earliest knownwrittenrecords of these shapes come from ClassicalGreekauthors, who also gave the first known mathematical description of them. The earlier Greeks were interested primarily in theconvex regular polyhedra, which came to be known as thePlatonic solids.Pythagorasknew at least three of them, andTheaetetus(circa 417 B.C.) described all five. Eventually,Eucliddescribed their construction in hisElements. Later,Archimedesexpanded his study to theconvex uniform polyhedrawhich now bear his name. His original work is lost and his solids come down to us throughPappus.[edit]Chinese and MuslimsBy 236 AD, in China Liu Hui was describing the dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids, using assemblages of these solids as the basis for calculating volumes of earth to be moved during engineering excavations.After the end of the Classical era, Islamic scholars continued to take the Greek knowledge forward. The ninth century scholar Thabit ibn Qurra gave formulae for calculating the volumes of polyhedra such as truncated pyramids. Then in the tenth centuryAbu'l Wafadescribed the convex regular and quasiregular spherical polyhedra.[edit]RenaissanceAs with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the ItalianRenaissance. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations intoperspective. Several appear in marquetry panels of the period. Piero della Francesca gave the first written description of direct geometrical construction of such perspective views of polyhedra. Leonardo da Vinci made skeletal models of several polyhedra and drew illustrations of them for a book by Pacioli. A painting by an anonymous artist of Pacioli and a pupli depicts a glassrhombicuboctahedronhalf-filled with water.As the Renaissance spread beyond Italy, later artists such as Wenzel Jamnitzer, Drer and others also depicted polyhedra of various kinds, many of them novel, in imaginative etchings.[edit]Star polyhedraFor almost 2,000 years, the concept of a polyhedron as a convex solid had remained as developed by the ancient Greek mathematicians.During theRenaissancestar forms were discovered. A marble tarsia in the floor ofSt. Mark's Basilica, Venice, depicts a stellated dodecahedron. Artists such as Wenzel Jamnitzer delighted in depicting novel star-like forms of increasing complexity.Johannes Keplerrealised thatstar polygons, typicallypentagrams, could be used to build star polyhedra. Some of these star polyhedra may have been discovered before Kepler's time, but he was the first to recognise that they could be considered "regular" if one removed the restriction that regular polytopes be convex. Later,Louis Poinsotrealised that starvertex figures(circuits around each corner) can also be used, and discovered the remaining two regular star polyhedra. Cauchy proved Poinsot's list complete, and Cayley gave them their accepted English names: (Kepler's) thesmall stellated dodecahedronandgreat stellated dodecahedron, and (Poinsot's) thegreat icosahedronandgreat dodecahedron. Collectively they are called theKepler-Poinsot polyhedra.The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process calledstellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push byH. S. M. Coxeterand others in 1938, with the now famous paperThe 59 icosahedra. This work has recently been re-published (Coxeter, 1999).The reciprocal process to stellation is calledfacetting(or faceting). Every stellation of one polytope isdual, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids.Bridge 1974listed the simpler facettings of the dodecahedron, and reciprocated them to discover a stellation of the icosahedron that was missing from the famous "59". More have been discovered since, and the story is not yet ended.See also: Regular polyhedron: History Regular polytope: History of discovery.[edit]Polyhedra in natureFor natural occurrences of regular polyhedra,seeRegular polyhedron: Regular polyhedra in nature.Irregular polyhedra appear in nature ascrystals.