Fructose - wikipedia Fructose Jump to : navigation , search D - Fructose D - Fructofuranose D - Fructose ( open - chain form ) Haworth projection of β - D - Fructofuranose Ball - and - stick model of D - fructose Names IUPAC name ( 3S , 4R , 5R ) - 1 , 3 , 4 , 5 , 6 - Pentahydroxyhexan - 2 - one Other names Fruit sugar , levulose , D - fructofuranose , D - fructose , D - arabino - hexulose Identifiers CAS Number 57 - 48 - 7 3D model ( JSmol ) Interactive image ChEBI CHEBI : 28645 ChEMBL ChEMBL604608 ChemSpider 388775 ECHA InfoCard 100.000. 303 EC Number 200 - 333 - 3 KEGG C02336 PubChem CID 5984 UNII 6YSS42VSEV InChI ( show ) InChI = 1S / C6H12O6 / c7 - 1 - 3 - 4 ( 9 ) 5 ( 10 ) 6 ( 11 , 2 - 8 ) 12 - 3 / h3 - 5 , 7 - 11H , 1 - 2H2 / t3 - , 4 - , 5 + , 6 - / m1 / s1 Key : RFSUNEUAIZKAJO - ARQDHWQXSA - N SMILES ( show ) O ( C@H ) 1 ( C@H ) ( O ) ( C@H ) ( O ( C @ ) 1 ( O ) CO ) CO Properties Chemical formula Molar mass 180.16 g mol Density 1.694 g / cm Melting point 103 ° C ( 217 ° F ; 376 K ) Solubility in water ~ 4000 g / L ( 25 ° C ) Magnetic susceptibility ( χ ) − 102.60 × 10 cm / mol Pharmacology ATC code V06DC02 ( WHO ) Hazards Lethal dose or concentration ( LD , LC ) : LD ( median dose ) 15000 mg / kg ( intravenous , rabbit ) Except where otherwise noted , data are given for materials in their standard state ( at 25 ° C ( 77 ° F ) , 100 kPa ) . N verify ( what is ? ) Infobox references Fructose , or fruit sugar , is a simple ketonic monosaccharide found in many plants , where it is often bonded to glucose to form the disaccharide sucrose . It is one of the three dietary monosaccharides , along with glucose and galactose , that are absorbed directly into blood during digestion . Fructose was discovered by French chemist Augustin - Pierre Dubrunfaut in 1847 . The name `` fructose '' was coined in 1857 by the English chemist William Allen Miller . Pure , dry fructose is a sweet , white , odorless , crystalline solid , and is the most water - soluble of all the sugars . Fructose is found in honey , tree and vine fruits , flowers , berries , and most root vegetables . Commercially , fructose is derived from sugar cane , sugar beets , and maize . Crystalline fructose is the monosaccharide , dried , ground , and of high purity . High - fructose corn syrup is a mixture of glucose and fructose as monosaccharides . Sucrose is a compound with one molecule of glucose covalently linked to one molecule of fructose . All forms of fructose , including fruits and juices , are commonly added to foods and drinks for palatability and taste enhancement , and for browning of some foods , such as baked goods . About 240,000 tonnes of crystalline fructose are produced annually . As for any sugar , excessive consumption of fructose may contribute to insulin resistance , obesity , elevated LDL cholesterol and triglycerides , leading to metabolic syndrome , type 2 diabetes and cardiovascular disease . The European Food Safety Authority stated that fructose is preferable over sucrose and glucose in sugar - sweetened foods and beverages because of its lower effect on postprandial blood sugar levels , and also noted that `` high intakes of fructose may lead to metabolic complications such as dyslipidaemia , insulin resistance , and increased visceral adiposity '' . Further , the UK 's Scientific Advisory Committee on Nutrition in 2015 disputed the claims of fructose causing metabolic disorders , stating that `` there is insufficient evidence to demonstrate that fructose intake leads to adverse health outcomes independent of any effects related to its presence as a component of total and free sugars . '' Contents ( hide ) 1 Etymology 2 Chemical properties 2.1 Reactions 2.1. 1 Fructose and fermentation 2.1. 2 Fructose and Maillard reaction 2.1. 3 Dehydration 3 Physical and functional properties 3.1 Sweetness of fructose 3.2 Fructose solubility and crystallization 3.3 Fructose hygroscopicity and humectancy 3.4 Freezing point 3.5 Fructose and starch functionality in food systems 4 Food sources 4.1 Carbohydrate content of commercial sweeteners ( percent on dry basis ) 4.2 Nutritional information 5 Fructose digestion and absorption in humans 5.1 Capacity and rate of absorption 5.2 Malabsorption 6 Fructose metabolism 6.1 Fructolysis 6.2 Metabolism of fructose to DHAP and glyceraldehyde 6.3 Synthesis of glycogen from DHAP and glyceraldehyde 3 - phosphate 6.4 Synthesis of triglyceride from DHAP and glyceraldehyde 3 - phosphate 7 Potential health effects 7.1 Weight gain 7.2 Cardiometabolic diseases 7.3 Compared with sucrose 8 See also 9 References 10 External links Etymology ( edit ) The word `` fructose '' was coined in 1857 from the Latin for fructus ( fruit ) and the generic chemical suffix for sugars , - ose . It is also called fruit sugar and levulose . Chemical properties ( edit ) Figure 1 : Relationship between the acyclic and the cyclic ( hemiketal ) isomers of fructose D - and L - isomers of fructose ( open - chain form ) Fructose is a 6 - carbon polyhydroxyketone . Crystalline fructose adopts a cyclic six - membered structure owing to the stability of its hemiketal and internal hydrogen - bonding . This form is formally called D - fructopyranose . In solution , fructose exists as an equilibrium mixture of 70 % fructo pyranose and about 22 % fructo furanose , as well as small amounts of three other forms , including the acyclic structure . Reactions ( edit ) Fructose and fermentation ( edit ) Fructose may be anaerobically fermented by yeast or bacteria . Yeast enzymes convert sugar ( glucose , or fructose ) to ethanol and carbon dioxide . The carbon dioxide released during fermentation will remain dissolved in water , where it will reach equilibrium with carbonic acid , unless the fermentation chamber is left open to the air . The dissolved carbon dioxide and carbonic acid produce the carbonation in bottled fermented beverages . Fructose and Maillard reaction ( edit ) Fructose undergoes the Maillard reaction , non-enzymatic browning , with amino acids . Because fructose exists to a greater extent in the open - chain form than does glucose , the initial stages of the Maillard reaction occur more rapidly than with glucose . Therefore , fructose has potential to contribute to changes in food palatability , as well as other nutritional effects , such as excessive browning , volume and tenderness reduction during cake preparation , and formation of mutagenic compounds . Dehydration ( edit ) Fructose readily dehydrates to give hydroxymethylfurfural ( `` HMF '' ) . This process , in the future , may become part of a low - cost , carbon - neutral system to produce replacements for petrol and diesel from plants . Physical and functional properties ( edit ) Sweetness of fructose ( edit ) See also : Sweetness § Examples of sweet substances The primary reason that fructose is used commercially in foods and beverages , besides its low cost , is its high relative sweetness . It is the sweetest of all naturally occurring carbohydrates . The relative sweetness of fructose has been reported in the range of 1.2 -- 1.8 times that of sucrose . However , it is the 6 - membered ring form of fructose that is sweeter ; the 5 - membered ring form tastes about the same as usual table sugar . Warming fructose leads to formation of the 5 - membered ring form . Therefore , the relative sweetness decreases with increasing temperature . However it has been observed that the absolute sweetness of fructose is identical at 5 ° C as 50 ° C and thus the relative sweetness to sucrose is not due to anomeric distribution but a decrease in the absolute sweetness of sucrose at lower temperatures . Figure 2 : Relative sweetness of sugars and sweeteners The sweetness of fructose is perceived earlier than that of sucrose or glucose , and the taste sensation reaches a peak ( higher than that of sucrose ) and diminishes more quickly than that of sucrose . Fructose can also enhance other flavors in the system . Fructose exhibits a sweetness synergy effect when used in combination with other sweeteners . The relative sweetness of fructose blended with sucrose , aspartame , or saccharin is perceived to be greater than the sweetness calculated from individual components . Fructose solubility and crystallization ( edit ) Fructose has higher solubility than other sugars as well as other sugar alcohols . Fructose is , therefore , difficult to crystallize from an aqueous solution . Sugar mixes containing fructose , such as candies , are softer than those containing other sugars because of the greater solubility of fructose . Fructose hygroscopicity and humectancy ( edit ) Fructose is quicker to absorb moisture and slower to release it to the environment than sucrose , glucose , or other nutritive sweeteners . Fructose is an excellent humectant and retains moisture for a long period of time even at low relative humidity ( RH ) . Therefore , fructose can contribute a more palatable texture , and longer shelf life to the food products in which it is used . Freezing point ( edit ) Fructose has a greater effect on freezing point depression than disaccharides or oligosaccharides , which may protect the integrity of cell walls of fruit by reducing ice crystal formation . However , this characteristic may be undesirable in soft - serve or hard - frozen dairy desserts . Fructose and starch functionality in food Systems ( edit ) Fructose increases starch viscosity more rapidly and achieves a higher final viscosity than sucrose because fructose lowers the temperature required during gelatinizing of starch , causing a greater final viscosity . Although some artificial sweeteners are not suitable for home - baking , many traditional recipes use fructose . Food sources ( edit ) This section needs additional citations for verification . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed . ( December 2017 ) ( Learn how and when to remove this template message ) Crystalline fructose Natural sources of fructose include fruits , vegetables ( including sugar cane ) , and honey . Fructose is often further concentrated from these sources . The highest dietary sources of fructose , besides pure crystalline fructose , are foods containing table sugar ( sucrose ) , high - fructose corn syrup , agave nectar , honey , molasses , maple syrup , fruit and fruit juices , as these have the highest percentages of fructose ( including fructose in sucrose ) per serving compared to other common foods and ingredients . Fructose exists in foods either as a free monosaccharide or bound to glucose as sucrose , a disaccharide . Fructose , glucose , and sucrose may all be present in a food ; however , different foods will have varying levels of each of these three sugars . The sugar contents of common fruits and vegetables are presented in Table 1 . In general , in foods that contain free fructose , the ratio of fructose to glucose is approximately 1 : 1 ; that is , foods with fructose usually contain about an equal amount of free glucose . A value that is above 1 indicates a higher proportion of fructose to glucose , and below 1 a lower proportion . Some fruits have larger proportions of fructose to glucose compared to others . For example , apples and pears contain more than twice as much free fructose as glucose , while for apricots the proportion is less than half as much fructose as glucose . Apple and pear juices are of particular interest to pediatricians because the high concentrations of free fructose in these juices can cause diarrhea in children . The cells ( enterocytes ) that line children 's small intestines have less affinity for fructose absorption than for glucose and sucrose . Unabsorbed fructose creates higher osmolarity in the small intestine , which draws water into the gastrointestinal tract , resulting in osmotic diarrhea . This phenomenon is discussed in greater detail in the Health Effects section . Table 1 also shows the amount of sucrose found in common fruits and vegetables . Sugarcane and sugar beet have a high concentration of sucrose , and are used for commercial preparation of pure sucrose . Extracted cane or beet juice is clarified , removing impurities ; and concentrated by removing excess water . The end - product is 99.9 % - pure sucrose . Sucrose - containing sugars include common table white granulated sugar and powdered sugar , as well as brown sugar . Table 1 . Sugar content of selected common plant foods ( g / 100g ) Food Item Total Carbohydrate including `` dietary fiber '' Total Sugars Free Fructose Free Glucose Sucrose Fructose / Glucose Ratio Sucrose as a % of Total Sugars Fruits Apple 13.8 10.4 5.9 2.4 2.1 2.0 19.9 Apricot 11.1 9.2 0.9 2.4 5.9 0.7 63.5 Banana 22.8 12.2 4.9 5.0 2.4 1.0 20.0 Fig , dried 63.9 47.9 22.9 24.8 0.9 0.93 0.15 Grapes 18.1 15.5 8.1 7.2 0.2 1.1 Navel orange 12.5 8.5 2.25 2.0 4.3 1.1 50.4 Peach 9.5 8.4 1.5 2.0 4.8 0.9 56.7 Pear 15.5 9.8 6.2 2.8 0.8 2.1 8.0 Pineapple 13.1 9.9 2.1 1.7 6.0 1.1 60.8 Plum 11.4 9.9 3.1 5.1 1.6 0.66 16.2 Vegetables Beet , Red 9.6 6.8 0.1 0.1 6.5 1.0 96.2 Carrot 9.6 4.7 0.6 0.6 3.6 1.0 77 Red Pepper , Sweet 6.0 4.2 2.3 1.9 0.0 0.0 Onion , Sweet 7.6 5.0 2.0 2.3 0.7 0.9 14.3 Sweet Potato 20.1 4.2 0.7 1.0 2.5 0.9 60.3 Yam 27.9 0.5 tr tr tr na tr Sugar Cane 13 -- 18 0.2 -- 1.0 0.2 -- 1.0 11 -- 16 1.0 high Sugar Beet 17 -- 18 0.1 -- 0.5 0.1 -- 0.5 16 -- 17 1.0 high Grains Corn , Sweet 19.0 6.2 1.9 3.4 0.9 0.61 15.0 ^ A The carbohydrate figure is calculated in the USDA database and does not always correspond to the sum of the sugars , the starch , and the `` dietary fiber '' . All data with a unit of g ( gram ) are based on 100 g of a food item . The fructose / glucose ratio is calculated by dividing the sum of free fructose plus half sucrose by the sum of free glucose plus half sucrose . Fructose is also found in the manufactured sweetener , high - fructose corn syrup ( HFCS ) , which is produced by treating corn syrup with enzymes , converting glucose into fructose . The common designations for fructose content , HFCS - 42 and HFCS - 55 , indicate the percentage of fructose present in HFCS . HFCS - 55 is commonly used as a sweetener for soft drinks , whereas HFCS - 42 is used to sweeten processed foods , breakfast cereals , bakery foods , and some soft drinks . Carbohydrate content of commercial sweeteners ( percent on dry basis ) ( edit ) Sugar Fructose Glucose Sucrose ( Fructose + Glucose ) Other sugars Granulated sugar 0 0 100 0 Caramel 97 HFCS - 42 42 53 0 5 HFCS - 55 55 41 0 HFCS - 90 90 5 0 5 Honey 50 44 5 Maple syrup 95 0 Molasses 23 21 53 Tapioca Syrup 55 45 0 0 Corn syrup 0 35 0 0 Data obtained from Kretchmer , N. & Hollenbeck , CB ( 1991 ) . Sugars and Sweeteners , Boca Raton , FL : CRC Press , Inc. for HFCS , and USDA for fruits and vegetables and the other refined sugars . Cane and beet sugars have been used as the major sweetener in food manufacturing for centuries . However , with the development of HFCS , a significant shift occurred in the type of sweetener consumption in certain countries , particularly the United States . Contrary to the popular belief , however , with the increase of HFCS consumption , the total fructose intake relative to the total glucose intake has not dramatically changed . Granulated sugar is 99.9 % - pure sucrose , which means that it has equal ratio of fructose to glucose . The most commonly used forms of HFCS , HFCS - 42 , and HFCS - 55 , have a roughly equal ratio of fructose to glucose , with minor differences . HFCS has simply replaced sucrose as a sweetener . Therefore , despite the changes in the sweetener consumption , the ratio of glucose to fructose intake has remained relatively constant . Figure 3 : Adjusted consumption of refined sugar per capita in the US Nutritional information ( edit ) Providing 368 kcal per 100 grams of dry powder ( table ) , fructose has 95 % the caloric value of sucrose by weight . Fructose powder is 100 % carbohydrates and supplies no nutrients in significant content ( table ) . Fructose , dry powdered Nutritional value per 100 g ( 3.5 oz ) Energy 368 kcal ( 1,540 kJ ) Carbohydrates 100 g Fat 0 g Protein 0 g Minerals Calcium ( 0 % ) 0 mg Iron ( 1 % ) 0.1 mg Phosphorus ( 0 % ) 0 mg Potassium ( 0 % ) 0 mg Sodium ( 1 % ) 12 mg Full Link to USDA Database entry Units μg = micrograms mg = milligrams IU = International units Percentages are roughly approximated using US recommendations for adults . Source : USDA Nutrient Database Fructose digestion and absorption in humans ( edit ) Figure 4 : Hydrolysis of sucrose to glucose and fructose by sucrase Figure 5 : Intestinal sugar transport proteins Fructose exists in foods either as a monosaccharide ( free fructose ) or as a unit of a disaccharide ( sucrose ) . Free fructose is absorbed directly by the intestine . When fructose is consumed in the form of sucrose , it is digested ( broken down ) and then absorbed as free fructose . As sucrose comes into contact with the membrane of the small intestine , the enzyme sucrase catalyzes the cleavage of sucrose to yield one glucose unit and one fructose unit , which are then each absorbed . After absorption , it enters the hepatic portal vein and is directed toward the liver . The mechanism of fructose absorption in the small intestine is not completely understood . Some evidence suggests active transport , because fructose uptake has been shown to occur against a concentration gradient . However , the majority of research supports the claim that fructose absorption occurs on the mucosal membrane via facilitated transport involving GLUT5 transport proteins . Since the concentration of fructose is higher in the lumen , fructose is able to flow down a concentration gradient into the enterocytes , assisted by transport proteins . Fructose may be transported out of the enterocyte across the basolateral membrane by either GLUT2 or GLUT5 , although the GLUT2 transporter has a greater capacity for transporting fructose , and , therefore , the majority of fructose is transported out of the enterocyte through GLUT2 . Capacity and rate of absorption ( edit ) The absorption capacity for fructose in monosaccharide form ranges from less than 5 g to 50 g ( per individual serving ) and adapts with changes in dietary fructose intake . Studies show the greatest absorption rate occurs when glucose and fructose are administered in equal quantities . When fructose is ingested as part of the disaccharide sucrose , absorption capacity is much higher because fructose exists in a 1 : 1 ratio with glucose . It appears that the GLUT5 transfer rate may be saturated at low levels , and absorption is increased through joint absorption with glucose . One proposed mechanism for this phenomenon is a glucose - dependent cotransport of fructose . In addition , fructose transfer activity increases with dietary fructose intake . The presence of fructose in the lumen causes increased mRNA transcription of GLUT5 , leading to increased transport proteins . High - fructose diets ( > 2.4 g / kg body wt ) increase transport proteins within three days of intake . Malabsorption ( edit ) Main article : Fructose malabsorption Several studies have measured the intestinal absorption of fructose using the hydrogen breath test . These studies indicate that fructose is not completely absorbed in the small intestine . When fructose is not absorbed in the small intestine , it is transported into the large intestine , where it is fermented by the colonic flora . Hydrogen is produced during the fermentation process and dissolves into the blood of the portal vein . This hydrogen is transported to the lungs , where it is exchanged across the lungs and is measurable by the hydrogen breath test . The colonic flora also produces carbon dioxide , short - chain fatty acids , organic acids , and trace gases in the presence of unabsorbed fructose . The presence of gases and organic acids in the large intestine causes gastrointestinal symptoms such as bloating , diarrhea , flatulence , and gastrointestial pain Exercise immediately after consumption can exacerbate these symptoms by decreasing transit time in the small intestine , resulting in a greater amount of fructose emptied into the large intestine . Fructose metabolism ( edit ) All three dietary monosaccharides are transported into the liver by the GLUT2 transporter . Fructose and galactose are phosphorylated in the liver by fructokinase ( K = 0.5 mM ) and galactokinase ( K = 0.8 mM ) . By contrast , glucose tends to pass through the liver ( K of hepatic glucokinase = 10 mM ) and can be metabolised anywhere in the body . Uptake of fructose by the liver is not regulated by insulin . However , insulin is capable of increasing the abundance and functional activity of GLUT5 in skeletal muscle cells . Fructolysis ( edit ) Main article : Fructolysis The initial catabolism of fructose is sometimes referred to as fructolysis , in analogy with glycolysis , the catabolism of glucose . In fructolysis , the enzyme fructokinase initially produces fructose 1 - phosphate , which is split by aldolase B to produce the trioses dihydroxyacetone phosphate ( DHAP ) and glyceraldehyde ( 1 ) . Unlike glycolysis , in fructolysis the triose glyceraldehyde lacks a phosphate group . A third enzyme , triokinase , is therefore required to phosphorylate glyceraldehyde , producing glyceraldehyde 3 - phosphate . The resulting trioses are identical to those obtained in glycolysis and can enter the gluconeogenic pathway for glucose or glycogen synthesis , or be further catabolized through the lower glycolytic pathway to pyruvate . Metabolism of fructose to DHAP and glyceraldehyde ( edit ) The first step in the metabolism of fructose is the phosphorylation of fructose to fructose 1 - phosphate by fructokinase , thus trapping fructose for metabolism in the liver . Fructose 1 - phosphate then undergoes hydrolysis by aldolase B to form DHAP and glyceraldehydes ; DHAP can either be isomerized to glyceraldehyde 3 - phosphate by triosephosphate isomerase or undergo reduction to glycerol 3 - phosphate by glycerol 3 - phosphate dehydrogenase . The glyceraldehyde produced may also be converted to glyceraldehyde 3 - phosphate by glyceraldehyde kinase or further converted to glycerol 3 - phosphate by glycerol 3 - phosphate dehydrogenase . The metabolism of fructose at this point yields intermediates in the gluconeogenic pathway leading to glycogen synthesis as well as fatty acid and triglyceride synthesis . Synthesis of glycogen from DHAP and glyceraldehyde 3 - phosphate ( edit ) The resultant glyceraldehyde formed by aldolase B then undergoes phosphorylation to glyceraldehyde 3 - phosphate . Increased concentrations of DHAP and glyceraldehyde 3 - phosphate in the liver drive the gluconeogenic pathway toward glucose and subsequent glycogen synthesis . It appears that fructose is a better substrate for glycogen synthesis than glucose and that glycogen replenishment takes precedence over triglyceride formation . Once liver glycogen is replenished , the intermediates of fructose metabolism are primarily directed toward triglyceride synthesis . Figure 6 : Metabolic conversion of fructose to glycogen in the liver Synthesis of triglyceride from DHAP and glyceraldehyde 3 - phosphate ( edit ) Carbons from dietary fructose are found in both the free fatty acid and glycerol moieties of plasma triglycerides . High fructose consumption can lead to excess pyruvate production , causing a buildup of Krebs cycle intermediates . Accumulated citrate can be transported from the mitochondria into the cytosol of hepatocytes , converted to acetyl CoA by citrate lyase and directed toward fatty acid synthesis . In addition , DHAP can be converted to glycerol 3 - phosphate , providing the glycerol backbone for the triglyceride molecule . Triglycerides are incorporated into very - low - density lipoproteins ( VLDL ) , which are released from the liver destined toward peripheral tissues for storage in both fat and muscle cells . Figure 7 : Metabolic conversion of fructose to triglyceride in the liver Potential Health effects ( edit ) Weight gain ( edit ) In a meta - analysis of clinical trials with controlled feeding -- where test subjects were fed a fixed amount of energy rather than being allowed to choose the amount they ate -- fructose was not an independent factor for weight gain ; however , fructose consumption was associated with weight gain when the fructose provided excess calories . Cardiometabolic diseases ( edit ) An expert panel of the European Food Safety Authority concluded that fructose is preferred in food and beverage manufacturing to replace sucrose and glucose due to the lower effect of fructose on blood glucose levels following a meal . However , as a common sweetening agent for foods and beverages , fructose has been associated with increased risk of obesity , diabetes , and cardiovascular disorders that are part of metabolic syndrome . Clinical research has provided no or only limited direct evidence that fructose itself is associated with elevated LDL cholesterol and triglycerides leading to metabolic syndrome , but rather indicates that excessive consumption of sugar - sweetened foods and beverages , and the concurrent increase in calorie intake , underlies metabolic syndrome . Similarly , increased consumption of sweetened foods and beverages raises risk of cardiovascular disease , including hypertension , but there is no direct cause and effect relationship in humans showing that fructose is the causative factor . Compared with sucrose ( edit ) Fructose is often recommended for diabetics because it does not trigger the production of insulin by pancreatic β cells , probably because β cells have low levels of GLUT5 , For a 50 gram reference amount , fructose has a glycemic index of 23 , compared with 100 for glucose and 60 for sucrose . Fructose is also 73 % sweeter than sucrose at room temperature , allowing diabetics to use less of it per serving . Fructose consumed before a meal may reduce the glycemic response of the meal . Fructose - sweetened food and beverage products cause less of a rise in blood glucose levels than do those manufactured with sucrose or glucose . See also ( edit ) Hereditary fructose intolerance Inverted sugar syrup References ( edit ) Jump up ^ `` Fructose -- Definition and More from the Free Merriam - Webster Dictionary '' . Mw4.m-w.com . Archived from the original on 5 June 2013 . Retrieved 10 December 2014 . Jump up ^ Levulose comes from the Latin word laevus , levo , `` left side '' , levulose is the old word for the most occurring isomer of fructose . D - fructose rotates plane - polarised light to the left , hence the name . `` Archived copy '' . Archived from the original on 2009 - 10 - 08 . Retrieved 2010 - 01 - 28 ... 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Jump up ^ Ushijima , K ; Fujisawa T ; Riby J ; Kretchmer N ( 1991 ) . `` Absorption of fructose by isolated small intestine of rats is via a specific saturable carrier in the absence of glucose and by the disaccharidase - related transport system in the presence of glucose '' . Journal of Nutrition. 125 ( 8 ) : 2156 -- 2164 . PMID 7643250 . Jump up ^ Ferraris , R ( 2001 ) . `` Dietary and developmental regulation of intestinal sugar transport '' . Journal of Biochemistry . 360 ( Pt 2 ) : 265 -- 276 . doi : 10.1042 / 0264 - 6021 : 3600265 . PMC 1222226 . PMID 11716754 . ^ Jump up to : Beyer , PL ; Caviar EM ; McCallum RW ( 2005 ) . `` Fructose intake at current levels in the United States may cause gastrointestinal distress in normal adults '' . J. Am. Diet . Assoc. 105 ( 10 ) : 1559 -- 1566 . doi : 10.1016 / j. jada. 2005.07. 002 . PMID 16183355 . Jump up ^ Ravich , WJ ; Bayless TM ; Thomas , M ( 1983 ) . `` Fructose : incomplete intestinal absorption in humans '' . Gastroenterology. 84 ( 1 ) : 26 -- 29 . PMID 6847852 . Jump up ^ Riby , JE ; Fujisawa T ; Kretchmer , N ( 1993 ) . `` Fructose absorption '' . American Journal of Clinical Nutrition. 58 ( 5 Suppl ) : 748S -- 753S . PMID 8213606 . Jump up ^ Rumessen , JJ ; Gudman - Hoyer E ( 1986 ) . `` Absorption capacity of fructose in healthy adults . Comparison with sucrose and its constituent monosaccharides '' . Gut. 27 ( 10 ) : 1161 -- 1168 . doi : 10.1136 / gut. 27.10. 1161 . PMC 1433856 . PMID 3781328 . Jump up ^ Skoog , SM ; Bharucha AE ( 2004 ) . `` Dietary fructose and gastrointestinal symptoms : a review '' . Am. J. Gastroenterol. 99 ( 10 ) : 2046 -- 50 . doi : 10.1111 / j. 1572 - 0241.2004. 40266. x . PMID 15447771 . Jump up ^ Fujisawa , T , T ; Mulligan K ; Wada L ; Schumacher L ; Riby J ; Kretchmer N ( 1993 ) . `` The effect of exercise on fructose absorption '' . Am. J. Clin. Nutr. 58 ( 1 ) : 75 -- 9 . PMID 8317393 . Jump up ^ Quezada - Calvillo , R ; Robayo CC ; Nichols BL ( 2006 ) . Carbohydrate Digestion and Absorption . Missouri : Saunders , Elsevier . pp. 182 -- 185 . ISBN 1 - 4160 - 0209 - X . Jump up ^ Hajduch , E ; Litherland GJ ; Turban S ; Brot - Laroche E ; Hundal HS ( Aug 2003 ) . `` Insulin regulates the expression of the GLUT5 transporter in L6 skeletal muscle cells '' . FEBS Letters. 549 ( 1 -- 3 ) : 77 -- 82 . doi : 10.1016 / S0014 - 5793 ( 03 ) 00773 - 7 . PMID 12914929 . Jump up ^ MA Parniak ; Kalant N ( 1988 ) . `` Enhancement of glycogen concentrations in primary cultures of rat hepatocytes exposed to glucose and fructose '' . Biochemical Journal . 251 ( 3 ) : 795 -- 802 . doi : 10.1042 / bj2510795 . PMC 1149073 . PMID 3415647 . ^ Jump up to : McGrane , MM ( 2006 ) . Carbohydrate metabolism : Synthesis and oxidation . Missouri : Saunders , Elsevier . pp. 258 -- 277 . ISBN 1 - 4160 - 0209 - X . ^ Jump up to : Sul , HS ( 2006 ) . Metabolism of Fatty Acids , Acylglycerols , and Sphingolipids . Missouri : Saunders , Elsevier . pp. 450 -- 467 . ISBN 1 - 4160 - 0209 - X . Jump up ^ Sievenpiper JL , de Souza RJ , Mirrahimi A , Yu ME , Carleton AJ , Beyene J , Chiavaroli L , Di Buono M , Jenkins AL , Leiter LA , Wolever TM , Kendall CW , Jenkins DJ ( Feb 21 , 2012 ) . `` Effect of Fructose on Body Weight in Controlled Feeding Trials : A Systematic Review and Meta - analysis '' . Ann Intern Med. 156 ( 4 ) : 291 -- 304 . doi : 10.7326 / 0003 - 4819 - 156 - 4 - 201202210 - 00007 . PMID 22351714 . Jump up ^ Bantle , J.P. ; Raatz , S.K. ; Thomas , W. ; Georgopoulos , A. ( 2000 - 11 - 01 ) . `` Effects of dietary fructose on plasma lipids in healthy subjects '' . The American Journal of Clinical Nutrition. 72 ( 5 ) : 1128 -- 1134 . ISSN 0002 - 9165 . PMID 11063439 . Jump up ^ Rippe , J.M. ; Angelopoulos , T.J. ( 2015 ) . `` Fructose - Containing Sugars and Cardiovascular Disease '' . Advances in Nutrition. 6 ( 4 ) : 430 -- 439 . doi : 10.3945 / an. 114.008177 . ISSN 2156 - 5376 . PMC 4496738 . PMID 26178027 . Jump up ^ Bray , George A. ( 2013 - 03 - 01 ) . `` Energy and fructose from beverages sweetened with sugar or high - fructose corn syrup pose a health risk for some people '' . Advances in Nutrition ( Bethesda , Md . ) . 4 ( 2 ) : 220 -- 225 . doi : 10.3945 / an. 112.002816 . ISSN 2156 - 5376 . PMC 3649102 . PMID 23493538 . Jump up ^ A.M. Grant ; M.R. Christie ; S.J. Ashcroft ( August 1980 ) . `` Insulin Release from Human Pancreatic Islets in Vitro '' . Diabetologia. 19 ( 2 ) : 114 -- 117 . doi : 10.1007 / BF00421856 . PMID 6998814 . Jump up ^ D.L. Curry ( 1989 ) . `` Effects of Mannose and Fructose on the Synthesis and Secretion of Insulin '' . Pancreas. 4 ( 1 ) : 2 -- 9 . doi : 10.1097 / 00006676 - 198902000 - 00002 . PMID 2654926 . Jump up ^ Y . Sato ; T. Ito ; N. Udaka ; et al. ( December 1996 ) . `` Immunohistochemical Localization of Facilitated - Diffusion Glucose Transporters in Rat Pancreatic Islets '' . Tissue Cell. 28 ( 6 ) : 637 -- 643 . doi : 10.1016 / S0040 - 8166 ( 96 ) 80067 - X . PMID 9004533 . Jump up ^ `` Glycemic index '' . Glycemic Index Testing and Research , University of Sydney ( Australia ) Glycemic Index Research Service ( SUGiRS ) . 2 May 2017 . Retrieved 23 February 2018 . Jump up ^ Patricia M. Heacock ; Steven R. Hertzler ; Bryan W. Wolf ( 2002 ) . `` Fructose Prefeeding Reduces the Glycemic Response to a High - Glycemic Index , Starchy Food in Humans '' . Journal of Nutrition. 132 ( 9 ) : 2601 -- 2604 . PMID 12221216 . Archived from the original on 2008 - 10 - 06 . External links ( edit ) Fructose and galactose metabolic intermediates Fructose Fructose - 1 - phosphate → DHAP / Glyceraldehyde → Glyceraldehyde 3 - phosphate Galactose Galactose - 1 - phosphate → Glucose 1 - phosphate → Glucose 6 - phosphate → Fructose 6 - phosphate Uridine diphosphate galactose Uridine diphosphate glucose Galactitol Iditol Mannose Mannose - 6 - phosphate → Fructose 6 - phosphate Inborn error of carbohydrate metabolism : monosaccharide metabolism disorders ( E73 -- E74 , 271 ) Including glycogen storage diseases ( GSD ) Sucrose , transport ( extracellular ) Disaccharide catabolism Congenital alactasia Sucrose intolerance Monosaccharide transport Glucose - galactose malabsorption Inborn errors of renal tubular transport ( Renal glycosuria ) Fructose malabsorption Hexose → glucose Monosaccharide catabolism Fructose : Essential fructosuria Fructose intolerance Galactose / galactosemia : GALK deficiency GALT deficiency / GALE deficiency Glucose ⇄ glycogen Glycogenesis GSD type 0 ( glycogen synthase deficiency ) GSD type IV ( Andersen 's disease , branching enzyme deficiency ) Adult polyglucosan body disease ( APBD ) Glycogenolysis Extralysosomal : GSD type III ( Cori 's disease , debranching enzyme deficiency ) GSD type VI ( Hers ' disease , liver glycogen phosphorylase deficiency ) GSD type V ( McArdle 's disease , myophosphorylase deficiency ) GSD type IX ( phosphorylase kinase deficiency ) Lysosomal ( LSD ) : GSD type II ( Pompe 's disease , glucosidase deficiency ) Glucose ⇄ CAC Glycolysis MODY 2 / HHF3 GSD type VII ( Tarui 's disease , phosphofructokinase deficiency ) Triosephosphate isomerase deficiency Pyruvate kinase deficiency Gluconeogenesis PCD Fructose bisphosphatase deficiency GSD type I ( von Gierke 's disease , glucose 6 - phosphatase deficiency ) Pentose phosphate pathway Glucose - 6 - phosphate dehydrogenase deficiency Transaldolase deficiency 6 - phosphogluconate dehydrogenase deficiency Other Hyperoxaluria Primary hyperoxaluria Pentosuria Aldolase A deficiency Metabolism : carbohydrate metabolism fructose and galactose enzymes Fructose / Fructolysis Hepatic fructokinase Aldolase B Triokinase Sorbitol Sorbitol dehydrogenase Aldose reductase Galactose / Galactolysis Galactokinase Galactose - 1 - phosphate uridylyltransferase / UDP - glucose 4 - epimerase Aldose reductase Lactose Lactose synthase Lactase Mannose Mannose phosphate isomerase Types of carbohydrates General Aldose Ketose Furanose Pyranose Geometry Anomer Cyclohexane conformation Mutarotation Monosaccharides Dioses Aldodiose Glycolaldehyde Trioses Aldotriose Glyceraldehyde Ketotriose Dihydroxyacetone Tetroses Aldotetroses Erythrose Threose Ketotetrose Erythrulose Pentoses Aldopentoses Arabinose Lyxose Ribose Xylose Ketopentoses Ribulose Xylulose Deoxy sugars Deoxyribose Hexoses Aldohexoses Allose Altrose Galactose Glucose Gulose Idose Mannose Talose Ketohexoses Fructose Psicose Sorbose Tagatose Deoxy sugars Fucose Fuculose Rhamnose Heptoses Ketoheptoses Mannoheptulose Sedoheptulose Above 7 Octoses Nonoses Neuraminic acid Multiple Disaccharides Cellobiose Isomaltose Isomaltulose Lactose Lactulose Maltose Sucrose Trehalose Turanose Trisaccharides Maltotriose Melezitose Raffinose Tetrasaccharides Stachyose Other oligosaccharides Acarbose Fructooligosaccharide ( FOS ) Galactooligosaccharide ( GOS ) Isomaltooligosaccharide ( IMO ) Maltodextrin Mannan - oligosaccharides ( MOS ) Polysaccharides Beta - glucan Oat beta - glucan Lentinan Sizofiran Zymosan Cellulose Chitin Chitosan Dextrin / Dextran Fructose / Fructan Inulin Galactose / Galactan Glucose / Glucan Glycogen Hemicellulose Levan beta 2 → 6 Lignin Mannan Pectin Starch Amylopectin Amylose Xanthan gum Sugar ( as food commodity ) List of sugars and sugar products Chemistry Fructose Galactose Glucose Lactose Maltose Sucrose Trehalose Xylose Monosaccharide Disaccharide Free sugar Reducing sugar Sources Sugar beet Sugarcane Agave nectar Birch Coconut Date Honeydew Maple Palm Products Syrups Candi sugar Cheong Maesil - cheong Mogwa - cheong Yuja - cheong Corn syrup Glucose syrup Golden syrup High - fructose corn syrup High - maltose corn syrup Honey Inverted sugar syrup Kuromitsu Maple syrup Molasses Pine honey Steen 's cane syrup Treacle Solid forms Brown Peen tong Chancaca Crystalline fructose Gelling Gula melaka Jaggery Misri Mizuame Molasses sugar Muscovado Nib Non-centrifugal cane sugar Panela Plantation Reserve Powdered Preserving Rock candy Sucanat Sugar candy Barley sugar Butterscotch Candy Hard Toffee Sugar glass Sugarloaf Wasanbon Other forms Caramel Cotton candy floss Maple sugar foods Rum Sugar alcohol Sugar confectionery Sugarcane juice Tuzemák Unrefined sweeteners Industry Production Boilery Plantation Casa grande Refinery Sugar bush Sugar cane mill Engenho Batey Zafra By region ( current ) Cuba Caribbean Philippines Sri Lanka U.S. Sugar Program By region ( historical ) Danish West Indies Fiji Hawaii History Amelioration Act 1798 Blackbirding Colonial molasses trade Demerara rebellion of 1823 Holing cane Molasses Act Reciprocity Treaty of 1875 Slavery in the British and French Caribbean Sugar Act Sugar Duties Act 1846 Sugar Intervention Taiwan Sugar Railways Triangular trade Culture Crop over Sugar house Sugaring Sugar nips Sugar packet Sucrology Sugar people Sugar tit Sugar sculpture Treacle mine Related Added sugar Australian Aboriginal sweet foods Bagasse Barley malt syrup Brown rice syrup Cane knife Flavored syrup Fruit syrup Date honey Grape Jallab Health effects Nectar Sugar addiction Sugars in wine Residual sugar Sugar substitute Sweetened beverage Sweetener Sweetness Vinasse Yacón syrup Production GND : 4155521 - 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Rotation - wikipedia Rotation Jump to : navigation , search This article is about movement of a physical body . For other uses , see Rotation ( disambiguation ) . This article needs additional citations for verification . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed . ( March 2014 ) ( Learn how and when to remove this template message ) A sphere rotating about an axis A rotation is a circular movement of an object around a center ( or point ) of rotation . A three - dimensional object can always be rotated around an infinite number of imaginary lines called rotation axes ( / ˈæksiːz / AK - seez ) . If the axis passes through the body 's center of mass , the body is said to rotate upon itself , or spin . A rotation about an external point , e.g. the Earth about the Sun , is called a revolution or orbital revolution , typically when it is produced by gravity . The axis is called a pole . Contents ( hide ) 1 Mathematics 2 Astronomy 2.1 Rotation and revolution 2.2 Retrograde rotation 3 Physics 3.1 Cosmological principle 3.2 Euler rotations 4 Flight dynamics 5 Amusement rides 6 Sports 7 Fixed axis vs. fixed point 8 Axis of 2 dimensional rotations 9 Rotation axis in 3 dimensions 10 Rotation plane 11 See also 12 References 13 External links Mathematics ( edit ) Main article : Rotation ( mathematics ) Rotation of a planar figure around a point Rotational Orbit v Spin Relations between rotation axis , plane of orbit and axial tilt ( for Earth ) . Mathematically , a rotation is a rigid body movement which , unlike a translation , keeps a point fixed . This definition applies to rotations within both two and three dimensions ( in a plane and in space , respectively . ) All rigid body movements are rotations , translations , or combinations of the two . A rotation is simply a progressive radial orientation to a common point . That common point lies within the axis of that motion . The axis is 90 degrees perpendicular to the plane of the motion . If the axis of the rotation lies external of the body in question then the body is said to orbit . There is no fundamental difference between a `` rotation '' and an `` orbit '' and or `` spin '' . The key distinction is simply where the axis of the rotation lies , either within or outside of a body in question . This distinction can be demonstrated for both `` rigid '' and `` non rigid '' bodies . If a rotation around a point or axis is followed by a second rotation around the same point / axis , a third rotation results . The reverse ( inverse ) of a rotation is also a rotation . Thus , the rotations around a point / axis form a group . However , a rotation around a point or axis and a rotation around a different point / axis may result in something other than a rotation , e.g. a translation . Rotations around the x , y and z axes are called principal rotations . Rotation around any axis can be performed by taking a rotation around the x axis , followed by a rotation around the y axis , and followed by a rotation around the z axis . That is to say , any spatial rotation can be decomposed into a combination of principal rotations . In flight dynamics , the principal rotations are known as yaw , pitch , and roll ( known as Tait -- Bryan angles ) . This terminology is also used in computer graphics . See also : curl ( mathematics ) , cyclic permutation , Euler angles , rigid body , rotation around a fixed axis , rotation group SO ( 3 ) , rotation matrix , axis angle , quaternion , and isometry Astronomy ( edit ) Star trails caused by the Earth 's rotation during the camera 's long exposure time . In astronomy , rotation is a commonly observed phenomenon . Stars , planets and similar bodies all spin around on their axes . The rotation rate of planets in the solar system was first measured by tracking visual features . Stellar rotation is measured through Doppler shift or by tracking active surface features . This rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the equator . One effect is that an object weighs slightly less at the equator . Another is that the Earth is slightly deformed into an oblate spheroid . Another consequence of the rotation of a planet is the phenomenon of precession . Like a gyroscope , the overall effect is a slight `` wobble '' in the movement of the axis of a planet . Currently the tilt of the Earth 's axis to its orbital plane ( obliquity of the ecliptic ) is 23.44 degrees , but this angle changes slowly ( over thousands of years ) . ( See also Precession of the equinoxes and Pole star . ) Rotation and revolution ( edit ) Main article : Orbital revolution While revolution is often used as a synonym for rotation , in many fields , particularly astronomy and related fields , revolution , often referred to as orbital revolution for clarity , is used when one body moves around another while rotation is used to mean the movement around an axis . Moons revolve around their planet , planets revolve about their star ( such as the Earth around the Sun ) ; and stars slowly revolve about their galaxial center . The motion of the components of galaxies is complex , but it usually includes a rotation component . Retrograde rotation ( edit ) Main article : Retrograde motion Most planets in our solar system , including Earth , spin in the same direction as they orbit the Sun . The exceptions are Venus and Uranus . Uranus rotates nearly on its side relative to its orbit . Current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history . Venus may be thought of as rotating slowly backwards ( or being `` upside down '' ) . The dwarf planet Pluto ( formerly considered a planet ) is anomalous in this and other ways . Physics ( edit ) Further information : Angular momentum See also : Speed § Tangential Speed , rotational energy , angular velocity , Centrifugal force ( fictitious ) , centripetal force , circular motion , circular orbit , Coriolis effect , spin ( physics ) , rotational spectroscopy , and Rigid body dynamics § Linear and angular momentum The speed of rotation is given by the angular frequency ( rad / s ) or frequency ( turn / s , turns per minute ) , or period ( seconds , days , etc . ) . The time - rate of change of angular frequency is angular acceleration ( rad / s2 ) , This change is caused by torque . The ratio of the two ( how heavy is it to start , stop , or otherwise change rotation ) is given by the moment of inertia . The angular velocity vector ( an axial vector ) also describes the direction of the axis of rotation . Similarly the torque is an axial vector . The physics of the rotation around a fixed axis is mathematically described with the axis -- angle representation of rotations . According to the right - hand rule , the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation , like a screw . Cosmological principle ( edit ) The laws of physics are currently believed to be invariant under any fixed rotation . ( Although they do appear to change when viewed from a rotating viewpoint : see rotating frame of reference . ) In modern physical cosmology , the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale , since the forces are expected to act uniformly throughout the universe and have no preferred direction , and should , therefore , produce no observable irregularities in the large scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang . In particular , for a system which behaves the same regardless of how it is oriented in space , its Lagrangian is rotationally invariant . According to Noether 's theorem , if the action ( the integral over time of its Lagrangian ) of a physical system is invariant under rotation , then angular momentum is conserved . Euler rotations ( edit ) Main article : Euler angles Euler rotations of the Earth . Intrinsic ( green ) , Precession ( blue ) and Nutation ( red ) Euler rotations provide an alternative description of a rotation . It is a composition of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant . Euler rotations are never expressed in terms of the external frame , or in terms of the co-moving rotated body frame , but in a mixture . They constitute a mixed axes of rotation system , where the first angle moves the line of nodes around the external axis z , the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves . These rotations are called precession , nutation , and intrinsic rotation . Flight dynamics ( edit ) Main article : Aircraft principal axes The principal axes of rotation in space In flight dynamics , the principal rotations described with Euler angles above are known as pitch , roll and yaw . The term rotation is also used in aviation to refer to the upward pitch ( nose moves up ) of an aircraft , particularly when starting the climb after takeoff . Principal rotations have the advantage of modelling a number of physical systems such as gimbals , and joysticks , so are easily visualised , and are a very compact way of storing a rotation . But they are difficult to use in calculations as even simple operations like combining rotations are expensive to do , and suffer from a form of gimbal lock where the angles can not be uniquely calculated for certain rotations . Amusement rides ( edit ) Many amusement rides provide rotation . A Ferris wheel has a horizontal central axis , and parallel axes for each gondola , where the rotation is opposite , by gravity or mechanically . As a result , at any time the orientation of the gondola is upright ( not rotated ) , just translated . The tip of the translation vector describes a circle . A carousel provides rotation about a vertical axis . Many rides provide a combination of rotations about several axes . In Chair - O - Planes the rotation about the vertical axis is provided mechanically , while the rotation about the horizontal axis is due to the centripetal force . In roller coaster inversions the rotation about the horizontal axis is one or more full cycles , where inertia keeps people in their seats . Sports ( edit ) `` Spin move '' redirects here . For other uses , see Spin move ( disambiguation ) . Rotation of a ball or other object , usually called spin , plays a role in many sports , including topspin and backspin in tennis , English , follow and draw in billiards and pool , curve balls in baseball , spin bowling in cricket , flying disc sports , etc . Table tennis paddles are manufactured with different surface characteristics to allow the player to impart a greater or lesser amount of spin to the ball . Rotation of a player one or more times around a vertical axis may be called spin in figure skating , twirling ( of the baton or the performer ) in baton twirling , or 360 , 540 , 720 , etc. in snowboarding , etc . Rotation of a player or performer one or more times around a horizontal axis may be called a flip , roll , somersault , heli , etc. in gymnastics , waterskiing , or many other sports , or a one - and - a-half , two - and - a-half , gainer ( starting facing away from the water ) , etc. in diving , etc . A combination of vertical and horizontal rotation ( back flip with 360 ° ) is called a möbius in waterskiing freestyle jumping . Rotation of a player around a vertical axis , generally between 180 and 360 degrees , may be called a spin move and is used as a deceptive or avoidance maneuver , or in an attempt to play , pass , or receive a ball or puck , etc. , or to afford a player a view of the goal or other players . It is often seen in hockey , basketball , football of various codes , tennis , etc . Fixed axis vs. fixed point ( edit ) The end result of any sequence of rotations of any object in 3D about a fixed point is always equivalent to a rotation about an axis . However , an object may physically rotate in 3D about a fixed point on more than one axis simultaneously , in which case there is no single fixed axis of rotation - just the fixed point . However , these two descriptions can be reconciled - such a physical motion can always be re-described in terms of a single axis of rotation , provided the orientation of that axis relative to the object is allowed to change moment by moment . Axis of 2 dimensional rotations ( edit ) 2 dimensional rotations , unlike the 3 dimensional ones , possess no axis of rotation . This is equivalent , for linear transformations , with saying that there is no direction in the place which is kept unchanged by a 2 dimensional rotation , except , of course , the identity . The question of the existence of such a direction is the question of existence of an eigenvector for the matrix A representing the rotation . Every 2D rotation ( of an angle θ ( \ displaystyle \ theta ) can be quite simply represented by the following matrix : A = ( cos  θ sin  θ − sin  θ cos  θ ) ( \ displaystyle A = ( \ begin ( bmatrix ) \ cos \ theta & \ sin \ theta \ \ - \ sin \ theta & \ cos \ theta \ end ( bmatrix ) ) ) A standard eigenvalue determination leads to the characteristic polynomial λ 2 − 2 λ cos  θ + 1 = 0 ( \ displaystyle \ lambda ^ ( 2 ) - 2 \ lambda \ cos \ theta + 1 = 0 ) , which has cos  θ ± i sin  θ ( \ displaystyle \ cos \ theta \ pm i \ sin \ theta ) as it 's eigenvalues . Therefore , there is no real eigenvalue , meaning that no real vector in the plane is kept unchanged by A . Rotation axis in 3 dimensions ( edit ) Every rotation in 3D space has an axis of rotation , that is , a line which is , as a whole , kept unchanged by the rotation . The direction of this axis corresponds to an eigenvector of the rotation matrix A ( \ displaystyle A ) . Therefore , the existence of such an axis corresponds to the existence of an eigenvalue of value 1 . We shall prove that there is one and only one such direction . The starting point of the proof it the fact that any n - dimensional matrix has n eigenvalues . It remains to determine whether these eigenvalues are distinct or not . The fact that A has only real coefficients imply that the complex conjugate of every complex eigenvalue is another ( different ) eigenvalue . This happens because , taking the hermitian conjugate of the eigenvalue equation A v = λ v ( \ displaystyle Av = \ lambda v ) , we have v  ̄ T A † = λ ∗ v  ̄ T ( \ displaystyle ( \ bar ( v ) ) ^ ( T ) A ^ ( \ dagger ) = \ lambda ^ ( * ) ( \ bar ( v ) ) ^ ( T ) ) , but A † ( \ displaystyle A ^ ( \ dagger ) ) is just the transpose A T ( \ displaystyle A ^ ( T ) ) , so v  ̄ T A T = λ ∗ v  ̄ T ( \ displaystyle ( \ bar ( v ) ) ^ ( T ) A ^ ( T ) = \ lambda ^ ( * ) ( \ bar ( v ) ) ^ ( T ) ) and we can transpose to obtain A v  ̄ = λ ∗ v  ̄ ( \ displaystyle A ( \ bar ( v ) ) = \ lambda ^ ( * ) ( \ bar ( v ) ) ) So λ ∗ ( \ displaystyle \ lambda ^ ( * ) ) is an eigenvalue and v  ̄ ( \ displaystyle ( \ bar ( v ) ) ) an eigenvector . Therefore , for every complex ( non-real ) eigenvalue , there corresponds another distinct complex eigenvalue ; it 's complex conjugate . The number of complex eigenvalue of a real matrix must then be pair . Since the rotation matrix in 3 dimensions has 3 eigenvalues , one of them must be real . Now , by the above discussion , we know that , if v is the eigenvector corresponding to this real eigenvalue , v  ̄ ( \ displaystyle ( \ bar ( v ) ) ) must also be an eigenvector corresponding to the same eigenvalue , and therefore their sum , v + b  ̄ ( \ displaystyle v+ ( \ bar ( b ) ) ) , is necessarily a real eigenvector . Finally , the fact that a rotation matrix is orthogonal reads , in matrix form A A T = I ( \ displaystyle AA ^ ( T ) = I ) . Then , if we take the scalar product of A v ( \ displaystyle Av ) with itself , we have A v 2 = ( A v ) T A v = v T A T A v = v T v = v 2 ( \ displaystyle Av ^ ( 2 ) = ( Av ) ^ ( T ) Av = v ^ ( T ) A ^ ( T ) Av = v ^ ( T ) v = v ^ ( 2 ) ) But A v = λ v ( \ displaystyle Av = \ lambda v ) implies A v 2 = λ 2 v T v = λ 2 v 2 ( \ displaystyle Av ^ ( 2 ) = \ lambda ^ ( 2 ) v ^ ( T ) v = \ lambda ^ ( 2 ) v ^ ( 2 ) ) Hence , λ 2 = 1 ( \ displaystyle \ lambda ^ ( 2 ) = 1 ) . If we choose λ = − 1 ( \ displaystyle \ lambda = - 1 ) , this corresponds to an orthogonal matrix of determinant - 1 , that is , a rotation followed by a reflection . Choosing λ = 1 ( \ displaystyle \ lambda = 1 ) corresponds to a rotation . Therefore , the line in this direction v is unchanged by the rotation ; it is a rotation axis . Rotation plane ( edit ) As much as every tridimensional rotation has a rotation axis , also every tridimensional rotation has a plane , which is perpendicular to the rotation axis , and which is left invariant by the rotation . The rotation , restricted to this plane , is an ordinary 2D rotation . The proof proceeds similarly to the above discussion . First , suppose that all eigenvalues of the 3D rotation matrix A are real . This means that there is an orthogonal basis , made by the corresponding eigenvectors ( which are necessarily orthogonal ) , over which the effect of the rotation matrix is just stretching it . If we write A in this basis , it is diagonal ; but a diagonal orthogonal matrix is made of just + 1 's and - 1 's in the diagonal entries . Therefore , we do n't have a proper rotation , but either the identity or the result of a sequence of reflections . It follows , then , that a proper rotation has some complex eigenvalue . Let v be the corresponding eigenvector . Then , as we showed in the previous topic , v  ̄ ( \ displaystyle ( \ bar ( v ) ) ) is also an eigenvector , and v + v  ̄ ( \ displaystyle v+ ( \ bar ( v ) ) ) and i ( v − v  ̄ ) ( \ displaystyle i ( v - ( \ bar ( v ) ) ) ) are such that their scalar product vanishes : i ( v T + v  ̄ T ) ( v − v  ̄ ) = i ( v T v − v  ̄ T v  ̄ + v  ̄ T v − v T v  ̄ ) = 0 ( \ displaystyle i ( v ^ ( T ) + ( \ bar ( v ) ) ^ ( T ) ) ( v - ( \ bar ( v ) ) ) = i ( v ^ ( T ) v - ( \ bar ( v ) ) ^ ( T ) ( \ bar ( v ) ) + ( \ bar ( v ) ) ^ ( T ) v-v ^ ( T ) ( \ bar ( v ) ) ) = 0 ) because , since v  ̄ T v  ̄ ( \ displaystyle ( \ bar ( v ) ) ^ ( T ) ( \ bar ( v ) ) ) is real , it equals it 's complex conjugate v T v ( \ displaystyle v ^ ( T ) v ) , and v  ̄ T v ( \ displaystyle ( \ bar ( v ) ) ^ ( T ) v ) and v T v  ̄ ( \ displaystyle v ^ ( T ) ( \ bar ( v ) ) ) are both representations of the same scalar product between v ( \ displaystyle v ) and v  ̄ ( \ displaystyle ( \ bar ( v ) ) ) . This means v + v  ̄ ( \ displaystyle v+ ( \ bar ( v ) ) ) and i ( v − v  ̄ ) ( \ displaystyle i ( v - ( \ bar ( v ) ) ) ) are orthogonal vectors . Also , they are both real vectors by construction . These vectors span the same subspace as v ( \ displaystyle v ) and v  ̄ ( \ displaystyle ( \ bar ( v ) ) ) , which is an invariant subspace under the application of A. Therefore , they span an invariant plane . This plane is orthogonal to the invariant axis , which corresponds to the remaining eigenvector of A , with eigenvalue 1 , because of the orthogonality of the eigenvectors of A . See also ( edit ) Absolute rotation Attitude ( geometry ) Balancing machine Circular motion Mach 's principle Rolling Rotation around a fixed axis Rotation formalisms in three dimensions Rotating locomotion in living systems Top Truck bolster References ( edit ) Jump up ^ `` An Oasis , or a Secret Lair ? '' . ESO Picture of the Week . Archived from the original on 11 October 2013 . Retrieved 8 October 2013 . External links ( edit ) Hazewinkel , Michiel , ed. ( 2001 ) ( 1994 ) , `` Rotation '' , Encyclopedia of Mathematics , Springer Science + Business Media B.V. / Kluwer Academic Publishers , ISBN 978 - 1 - 55608 - 010 - 4 Product of Rotations at cut - the - knot . cut-the-knot.org When a Triangle is Equilateral at cut - the - knot . cut-the-knot.org Rotate Points Using Polar Coordinates , howtoproperly.com Rotation in Two Dimensions by Sergio Hannibal Mejia after work by Roger Germundsson and Understanding 3D Rotation by Roger Germundsson , Wolfram Demonstrations Project . demonstrations.wolfram.com Retrieved from `` https://en.wikipedia.org/w/index.php?title=Rotation&oldid=835678172 '' Categories : Euclidean geometry Classical mechanics Orientation ( geometry ) Rotation Kinematics Hidden categories : Articles needing additional references from March 2014 All articles needing additional references Pages using div col without cols and colwidth parameters Talk Contents About Wikipedia Asturianu বাংলা भोजपुरी Български བོད ་ ཡིག Bosanski Català Čeština ChiShona Dansk Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Galego 한국어 हिन्दी Hrvatski Ido Bahasa Indonesia IsiXhosa Italiano Basa Jawa ಕನ್ನಡ Қазақша Latina Magyar Македонски मराठी Bahasa Melayu Nederlands 日本 語 Norsk nynorsk ਪੰਜਾਬੀ Plattdüütsch Polski Português Română Русский Scots Sicilianu Simple English Slovenščina Soomaaliga Српски / srpski Srpskohrvatski / српскохрватски Basa Sunda Suomi தமிழ் తెలుగు ไทย Türkçe Українська Tiếng Việt 中文 51 more Edit links This page was last edited on 10 April 2018 , at 03 : 19 . 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Question: an object that moves around an external axis is said to be