Fibonacci Series And The Golden Ratio Engineering Essay

Fibonacci Series And The Golden Ratio Engineering Essay The research question of this extended essay is, Is there a relation between the Fibonacci series and the Golden Ratio? If so be the reason, what is it and explain it. The Fibonacci series, which was first introduced by Leonardo of Pisa (Fibonacci), was found to have had a close connection with the Golden Ratio. The relation found was that the limit of the ratios of the numbers in the Fibonacci sequence converges to the golden mean/golden ratio. I decided to carry out a few set of experiments that involved individual concepts of both: the Fibonacci series and the Golden Ratio. Using their individual applications such as the Golden Rectangle, a computerized calculation supported by a sketched graph, I found that I could arrive at a conjecture that linked the two concepts. I also used the Fibonacci spiral and Golden spiral to find the limit where the values would tend to meet. After carrying out the experiments, I decided to find the proof of the relation using the Binets formula which is essentially the formula for the nth term of a Fibonacci sequence. However, the Binets formula was interesting enough to make me find its proof and solve it myself. From there, I proceeded on to the proof of the relation between the Fibonacci series and the Golden Ratio using this formula. The Binet formula is given by ; . Following the proof, I carried out steps to verify it by substituting different values to check its validity. After proving the validity of the conjecture, I arrived at the conclusion that such a relation does exist. I also learned that this relation had applications in nature, art and architecture. Apart from these, there is a possibility that there are other applications which can be subjected to further investigation. Table of Contents Sl. No. Contents Page No. 1. Introduction to the Fibonacci Series 4 2. Introduction to the Golden Ratio 5 3. The Relationship between them 6 4. Forming the conjecture 6 5. Testing the conjecture 7 6. The proof 15 7. Verification of the proof 20 8. Conclusion 22 9. Further Investigation 22 10. Bibliography 23 Introduction The Fibonacci Series The Fibonacci series is that sequence where every term is the sum of the two terms that precedes it (in the Hindu-Arabic system) where the first two terms of the sequence are 0 and 1. The Fibonacci series is shown below 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 à ¢Ã¢â€š ¬Ã‚ ¦ Where the first two terms are 0 and 1 and the term following it is the sum of the two terms preceding it, which in this case are 0 and 1. Hence, 0 + 1 = 1 (third term) Similarly, Fourth term = third term + second term Fourth term = 1 + 1 = 2 And so the sequence follows. The series was first invented by an Italian by the name of Leonardo Pisano Bigollo (1180 1250) in 1202. He is better known as Fibonacci which essentially means the son of Bonacci. In his book, Liber Arci, there was a puzzle concerning the breeding of rabbits and the solution to this puzzle resulted in the discovery of the Fibonacci series. The problem was based on the total number of rabbits that would be born starting with a pair of rabbits first followed by the breeding of new rabbits which would also start giving birth one month after they were born themselves.  [1]   The problem was broken down into parts and the answer that was obtained gave rise to the Fibonacci series. The Fibonacci series gained a worldwide acceptance soon as after its discovery and was used in many fields. It had its uses and applications in nature (such as the petals of a sunflower and the nautilus shell). Shown below is the application of the series on the whirls of a pine cone.  [2]   http://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/fib2.jpghttp://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/fib3.jpg The Golden Mean / Golden Ratio The golden mean, also known as the golden ratio, as the name suggests is a ratio of distances in simple geometric figures  [3]  . This is only one of the many definitions found for the term. It is not solely restricted to geometric figures but the proportion is used for art, nature and architecture as well. From pine cones to the paintings of Leonardo Da Vinci, the golden proportion is found almost everywhere. Another definition of the golden ratio is a precise way of dividing a line  [4]   There has never been one concrete definition for the golden ratio which makes it susceptible to different definitions using the same concept. First claimed to be known by Pythagoreans around 500 B.C., the golden proportion was established in print in one of Euclids major works namely, Elements, once and for all in 300 B.C. Euclid, the famous Greek mathematician was the first to establish what the golden section really was with respect to a line. According to him, the division of a line in a mean and extreme ratio  [5]  such a way that the point where this division takes place, the ratio of the parts of the line would be the Golden proportion. He determined that the Golden Ratio was such that The golden ratio is denoted by the Greek alphabet which has a value of 1.6180339à ¢Ã¢â€š ¬Ã‚ ¦ Since then, the golden ratio has been used in various fields. In art, Leonardo Da Vinci coined the ratio as the Divine Proportion and used it to define the fundamental proportions of his famous painting of The Last Supper as well as Mona Lisa. http://goldennumber.net/images/davinciman.gif Finally, it was in the 1900s that the term Phi was coined and used for the first time by an American mathematician Mark Barr who used the Greek letter phi to name this ratio.  [6]  Hence, the term obtained a chain of different names such as the golden mean, golden section and golden ratio as well as the Divine proportion.   The Relation between the Fibonacci series and the Golden Ratio After the discovery of the Fibonacci series and the golden ratio, a relation between the two was established. Whether this relation was a coincidence or not, no one was able to answer this question. However, today, the relation between the two is a very close one and it is visible in various fields. The relation is said to be The limit of the ratios of the numbers in the Fibonacci sequence converges to the golden ratio. This means that as we move to the nth term in the Fibonacci sequence, the ratios of the consecutive terms of the Fibonacci series arrive closer to the value of the golden mean ().  [7]   Forming the Conjecture The Fibonacci series and the golden ratio have been linked together in many ways. Hence, I shall now produce the same statement as a conjecture as I am about to prove the relation through a set of experiments and eventually proving the conjecture (right or wrong). The conjecture is stated below The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ , where n is the nth term of the Fibonacci sequence. In order to prove this conjecture, I have carried out a few experiments below that shall attribute to the result of the above conjecture. Testing the Conjecture Experiment No. 1: The first set of experiments deal with the Golden Rectangle. The golden rectangle is that rectangle whose dimensions are in the ratio (where y is the length of the rectangle and x is the breadth of the rectangle), and when a square of dimensions is removed from the original rectangle, another golden rectangle is left behind. Also, the ratio of the dimensions ( is equal to the golden mean (). I have used the concept of the Golden Rectangle to test whether the ratios of the dimensions of the two golden rectangles, when equated to each other, give the value of the golden ratio or not which is also said to be the formula for the nth term of the Fibonacci series. The latter part of the statement is in accordance with Binets formula. The following experiment shows how this works. Let us consider a rectangle with dimensions . The dotted line is the line that has divided the rectangle in such a way that the square on the left has dimensions of . Now, the rectangle on the right has the dimensions of where x is now the length of the new golden rectangle formed and (y-x) is the breadth. Golden Rectangle 1: y x y-x The reason why this rectangle is called a Golden Rectangle is because the ratio of its dimensions gives the value of à Ã¢â‚¬  . Hence, the information we can gather from the above figure is that (1) The new golden rectangle formed from the above one is shown below with dimensions Golden Rectangle 2: y x x The above new golden rectangle shown must thus also have the same property as that of any other golden rectangle. Therefore, From the above experiments we can establish the following relation (2) For convenience sake, I have decided to take so as to make y the subject of the equation. Hence, the above equation can now be re-written as On cross-multiplying the terms above we get Writing the above equation in the form of a quadratic equation, we get Using the quadratic formula, , we get Hence, the two roots obtained are However, the second root is rejected as a value as y is a dimension of the rectangle and hence cannot be a negative value. Hence we have, Evaluating this value we have But, from equation 1, we know that However, the value of x was restricted to 1 in the above test. So as to eliminate the variable in order to keep only y as the subject, I carried out the calculations below that help in doing so Rewriting the equation Cross-multiplying the variables Dividing the equation by , we get But we know that . Thus, using this substitution in the above equation we have This is the same quadratic that we obtained earlier and hence the doubt for the presence of x clears out. Experiment No. 2: For my second experiment, I have decided to use the concept of the Fibonacci spiral and that of the Golden Spiral. The steps on how to draw these spirals are given below A Fibonacci spiral is formed by drawing squares with dimensions equal to the terms of he Fibonacci series. We start by first drawing a 1 x 1 square 1 x 1 Next, another 1 x 1 square is drawn on the left of the first square. (every new square is bordered in red) Now, a 2 x 2 square is drawn below the two 1 x 1 squares. Next, a 3 x 3 square is drawn to the right of the above figure. Now, a 5 x 5 square is adjoined to the top of the figure. Next, a 8 x 8 square is adjoined to the left of the figure. And so the figure continues in the same manner. The squares are adjoined to the original shape in a left to right spiral (from down to up) and each time the square gets bigger but with dimensions equal to the numbers in the Fibonacci series. Starting from the inner square, a quarter of an arc of a circle is drawn within the square. This step is repeated as we move outward, towards the bigger square. The spiral eventually looks like this http://library.thinkquest.org/27890/media/fibonacciSpiralBoxes.gif The shape shown below is the Fibonacci spiral without the squares http://library.thinkquest.org/27890/media/fibonacciSpiral2.gif A similar process is followed for forming the golden spiral. However, the only difference is that we draw the outer squares first and then draw the arcs starting from the larger squares. Hence, the spiral turns inwards all the way to the inner squares. Golden Spiral The Golden spiral eventually looks like this Golden Spiral On comparing the two spirals, it can be seen that they overlap as the arcs occupy the squares with dimensions of the latter terms of the Fibonacci series. An image of how the two spirals look is shown below http://library.thinkquest.org/27890/media/spirals.gif From the above experiment, it can be seen that there is a connection between the Fibonacci series and the Golden Mean as their individual spirals overlap each other as the n (which is the nth term in the series) tends to infinity. Experiment No. 3: My third experiment involves technology. In this experiment, I decided to use a program of Microsoft Office, namely, Microsoft Excel in order to record the values obtained on calculating the ratio of the consecutive terms of the Fibonacci series. In the table below, I have recorded the terms of the Fibonacci series in the first column, the value of the ratio of the consecutive terms in the Fibonacci sequence in the second column, the value of  [8]  in the third column and the variation of the value of the ration from the value of à Ã¢â‚¬   in the last column. Term of Fibonacci Series Value of ratio of consecutive terms value of variation of value calculated from value of 0 1 1 1.00000000000000 1.61803398874989 0.61803398874989 2 2.00000000000000 1.61803398874989 -0.38196601125011 3 1.50000000000000 1.61803398874989 0.11803398874989 5 1.66666666666667 1.61803398874989 -0.04863267791678 8 1.60000000000000 1.61803398874989 0.01803398874989 13 1.62500000000000 1.61803398874989 -0.00696601125011 21 1.61538461538462 1.61803398874989 0.00264937336527 34 1.61904761904762 1.61803398874989 -0.00101363029773 55 1.61764705882353 1.61803398874989 0.00038692992636 89 1.61818181818182 1.61803398874989 -0.00014782943193 144 1.61797752808989 1.61803398874989 0.00005646066000 233 1.61805555555556 1.61803398874989 -0.00002156680567 377 1.61802575107296 1.61803398874989 0.00000823767693 610 1.61803713527851 1.61803398874989 -0.00000314652862 987 1.61803278688525 1.61803398874989 0.00000120186464 1597 1.61803444782168 1.61803398874989 -0.00000045907179 2584 1.61803381340013 1.61803398874989 0.00000017534976 4181 1.61803405572755 1.61803398874989 -0.00000006697766 6765 1.61803396316671 1.61803398874989 0.00000002558318 10946 1.61803399852180 1.61803398874989 -0.00000000977191 17711 1.61803398501736 1.61803398874989 0.00000000373253 28657 1.61803399017560 1.61803398874989 -0.00000000142571 46368 1.61803398820532 1.61803398874989 0.00000000054457 75025 1.61803398895790 1.61803398874989 -0.00000000020801 121393 1.61803398867044 1.61803398874989 0.00000000007945 196418 1.61803398878024 1.61803398874989 -0.00000000003035 317811 1.61803398873830 1.61803398874989 0.00000000001159 514229 1.61803398875432 1.61803398874989 -0.00000000000443 The aim of the table is to find out whether the value of the ratio reaches the value of à Ã¢â‚¬   or not, as the number of terms increases infinitely. Observation: From the above table, it can be seen that as we reach the nth term of the Fibonacci series, the variation in the value of the ratios from the value of à Ã¢â‚¬  , decreases. This observation is in agreement with the conjecture The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ , where n is the nth term of the Fibonacci sequence. Inference: From the above 3 experiments, I have found that the conjecture holds true for them all. Hence, I would like to state that the tests for the conjectures have been significantly successful. The Proof In order to find the relation between the Fibonacci series and the Golden Ratio, I followed the proof below that uses calculus to establish the required relation. The Fibonacci series is given by, Assuming that 0, 1, and 1 are the first three terms of the sequence: (3) This eventually goes on to form the well known sequence: 0, 1, 1, 2, 3, 5, 8, 13à ¢Ã¢â€š ¬Ã‚ ¦ Dividing the Left Hand Side (or LHS) and the Right Hand Side (or RHS) of equation 3 by F(n), gives (By taking the numerator as the denominator of F(n)) By substituting the limit of the ratios of the terms (as n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ ) of the Fibonacci series with A, the limit is taken on both sides such that n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ The above is true as the ratio Hence, the below quadratic equation is formed We can find the roots of A by using the quadratic formula, . or From this we find that This value of is easily attainable using the Binet formula. The Binet formula is that formula which gives the value of by substituting the variable x with one of the n terms of the Fibonacci series. Using the concept of the golden rectangle, the quadratic that was obtained earlier Gave the value of . The proof of the Binet formula shows another possibility to arrive at the relation between the Fibonacci series and the Golden Ratio. The beauty of this proof is that the quadratic first arose from the Fibonacci series calculation and the root that was obtained gave the value of phi. This is from the proof that was written above. Under the heading Testing the Conjecture that was done earlier, the quadratic arose from the dimensions of the Golden Rectangle and the equation thus obtained gave the value of phi. Using this concept, I have followed the proof below which was solved by older mathematicians. The Binet formula is given by Now, from the above tests, we got However, there were 2 values that were obtained on calculating the value of y. The value of y that was negative was rejected then as it was incorrect to consider it a valid answer for a dimension of a geometric figure. Calling this negative root as , we can rewrite the Binet formula as Going back to the quadratic equation, we can substitute in place of y and so the quadratic equation is (4) This quadratic was obtained from the Golden Rectangle. In order to arrive at the Fibonacci sequence, a series of algebraic manipulations will help us reach that step. To start off with, we have the value of in terms of . Now, to get the value of in terms of , we multiply equation (4) into . Using equation (4), we substitute for and we get Using the same method to find the value for raised to higher powers, we have Similarly, Writing the various values for raised to higher powers (5) à ¢Ã¢â€š ¬Ã‚ ¦ Now if we look at the coefficients closely, we see that they are the consecutive terms of the Fibonacci series. This can be written as (6) However, the above trend is not enough proof for generalizing the above statement. Hence, I decided to prove it by using the principle of mathematical induction. Step 1: Step 2: To prove that P(1) is true. Hence, P(1) is true (from equation 5) Step 3: Hence, P(k) is true where Step 4: To prove that P(k+1) is true. Starting from the RHS, (from equation 3) (from equation 4) (from P(k)) = RHS Hence, P(k+1) is true. Therefore, P(n) is true for all Now that we have proved that P(n) is true is true in its generalized form. Also, we know that is the other root of the quadratic equation and so the above general equation can be written in the above form as well (7) In order to obtain the Binet formula in the form of We can subtract equation (7) from equation (6) to get Substituting the original values of and in denominator of the above equation, we get Substituting the value of and in the above equation, we get This is the Binet formula which we started to prove. Hence, the formula is valid. Verifying the Proof In order to validate a proof, it must be tested in order to check whether the conjecture is valid and can be generalized. For this reason, I have decided to use the Binet formula (that was proved above) to check the validity of the relation between the Fibonacci series and the Golden Ratio by substituting values for x in the equation Using Case 1: , Which is the first term of the Fibonacci series. Case 2: , Which is the second term of the Fibonacci series. Case 3: , Which is the third term of the Fibonacci series. Case 4: , Which is the fourth term of the Fibonacci series. From these substitutions it is clear that the formula is a valid one which gives the desired result. Also, the above calculations have proved to be substantial examples for proving the validity of the proofs shown above. However, an important note to remember in the Binet formula is that the value of x starts from 0 and increases. So it can be said that (x belongs to the set of whole numbers). This is to account for the fact that the Fibonacci series starts from 0 and then continues. Hence, the conjecture is true and can be generalized. Hence the conjecture below can be considered true. The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ , where n is the nth term of the Fibonacci sequence. Conclusion From the above tests and verifications, it is clear that a relation between the Fibonacci series and the Golden Ratio does truly exist. The relation being The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ , where n is the nth term of the Fibonacci sequence. The Fibonacci series as well as the Golden Ratio have their individual applications as well as combined applications in various fields of nature, art, etc. As mentioned earlier, the Fibonacci series was used to find a solution to the rabbit problem. The relation between the two concepts was an integral part of the central idea in the novel The Da Vinci Code. Along with these well known ideas, other applications of the two concepts are present in the whirls of a pine cone, the paintings of Leonardo Da Vinci, the spiral of the nautilus shell, the petals of the sunflower. These are only very few examples regarding the applications of the two concepts. However, this relation has proved to be useful to environmentalists, artists and many other researches. For example, artists were able to use the study of the concept in the paintings of Leonardo Da Vinci and decipher old symbols. It also has given them the chance to create art of their own that by using this concept in their procedure of creating. Further Investigation With the great number of applications that were found regarding the Fibonacci series and the Golden Ratio, there is a possibility that there are other applications of the concept as well. The convergence of the ratios of the values to the value of phi may prove to be of great significance if applied to another theory that has boggled minds of mathematicians for years. Possibilities such as these give rise to the question of further investigation in this aspect of the relationship between the two concepts.

The Effects of Thc, Spice, and Opiates on the Human Body

Ben Vu The effects of THC, Spice, and Opiates on the human body In the modern generation, many kids have experimented with all sorts of drugs. They are looking for a euphoric feeling that nothing else gives them. Some of these drugs have been used for medical reasons and recreational use since the beginning of man. Humans naturally are curious and try many different things to give them the high they are looking for. The most common of these drugs are marijuana, spice, and opiates. Cannabis, also known as marijuana, has over hundreds of different chemicals that affect the human body.Delta-9 tetrahydrocannabinol or THC is the main chemical that gives the user the euphoric high they are looking for. Along with the euphoric feeling, the user also receives relaxation. Marijuana can be used in many different ways such as smoking it in a joint or cooking it in to food or sweets. The most common way of getting high is by smoking it. This can either be achieved by rolling a simple joint or bl unt to smoking it out of a bong or a vaporizer. Marijuana has been so common for many generations that the brain has adapted to have cannabinoid receptors.When ingested, the THC flows from the lung into the bloodstream. Once in the bloodstream it goes to the brain attaching to the cannabinoid receptors. The most common receptor, CB1, is mainly found in the part of the brain that associates with movement and memory. This explains why marijuana affects the users balance and coordination. While THC is in the brain, it makes the user have short-term memory lost; this is why it is used to treat nausea, pain, and lost in appetite. Other effects on the user’s body include laughter, altered perception of time, increased appetite and heart rate, paranoia and panic attacks.When marijuana is ingested or inhaled the effects appear as soon as the chemicals enter the brain and may last from one hour up to two hours. If it is ingested in food the short-term effects such as short-term memory lost and coordination begin more slowly usually taking up to half an hour to an hour to kick it. The benefits of ingesting it would be that the effects of marijuana would last longer, up to four hours. When smoking marijuana more of the chemicals are being deposited in the bloodstream.It only takes a few minutes after smoking that the user’s heart rate begins to beat more rapidly resulting in the blood vessel to expand resulting in bloodshot eyes. As the THC enters the brain, it causes the brain to release a chemical called dopamine, which in turns gives the user the high feeling. When high the user may experience pleasant sensations, colors and sounds may seem more intense, time appears to pass by a lot more slowly. The user’s mouth begins to feel dry this is known as cottonmouth causing the user to become suddenly very hungry or thirsty.After the euphoric high goes away the user begins to feel drowsy. There are two different strains of marijuana, indica and sativa. Indica strains have more cannabinoids than THC. This affects the user in more of a physical high resulting in an influx of appetite, laughter and a couch lock feeling. This is when the user gets the stoned effect where they do not want to move around. Indica is the most common strain of marijuana that is smoked by the average person. As a result of the higher CBDs indica strains are helpful to those who need sleeping aid and people wit insomnia.Whereas sativa which has a lot more of the chemical THC which gives the user more of a head high than a body high. This strain is prescribed to patients who have anxiety and depression. It gives the user more of a jubilant feeling without the feeling of being couch locked. The smoke of marijuana contains of a toxic mixture of gases and other particulates that are harmful to the lungs. The lungs of someone who smokes marijuana on a daily basis resembles those of a tobacco smoker. Marijuana has the potential to promote lung cancer because of th e carcinogens in the smoke.Nowadays most companies, businesses, and schools drug test for THC. This has caused people to move on to another drug even more dangerous and marijuana, spice. Spice is synthetic marijuana, made to give the user the same high as marijuana. Most people are not aware of what is actually in the spice and just smoke it because they cannot smoke marijuana. The most dangerous thing about spice is the fact that it is not regulated; anybody can make spice in his or her own privacy. Spraying chemicals such as JWH-018 on herbs and plants makes spice.When the government decides to make the chemical illegal producers tweak the chemical just a bit to create a whole knew molecule. Spice affects each user different; one smoker may a certain way and the other another way. Spice is a rarely new drug and scientist are still not certainly sure about how it affects the human body. A few minutes after smoking spice the user have side effects such as nausea/vomiting, severe par anoia, involuntary movements, hallucinations, and prolonged headaches lasting up to days after use.Some cases have shown that teenagers under the influence of spice are unable to speak or move; they are conscious but respond to normal situations in a weird way. User who smoke spice on a daily basis have a high risk of becoming addicted an addiction, which is similar to those of meth, cocaine, and opiates. The effects of spice usually wear off in about thirty minutes to an hour and this causing the user to want to keep smoking more and more. Scientists do not know the long-term effects of smoking spice; this makes it more dangerous because teens are smoking something that has the potential to harm their bodies.Opiates have been used as a pain reliever for over hundreds of generations. Opiates are the common name for any narcotic that was derived from opium. Painkillers such as morphine, codeine, hydrocodone, and oxycodone are all obtained from opium. Opiates the most additive an abus ive substances in the world today. Opiates obtain their powerful effects by attaching themselves in opiate receptors in the brain and body. The effects of opiates include extreme relaxation, euphoria, fatigue, confusion, decreased feeling of pain, and decreased sexual drive.While on opiates they cause the pupil of the eyes to dilate. Other effects include nausea and vomiting if too much of the opiate is taken. Opiates attach themselves to the neurotransmitters in the brain, which control body movement, moods, digestion, body temperature and breathing. They cause to the neurotransmitter to work at a very high rate. The short-term effects can show soon after a dose and lasting up to a few hours. Regular use of opiates leads to a higher tolerance, meaning the user needs more of the opiate to achieve the same effect as before.After a time of increasing tolerance the body becomes addicted to the drug, developing dependent on the opiate to function properly. Death from an opiate overdose usually occur when a user who has been off of opiates for some time again starts to take the same amount of the opiate as they are used to and because the user’s tolerance has gone done the result is an overdose. After repeated use of opiates, long-term effects soon appear. Most addicts who have been using for a long time seem to just ignore their health because they are only concerned about obtaining the opiate.Longtime users may develop collapsed veins, infection in their heart and valves, and liver disease. Due to the fact that opiates decrease reparation rate, pneumonia may occur in longtime users due to respiratory depression and the poor health of the user. When trying to stop the use of opiates, the withdrawal is very dangerous and painful. Withdrawl symptoms can occur as soon as a few hours after the last does. Symptoms include intense craving for the opiate, restlessness, body pain, insomnia, diarrhea, vomiting, and cold flashes.

Process Analysis Essay for Writing Papers - 648 Words

Process Analysis Essay When writing a paper it can be very difficult unless you break it up into sections. When I had to write my first paper I felt like a man on a desert island all alone without a clue on how to do anything. But with the help of a few teachers they taught me how to survive on the island of writing papers. What the teachers taught me was that just like everything in life it needs to be taken in steps. The steps they taught me still apply to the papers I write to this day. The first thing you need when you write a paper is a strong foundation. Everything must start at the bottom and be built up. This applies to everything. It all starts with an idea. So start your paper with an idea, write them all down and†¦show more content†¦This helps you organize and structure the paper. So you don’t put roof on before the walls are up. Now you are ready to start writing the paper. Make sure when you do start to be in a comfortable environment. Don’t go grab your laptop and sit-down in front of the TV. Go to your desk and put some music on then write the paper. It will help your the quality of your paper trust me. Now that you’ve started the construction of your paper make sure to use complete sentences and correct your grammar. There’s nothing worse than getting a B when you could’ve took then extra 5 minutes to look back though what you’ve just written. This will also help you with making the paper flow well. Make sure you have a complete thesis with support. After you are done with the paper take it to someone and have him or her inspect your paper for any mistakes you may have missed. The draft they are correction is called the rough draft. After they check your paper fix any mistakes they may have found. Repeat this process with a couple of friends or teachers and you should be ready for the final draft. 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Profile of The Beatles

The Beatles were an English rock group that  shaped not only music but also an entire generation. With 20 songs that hit #1 on Billboards Hot 100 chart, the Beatles had a large  number of ultra-popular songs, including Hey Jude, Cant Buy Me Love, Help!, and Hard Days Night. The Beatles  style and innovative music set the standard for all musicians to follow. Dates: 1957 -- 1970 Members: John Lennon, Paul McCartney, George Harrison, Ringo Starr (stage name of Richard Starkey) Also Known As Quarry Men, Johnny and the Moondogs, Silver Beetles, Beatals John and Paul Meet John Lennon and Paul McCartney first met on July 6, 1957, at a fete (fair) sponsored by St. Peters Parish Church in Woolton (a suburb of Liverpool), England. Although John was only 16, he had already formed a band called the Quarry Men, who were performing at the fete. Mutual friends introduced them after the show and Paul, who had just turned 15, wowed John with his guitar playing and ability to remember lyrics. Within a week of meeting, Paul had become part of the band. George, Stu, and Pete Join the Band In early 1958, Paul recognized talent in his friend George Harrison and the band asked him to join them. However, since John, Paul, and George all played guitars, they were still looking for someone to play bass guitar and/or the drums. In 1959, Stu Sutcliffe, an art student who couldnt play a lick, filled the position of bass guitarist and in 1960, Pete Best, who was popular with the girls, became the drummer. In the summer of 1960, the band was offered a two-month gig in Hamburg, Germany. Re-naming the Band It was also in 1960 that the Stu suggested a new name for the band. In honor of Buddy Hollys band, the Crickets—of whom Stu was a huge fan—he recommended the name of The Beetles. John changed the spelling of the name to Beatles as a pun for beat music, another name for rock n roll. In 1961, back in Hamburg, Stu quit the band and went back to studying art, so Paul took up the bass guitar. When the band (now only four members) returned to Liverpool, they had fans. The Beatles Sign a Record Contract In the fall of 1961, the Beatles signed a manager, Brian Epstein. Epstein succeeded in getting the band a record contract in March 1962. After hearing a few sample songs, George Martin, the producer, decided he liked the music but was even more enchanted with the boys witty humor. Martin signed the band to a one-year record contract but recommended a studio drummer for all recordings. John, Paul, and George used this as an excuse to fire Best and replace him with Ringo Starr. In September 1962, the Beatles recorded their first single. On one side of the record was the song Love Me Do and on the flip side, P.S. I Love You. Their first single was a success but it was their second, with the song Please Please Me, that made them their first number-one hit. By early 1963, their fame began to soar. After quickly recording a long album, the Beatles spent much of 1963 touring. The Beatles Go to America Although Beatlemania had overtaken Great Britain, the Beatles still had the challenge of the United States. Despite already having achieved one number-one hit in the U.S. and had been greeted by 5,000 screaming fans when they arrived at the New York airport, it was the Beatles February 9, 1964, appearance on The Ed Sullivan Show that ensured Beatlemania in America. Movies By 1964, the Beatles were making movies. Their first film, A Hard Days Night portrayed an average day in the life of the Beatles, most of which was running from chasing girls. The Beatles followed this with four additional movies: Help! (1965), Magical Mystery Tour (1967), Yellow Submarine (animated, 1968), and Let It Be (1970). The Beatles Start to Change By 1966, the Beatles were growing weary of their popularity. Plus, John caused an uproar when he was quoted as saying, Were more popular than Jesus now. The group, tired and worn out, decided to end their touring and solely record albums. About this same time, the Beatles began to shift to psychedelic influences. They started using marijuana and LSD and learning about Eastern thought. These influences shaped their Sgt. Pepper album. In August 1967, the Beatles received the terrible news of the sudden death of their manager, Brian Epstein, from an overdose. The Beatles never rebounded as a group after Epsteins death. The Beatles Break Up Many people blame Johns obsession with Yoko Ono and/or Pauls new love, Linda Eastman, as the reason for the bands break up. However, the band members had been growing apart for years. On August 20, 1969, the Beatles recorded together for the very last time and in 1970 the group officially dissolved. John, Paul, George, and Ringo went their separate ways. Unfortunately, John Lennons life was cut short when a deranged fan shot him on December 8, 1980. George Harrison died on November 29, 2001, from a long battle with throat cancer.