Fibonacci Sequence

Fibonacci Sequence How Does the Fibonacci Sequence Relate to Nature and Other Math Processes? Nature is all around us, and because I spend a lot of time outside I have been able to enjoy and observe all that nature has to offer. Due to the fact that I love science and discovering how everything around me functions and relates to everything else, I decided to investigate the relation that Fibonacci has with other math processes—as well as with the environment. I wanted to understand how plants know the best way to form their seeds or outer shell, and why some patterns may repeat in nature in different plants and organic materials. Thus, this exploration looks at two seemingly unrelated topics—Fibonacci and the golden ratio—both of which produce the same number, phi. While this could be mere coincidence, that possibility is negated when the fact that the number produced is irrational is introduced. It was this peculiar discovery, as well as the abundant appearances of Fibonacci in nature, that led me to choose this exploration topic. To begin, I should start by identifying what initially sparked my curiosity in this subject: a pinecone. As with many other plants, as well as fruits and vegetables, pinecones display the golden ratio. In order to better understand what I am talking about I have included a picture of a pinecone similar to the one that I first inspected. Labeled below is the noticeable spiral pattern on the pinecone. Counting the number of spirals in that direction produces the number eight, and in the other direction it produces the number thirteen while a third and tighter spiral produces twenty-one. These numbers are situational to the pinecone in the pictures, but the Fibonacci numbers as a whole are far more complex than they first appear to be. To understand the importance of these numbers it is crucial to understand the fundamentals of the Fibonacci sequence itself. The sequence usually begins with the numbers 1, 1, 2, 3, 5, 8, 13 and follows an easily definable pattern. 1, 1, 2, 3, 5, 8, 13 Start with the number 5, or the nth number in the sequence. We’ll call it n. 5 equals the two numbers before it added together: 2 + 3. Or, in broader terms, a number in the sequence is the sum of the two numbers preceding it. 1, 1, 2, 3, 5, 8, 13n = n-1 + n-2 An interesting idea comes up at the mention of this formula though. = This ratio just so happens to equal a number often notated as, or phi. > 1/11Phi is greater than one, < 2/12but less than two. > 3/21.5Phi is greater than three halves, < 5/31.666but less than five thirds. > 8/51.6Phi is greater than eight fifths, < 13/81.625but less than thirteen eights. 1.6180339988†¦ You’ll notice that each fraction listed above is made up of numbers from the original seven number sequence, in other words, each pair of Fibonacci numbers creates a ratio that gets closer and closer to phi as the numbers increase. This is better shown on a graph I created, displayed below. The ratio created by these sequences as they approach phi is called the golden ratio. The golden ratio, however, is not as important to this study as the lesser known concept of the golden angle. Below is a representation of the golden ratio in relation to the golden angle, the smaller portion of the circle notated using alpha, or ÃŽ ±. ÃŽ ± = 137.507764 ° 137.5 ° The reason this conversion is necessary is because the golden angle is present in the next discussion topic: sunflowers. Or, more specifically, their seeds. Sunflowers are another great example of the appearance of Fibonacci in nature, and also led me to an interesting discovery. In order to plot the distribution of a sunflower’s seeds we need an X and a Y coordinate pair. Using the square roots from an index numbered from one to one thousand and multiplying them by the cosine of the radian of the angle alpha gives us a formula to find x, dependent on the index number used. Y can be calculated with a very similar formula, using sine instead of cosine. The equations are listed in their entirety below. When these formulas are used and input into Microsoft Excel they produce a graph similar to the following. Wow! That graph bears a striking resemblance to the original Fibonacci spirals that appeared in the pinecones, and as mentioned earlier it is not mere coincidence. While the use of the golden ratio is apparent, there is another aspect of it that I wish to address, the golden spiral. Its formulae are given by the following equations, and are readily apparent in nature as well (nautilus shells for example). In these equations is the undetermined scaling factor and is the growth factor of the spiral. In the instance of the golden spiral, is equal to the operation below. At first, these formulae appeared to be a strange smattering of numbers, and one I didn’t understand at all. However, upon noticing the appearance of a natural log in the formula for I made a connection to the letter , better known as Euler’s number, that is present in both the X and Y formulae. After thorough searches of many sources I discovered another math process that bares resemblance to the above formulae. This is Euler’s formula. It becomes increasingly apparent that its resemblance is not coincidental when the formula is transformed into the final formula shown below. While the visual similarities may be obvious when the formula is displayed as it is above, the importance of each variable can be clarified with simple explanations. is the arbitrary scaling factor, responsible for determining the scale of the spiral. dictates the rotation of the spiral, and remains constant. The in dictates the growth of the spiral, and the dictates the speed—together representing the speed of the growth of the spiral. More simply put, any given ordered pair can be found by multiplying the growth of the spiral by its rotation (as shown in the originally given formulae for finding said coordinates.) What is produced, however, after inputting over two thousand pieces of data, derived from the coordinates calculated using the formulae above, into Microsoft Excel, is shown in the graph below. After putting in the Fibonacci squares (using the original golden ratio) into the spiral its appearance and relation to Fibonacci become even clearer. Very simply put, my investigation yielded the result that the Fibonacci sequence, the golden spiral, and Euler’s number are all related to one another in nature. The results are eye opening for me, as I am beginning to realize just how much of the world is made up of math—rather than my previous belief that everything natural occurred randomly. My exploration only stemmed into plants, and while that may only have practical use in fields such as botany, all three have great value in many fields. To begin with, Fibonacci appears in bee populations, proportions of the human body, formation of cells, and possibly more practically in code and the stock market. Any of these fields could present an interesting extension to my exploration, and because they all stem from Fibonacci they all have roots in combinatorics and number theory. The implications of this are staggering! Simply the thought that all of these vastly different fields are related to one another by one sequence of numbers discovered by Leonardo of Pisa, better known as Fibonacci himself, is baffling considering that he discovered them while looking at the breeding patterns of rabbits. There are so many other areas in nature that Fibonacci appears in, and I’m so excited that I have the opportunity to discover and study them now that I know more about them. Works Cited Azad, Kalid. Intuitive Understanding Of Euler’s Formula. Better Explained. N.p., n.d. Web. 23 Feb. 2015. . Nature by Numbers. Eterea. N.p., n.d. Web. 3 Feb. 2015. . â€Å"Spirals.† < http://faculty.smcm.edu/sgoldstine/pinecone/spirals2.gif> Wolverson, Tim. Plot a Fibonacci Spiral in Excel. Reviews and Guides. WordPress, 08 Feb. 2014. Web. Jan.-Feb. 2015. . McDonald 1

Essay on the Evils of Capitalism Exposed in Catch-22 -- Catch-22

Evils of Capitalism Exposed in Catch-22  Ã‚  Ã‚  Ã‚  Ã‚      "That's some catch, that Catch-22" (47). Some catch indeed, for Catch-22 "is the best there is" (47). A strange paradox preventing men from being grounded under any circumstances, Catch-22 eventually evolves into a justification for doing virtually anything. After all, it "says [anyone] can do anything [that] we can't stop them from doing" (416). A less obviously stated, but equally powerful, validation for one's actions is the guarantee of profit. "It [is] odd how many wrongs leaving money [seems] to right" (418), for the promise or presence of some form of profit, rights even the wrongs warranted by Catch-22. Milo Minderbinder takes full advantage of this powerful reasoning and uses it extremely well. Yet, rather than using it to right wrongs, Milo uses it to justify his own dastardly deeds. Therefore, throughout Catch-22, Milo's capitalistic greed leads him to be an emblem evil. Milo spends most of his time in the army traveling Europe, the Middle East, and Africa in search of the best deal. With the use of "donated army equipment" (239) he buys and sells various items in order to make the highest profit. Rather than fly missions, Milo seeks to make money, capitalizing on his time abroad. After all, Milo "didn't start this war...[he's] just trying to put it on a businesslike basis" (262). This attitude leads Milo to begin a syndicate, one in which "everybody has a share" (238-239). This proposed arrangement keeps everyone at ease, so much so it leads to general sloth. Because "everybody [has] a share, ...men [get] fat and [move] about tamely with toothpicks in their greasy lips" (259). One by one, the men succumb to the charms of plenty as well as to their internal greed... ...avoids and resists many things that bring him closer to death or to violating his morals. Be it Aarfy, Colonel Cathcart, or war itself, Yossarian distances himself from the evil in question. Ironically, defiant as he is, Yossarian fails to break through the limitations of the syndicate and actually befriends its proprietor. Yossarian doesn't even attempt to do otherwise, for even he "sagged back in a contented stupor, his mouth filmy with a succulent residue" (22); Yossarian had become one of the men who "got fat and moved about with toothpicks in their greasy lips" (259). Therefore, even he who refuses to conform to other standards is corrupted by the sweet blandishments and innocent appearance of capitalism. The novel thus ends on the note that no one remains free from so powerful a force. Works Cited Heller, Joseph. Catch-22. New York: Knopf, Inc., 1995.

Roots of Education

Have you ever thought of why such quotes becomes so special and popular? If one tries to notice, it can easily be seen that famous quotes are pithy that are capable of grabbing the interests of the readers. While writing a personal essay it is useful to link a thesis statement with a famous quote. Including one such quote that resembles your thesis sentence intrigues a writer to read the whole content. The other thing is, using quotes can make your thesis statement clear and adds value to it. A quote cannot be used as a thesis sentence instead; an individual should try to write meaningful concise sentences that can work as quotes. An original thesis statement leads towards many ideas that encompasses to other distinctive, but creative ideas. For example, here we are including the famous quote of Winston Churchill, who have said that â€Å"Success consists of going from failure to failure without loss of enthusiasm† Now, here we can see that it sets up an argument that is curt, but interesting, we can use following ideas that can work as a thesis statement and links with the quote too: -A failure can be overcome -Learn from your failure -Why everybody fails in a life time Now, these ideas can act as body paragraphs arguments, which can be followed up by thesis statement. It requires a brief sentence which can engage the sentences and can help making a quality thesis statement. One can easily find good quotes of the famous philosophers. It would help in developing a unique, interactive and informative thesis sentence writing to which a body can be easily constructed. A few are mentioned here that can be linked easily. â€Å"Attempt the impossible in order to improve your work† by Bette Davis â€Å"If you can find humour in anything, even poverty, you can survive it† by Bill Cosby

- Free Essay Example

Sample details Pages: 6 Words: 1780 Downloads: 1 Date added: 2017/06/26 Category Statistics Essay Did you like this example? Critically appraise the circumstances where an impairment loss is deemed to have occurred Introduction IAS 36 Impairment of assets published in 1998 and subsequently amended in 2004 and in 2008, seeks to ensure that an asset is not carried on the statement of financial position at a value that is greater that its recoverable amount. This paper aims to critically appraise the circumstances where an impairment loss is deemed to have occurred and explain when companies should perform an impairment review of assets, while discussing the effects of impairment decisions on the firms financial position and performance. . Don’t waste time! Our writers will create an original "" essay for you Create order Previously there was little authoritative guidance on the accounting for asset impairments. The absence of explicit guidance for many assets permitted substantial discretion in terms of amounts and timing of write offs (Francis et al, 1996). Over time accounting standards have moved towards presenting more items at fair value on the Balance Sheet. In doing so IAS 36 tries to remove as much discretion as possible. The primary objective of IAS 36 Impairment of Assets is to ensure that an entitys assets are carried at no more than their recoverable amount and the standard sets out the criteria for defining how the recoverable amount is determined. Entities are required to conduct impairment tests where there is an indication of impairment of an asset, with the exception of goodwill and certain intangible assets for which an annual impairment test is required. Intangible assets with an indefinite useful life, an intangible asset not yet available for use and acquired goodwill should all be measured annually whether or not there is any indication of impairment. Impairment is deemed to occur when the carrying amount is higher than the recoverable amount (i.e. the value in use. the assets net selling price or the fair value as determined in accordance with IFRS 13). At the end of each reporting period an entity is required to assess whether there is any indication of impairment. If an indication of impairment is evident then the assets recoverable amount must be calculated [IAS 36.9]. An impairment loss is recognised where the recoverable amount is below the carrying amount [IAS 36.59]. The impairment loss should be immediately recognised, generally as an expense unless it relates to a revalued asset where the impairment loss is treated as a revaluation decrease [IAS 36.60]. In the case of goodwill, a cash-generating unit to which goodwill has been allocated shall be tested for impairment at least annually by comparing the carrying amount of the unit, including the goodwill, with the recoverable amount of the unit: [IAS 36.90] In each situati on, if the carrying amount of the unit exceeds the recoverable amount of the unit, the entity must recognise an impairment loss. This is a radical change in accounting for goodwill. Previously, International Accounting Standards required recognition of Goodwill subject to amortisation over its useful lifetime. Indicators of impairment are set out in IAS 36 with a view to making the decision less subjective than previously was the case. Negative changes in technology, markets, economy and law could all have adverse impact on the value of an entitys assets. Indicators of impairment could be as a result of internal or external sources. The market value of an asset may decline as a result of usage or the passage of time. Other external indicators of a decline in value could be the result of significant technological, market, economic, or legal changes which occur and have an adverse effect on the asset or entity. Market interest rates may impact the discount rate used in calculating the value in use of an asset and therefore decreasing its recoverable amount. Internal indicators of an impairment review could be the result of obsolete or physically damage assets, or if an asset is part of a restructure or held for sale, or where the economic performance of an asset is worse than expected. Despite the standard being objectively set, it can be difficult in determining the measurement of value attributable in assessing impairment options. The timings and measurement of asset write-downs rely heavily on estimates. A number of features of impairment testing and measurement process make implementation a challenge. Triggering events to indicate impairment are many and vary greatly in significance and severity. Different valuation models are used and there is little conformity in the selection of discount rates. (Comiskey and Mulford, 2010). A difference in nature continues to exist between fair values disclosed by management. While the standard seeks to increase transparency and eliminate the subjectivity of accounting for impairments, the exercise for determining if an asset is impaired and by how much remains at managements discretion. It was felt that previously management took advantage of the discretion afforded by accounting rules to manipulate earnings either by not r ecognising impairment when it has occurred or by recognising it only when it is advantages (to them) to do so (Francis et al, 1996). The standard now seeks to address this discretion by requiring annual impairment (Goodwill and intangibles) or impairment reviews to be carried out whenever there is an indication of impairment. Even still, there is an element of discretion afforded to the calculation of impairments and so management incentives to manage earnings can still play a part in any impairment decisions. These estimates might be managed to alter or avoid impairments, limiting the comparability across firms. A goodwill impairment loss, for example, is estimated in most cases from managements projections of future cash flows (Z Li et al, 2011). This is problematic to the investors who are unable to see through these potential manipulations. Indeed, investors and analysts have the option to adjust, or indeed totally ignore, reported accounting numbers, therefore it is far less ce rtain whether this reporting behaviour actually misleads users or reduces reliability and relevance (Lhaopadchan, 2010). Additionally, financial statements differ from the management accounts used by an entity and the effect of any impairment further widens a gap already existing between management information accounts used by the board and the financial statements audited and published. Earnings manipulation is one such concern given the judgemental approach to the indication of and calculation of impairment. One of the most widely cited papers that investigate the effect of executive compensation plan on accounting choice is Healy (1985). Healy hypothesizes that managers have an economic incentive to manipulate earnings in order to increase their cash compensation, this being the case certain accounting standards allow for this more than others and IAS 36 still allows for an element of judgement in the calculation of impairments. Furthermore, papers have cited the nature behind recognition (or lack of recognition) of impairments and IAS 36 permits an impairment loss on a long lived asset to be reversed if the economic value of the asset recovers. This has been seen to have a direct impact on the practice of impairments whereby reversibility has a positive effect on a managers decision to record asset impairments. Permitting reversals significantly increases the lik elihood that a manager will record the impairment (Trottier 2013) thus highlighting the discretion that management can withhold towards the treatment of impairments. Volatile financial markets and shifting economic conditions can impact the value of a companys assets across the Balance Sheet. The recent global meltdown of financial markets was accompanied by highly publicised asset write-downs (Spear and Taylor 2011) and so the standard seeks to address the transparency of the financial statements by ensuring that impairments are directly reflected through the profit and loss account and statement of comprehensive income, disclosed by class of asset. It is not surprising that the most frequent write down activities took place during periods of economic recession confirming the strong relationship between asset write-downs and economic conditions. In 2013 the consolidated results of PSA Peugeot Citroen saw a à ¢Ã¢â‚¬Å¡Ã‚ ¬1,101 million impairment charge, mainly recognised with respect to the assets of the Automotive Division, primarily to reflect the deteriorating automobile markets and adverse exchange rate movements in Russia and Latin America. Additionally, in 2014 Vodafones end of year profits dropped after a  £6.6bn impairment relating to the value of European operations whereby lower than expected cash flows were the result of a tougher macroeconomic environment and heavy price competition contributing to a total decline in revenues. Both investors and financial analysts revise their expectations downward on the announcement of an impairment loss. The negative impact of the loss serves as a leading indicator of a decline in the future profitability of an entity. (Z Li et al, 2011). Conclusion In conclusion, despite the presumed benefits associated with Fair Value accounting, it is shown that in practice managerial self-interests and earnings management concerns appear to motivate many impairment decisions (Lhaopadchan, 2010). IAS 36 goes further than any previous standard and subsequent amendments to eliminate any subjectivity involved in highlighting and calculating an impairment loss. While goodwill should be assessed annually for impairment other potentially impaired assets are only reviewed in detail for impairment if there is an indication of impairment, some of which are highlighted by the standard itself, however the indicator of impairment could go unidentified resulting in misleading financial statements. Additionally many calculations of impairment use management projections which could include error or contain an element of managerial self-interest and manipulation. Generally speaking the reaction of market participants to any impairment disclosed in the financ ial statements is of a negative nature with the exception of restructuring costs for which highlight future spend. While the standard seeks to provide a truer and fairer representation of asset value it should be noted with caution the subjective nature of any calculations. Even with an unqualified audit report on the financial statements the audit opinion on impairment is only as good as the information provided and made available to the external auditors. Hence, it can be concluded that IAS 36 Impairment of assets has come far to contribute to improve the transparency of the financial statements by successfully determining when and how impairment reviews should be conducted, however there will remain an element of managerial judgement for which caution should be taken by all users of the financial statements. References Causes and effects of discretionary asset write-off, Francis, J.. J. D. Hanna, and L. Vincent. Journal of Accounting Research Volume 34, 1996. Goodwill, Triggering Events and Impairment Accounting, Eugene E Comiskey; Charles W. Mulford. Managerial Finance, Volume 36 (9): 22 à ¢Ã¢â€š ¬Ã¢â‚¬Å" August 10 2010 Causes and Consequences of Goodwill Impairment Losses Z Li et al. Review of Accounting Studies, Volume 16 (4) à ¢Ã¢â€š ¬Ã¢â‚¬Å" Dec 1, 2011 à ¢Ã¢â€š ¬Ã¢â‚¬Å" May 11 à ¢Ã¢â€š ¬Ã¢â‚¬Å" 2010. Fair Value Accounting and Intangible Assets. Lhaopadchan. Journal of Financial Regulation and Compliance, Volume 18 (2) The Effect of Reversibility on a Managers Decision to Record Asset Impairments, Trottier, Kim. Journal Accounting Perspectives , Volume 12 (1) à ¢Ã¢â€š ¬Ã¢â‚¬Å" Mar 1, 2013 The Effect/Decisions of Bonus Schemes on Accounting Choices, Healy, P M. Journal of Accounting and Economics Volume 7 (1), 1985 Asset Write Downs: Evidence from 2001-2008, Spear, Nasser A; Taylor, Alexandra M. Australian Accounting Review, Volume 21 (1) à ¢Ã¢â€š ¬Ã¢â‚¬Å" Mar 1, 2011