The Concept of Majority Rule Assignment Example | Topics and Well Written Essays - 500 words

The Concept of Majority Rule - Assignment Example The relationship between the two is that the principles of majority rule and minority rights identically support an egalitarian form of government in the United States of America. An example of this is the â€Å"Title VII of the Civil Rights Act of 1964 which make illegal employment discrimination on the grounds of race, color, religion, sex, or national origin. It applies to all colleges and universities, public or private that have at least fifteen employees. The U.S. Constitution prohibits state employment in violation of the equal protection of the laws and applies to all public institutions of higher education. Title VII and the Constitution jointly establish what is required and permitted with respect to affirmative action in higher education† (The U.S. Equal Employment Opportunity Commission, 1964). Another example which exhibits the duality of the constitution is the No Child Left Behind Act of the NCLB, a constitutional provision ensuring the right of all children to be studying in school (Henley, Mcbride, Milligan, and Nichols, 2007). Practically, this constitutional provision promotes the need for every citizen of the United States of America to gain the appropriate knowledge for wisdom in any chosen field. This must also be inconsistent with the current global trend towards service industry.

Marketing Essay Example | Topics and Well Written Essays - 2000 words - 21

Marketing - Essay Example In this paper, we will define this product category as clearly as possible and we will then examine it thoroughly. As mentioned in the introduction section, the product category chosen is apparel. Apparel is clothing in general. This could range from t-shirts to dresses to cardigans to skirts to denim wear. Some people may even include under garments in this category but for our purpose, we will only consider outer garments or attire. We will later consider companies who make these, other products and product categories that compete with these also. Outer garments or clothing may range from casual, inexpensive t-shirts to sophisticated, costly and maybe even branded suits. Companies who sell (and/or make) casual clothing include Abercrombie and Fitch, Gap, Guess, Old Navy etc. Companies or designers who sell expensive branded clothing are Armani, Prada, Chanel, etc. There are several others that sell both types of clothing or only one of them that are not mentioned above. The point that I am trying to make is that clothes as a product category is not singular. It has many different branches, as mentioned above. We will mainly consider casual apparel retail products. Another way of further categorizing the clothes category is in terms of gender and also age. For example, some companies provide apparel for women only such as Ann Taylor. Others provide clothing for only men or most retailers sell both men and women apparel. Another sub-category of the apparel product category is clothing in terms of age, for example, Toddlers, Children, Teenagers. There are retailers that provide one of them or all of them. Some retailers may specialize in providing only apparel for new born babies, toddlers and pregnant women (maternity apparel) such as Mother Care. There are many different retailers in the apparel retail industry in the US. According to a report by Nielsen Net Ratings,

Compare contrast and analyze the coastal management and its effect on Research Paper

Compare contrast and analyze the coastal management and its effect on the population in both ghana and South Africa - Research Paper Example The destructive activities performed by individual lead to destroying of the coastline resources of the countries. In this paper, the coastal management systems of South Africa and Ghana have been discussed precisely by considering all the significant facts about its coastal areas. The reasons for the destruction of the coastal areas of Ghana and South Africa have been explained and how those have impacted on the people and business located there, are also discussed. 2.0 Coastal Management Coastal management comprises of the coastal resource management, coastal zone management and integrated coastal zone management that relates to formal or informal practices, economies, rules, technologies as well as interaction among natural resources and humans who are located near sea coast. The coastal management also includes the way through which the resources are utilized and protected1. In the coastal management process, there are five phases such as identification of issue as well as evalua tion of baseline. The second phase includes the preparation as well as adoption of coastal resource management, in the third phase there is an action plan along with implementation of project. The fourth phase is the monitoring and evaluation, and the last phase comprises of information management, outreach and education1. The term ‘Coastal Management’ came into being after United States Coastal Zone Management Act of 1972 was implemented. According to this act, sectoral management approach was not operating effectively, that was developed to focus on the individual resources such as fisheries, and other activities such as transport. Then the idea came to develop a coastal management approach with definite plans that primarily focus on particular issues related to national along with local importance. From that period new coastal management approach was applied in various situations all around the world2. The activities of coastal management takes place both in the dire ct land-sea border as well as in wider area of indirect influence of sea on land as well as land on sea. It also involves organizing the production of intertidal organisms and co-ordination of various sectors along with management agencies that have a direct along with indirect influence on the coast. The management of coast facilitates synchronization between the sectors and agencies. It also helps to develop the capability for integration of coastal consideration to the particular work2. Integrated Coastal Management (ICM) is considered as inter as well as trans-disciplinary approach of management, which was first initiated from the year 1970. The main objective was settlement of the conflicts of coastal stakeholders and to attain sustainable coastal development. Integrated coastal management is referred as the process by which several marine as well as coastal environments are managed in order to provide broad range of needs of bio-diversity protection along with sustainable use by facilitating the stakeholders to participate in the management programs3. MPA (Marine Protected Areas) is considered as an essential component of integrated coastal management program. They help in the protection of biodiversity as well as ecological

Logistics within the Business Organization Coursework

Logistics within the Business Organization - Coursework Example One of the most important strategies that can be used in encouraging management to pay closer attention to the supply chain management is to call the attention of senior management to the effective â€Å"sales† coming out of their competition and other business models that utilize a concentration on the supply chain. According to Dutton, this is a highly effective strategy because it demonstrates the better supply/logistics processes or use of technology that can result in reduced costs for inventory items. It is best that the manager knows how to frame his thoughts and words in such a manner that will make the supply and logistics discussion an interesting and important topic for the managers. Rudski agrees with the observations and suggestions of Dutton. But he adds that the supply management must be able to favorably impact 5 areas of supply management. These include: (1) exceeding performance and earnings expectations, (2)growing revenues year on year, (3) reducing risk an d the  volatility in revenues and earnings, (4) continuously improve the return on investment capital and return on equity, and finally (5) will it create a unique business model that others cannot copy? Furthermore, Dutton calls attention to some additional tips that can help interest business managers in the demands of supply chain management.

The Past and Present of Ethical Cultures Essay Example | Topics and Well Written Essays - 1000 words

The Past and Present of Ethical Cultures - Essay Example Therefore, Martin Luther King finds a reason to break laws as long as he believes the laws are unjust according to what his conscience tells him. As such, he was willing to break laws as long as he felt that the laws were unjust to his morality and ethical standing, and by breaking such laws, he felt that that was the highest level of respect for the law altogether (Redner 2001, 191). Therefore, as much as Martin Luther King was expected to respect the laws of the land, he also found it his moral duty to seek justice by breaking an unjust law, thereby finding a reason to be willing to break the law. Martin Luther King’s arguments for being willing to break the law calls for the willing individual to be ready to willingly take up the corresponding penalty and carry it to its maturity. This means that breaking a law does not just end there, but incorporates the ability to be able to be responsible for your actions to the level of going through the penal system should the need arise (Shah 2007, p. 21). Consequently, this highlights the full depth of what willingness to break the law entails. J.A. Boss speculated that no one could rightfully accuse David Duke of being insincere since there were technically no laws that could be used in support of such an accusation. This was at a time when discriminatory laws were the order of the day and most instances of injustice, particularly racial, went unchecked. In such a scenario, it was increasingly difficult to counter such instances, as it would basically necessitate the changing of the constitution and a variety of laws to support a counter accusation. In addition, it would require drumming up the support of a significant number of individuals, and this was just not an easy feat to pull through.  

Barriers to entry into foreign markets Essay Example for Free

Barriers to entry into foreign markets Essay It is completed near the end of your first year of entry into the country market. One must identify and prepare for Trade Events. Trade shows, international buyer programs, matchmaker trade delegations programs or a catalog exhibition program can lead to tremendous international opportunities . Methods of foreign market entry Methods of foreign market entry include exporting, licensing, joint venture and off-shore production. The method you choose will depend on a variety of factors including the nature of your particular product or service and the conditions for market penetration which exist in the foreign target market . Exporting can be accomplished by selling your product or service directly to a foreign firm, or indirectly, through the use of an export intermediary, such as a commissioned agent, an export management or trading company. International joint ventures can be a very effective means of market entry. Joint ventures overseas are often accomplished by licensing or off-shore production. Licensing involves a contractual agreement whereby you assign the rights to distribute or manufacture your product or service to a foreign company. Off-shore production requires either setting up your own facility or sub-contracting the manufacturing of your product to an assembly operator. Barriers to entry into foreign markets The main trade barriers to any foreign market include: Psychological barriers in foreign exchange markets Traders adjust their anchors in two ways. Some believe that exchange rates move toward (perceived) fundamentals, while others bet on a continuation of the current exchange rate trend. The behavior of the traders causes complex dynamics. Since the exchange rate tends to circle around its perceived fundamental value, the foreign exchange market is persistently misaligned. Central authorities have the opportunity to reduce such distortions by pushing the exchange rate to less biased anchors, but to achieve this; they have to break psychological barriers between anchors. High import tariffs inclusive of restrictions related to national security Tariffs are taxes that raise the price of a good when it is brought into another country. Tariffs and import quotas form the toughest barriers. Seventy percent of respondents say tariffs on goods and services are the most effective form of protectionism, followed closely by import quotas (68%). But this is by no means the whole story: 45% say that artificially undervalued exchange rates do much to boost the competitiveness of local firms, while 59% cite subsidized competitors as a major barrier. Many also noted the challenges of informal protectionism, such as local firms convincing government officials to block the approval of licenses. Quota systems in Japan: The tariff quota system charges a lower duty rate (primary duty rate) on imports of specific goods up to a certain quantity, but a higher duty rate (secondary duty rate) on quantities exceeding that volume. This system protects domestic producers of similar goods but also benefits consumers with the lowest tariff rates possible. The tariff quota volume for each allocation can be applied in one of two ways: according to the order in which the request was received, or according to prior allocations. Japan utilizes the prior allocation method. The tariff quota system does not restrict direct imports, since imports can be made without a tariff quota certificate, provided high duty is paid. Regarding footwear, quota allocations to individuals or companies are based on historical trade performance in the importation of footwear. Japan has allocated quota not to quota traders but to footwear importers, so business can take place as per footwear importers requirements. At the same time, new importers can acquire special quota for new importers. The Government of Japan implements this system in accordance with governmental regulation. Therefore, Japan believes that new importers have opportunities to obtain quotas under the current quota allocation system. Unfavorable foreign rules regulations Voluntary export restraints limit the quantity of a good brought into a country, but they are initiated by the country producing the good, not the country receiving the good. Federal, state, and local governments sometimes restrict entry into markets by requiring firms to have licenses. The Federal Communications Commission, for example, grants licenses to radio and television stations; there simply arent enough frequencies for an unlimited number of firms to broadcast in any area. For safety reasons, all nuclear power plants are licensed as well. Governments also bar entry by giving firms exclusive rights to a market. The U. S. Postal Service, for example, has an exclusive right to deliver first class mail. Firms are sometimes given exclusive rights to do things like operate gas stations along toll roads, produce electricity, or collect garbage in a city. Exclusive rights are granted if a government believes that there is room for only one firm in a market. Until the 1980s, the federal government also restricted entry into the airline, trucking, banking, and telecommunications industries. Many of the laws that restricted entry into these industries were put into place in the 1930s, when many people believed that large firms needed to be protected from cutthroat competitors. Many economists now believe that these laws did more harm than good. In 1938, for example, the Civil Aeronautics Board, or CAB, was established to regulate the airline industry for interstate flights. For the forty years that it existed, it didnt allow a single new firm to enter the market, although it received over 150 applications for routes. In 1978, despite protests from the airlines, President Carter ordered the deregulation of the industry and the phasing out of the CAB. Within five years, 14 new firms entered the industry. Many experts believe that airline fares after deregulation were well below what they would have been had regulation continued. For instance, take China as an example. China’s government has set policies that are posing great challenges for foreign investors. China’s regulatory framework for cross-border remains a complex and incomplete patchwork of laws, regulations and policy decisions made by various ministries and government agencies. A lack of transparency, coupled with low standards of corporate transparency and disclosure, makes it difficult for potential investors to carry out due diligence to accepted international standards. Valuing the potential liabilities of a firm is especially difficult. At the same time, the Chinese government continues to close off so-called â€Å"strategic assets† to cross-border without specifying which sectors are defined as strategic, or why. To address these issues and remove other obstacles to cross-border deals, it is recommended for China to: †¢ Streamline the approval process for cross-border and make it more transparent; †¢ Put in place a sound competition framework; †¢ Further open its capital markets to foreign investors; †¢ Encourage its firms to increase corporate transparency and provide more up to date and accurate financial information to make it easier to value a potential acquisition, especially regarding a firm’s liabilities; Relax foreign ownership restrictions. In particular, revise existing catalogues that list the type of firms that can or cannot be acquired by foreign investors. The report also recommends that China pilot these recommendations in the North-East of the country before rolling them out nationwide. This region, China’s historical industrial heartland, has a high concentration of state-owned firms in need of restructuring and technological upgrading, as well as high unemployment and low productivity. Cross-border could help rejuvenate the region’s economy. Free Trade Policy Policy in which a government does not discriminate against imports or interfere with exports. A free-trade policy does not necessarily imply that the government abandons all control and taxation of imports and exports, but rather that it refrains from actions specifically designed to hinder international trade, such as tariff barriers, currency restrictions, and import quotas. The theoretical case for free trade is based on Adam Smiths argument that the division of labor among countries leads to specialization, greater efficiency, and higher aggregate production. The way to foster such a division of labor, Smith believed, is to allow nations to make and sell whatever products can compete successfully in an international market.

Fibonacci And The Golden Ratio Mathematics Essay

Fibonacci And The Golden Ratio Mathematics Essay Some aspects of mathematics can be dull and tedious from start to end, much of it however is intriguing and inspiring, when you truly see the beauty and the relevance. This is why I would like to bring to your attention the magic of the Fibonacci numbers. If you have ever looked at a sheet of paper and wondered Why do we use those dimensions? or looked at the leaf or an attractive plant and wondered Why can I never find a four leaved clover? then this may be of some interest. Many of these things are quite interconnected in a way you would not realise, and most of them are connected by the Fibonacci sequence. If I return to one of my original questions Why can I never find a four leaved clover? it seems reasonable, that if you can find 3 leaved clover and 5 leaved clover, you would be able to find the more symmetrical 4 leaved clover. Why then is it so rare to find one? If we look closely at other examples of nature, we can perhaps find the answer. If you were to search through your average garden, you would find the majority of flowers have 5 petals, many have 3 or 8 or more but if you look closely, you will always find more of certain numbers, compared to others. These numbers just so happen to be part of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 Although, why does nature choose these numbers over others? In addition, the connection between the real world and this sequence does not just end there; it can be found almost everywhere we look: spirals on a snail shell, the core of an apple, geometry, art, architecture, the stock market and even the human body. So what makes it so useful? Why is it so special? My project intends to answer these questions and along the way discover new applications and more examples. I will be delving into the mathematical concepts behind the nature we see every day, the regular objects we rely on, the human body and the stock market. I shall also investigate aspects of the golden ratio and how the Fibonacci sequence is related to this. The Fibonacci sequence is found by adding the previous term to the term before that. For example: 0, 1, 1, 2, ? 0 +1=1 1+1=2 1+2=3 and so on.. Overall equation for next term: a_(n+1)= a_n+ a_(n-1) This creates an infinite sequence of numbers and is known as a recursive sequence, as each number is a function of the previous two. Also, as the sequence progresses the ratio between each consecutive term seems to converge upon a single number. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 2/1=2 3/2=1.5 5/3=1.667 8/5=1.6 13/8=1.625 21/13=1.615 F=1.618034 Eventually, it converges to 1.618034 This number has a specific interest to many mathematicians and is known as the golden ratio. It is also useful when we consider where it is found. If you were to take your hand and bend the index finger as full as possible, measuring the dimensions of the rectangle created, you would find what is known as a golden rectangle. The average height (of the intermediate phalange) would be around 3cm and the average length (of the proximal phalange) would be 5cm. As we can see from left this creates a shape of ratio 5:3 or simply 1.667:1 (the golden ratio). This is only one of the many examples of golden ratio in the body. There are many, many more some of which have been known for hundreds of years (see Da Vinci s Vitruvian man right). Also, the golden ratio is not just confined to the human body. Rather than cutting and apple from pole to pole, if you were to slice in a horizontal fashion, you would find a simple five pointed star. However, it is much more complex than meets the eye. If you were to take the distance AB as 1 unit, the distance AC would be 1.618, the golden ratio. But why does this happen, what make this ratio so efficient and so appealing, and why has nature adopted it? History of the Sequence and Ratio From the start of the Palaeozoic era, 400 million years ago, animals of divine proportions have been roaming the earth. The most notable is the nautilus shell (right) which follows a logarithmic spiral based on the golden ratio in rectangles. The earliest written documentation of a special ratio belongs to the Rhind papyrus. A scroll about 6 metres long and 1/3 of a metre wide, it is one of the first mathematical handbooks. It was discovered by Scottish Egyptologist Henry Rhind in 1858 and is believed to have been written by Egyptian scribe, Ahmes in 1650 BC. He is believed to have copied it, from a document 200 years older, dating the first notation of the sequence to 1850 BC. However, the pyramids, built 1000 years previous, show many examples of the use of golden ratio, although many scholars believe it is merely coincidence created by the need for right angles. Between the 6th and 3rd centuries, Greek philosophers, mathematicians and artists used and analysed the golden ratio. It is visible in pentagons and pentagrams throughout the period and was attributed to Pythagoras and his followers. It was used as part of his symbol (a pentagram with a pentagon within) and it was he, who first suspected the proportion was the basis of the human figure. Plato also studied the ratio naming it most blinding of mathematical relations, the key to the physics of the cosmos. and from his lectures so did Eudoxus, whose work was used by Euclid in his book of elements II. Here he writes one of the first definitions A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. During his work he creates problems based on the ratio in pentagons, equilateral triangles and some of his prepositions show the ratio to be an irrational number. The first person to apply numbers and sequence to construct the golden ratio was Leonardo of Pisa (full name, Leonardo Pisano Bigollo, lived 1180-1250). He was the son of an Italian businessman from the city of Pisa and grew up within a trading colony in North Africa. At the time, Italy and the majority of Europe was using the Roman numeral system of counting, this was quite complex and meant most calculations required an abacus. While growing up in Algeria he learned the Hindu-Arabic system of calculation (the familiar 0, 1, 2). After returning to Pisa as a young man in the thirteenth century, he recognised the superiority of this new structure and began to spread it throughout Europe. He did this through his book the Liber Abaci (book of abacus) published in 1202 under the nickname, Fibonacci (a contraction of filius Bonacci, meaning son of Bonacci). To explain the system he used the Fibonacci sequence in his famous immortal rabbits problem (see next section of more detail). This allowed him to explain addition, subtraction and division using the Hindu- Arabic system and in turn allowed him to popularise it through Western Europe. Due to this he was later known as the founder of western mathematics and the greatest European mathematician of the middle ages. He introduced concepts such as algebra, geometry, the common fraction and even the square root symbol. He also considered the possibility of negative numbers and related it to merchant problems which began with a debt. There was very little significant work done upon the topic until 1509, when Luca Pacioli published De Divina Proportione with the help of illustrations by Leonardo Da Vinci, who later used this within his famous work the Vitruvian man . In 1611, German astronomer Johann Kepler discovered the numbers within his own work on planetary motion saying as 5 is to 8, so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost in relation to the rings around Saturn. It was later found that the ratio of mean distance between planets was in fact the golden ratio. Over the next two centuries many scholars investigated the sequence, deriving formulas and functions. In 1830, A. Braun first applied the sequence to the arrangement of bracts on a pinecone. A decade later and J.P.M. Binet derived a formula for the value of any Fibonacci number without the need for the previous two. nth number= 1/(v5) ((1+v5)/2)^n- 1/(v5) ((1-v5)/2)^n In 1920, Oxford Botanist A.H Church discovered spirals on sunflower heads corresponded to the numbers in Fibonacci s rabbit problem (see next section). This discovery inspired botanists to look for Fibonacci numbers elsewhere, teams then began to realise that many phyllotactic ratio s are golden ratio s (see flower patterns and primorda). In the 1930 s, Joseph Schillinger consciously composed a piece of music using Fibonacci intervals and Ralph Elliot began predicting the stock market in Fibonacci periods. By the 1960 s, a lively interest had been aroused and to this day mathematicians around the world are investigating the uses and problems connected with the sequence. The Immortal Rabbits Problem To explain his mathematical theorems, Fibonacci liked to create problems to allow his audience to use the maths he tried to describe. The immortal rabbits problem is one such challenge. It was first described within his famous Liber abaci and was later adopted as an explanation for the Fibonacci sequence. Imagine if you will a large enclosure and within it a pair of rabbits. The immortal rabbit problem asks if there is one pair to begin with, how many rabbits will there be after a certain length of time if: Each rabbit is immortal They stay within their pairs They breed once per month and produce a pair each time Each new pair takes 1 month to mature, and then breeds to form a new pair the next month January, we start with 2 rabbits, these then take one month to breed.. February, there is now one adult pair and a new born pair of immature rabbits. March, the new born pair have now matured, and the adult pair have reproduced April, the new born pair from March have now developed, the first pair reproduce again and the second pair reproduce for the first time.. The pattern continues until Month Pairs of mature rabbits Pairs of immature rabbits Overall Number of Pairs January 1 0 1 February 1 1 2 March 2 1 3 April 3 2 5 May 5 3 8 June 8 5 13 July 13 8 21 August 21 13 34 September 34 21 55 October 55 34 89 November 89 55 144 December 144 89 233 After a while, we begin to notice a pattern, the total number of rabbits in any given month is a Fibonacci number. This is because the total is formed from the number of immature rabbits (the same as the number of mature rabbits the last month) and the number of mature rabbits (the total from the previous month) i.e. a_(n+1)= a_n+ a_(n-1) Another interesting note is the rate of growth in the population. 2/1 = 2 3/2= 1.5 5/3= 1.666 8/3= 1.625 . this continues until we reach a_(n+1)/a_n =1.618034.. i.e. the Golden Ratio. Flower patterns and primorda As we have seen in the introduction, nature has applied the Fibonacci sequence and golden ratio from the number of petals on a flower, to the core of an apple and the spirals of a sunflower. On the face of it, this seems to be a fortunate and appealing coincidence, but since the 1920 s botanist have searched and found more and more of these coincidences . This leads us to believe that perhaps, they have a much deeper and more interesting meaning for the life of your average plant. Maybe these numbers and ratios were chosen for a reason. Even from Egyptian times it was noted that most flowers had 5 petals, the rest by majority also have Fibonacci numbers of petals. Also, if you examine the many plant stems you will find the regular pattern or 1, 2, 3, 5, 8 stems at standard heights. Another interesting phenomenon, and one which may reveal the mystery of why plants behave so regularly in conjunction with the Fibonacci sequence, are the spirals shown by plants. Look carefully at the picture of the pineapple left. As you move from the top right to the bottom left you may begin to see a set of spirals, curving round the pineapple in a diagonal fashion. Upon closer inspection you may also find a similar on from top left to bottom right and less obvious, from top to bottom. If we count the number of spirals we (fortunately for this topic) seem to find only Fibonacci numbers. In fact in a study of over 2000 pineapples not a single on differed from the pattern. The same principle applies to the pinecone. Upon close inspection, you will find two different spirals, one vertically and another horizontally, all of which come in Fibonacci numbers. A separate study to that of the pineapples showed that this was the case 99% of the time. The sunflower however, has its own unique spiral display. Starting from the centre and continuing in a clockwise fashion to the outside, the number of spirals again adds to a Fibonacci number. Although, if you look in the opposite (anticlockwise) direction you will find yet another spiral and adding the number of these gives the consecutive Fibonacci number. The majority of the time this is the case, however from time to time there are variations; with larger sunflowers the number of spirals can be double Fibonacci numbers (i.e. 2, 4, 6, 10, 16, 26.). These spirals may be interesting and attractive to look at, but hold much more value than just aesthetics; they allow us to show just why Fibonacci numbers are so widely used in nature and give us an insight into how nature uses maths at its very core. To understand the maths behind the growth of plants we must look deep into the way it grows. As the plant grows taller the interesting components (i.e. petals, sepals, stamens, leaves) all grow from small clumps of tissue called primorda. As these begin to grow they aim to have the largest distance between leaves as possible, this means they have the maximum amount of space and light to grow, ultimately making the plant stronger and more likely to survive. This distance has been decided through evolution to allow the maximum about of light to hit the plant and it turns out this maximum point of efficiently is related to the golden ratio. It just so happens that the Golden angle is the angle one golden ratio away from the starting position. 360 1.618.. 582.5 i.e. 582.5 -360 = 222.5 away clockwise (or 137.5 anticlockwise). As they grow at their angles the leaves have enough light and space to grow. However, when the 6th leaf begin to grow the angle means it is only 32.5 from the first, this leaves it in the shade meaning it is less likely to grow and develop; this is the reason many plants use the number 5 in some areas (i.e. in the number of petals) as the 6th would have less room and is less likely to grow. Sometimes called the phyllotactic ratio, the connection between this and efficiency in plants does not just end there. If we take ourselves back to the sunflower and its spirals we can see that this also has connections to the same ratio. As it begins to grown from the centre outwards each primorda (and therefore each seed head) grows on golden angle away from the previous. As the ratios between consecutive Fibonacci numbers are approximations to the golden ratio (and therefore used to create approximations to the golden angle) we begin to see them within the spirals. This is the main reason Fibonacci numbers are present in so many places; they form the best approximations of the golden ratio. Although, the actual number of spirals that arises depends upon the size of the seed head and slight variations in the rate at which the primorda migrate away from the tip of the growing shoot. As we saw from the rotations in plant leaves above, the golden angle is used to give the most space and therefore the most light. In the seed head however this is not a problem so why has evolution adapted to use it? The answer to this was first discovered by Professor H. Vogel in 1979. He noticed that using the golden angle allowed the seed head to pack together with hardly any missing space. This meant it was very efficient as more seeds could fit in a small area and also much stronger. In turn it meant there would be more seeds and better chance of offspring. This was later supported by French physicists Yves Couder and Stephan Douady, who found the choice of angle the natural consequence of the dynamics of growing a plant shoot . They stated that each primorda gets pushed into the largest available space, so they pack more efficiently, making the golden angle the most likely choice. They also discovered that the next best choice for packing an angle created by a second very similar sequence called the anomalous series (4, 7, 11, 18, 29). After inspection of more spirals and more plant this was found to be the 2nd most common choice after the Fibonacci sequence. Overall, nature has evolved and adapted to use Fibonacci numbers and the golden ratio they approximate, as it gives the most efficient method for survival. Over the years this had been pondered by many people and its frequency in nature has been described as many to be proof of intelligent design and higher power . Shapes of the Golden ratio Although undeniably stunning, the sources of the golden ratio and Fibonacci numbers in nature are only half the applications of these phenomena in the real world. As humans, along with the rest of nature, are hotwired to apply the golden proportions, some of the human applications are some of the most remarkable. As a species we are attracted to the shapes they make and therefore adapt it to the structures we built, the way we think and the art we create. One of the most common shapes is that of the golden rectangle. It is formed from a ratio of length to width of 1.168 : 1 (i.e. the golden ratio). This alone is not that interesting, but remove a square with the same width and height as the width of the golden rectangle (a square ratio 1:1) and you are left with another rectangle. If you take the measurements of this you once again find the ratio 1.168 : 1 the golden rectangle. Repeat the process and the same happens again and again and again; removing a square ratio 1:1 leaves a smaller golden rectangle. The pattern continues indefinitely and is known in mathematics as a fractal (a geometric pattern that is repeated at every scale). Look at most regular paper sizes, credit cards and company logo s you will find an abundance of golden rectangles. However its man-made applications are not its only uses, it can be applied to create another, much more stunning shape the logarithmic spiral. Visually, it can be described as a long, slow spiral and is known as a logarithmic or equiangular spiral. It is known as this as each radii from the centre intercepts the curve at exactly the same angle. It is created by constructing an arc from the furthest corner of each square in the golden rectangle to the opposing corner of that square. The pattern continues and repeats the further you zoom toward the centre making this yet another Fibonacci fractal. The most stunning example of this is the chambered nautilus (see the image of its shell right). As it grows it must produce more room within its shell, while keeping its original shape. To do this it adds a chamber larger than its previous, with each radii intercepting the curve at the same angle (remaining equiangular), keeping the original shape. There are also numerous other examples including; a rams horn, a galaxy spiral, a sea horse and many more. Last but not least, the pentagon and pentagram are found to have Fibonacci connections. These shapes have interested humans for many years and have been the insignia of many religious and political groups. The explanation for its popularity however lies with our desire to search for the golden ratio. From the diagram (left), we can see how the ratio 1:F connects the length of the side of the pentagon to the distance between corners of the pentagram. There is however another ratio, the distance between a vertex and the corner of the inscribed pentagon is 1: 1/F. These ratios mean that many pentagons in nature, art and architecture have Fibonacci numbers present in the lengths. Overall, we can see how many of the regular shapes found both in nature and modern life have been dictated by the Fibonacci sequence. There are thousands of examples of these proportions in the real world and more regular shapes than have been divulged here. As interesting as finding them in the real world is, it doesn t come close the intrigue which lies behind the way we can use them to our own advantage. Art and Architecture It is said that renaissance art was inspired by a sense of beauty and proportion . It seems fitting therefore that the dimensions for such art would lie in the ratio s and sequences of the most elaborate and efficient set of numbers known to maths. The use of the series in art has however been known long before this period with Luca Pacioli stating without mathematics there is no art upon the completion of his work with Leonardo Da Vinci on De Divina Proportione (you may recall this from History of the Sequence). Legend also has it that long before this, Greek mathematician Eudoxus studied human affinity to this proportion by asking a group of his followers to divide sticks into the ratios they found most pleasing. This experiment was later adapted by German psychologist, Gustav Fechner in the 1860 s. He took a series of ten rectangles of different proportion and asked subjects to choose which they found to be the most pleasing, 76% of all participants chose the three rectangles closest to the golden rectangle. It is clear from this then that we have known for many years that the golden or divine proportion has visually pleasing qualities and unknown to us, it can be found almost everywhere we look as a direct result. One of the earliest and most obvious sightings of this was in the Great Pyramids of 4700BC. Here F is found extensively in its construction but most scholars now believe that this is more coincidence than design, it is however interesting to note that the exact height of the structure is 5813 inches (numbers of the Fibonacci sequence. 1,400 years later and the Tomb of Ramses IV was built, this was later discovered to have several approximations to the golden rectangle as its centre. It had been constructed with a double square (approximation to the golden rectangle, a golden rectangle and a double golden rectangle. The first people to consciously apply the maths of the golden ratio to their art and architecture were the Greeks. The Parthenon of Greece 440BC is the single finest example of this. The whole structure fits within the golden rectangle proportions as well as each pair of columns and even the sections of sculpture that run above the columns. The designer, Phidias was said to be the greatest and most prolific sculptor of his age. His work was dependent upon extensive use of the golden proportion. Its abundance in his work later meant the ratio was named Phi in his honour. Many artefacts of the era from urns and vases to Afroditas sculpture (right) and temples all extensively used the proportion. It is believed that as Pythagoras linked it to the human body (see next section) it was generally associated with the divine and beautiful, making many associate it with the Gods and good. One of the most interesting instances of the Fibonacci sequence at work is in the operation of the stock market.