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.

JC Penney Company, Inc. :: Marketing Research

JC Penney Company, Inc. J. C. Penney Company, Inc. Is one of America’s largest department store, drugstore, catalog and e-commerce retailers. Providing merchandise and services through department stores, catalogs, and the Internet. Their targeted customers are â€Å"Modern Spenders† and â€Å"Starting Outs†, who shop for apparel, accessories, and home furnishings through the centers where JCPenney is located and through the convenience of catalog and the Internet. Starting Outs  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚    ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Less than 35 years of age  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚    ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Singles, young families  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  0-1 children  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Shopping patterns & relationships emerging  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  No strong retail loyalties  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  28% of U.S. households.  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Currently 16% of sales  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Potentially 30% of sales  Ã‚  Ã‚  Ã‚  Ã‚   Modern Spenders  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  35-54 years of age  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Dual-earner households  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  0-2 children (often includes teenagers)  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Consumption oriented  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  No strong retail loyalties & relationships  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Retail loyalties more likely  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Established shopping patterns  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Time-starved  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  27% of U.S. households  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Currently 43% of sales  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Potentially 50% of sales Distribution *Catalogs J.C. Penney is the nation’s largest catalog merchant, with the most modern facilities and the largest privately owned telemarketing network in America. Serving this $4 billion catalog business are nearly 2,000 catalog departments in JCPenney department stores, Eckerd drugstores, freestanding sales centers and independent catalog merchants. *Internet J.C. Penney is in only its second year of Internet sales, and its going strong and growing. Sales jumped from $15 million to $102 million since the beginning of jcpenney.com. *Department Stores JCPenney has more retail space in major regional shopping centers than any other department store retailer in America, with about 1,140 department stores located in all 50 states. JCPenney’s drugstore ECKERD has over 2,600 stores in operation in 23 states. PROMOTIONAL OFFERS  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Direct mail. An invitation to shop mailed to selected catalog customers. These promotions may be associated with a holiday or other special savings event, including many of our storewide events  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Sunday Supplement. JCPenney color inserts that are delivered with your Sunday or late-week newspaper.  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Newspaper Ads. Promotional offers are often supplemented by ads in your local newspaper.  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Catalog Inserts. Many of our Sale and JCPenney â€Å"Signature Series† catalogs contain special offers for limited-time savings that are bound into mailed copies.  ·Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  E-mail Promotions.

An Old Man’s Winter Night Analysis

An Old Man’s Winter Night This is a very haunting poem about an old man who stands alone dying in a dark house in winter. His memory is failing him and because of that he doesn’t know who he is or why he is in the house but he stays there inside the house because of the gruelling winter weather outside. There is no sense that the old man is existing for anyone or anything, he is purely alone. He is alone not only because no one is with him, but also because there will be no one to remember him after he dies.He develops a fear of the cellar beneath him and the darkness that lies outside so he strikes the ground in an attempt to frighten the unknown rather than confronting his fears. Finally, he falls asleep in front of the fire only to be disturbed by a log that has shifted in the fire but in due course, falls into a deep sleep. Frost uses the dying fire as a symbol to his fading life. As the night goes on, the fire dims and the old man grows closer to death. He knows th at eventually the darkness will consume him.The piece does not stray from the subject matter from the beginning to the end, continuously conveying the extent of how scared and lonely he is. Frost’s intention is clearly to portray the depth of loneliness that the old man is feeling in his old age and the emotions that accompany this. In terms of form, the poem does not have a traditional rhyme scheme and the lines vary in length. Frost uses many different literary devices throughout the poem such as imagery which appeals to our sight, touch and hearing senses.Frost has used Imagery such as â€Å"In clomping there, he scared it once again† which appeals to our touch because you can almost feel how he has stomped the floor to try and frighten off the unknown. He has appealed to our hearing senses by using personification, â€Å"like the roar of trees† lets you almost hear how the trees were thrashing around on the cold winter night. â€Å"That brought him to that creaking room was age. He stood with barrels round him – at a loss† appeals to our sight and paints a vivid eerie image of him standing alone in the dark house.Frost’s use of personification, â€Å"like the roar of trees† is used to give a more humanistic quality to the trees to create a more eerie surrounding. Onomatopoeia is used â€Å"crack of branches† to make you think about the sound and to give a realistic feel to the poem, but more significantly alliteration is used, â€Å"doors darkly†, â€Å"beating box† and â€Å"separate stars†, this makes the poem sound more pleasant to the readers. There is also evidence of internal rhyme on the tenth line â€Å"In clomping there, he scared it once again† An internal rhyme puts emphasis on the two words that rhyme and quickens the pace of the line.On the twenty third line, he used caesura to form important thoughts rather than breaking it â€Å"And slept. The log that shift ed with a jolt†. There are eight strong enjambments throughout the poem helping it to run on and flow into the next line and continue momentum instead of the usual rhythm a poem would have. The mood of the poem is sad and disheartening. Frost’s use of imagery creates a sad setting. â€Å"All out of doors looked darkly in at him† could almost mean that people know and see that he is alone in the house but yet they choose to ignore it.The tone of the poem is candid, almost as if Frost is just telling a story without any feeling or emotion being put into it. From reading the poem, we realise that the old man is alone but the writer never clarifies the reason why, he only repeats that he is completely isolated and beyond the comfort of another human being. The most poignant aspect of this poem is the old man’s loss of memory and the frost forming on the windows because it’s so cold, â€Å"Through the thin frost, almost in separate stars, that gathers o n the pane in empty rooms. He has no recollection of his purpose or identity and simply finds himself standing â€Å"with barrels round him — at a loss. † Not only is the old man isolated in body, he is isolated in mind. His memories of his past happiness cannot comfort him now. Although the old man is in a state of utter isolation, he still has the bravery to fight for his existence and attempt to scare away his fears that creep through the night. Although the old man is unaware of what exactly he is afraid of in the cellar or the dark of night, he clutches to the act of â€Å"clomping† as a familiar and unfamiliar comfort.The devastating sense of loneliness and fear is accentuated by the noises all around the old man, the cracking of branches, the roar of the trees – this use of personification is used to make the scene more disturbing. However, the old man himself remains silent throughout the poem. When he does make sounds, he resorts to the more anim alistic action of stomping his feet rather than trusting his voice. In reading the title of the poem it suggests there should be a pleasant setting of an old man inside house beside a fire on a cold winter’s night but instead the writer has denied the readers any comforting expectations. Instead the writer conveys that he is slowly dying alone in the house on a devastatingly cold frosty night but he wants to live and fight death until the end even though he is losing his mind he still knows he doesn’t want to die. The old man’s isolation keeps the reader at a distance so they are not able to feel a sense of empathy with the old man.If Frost divulged the old man’s thoughts it would be easier for the readers to form some kind of connection with him but Frost wants the readers to feel the same lonely, isolated feeling that the old man has and does this by rendering the old man mute. The reader is forced to remain a silent onlooker who cannot connect to the i nner workings of the old man’s mind. This poem could be interpreted as how Frost feels about his life at this point in time. â€Å"All out of doors looked darkly in at him through the thin frost almost in separate stars† This could be Frost’s way of expressing his feelings that he thinks nobody cares about him anymore.The poem does not end on a completely desperate note. Although the man is frightened of what he does not know, he still succeeded in â€Å"scaring† off the unknown when he was alone and frightened. Frost suggests that even a person in the depths of isolation and loneliness is still capable of maintaining a presence and â€Å"keeping† a house. The old man’s behavior in the house is not ideal or necessarily human, and he is still destined to face death and constant loneliness, and yet his house is still his own because of his insistent grasp on it and his refusal to abandon himself completely.

Abortion Issues - How They Affect American Politics

Abortion issues surface in almost every American election, whether its a local race for school board, a statewide race for governor or a federal contest for Congress or the White House. Abortion issues have polarized American society since the U.S. Supreme Court legalized the procedure. On one side are those who believe women are not entitled to end the life of an unborn child. On the other are those who believe women have the right to decide what happens to their body. Often there is no room for debate between the side. Related Story: Is Abortion the Right Thing to Do? In general, most Democrats support a womans right to have an abortion and most Republicans oppose it. There are notable exceptions, though, including some politicians who have waffled on the issue. Some Democrats who are conservative when it comes to social issues such oppose abortion rights, and some moderate Republicans are open to allowing women to have the procedure. A 2016 Pew Research Survey  found that 59 percent of Republicans believe abortion should be illegal, and 70 percent of Democrats believe the procure should be allowed. Overall, though, a narrow majority of Americans — 56 percent in the Pew poll  Ã¢â‚¬â€Ã‚  support legalized abortion and 41 percent oppose it.  In both cases, these figures have remained relatively stable for at least two decades, the Pew Researchers found. When Abortion Is Legal In the United States Abortion refers to the voluntary termination of a pregnancy, resulting in the death of the fetus or embryo. Abortions performed prior to the third trimester are legal in the United States.Abortion-rights advocates believe a woman should have access to whatever health care she needs and that she should have control over her own body. Opponents of abortion rights believe an embryo or fetus is alive and thus abortion is tantamount to murder.   Current Status The most controversial of abortion issues is the so-called partial birth abortion, a rare procedure. Beginning in the mid-90s, Republicans in the U.S. House of Representatives and U.S. Senate introduced legislation to ban partial birth abortions. In late 2003, Congress passed and President George W. Bush signed the Partial-Birth Abortion Ban Act.This law was drafted after the Supreme Court ruled Nebraskas partial birth abortion law unconstitutional because it did not allow a doctor to use the procedure even if it were the best method to preserve the health of the mother. Congress attempted to circumvent this ruling by declaring that the procedure is never medically necessary. History Abortion has existed in almost every society and was legal under Roman law, which also condoned infanticide. Today, almost two-thirds of the women in the world may obtain a legal abortion.When America was founded, abortion was legal. Laws prohibiting abortion were introduced in the mid-1800s, and, by 1900, most had been outlawed. Outlawing abortion did nothing to prevent pregnancy, and some estimates put the number of annual illegal abortions from 200,000 to 1.2 million in the 1950s and 1960s.States began liberalizing abortion laws in the 1960s, reflecting changed societal mores and, perhaps, the number of illegal abortions.  In 1965, the Supreme Court introduced the idea of a right to privacy in Griswold v. Connecticut as it struck down laws that banned the sale of condoms to married people.Abortion was legalized in 1973 when the U.S.Supreme Court ruled in Roe v. Wade that during the first trimester, a woman has the right to decide what happens to her body. This landmark decision rested on the right to privacy which was introduced in 1965. In addition, the Court ruled that the state could intervene in the second trimester and could ban abortions in the third trimester. However, a central issue, which the Court declined to address, is whether human life begins at conception, at birth, or at some point in between.In 1992, in Planned Parenthood v. Casey, the court overturned Roes trimester approach and introduced the concept of viability. Today, approximately 90% of all abortions occur in the first 12 weeks.In the 1980s and 1990s, anti-abortion activism -- spurred on by opposition from Roman Catholics and conservative Christian groups -- turned from legal challenges to the streets. The organization Operation Rescue organized blockades and protests around abortion clinics. Many of these techniques were prohibited by the 1994 Freedom of Access to Clinic Entrances (FACE) Act. Pros Most polls suggest that Americans, by a slim majority, call themselves pro-choice rather than pro-life. That does not mean, however, that everyone who is pro-choice believes that abortion is acceptable under any circumstance. A majority support at least minor restrictions, which the Court found reasonable as well under Roe.Thus the pro-choice faction contains a range of beliefs -- from no restrictions (the classic position) to restrictions for minors (parental consent) ... from support when a womans life is endangered or when the pregnancy is the result of rape to opposition just because a woman is poor or unmarried.Principle organizations include the Center for Reproductive Rights, The National Organization for Women (NOW), National Abortion Rights Action League (NARAL), Planned Parenthood, and the Religious Coalition for Reproductive Choice. Cons The pro-life movement is thought of as more black-and-white in its range of opinions than the pro-choice faction. Those who support life are more concerned with the embryo or fetus and believe that abortion is murder. Gallup polls starting in 1975 consistently show that only a minority of Americans (12-19 percent) believe that all abortions should be banned.Nevertheless, pro-life groups have taken a strategic approach to their mission, lobbying for mandated waiting periods, prohibitions on public funding and denial of public facilities.In addition, some sociologists suggest that abortion has become a symbol of the changing status of women in society and of changing sexual mores. In this context, pro-life supporters may reflect a backlash against the womens movement.Principle organizations include the Catholic Church, Concerned Women for America, Focus on the Family, and National Right to Life Committee. Where It Stands President George W. Bush supported and signed the constitutionally questionable partial-birth abortion ban and, as Governor of Texas, vowed to put an end to abortion. Immediately after taking office, Bush eliminated U.S. funding to any international family planning organization that provided abortion counseling or services -- even if they did so with private funds.There was no easily-accessed issue statement about abortion on the 2004 candidate web site. However, in an editorial entitled The War Against Women the New York Times wrote: The lengthening string of anti-choice executive orders, regulations, legal briefs, legislative maneuvers, and key appointments emanating from his administration suggests that undermining the reproductive freedom essential to womens health, privacy and equality is a major preoccupation of his administration - second only, perhaps, to the war on terrorism.