Number of Folds in Paper: Thickness of Earth to Sun

Number of Folds in Paper: Thickness of Earth to Sun Calculating the number of folds and the hypothetical size of a piece of paper so that its thickness equates to the distance from the Earth to Saturn. As a physics student doing the option on astrophysics, I have become very interested in the immensity of the universe. I decided it would be intriguing to combine it with my personal hobby of doing origami which helps me to relieve stress. As a result, I investigated the amount of times a piece of paper would need to be folded in half so that its thickness achieves a length which equates to the distance from the Earth to Saturn. I have chosen the planet Saturn because it is one of the biggest planets that can be observed by the naked eye and I have been able to see it last summer from the top of a high hill in La Pedriza in the Guadarrama mountain range near my hometown (Collado Villalba, Madrid, Spain). Popular claims suggested that it is impossible to fold a piece of paper in half more than seven times no matter its size or thickness.   Previous mathematicians have worked out the number of folds required to reach the moon from the Earth which was worked out to be 42 folds[1] using a m thick paper. The size of the paper however, was not found. In theory, the average folds for a normal size A4 (m) paper is seven. This is an interesting topic because it is incredible how, by exponential growth, a miniature of m thick like a piece of paper can be folded in half to reach the distance of the planet Saturn. Britney Gallivan[2] was able to fold a piece of toilet paper of 1200 meters a number of 12 folds. She derived two mathematical expressions based on geometrical sequences, taking into consideration the amount of paper lost in every fold. These formulas make it possible to calculate the hypothetical length L and width W of a piece of paper that would be folded n times to equate the distance from the Earth to Saturn. Gallivan established some rules that would need to be followed when folding a sheet of paper in half:       A single rectangular sheet of paper of any size and uniform thickness can be used. The fold line has to be in the same direction each time. The folding process must not tear the paper. When folded in half, the portions of the inner layers which face one another must almost touch one another. The average thickness or structure of material of paper must remain unaffected by the folding process. A fold is considered complete if portions of all layers lie in one straight line. Hence, the length L of the paper influences the number of times it can be folded in half. Hypothesis The distance from the Earth to Saturn will be obtained by folding a piece of paper ≠¥50 times with a hypothetically large enough paper. This exploration used exponential growth and logarithms in order to find out the number of folds required to reach Saturn. Before any calculation was done, it was indispensable to collect all the data required for the investigation. All the values were used were in the international system of units (metres) and standard form in order to keep the exploration standardized. Taking into consideration the elliptical orbits of the planets, sometimes they are closer to Earth and sometimes they are further away. Therefore, the value used during this mathematical exploration was the mean value of when they are the furthest apart and the closest together. Astronomical units (AU) are the standard unit measure used when dealing with distances within the Solar System. 1 AU is equal to the distance from the Earth to the Sun which is equivalent tom. Distance from the Earth to Saturn when closer together 8.00 AU[3] Distance from the Earth to Saturn when further apart 11.0 AU3 Mean distance from the Earth to Saturn    AU   Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   m The thickness of a normal A4 paper (0.210 297m) was calculated by taking a measurement of an office pack of 500 pages (80gsm) which was found to be about 0.05m. The thickness of each individual page is calculated by dividing the total thickness (0.05m) by the amount of sheets (500); giving a result of m. Whenever a paper is folded in half, the number of layers is doubled so the thickness increases by two. When there is only one layer of paper (not folded), its thickness is m. Once it is folded in half for the first time, its thickness will be multiplied by 2 hence, m Folding it one more time means multiplying it by two again, m Thus, an expression can be established, showing the exponential growth; The expression can be represented in a graph to illustrate graphically the exponential growth because of folding. Graph 1: From Graph 1, it is possible to visualise how, something that seems unrealistic like folding a sheet of paper to reach Saturn becomes possible. The graph also illustrates how rapidly exponential growth occurs. Since the expression needs to be equal to the distance from the Earth to Saturn to work out (the number of folds), an equation to find can be solved: Folds To find, the rules of logarithms were put in place due to the exponential nature of the equation. The answer has been rounded up to 54 because it is not possible to have a half fold. Gallivan derived the following formula for the minimum length of a piece of paper of thickness t to be folded n times in a single direction To prove this formula, it is neccesary to understand that after each fold, some part of the paper is lost and becomes a rounded edge. I folded an A4 sheet of paper seven times in order to illustrate this: As you can see from the picture, there is a rounded edge on the side which is paper being lost and is not contributing to the real thickness but just joining the layers. The curved portion becomes bigger in correlation with the number of folds and begins to take a greater area of the volume of the paper. At the first fold, a semicircle of radius t (thickness) is formed, which has a perimeter . Thus, units of the paper are being used in the fold.   A paper smaller than this cannot be folded since there is not enough paper to form the fold. After the fold, there is a two-layer sheet of paper with a thickness of 2t. Another fold results in folding the second layer over the first layer. The second layer has a radius of , so it uses units of paper. The total amount of paper used by the second fold, for both layers, is resulting in a four-layer piece of paper. The ith fold begings with layers, and folding the jth layer uses units of paper. Hence, the total length of paper used for the ith fold is given by Therefore, to obtain the total length of paper required for n number of folds, sum this over i from 1 to n, which gives Gallivans formula: The thickness can be substituted into t and the number of folds can be substituted into n which gives L to be equal to 1.70-1028m. The other expression proposed by Gallivan can be used to calculate the width of the paper. If the length lost in the radii of earlier folds is not considered, the length lost must be considered in the last fold. At the final fold n, the side of the square must be at least equal to the length lost in the final fold which is (amount of length lost in each fold. Taking into consideration that the total area of the sheet (area = nb of sheets in penultimate step area of square in penultimate step) is preserved, Gallivans equation can be derived: Again, the thickness and number of folds can be substituted and an answer for W can be found giving W to be 2.69-1020m. In conclusion, the initial hypothesis was right since the number of folds was 54 which is, indeed, greater than 50. The hypothetical paper that could, in theory, be folded 54 times so that its thickness equates the distance from the Earth to Saturn of m would be long and wide (taking into consideration that its thickness would bem). The dimensions of this paper would be bigger than the actual distance from the Earth to Saturn so, unfortunately, we do not have a paper that big that would allow us to reach to Saturn just by folding it in half. This mathematical exploration used logarithms to find out the number of folds needed to reach Saturn with a m thick paper. However, the dimensions of this sheet of paper would be too big and hence, impossible to find in the Earths surface. Nevertheless, the exploration could have looked at using a thinner piece of paper to see if its dimensions would have been smaller and perhaps, we would have been able to find it in the surface of the Earth and we would have been able to reach Saturn. References       Astronomy, S. (2012). How Far away is Saturn? [Online] Space.com. Available at: http://www.space.com/18477-how-far-away-is-saturn.html [Accessed 16 Jan. 2017]. IFLScience. (2016). Fold A Piece of Paper in Half 103 Times and It Will Be As Thick As the UNIVERSE. [Online] Available at: http://www.iflscience.com/space/fold-piece-paper-half-103-times-and-it-will-be-thick-universe/ [Accessed 16 Jan. 2017]. Pomonahistorical.org. (2002). Folding Paper in Half Twelve Times. [Online] Available at: http://pomonahistorical.org/12times.htm [Accessed 16 Jan. 2017]. [1] IFLScience. (2016). Fold A Piece of Paper in Half 103 Times and It Will Be As Thick As the UNIVERSE. [online] Available at: http://www.iflscience.com/space/fold-piece-paper-half-103-times-and-it-will-be-thick-universe/ [Accessed 16 Jan. 2017]. [2]Pomonahistorical.org. (2002). Folding Paper in Half Twelve Times. [online] Available at: http://pomonahistorical.org/12times.htm [Accessed 16 Jan. 2017]. [3] Astronomy, S. (2012). How Far away is Saturn? [online] Space.com. Available at: http://www.space.com/18477-how-far-away-is-saturn.html [Accessed 16 Jan. 2017].

Office Ergonomics

Office Ergonomics Contents 1. Introduction 2. Office Computer Workstations 3. Computer Workstation Evaluation Checklist 4. Appendix A 1. Introduction Ergonomics is the process of designing the work environment to fit the worker, rather than fitting the worker to the work environment. The goal of this ergonomic program is to minimize accidents and illnesses due to chronic physical and psychological stresses, while maximizing productivity and efficiency.Cumulative trauma disorders (CTD) or repetitive motion strain injuries are musculoskeletal disorders that result from repeated exposure to physical stressors. Stressors affect tendons, ligaments, nerves, muscles and bones. Physical stressors in the office environment are caused by sustained awkward postures, repetitive motions, using excessive force or compression. 2. Office Computer Workstations The workforce population varies greatly in physical size and stature. The idea of the average size person is obsolete.Adjusting office furnitu re and office equipment help employees make changes in the office to ensure proper posture is maintained throughout the day. There is no single â€Å"correct† posture or arrangement of components that will fit everyone. However, there are basic design goals to consider when setting up a computer workstation or performing computer-related tasks. Consider your workstation as you read through this guide and see if you can identify areas for improvement in posture, component placement, or work environment.This guide provides suggestions to minimize or eliminate identified problems, and allows you to create your own â€Å"custom-fit† computer workstation. Good Working Positions: To understand the best way to set up a computer workstation, it is helpful to understand the concept of neutral body positioning. This is a comfortable working posture in which your joints are naturally aligned. Working with the body in a neutral position reduces stress and strain on the muscles, te ndons, and skeletal system and reduces your risk of developing a musculoskeletal disorder (MSD).The following are important considerations when attempting to maintain neutral body postures while working at the computer workstation: †¢ Hands, wrists, and forearms are straight, in-line and roughly parallel to the floor. †¢ Head is level, or bent slightly forward, forward facing, and balanced. †¢ Shoulders are relaxed and upper arms hang normally at the side of the body. †¢ Elbows stay in close to the body and are bent between 90 and 120 degrees. †¢ Feet are fully supported by floor or footrest. †¢ Back is fully supported with appropriate lumbar support when sitting vertical or leaning back. Thighs and hips are supported by a well-padded seat and generally parallel to the floor. †¢ Knees are about the same height as the hips with the feet slightly forward. Regardless of how good your working posture is, working in the same posture or sitting still for prolonged periods is not healthy. You should change your working position frequently throughout the day in the following ways: †¢ Stretch your fingers, hands, arms, and torso. †¢ Stand up and walk around for a few minutes periodically.

Health Care Spending Essay

Healthcare is a very important sector in the countries around the world. The countries and the government always keep some money in their budget in relation to the health care activities in their countries. In the United States of America also, there are certain provisions in relation to the health care. The paper shall discuss about the health care spending in relation to United States of America. It shall discuss about the current level of health care expenditure in United States of America. There are shall be a lot of discussion in relation to the fact that whether the expenditure on health care in United States of America is appropriate or not. It shall also discuss about the manner in which the health care needs of the people in United States of America are paid by the government. In the end of the paper, there shall be a forecast in relation to the health care spending in United States of America. All the above said things shall be discussed in detail in the current paper. The government of United States of America has made a lot of provisions in relation to the expenditure of health care facilities in United States of America. In the United States of America, the health care spending has been increasing continuously (Hawe, 2008). There have been a lot of funds kept by the government of United States of America in its budget in relation to the health care services. In the year 1990, the expenditure made by the government of United States of America in relation to the health care spending was $714 million. Since then, the amount of expenditure on health care has increase to $2. 3 trillion. This is around 3 times of what it was in the year 1990. The government of United States of America is spending around 16% of its total gross domestic production on providing the health care facilities to the people. It is estimated that around $7600 are spent by the government of United States of America for each and every resident in the country. In this way, it can be said that the expenditure in relation to health care in United States of America is quite good and the country is taking adequate steps for making sure that adequate health care facilities are provided to the people in a proper and appropriate manner. The health care spending of the United States of America is the highest all around the world. There is no problem with the extent of money spend by the country towards the health care. Other countries around the world are also spending some money towards the health care but the expenditure made by the government of United States of America is the highest. Currently the expenditure incurred in United States of America towards is adequate and appropriate. It has to increase only with the increase in population. According to the per capita expenditure, the current expenditure in the country towards the healthcare is appropriate. When the country makes the budget in relation to the health care expenditure, then it has to take care that adequate expenditure is incurred on each and every activity related to health care. The government of United States of America should make sure that it spends appropriate expenditure on each and every area of the health care. The expenditure should be more on the critical areas and it should be reduced on the areas which are not very difficult to handle for the government. The government of United States of America should make sure that it increases the expenditure of the health insurance of those people which are not in a position to afford the costs for themselves (U. S. Health Care Costs, n. d. ). The government should also increase the expenditure in relation to the funds provided to the health care organizations for providing the benefits to the poor people in the United States of America. The government of United States of America should also increase the expenditure in relation to some specific and high risk diseases such as Cancer, Aids etc. If the expenditure shall be reduced in relation to the above said activities then, there is a possibility that the people in United States of America shall be benefitted to a very high extent. Other than the above areas, there are certain areas in which the government of United States of America should look to reduce the expenditure. The cost on some of the areas in the health care should not be reduced to the extent it hampers the health care services. If there is a requirement to reduce the health care costs then, these should be reduced only till the services are not affected in a negative manner. The areas in which unnecessary costs are incurred by the government should be recognized and the costs should be reduced in those areas. In the United States of America, there are adequate sources used for the purpose of financing the health care services in the country. In the country, the funding for the health care services is done together by the government and the non government agencies in the country. While preparing the budget, the government of United States of America keeps a percentage of funds in relation to providing the health care services to the people who are in need and also the people which are not able to have the health care insurance for themselves. Other than the government of United States of America, there are various other forms of associations which work for the purpose of providing the health care services to the people. They not only provide the health care services but also provide the financial for the insurance of the patients. There are few programs also in relation to the same. The programs include Healthcare Facilities Accreditation Program, Community Health Accreditation Program and various other programs. The health care finance in the United States of America are funded by the above said facilities and things. In the United States of America, the government and the non government associations share 50% of the expenditure each. In this way, there is a proper sharing the expenditure. For the last few years, the healthcare expenditure in United States of America has been increased considerably at an increasing rate. It is expected that the healthcare expenditure in United States of America shall increase in the same rate as it was increasing earlier (Scambler, 2008). In the year 2007, the health care expenditure in United States of America was $2. 2 trillion. This amount increased to $2. 3 trillion in the financial year 2008. In this way, there was a definite increase in the health care expenditure in the two years. It is expected that the health care expenditure will grow even more. In the year 2013, the healthcare expenditure is expected to reach the figure of $3 trillion and in the year 2017, it is expected that the health care expenditure shall reach the figure of $4. 3 trillion. In this way, it can be said that there is a forecast of increased expenditure on health care in the years to come. The chart below provides the forecast for health care in a proper manner. This chart is showing the healthcare expenditure in United States of America. The government of United States of America is spending a lot of money on the healthcare services in the country. With the kind of expenditure, the government of United States of America is spending on the health care facilities in the country, it can be said that the government is spending sufficient amount of money on the health care facilities.

Bulgars, Bulgaria and Bulgarians

The Bulgars were the early people of eastern Europe. The word bulgar derives from an Old Turkic term denoting a mixed background, so some historians think they may have been a Turkic group from central Asia, made up of members of several tribes. Along with the Slavs and the Thracians, the Bulgars were one of the three primary ethnic ancestors of present-day Bulgarians.   The Early Bulgars The Bulgars were noted warriors, and they developed a reputation as fearsome horsemen. It has been theorized that beginning in about 370,  they moved west of the Volga River along with the Huns. In the mid-400s, the Huns  were led by Attila, and the Bulgars apparently joined him in his westward invasions. After Attilas death, the Huns settled in the territory north and east of the Sea of Azov, and once again the Bulgars went with them.   A few decades later, the Byzantines hired the Bulgars to fight against the Ostrogoths. This contact with the ancient, affluent empire gave the warriors a taste for wealth and prosperity, so in the 6th century, they began to attack the nearby provinces of the empire along the Danube in hopes of taking some of that wealth. But in the 560s, the Bulgars themselves came under attack by the Avars. After one tribe of Bulgars was destroyed, the rest of them survived by submitting to yet another tribe from Asia, who departed after about 20 years. In the early 7th century, a ruler is known as Kurt (or Kubrat) unified the Bulgars and built a powerful nation that the Byzantines referred to as Great Bulgaria. Upon his death in 642, Kurts five sons split the Bulgar people into five hordes. One remained on the coast of the Sea of Azov and was assimilated into the empire of the Khazars. A second migrated to central Europe, where it merged with the Avars. And a third disappeared in Italy, where they fought for the Lombards. The last two Bulgar hordes would have better fortune in preserving their Bulgar identities. The Volga Bulgars The group led by Kurt’s son Kotrag migrated far to the north and eventually settled around the point where the Volga and the Kama rivers met. There they split into three groups, each group probably joining with peoples who had already established their homes there or with other newcomers. For the next six centuries or so, the Volga Bulgars flourished as a confederation of semi-nomadic peoples. Although they found no actual political state, they did establish two cities: Bulgar and Suvar. These places benefited as key shipping points in the fur trade between the Russians and Ugrians in the north and the civilizations of the south, which included Turkistan, the Muslim caliphate at Baghdad, and the Eastern Roman Empire. In 922, the Volga Bulgars converted to Islam, and in 1237 they were overtaken by the Golden Horde of the Mongols. The city of Bulgar continue to thrive, but the Volga Bulgars themselves were eventually assimilated into neighboring cultures. The First Bulgarian Empire The fifth heir to Kurts Bulgar nation, his son Asparukh, led his followers west across the Dniester River and then south across the Danube. It was on the plain between the Danube River and the Balkan Mountains that they established a nation that would evolve into what is now known as the First Bulgarian Empire. This is the political entity from which the modern state of Bulgaria would derive its name. Initially under the control of the Eastern Roman Empire, the Bulgars were able to found their own empire in 681, when they were officially recognized by the Byzantines. When in 705 Asparukh’s successor, Tervel, helped restore Justinian II to the Byzantine imperial throne, he was rewarded with the title Caesar. A decade later Tervel successfully led a Bulgarian army to assist Emperor Leo III in defending Constantinople against invading Arabs. At about this time, the Bulgars saw an influx of Slavs and Vlachs into their society. After their victory at Constantinople, the Bulgars continued their conquests, expanding their territory under the khans Krum (r. 803 to 814) and Pressian (r. 836 to 852) into Serbia and Macedonia. Most of this new territory was heavily influenced by the Byzantine brand of Christianity. Thus, it was no surprise when in 870, under the reign of Boris I, the Bulgars converted to Orthodox Christianity. The liturgy of their church was in Old Bulgarian, which combined Bulgar linguistic elements with Slavic ones. This has been credited with helping to create a bond between the two ethnic groups; and its true that by the early 11th century, the two groups had fused into a Slavic-speaking people who were, basically, identical to the Bulgarians of today. It was during the reign of Simeon I, the son of Boris I, that the First Bulgarian Empire achieved its zenith as a Balkan nation. Although Simeon evidently lost the lands north of the Danube to invaders from the east, he expanded Bulgarian power over Serbia, southern Macedonia and southern Albania through a series of conflicts with the Byzantine Empire. Simeon, who took for himself the title Tsar of All the Bulgarians, also promoted learning and managed to create a cultural center at his capital of Preslav (present-day Veliki Preslav). Unfortunately, after Simeons death in 937, internal divisions weakened the First Bulgarian Empire. Invasions by Magyars, Pechenegs, and Rus, and reignited conflict with the Byzantines, put an end to the sovereignty of the state, and in 1018 it became incorporated into the Eastern Roman Empire. The Second Bulgarian Empire In the 12th century, stress from external conflicts reduced the Byzantine Empires hold on Bulgaria, and in 1185 a revolt took place, led by the brothers Asen and Peter. Their success allowed them to establish a new empire, once again led by Tsars, and for the next century, the house of Asen reigned from the Danube to the Aegean and from the Adriatic to the Black Sea. In 1202 Tsar Kaloian (or Kaloyan) negotiated a peace with the Byzantines that gave Bulgaria complete independence from the Eastern Roman Empire. In 1204, Kaloian recognized the authority of the pope and thus stabilized the western border of Bulgaria. The second empire saw increased trade, peace, and prosperity. A new golden age of Bulgaria flourished around the cultural center of Turnovo (present-day Veliko Turnovo). The earliest Bulgarian coinage dates to this period, and it was around this time that the head of the Bulgarian church attained the title of patriarch. But politically, the new empire wasnt particularly strong. As its internal cohesiveness eroded, external forces began to take advantage of its weakness. The Magyars resumed their advances, the Byzantines took back portions of Bulgarian land, and in 1241, Tatars began raids that continued for 60 years. Battles for the throne among various noble factions lasted from 1257 to 1277, at which point peasants revolted due to the heavy taxes their warring overlords had imposed on them. As a result of this uprising, a swineherd by the name of Ivaylo took the throne; he wasnt ousted until the Byzantines lent a hand.   Only a few years later, the Asen dynasty died out, and the Terter and Shishman dynasties that followed saw little success in maintaining any real authority. In 1330, the Bulgarian Empire reached its lowest point when Serbs slew Tsar Mikhail Shishman at the Battle of Velbuzhd (present-day Kyustendil). The Serbian Empire took control of Bulgarias Macedonian holdings, and the once-formidable Bulgarian empire began its last decline. It was on the verge of breaking apart into lesser territories when the Ottoman Turks invaded. Bulgaria and the Ottoman Empire The Ottoman Turks, who had been mercenaries for the Byzantine Empire in the 1340s, began attacking the Balkans for themselves in the 1350s. A series of invasions prompted the Bulgarian Tsar Ivan Shishman to declare himself a vassal of Sultan Murad I in 1371; yet still, the invasions continued. Sofia was captured in 1382, Shumen was taken in 1388, and by 1396 there was nothing left of Bulgarian authority.   For the next 500 years, Bulgaria would be ruled by the Ottoman Empire in what is generally viewed as a dark time of suffering and oppression. The Bulgarian Church, as well as the empires political rule, was destroyed. The nobility either was killed, fled the country, or accepted Islam and was assimilated into Turkish society. The peasantry now had Turkish lords. Every now and then, male children were taken from their families, converted to Islam and raised to serve as Janissaries. While the Ottoman Empire was at its height of power, the Bulgarians under its yoke could live in relative peace and security, if not freedom or self-determination. But when the empire began to decline, its central authority could not control local officials, who were sometimes corrupt and at times even downright vicious.   Throughout this half a millennium, Bulgarians held stubbornly to their Orthodox Christian beliefs, and their Slavic language and their unique liturgy kept them from becoming absorbed into the Greek Orthodox Church. The Bulgarian peoples thus retained their identity, and when the Ottoman Empire began to crumble in the late 19th century, the Bulgarians were able to establish an autonomous territory.   Bulgaria was declared an independent kingdom, or tsardom, in 1908.