Magic Squares free essay sample

Magic squares have intrigued people for thousands of years and in ancient times they were thought to be connected with the supernatural and hence, magical. Today, magic squares are considered magical because there are so many relationships between the sums of the numbers in the squares. So, what is a magic square? A magic square is an arrangement of the numbers from 1 to n2 in an n x n matrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same (Alejandre), called the magic constant or sum. The magic sum can be found using the formula (n(n2 + 1))/2 for any n x n matrix. There are many different variations of magic squares, of which, some have more ways of finding the magic sum in the square and others use geometric shapes or number words. The earliest known magic square was found in a Chinese book, Yih King, in which the legend of â€Å"Lo Shu† is told. We will write a custom essay sample on Magic Squares or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page This magic square is a 3 x 3 matrix, with numbers 1-9, and the magic sum is 15. â€Å"The legend of â€Å"Lo Shu† or â€Å"scroll of the river Lo† tells the story of a huge flood that destroyed crops and land. The people offered a sacrifice to the river god to calm his anger. Every time the river flooded, there emerged a turtle that would walk around the sacrifice. It wasn’t until a child noticed a unique pattern on the turtle’s shell that told the people how many sacrifices (15) to make for the river god to accept their sacrifice.† (http://plaza.ufl.edu/ufkelley/magic/index.htm) This kind of magic square is known as the â€Å"traditional† magic square since it has no other special properties besides the ones noted above. The magic square is still common in China today. It is found on buildings and in artistic designs, and fortune tellers uses them in their trade. The magic squares then found their way to India. Here, the magic squares were not only used to spread mathematical knowledge but also had spiritual purposes. For example, a magic square was found in a medical book as a way to ease childbirth. The oldest magic square of order four (4 x 4) was found inscribed in Khajuraho, India dating back to the 11th or 12th century. http://www.markfarrar.co.uk/graphics/msq004.gif This kind of magic square has many more properties than a traditional magic square. In addition to the rows, columns, and diagonals, but also the broken diagonals (4+6+13+11=34), of any 2 x 2 block of number (9+6+4+15=34), the four corners (9+16+7+2), the corners of any 3 x 3 block (9+3+14+8=34) , and the sum of the middle two entries of the two outer columns and rows (6+3+ 12+13 = 34) all have the same sum. The first recorded magic square in Europe and most famous 4 x 4 magic square is found in the famous painting, Melancholia, by German artist Albrecht Duerer (1471 – 1528). The painting is said to â€Å"depict the indecision of the intellectual† (Britton). A unique â€Å"magic† property of this magic square is that the center entries of the bottom row are 15 and 14, which is the year the painting was made (1514). This magic square shares the same properties as the other mentioned 4 x 4 magic square only the rows and columns are interchanged in such a way to remain the constant sum of 34. We can use some properties of magic squares to construct more squares from other manufactured squares; 1. Magic square will remain magic if any number is added to every number of a magic square. 2. A magic square will remain magic if any number multiplies every number of a magic square. 3. A magic square will remain magic if two rows, or columns, equidistant from the center are interchanged. 4. An even order magic square ( n x n where n is even) will remain magic if the quadrants are interchanged. 5. An odd order magic square will remain magic if the partial quadrants and the row is interchanged.† (Hawley) In the seventeenth century, a Frenchman named Antoine de la Loubere created a method for constructing any odd n x n magic squares using consecutive numbers starting with 1. The general rule is that you move diagonal upwards and to the right. However, there are two exceptions. If your move is outside of the magic square, you put your entry on the opposite side of the row or column and if when you move, you land on an occupied space, you put the entry under your last entry. Many ingenious methods for constructing magic squares have been devised but all methods are only for specific cases or different types of magic squares. There are many famous names associated with magic squares including Martin Gardner, Leonard Euler, and Benjamin Franklin. Benjamin Franklin is known for his invention of large ordered magic squares. In his youth, he created an 8 x 8 magic square with magic sum of 260 and a 16 x 16 magic square with sum 2,056. These magic squares have the same properties as the traditional magic square except the diagonals do not add up to the magic sum but, other special properties exist. For example in the 8 x 8 magic square, the sum of half of a row or column is equal to half of 206 and each of the â€Å"bent† rows (as Franklin called them) of 8 numbers total to 206. Many more variations of magic squares and their constructions exist as well as different properties. Attractive patterns are seen by connecting consecutive numbers in a magic square. Alphamagic squares are constructed by using the number of letters in the word for each number, which generates another magic square.

Aether Definition in Alchemy and Science

Aether Definition in Alchemy and Science There are two related science definitions for the term aether, as well as other non-scientific meanings. (1) Aether was the fifth element in alchemical chemistry  and early physics. It was the name given to the material that was believed to fill the universe beyond the terrestrial sphere. The belief in aether as an element was held by medieval alchemists, Greeks, Buddhists, Hindus, the Japanese, and the Tibetan Bon. Ancient Babylonians believed the fifth element to be the sky. The fifth element in the Chinese Wu-Xing was metal rather than aether.(2) Aether was also considered the medium that carried light waves in space by 18th and 19th Century scientists. Luminiferous ether was proposed in order to explain the capacity of light to propagate through apparently empty space. The Michelson-Morley experiment (MMX) led scientists to realize there was no aether and that light was self-propagating. Michelson-Morley Experiment and Aether The MMX experiment was performed at what is now Case Western Reserve University in Cleveland, Ohio in 1887 by Albert A. Michelson and Edward Morley. The experiment used an interferometer to compare the speed of light in perpendicular directions. The point of the experiment was to determine the relative motion of matter through the aether wind or luminiferous aether. It was believed light required a medium in order to move, similar to the way sound waves require a medium (e.g., water or air) to propagate. Since it was known light could travel in a vacuum, it was believed the vacuum must be filled with a substance called aether. Since the Earth would revolve around the Sun through the aether, there would be a relative motion between the Earth and the aether (the aether wind). Thus, the speed of light would be affected by whether the light was moving in the direction of the Earths orbit or perpendicular to it. The negative results were published in the same year and followed up with exp eriments of increased sensitivity. The MMX experiment led to the development of the theory of special relativity, which does not rely on any aether for the propagation of electromagnetic radiation. The Michelson-Morley experiment is considered to be the most famous failed experiment. (3) The word aether or ether may be used to describe apparently empty space. In Homeric Greek, the word aether refers to the clear sky or pure air. It was believed to be the pure essence breathed by gods, while man required air to breathe. In the modern usage, aether simply refers to invisible space (e.g., I lost my email to the aether.) Alternate Spellings: Æther, ether, luminous aether, luminiferous aether, aether wind, light-bearing ether Commonly Confused With: Aether is not the same thing as the chemical substance, ether, which is the name given to a class of compounds containing an ether group. An ether group consists of an oxygen atom connected to two aryl groups or alkyl groups. Aether Symbol in Alchemy Unlike many alchemical elements, aether does not have a commonly accepted symbol. Most often, it was represented by a simple circle.

Kidney infection, cystitis caused by Escherichia Coli (E-Coli) Research Paper - 1

Kidney infection, cystitis caused by Escherichia Coli (E-Coli) - Research Paper Example These infections are usually caused by Escherichia coli (E. coli). This bacteria is present in bowel and intestine. The infection takes place when this bacteria enters into urethra during intercourse or cleaning from back to front after excretion. Entering into urethra and reaching bladder results in cystitis. When this bacteria reaches kidneys, it causes kidney infection i.e. Pyelonephritis. The kidney infection can occur in one kidney or both at the same time. The statistics have shown that ratio of women suffering from these infections is higher than men since it is easier for bacteria to transfer to urethra from surrounding skin. Secondly, the length of urethra is lesser in women which make it easier for bacteria to reach bladder. The kidney infection due to E. Coli, can also take place without cystitis which usually happens when the kidney is blocked or the patient has other health disorders i.e. kidney stone, diabetes, low immunity etc (Gupta et al., 2011; Madappa et al., 2011). Kidney infection can result in further complications such as abscess, kidney inflammation and blood poisoning. In young male patients, posterior urethral valves are the main sites for uncomplicated urinary tract infection whereas older men with prostatic hypertrophy are more prone to it (Mayfield-Blak, 2012). Possible methods of examination may include urinalysis and urine culture where first one will help in examining blood components along with chemicals like nitrites in the urine sample, the later one is intended to examine the nature and content of bacteria for prescription of correct antibiotic. Cytoscopy and imaging tests can also be used (Gupta et al., 2011). Treatment mainly includes intake of antibiotics by mouth or direct insertion into urethra. The intake may continue up to 14 days depending on the nature and severity of the infections. It is important to