Hypotheses

**I. Hypotheses**
 * Wow 5pts[[image:girl_with_pearl.gif width="87" height="134"]] **

When hypothesizing you are giving a possible solution to a problem or situation. Please visit the following link so that you can learn how to write hypotheses and when to use them. [|http://www.accessexcellence.org/LC/TL/filson/writhypo.php]


 * As you could see in the link above, hypotheses are written using modal verbs, like may, could, should. would, and if conditional structures. They can also be written using expresions (__key words__) as probably, possibly, and verbs such as: think, assume, hypothesize, imagine, suppose, guess, believe, among others. When reading a text, the indicators of hypotheses are the previously mentioned grammatical structures and key words. **


 * Read the following information extracted from the web page**: [] on Dec 27th, 2008
 * Hypotheses and mathematics**

So where does mathematics enter into this picture? In many ways, both obvious and subtle: Very often, the situation under analysis will appear to be complicated and unclear. Part of the mathematics of the task will be to impose a clear structure on the problem. The clarity of thought required will actively be developed through more abstract mathematical study. Those without sufficient general mathematical skill will be unable to perform an appropriate logical analysis. (Taken from [] on Dec 27th, 2008)
 * A good hypothesis needs to be clear, precisely stated and testable in some way. Creation of these clear hypotheses requires clear general mathematical thinking.
 * The data from experiments must be carefully analysed in relation to the original hypothesis. This requires the data to be structured, operated upon, prepared and displayed in appropriate ways. The levels of this process can range from simple to exceedingly complex.

There is often confusion between the ideas surrounding proof, which is mathematics, and making and testing an experimental hypothesis, which is science. The difference is rather simple: Of course, to be good at science, you need to be good at deductive reasoning, although experts at deductive reasoning need not be mathematicians. Detectives, such as Sherlock Holmes and Hercule Poirot, are such experts: they collect evidence from a crime scene and then draw logical conclusions from the evidence to support the hypothesis that, for example, Person M. committed the crime. They use this evidence to create sufficiently compelling deductions to support their hypotheses //beyond reasonable doubt//. The key word here is 'reasonable'. There is always the possibility of creating an exceedingly outlandish scenario to explain away any hypothesis of a detective or prosecution lawyer, but judges and juries in courts eventually make the decision that the probability of such eventualities are 'small' and the chance of the hypothesis being correct 'high'. (Taken from [] on Dec 27th, 2008)
 * Using deductive reasoning in hypothesis testing**
 * Mathematics is based on //deductive reasoning// : a proof is a logical deduction from a set of clear inputs.
 * Science is based on //inductive reasoning// : hypotheses are strengthened or rejected based on an accumulation of experimental evidence.

**II. Assignment** 1. Check the following links and explain what deductive reasoning is and inductive reasoning is. []

Deductive reasoning is reasoning based on logic, on facts, on evidence, sometime following obvious deductions, which tend to high posibilities of being rigth.Deductive reasoning is used when it know that something is true whe other statements is true; when there are given relations that are always valid or not at the same time.

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Inductive reasoning is reasoning ased on general facts and trying to create theories explaining relationships between what is being studied and trying to predict its behavior or results. It is usually based on observation and experience, sometimes it also include logical deductions to make conclusions more probable to be true.

2. Please visit the following page and read the text **"Geometrical proportions of the Egyptian Pyramids"** then find and extract the hypotheses in it. There are 6 hypotheses in the text extract 5 and explain how you found them. Geometrical proportions of the Egyptian Pyramids.doc ( I will underline the hypotheses)

__The heptagonal geometrical network of lines is a universal figure which during development of the human civilization was used for measurement of proportional ratio and for creation of objects of world around in which people aspired to fix principles of harmony. Or it is possible to tell, that people aspired to cipher knowledge of world around in the created objects of human culture for what used proportional parities of a heptagon which expressed absolute knowledge. //(//__//I recognized it is an hypotesys becauses it is trying to explaing something it not completely knonw, and also because i found the "key" word, possible and and the sentence is written in conditional structure)// Among many other objects of the world, the most significant monument of human knowledge are the Egyptian Pyramids (Great Pyramid of Giza), and in particular the Great Pyramid of Cheops (Pyramid of Khufu). //I// can not result the detailed analysis of all Egyptian Pyramids and consequently I bring the description only for pyramid of Cheops as this pyramid the most significant. But nevertheless the analysis of geometrical proportions of other Egyptian Pyramids can be made in comparison with pyramid of Cheops. It agrees to different sources in the basic sizes of Cheops' pyramid are: length of the side basis 500 "elbow" (cubits), height 318 cubits, angle of incline of lateral sides 51 degrees 50 minutes (the Egyptian cubit is approximately equal 466 millimeters). __In different sources of the information there are different data on size of the Egyptian cubit, but I think that the size of the Egyptian cubit is equal to 466 millimeters that is taken from sources of the information which the authentic from my point of view, as it is anthropometrical size of a human "elbow" (forearm + palm + fingers).__ //(This one is an hypothesys because it it telling there are differen ideas about a topic, and it is chosing one of them and explainig why).// According to the listed sizes the main proportions of the Cheops' pyramid consist in ratio of the triangle which is formed by height (OP), half of length of the basis (PR) and an apothem (OR) which is length of the lateral side, that is shown on the chart: Angle PRO with top in point R is the angle of incline of lateral sides, and angle PSO with top in point S is the angle of incline of diagonal edges. The angle of incline of lateral sides and the angle of incline of diagonal edges of a pyramid have different magnitudes. Angle PRO and angle PSO are key parameters of the Cheops' pyramid which allow to compare proportions of a pyramid to proportions of a heptagon. || __Many researchers of the Pyramid of Cheops assume, that to builders (architects) of the Egyptian Pyramids knew the number of golden section and number "Pi" but actually in this knowledge there is no necessity, though it is obvious that builders of pyramids knew about "golden numbers" which are ciphered in pyramids.__ //(I recognized this is an Hypothesys because of the "key" word "asume", meaning it is not know at all, but it is believe is true)// For construction of pyramids it is enough to know proportions of a heptagon and to use ratio of lines which exist in the geometrical figure of a heptagon, that is shown in the following chart: The shown silhouette of lateral sides is approximate as the angle of heptagon AEK with top in the point K is equal 360 / 7 = 51,429 degrees (51 degrees 25,71 minutes), and the angle of incline of lateral sides of the Cheops' pyramid is equal 51 degree 50 minutes. Builders compensated an existing difference by means of that: to height of triangle AEK have added magnitude of human growth AX. Namely builders of the Cheops' pyramid have placed a figure of the man in top of the triangle and as a result have received the angle EKX with top in the point K which equal 51 degrees 50 minutes, that differed from the exact angle of a heptagon which is equal 51 degrees 25,71 minutes. Namely if the height of triangle XEK is equal 318 cubits then height of triangle AEK approximately 314 cubits provided that the height of human growth is little bit more than 4 cubits (the detailed information on size of the Egyptian cubits look at the end of this page). || Builders of the Cheops' pyramid have increased the correct angle of a heptagon, as if at top of the pyramid there is a man, and as a result in ratio of lines EK/KX have ciphered number of the golden section and in proportions of the pyramid have ciphered proportions of a human body, that was the project of the pyramid. In essence builders of the Cheops' pyramid have entered the heptagonal network of lines in a living circle in which the size of vertical diameter differed from size of horizontal diameter in relative size of human growth that is shown in the following chart: //__Probably, the concrete ratio of diameters of a living circle in the geometrical drawing of the Cheops' pyramid has other sizes about which I can not tell anything certain as more exact calculations are necessary for this purpose__. (this is an hypothesys, it is using the key word "probbaly", so, it not known for sure) __It is possible to assume, that the ratio of diameters of a living circle in the geometrical drawing of the Cheops' pyramid turns out as a result of transformation of the living circle when size of the line TA is precisely equal to size of lines CE, DF, LJ, MK__. (this is an hypothesys, it starts with"is it possible to asume", meaning it could be true, but it is not sure)// The dark blue contour in the chart specifies an approximate silhouette of lateral sides of the pyramid, and the white contour specifies an approximate silhouette of diagonal edges. Angle DLA with top in point L is the angle of incline of diagonal edges of the Cheops' pyramid which valid magnitude is equal 40 degrees 59 minutes. The magnitude of angle DLA, the same as magnitude of angle EKA with top in the point K (the angle of incline of lateral sides), differs from angular magnitudes of the correct diheptagon in relative size of human growth according to which the living circle is transformed. || The shown geometrical drawings are approximate as for exact geometrical calculations many numbers and formulas are necessary, that is inexpedient in a context of gallery of phantom images, but according to the shown drawings it is possible to do exact geometrical calculations if it is necessary. The given geometrical concepts allow to understand the basic proportions of the Egyptian Pyramids, but besides in geometrical measures of pyramids many other proportional geometrical laws are ciphered, according to which the world around is arranged. //__It is possible to speak that magnitudes of the Egyptian Pyramids have fixed sizes of measurements which allow to understand structure of world around, and allow to apply "Great Egyptian Measures" to designing environmental space and for an arrangement of the objects of the human world created by people__. (once more, __t__his is an hypothesys, it starts with"is it possible ", meaning it is not sure)//
 * || In ratio of lines OR/PR the size of the golden section is ciphered (this is the famous Golden Ratio solution, or Golden Section, also known as the Divine Ratio), and in ratio of lines PR/PO the number "Pi" is ciphered (the famous Pi ).
 * || In the chart triangle AEK is an approximate silhouette of lateral sides of the Cheops' pyramid.
 * || In the chart the diheptagonal network of lines is entered within the framework of a living circle which has a ratio of vertical and horizontal diameters approximately 15 : 14 that corresponds to parities of proportions of a male and female body.

The basic size of measurement in ancient Egypt is the cubit which consist of seven palms, and each palm was subdivided into four fingers. Total the cubit consist of 28 fingers that corresponds to 28 days of lunar month. //Pay attention, that the diheptagonal network of lines has 14 tops, and the quadraheptagonal network of lines has 28 tops, that also is equal to number of days of lunar month.// In a modern science about ancient Egypt there are different parameters of the Egyptian cubit: 1 ordinary Egyptian cubit = 6 palms = 24 fingers = 450 millimeters; 1 royal Egyptian cubit = 7 palms = 28 fingers = 525 millimeters. I have resulted the size of the Egyptian cubit which is equal 466 millimeters that has mathematical sense as it is anthropometrical size of a human elbow. Also other sizes according to which parameters of the Cheops' pyramid are calculated by cubits, correspond to size of the cubit which is equal 466 millimeters. The found panels have images on both sides, that in the sum makes 22 arcane. //There is hypothesis that the found 22 arcane became the reason of an esoteric legend that predictive cards of Tarot have the Egyptian origin.// The measuring rod in a hand of the land surveyor is equal to length from middle of a body up to a line of eyebrows that makes two cubits. If to project the given measuring rod on the diheptagonal network of lines then the rod is equal to line AT, and also it is approximately equal to length of the side of a heptagon that is shown in the previous chart. Namely the measuring rod is equal to lines CE, DF, LJ, MK. Hence, the Egyptian cubit is equal to length of the side of a diheptagon (to distance between two near tops), and any other sizes of measurements are derivatives from the diheptagonal geometrical network in which the figure of a human body is entered, that allows to do the statement about a ratio of the Egyptian cubit with proportions of the heptagonal network and about a ratio of the Egyptian Pyramids with proportions of a human body. But provided that the diheptagonal network is entered in a living circle which forms "an ellipse with a focal length proportional to human growth taken in the attitude to height of the pyramid". Otherwise it is possible to tell that the difference of diameters (difference of the big and small axes of an ellipse) corresponds to the attitude of human growth to height of the Pyramid. //If exact geometrical calculations are not required, then it is possible to count that approximately the cubit is equal to the side of a correct diheptagon which is entered within the framework of a correct circle.// || By means of the measuring rod builders have calculated the sides of pyramids, and in the project of the Khafre's Pyramid (The Pyramid of Chephren) have fixed the Sacred Egyptian Triangle with the attitude of the sides 3 : 4 : 5 which differently name an integral Pythagorean triangle, and in the project of the Cheops' pyramid have fixed "the golden triangle" with the attitude of the sides which answers to the golden section, but provided that The Golden Triangle of the Pyramid of Cheops is calculated by means of the measuring rod according to lines of the diheptagonal geometrical network. The project of the Pyramid of Chephren and the project of the Pyramid of Cheops fix different methods of calculations and consequently pyramids have different esoteric value. // The additional information on the Golden Section (Golden Ratio or Divine Ratio) and on the Egyptian Triangles look in other sources of the information. // //The following page in detail tells about measuring tools which can be derivatives from lines of the heptagonal geometrical network.// Nov 2008 http://www.phantomgallery.64g.ru/pyramid/pyr2en.htm
 * || The photo at the left shows "Arcane" where the land surveyor with the measuring rod (measuring wand) in a hand is represented, namely the photo shows one of 11 wooden carved panels which were found in the tomb of Hesi-Ra who is considered the architect of Pyramids.

3. Look for any mathematical hypothesis and put it in your wiki. Please make sure you cite the source properly so that you do not commit plagiarism. Explain whether the hypothesis you are explaining is deductive or inductive and give reasons to your explanation.

This is actually a very old theorem (strictly speaking, before proof it was a [|conjecture]), known to the ancient Greeks (appearing in a textbook by [|Diophantus]). It goes like this:
 * " [|Fermat's Last Theorem]**

For numbers //n// greater than 2, the equation: a//n// + b//n// = c//n// has no solutions in nonzero integers a, b and c.

By writing a perplexing note in the margin of a textbook, [|Pierre de Fermat] made this theorem his own. Here is Fermat's note (translated from the Latin):

//"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."// Fermat's Last Theorem has a storied history. Because of its simple statement coupled with the difficulty of solution, it has probably created more false proofs than any other mathematical idea. According to a story, the Theorem attracted so many amateur mathematicians and useless correspondence that a university created a form letter for replies, saying:

//Thank you for your submission of a proof of Fermat's Last Theorem. The first error in the proof can be found on line of page. Sincerely,// In early attempts to prove this theorem, many workers attacked it piecemeal (reminiscent of the four-color map theorem), proving it for specific exponents or ranges of exponents, but without attempting to construct a general proof. Finally, in 1993 [|Andrew Wiles] solved the problem with an unconventional approach involving elliptic curves. After a few corrections, his result has come to be regarded as a formal proof of the Theorem. In this case as with the others, a conjecture is made (equivalent to a hypothesis in normal science), evidence is gathered and tested, and a solution is produced that must survive the most detailed scrutiny. Unlike normal science, a formal mathematical proof answers a conjecture in a conclusive way, without the possibility of later refutation, although in some cases a more elegant, shorter proof may supersede an earlier approach on aesthetic grounds. "

(Taken from [] )

Fermat's last theorem proposes that the equation a//n// + b//n// = c//n// can not be solved with integrers when n >2. Whe he postulated this, he said the proof was simple but he did not have space on that sheet of paper to write it down. It took more than a hundred years to find a proof, and when it was finally solved, the proof was based in mathematical tools that where not available at Fermat's time. But, even thought, what Fermat's postulated results to be true. Some people believe he never had a proof when he published it, but his idea got true anyway. So, if he did not have a proof, this was an hypothesis by inductive reasoning, he could probably write down what he thought was right; maybe he made a general proposal about some cases he actually knew. But know that the theorem is proved, it is no longer an inductive hypothesis, because it was pproved using logical trustable proceedures.