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| The '''Huzita–Hatori axioms''' or '''Huzita–Justin axioms''' are a set of rules related to the [[Mathematics of paper folding|mathematical principles of paper folding]], describing the operations that can be made when folding a piece of paper. The axioms assume that the operations are completed on a plane (i.e. a perfect piece of paper), and that all folds are linear. These are not a minimal set of axioms but rather the complete set of possible single folds.
| | , 72% of Americans have tried to get rid of weight at least when. However, various individuals that try to get rid of fat fail. Why is the fact that? Perhaps advertising got in the way. Though there are programs that promote healthy weight reduction, a few of the items advertisers want we to believe will really prevent healthy fat control. Here are 3 of the many usual lies advertisers need you to believe.<br><br>Over the last hundred years the average person has gotten taller and seems to carry more muscle mass. The result is the fact that BMI calculations tend to be a little bit off, many individuals will read high than they actually are. Nevertheless for most folks the results are nevertheless fairly accurate. If you are especially tall or you may be carrying a lot of muscle be prepared for the charts to tell we you are overweight.<br><br>As of 2010 the percentage of Americans with obesity is over 35% plus steadily growing. In 1985 less than 15% of Americans were fat. Obesity is defined has having a BMI (Body Mass Index) of 30 or high. For someone that is 5'9" that is 203 pounds or higher. For someone that is 5'5" which is regarding 180 pounds or high.<br><br>A [http://safedietplansforwomen.com/bmi-calculator bmi calculator] is important tool to have specifically for females since they are more prone to get fat deposits than guys. Hormonal changes are the many significant reason. Women are moreover proven to emotionally devour food than guy. When ladies are depressed, happy or tired they tend to look for comfort foods. Women are structurally different than man to accommodate the growing baby inside the abdominal area. Plus a healthy woman (21-36%) would have twice more fat compared to a healthy man (8-25%). Since women start off with a better fat percentage than men, then it's not surprising which they are more liable to be overweight.<br><br>This really is the case with a individual i learn. She's 63, 5'9" and pretty close to 300 lbs. While she will walk, inside a limited way, her efforts to get rid of fat are really not functioning. She has other problems too like CHF plus osteoporosis. Her target weight is about 180, thus thats over 100 lbs she must loose. I devised this program for her!<br><br>This means which the question of the ideal weight range for athletic females in their 40's is a tough nut to crack. The body is the body, regardless what age you may be and regardless what amount of bodily activity you do. Ladies, if you need to have the body of a 21-year-old, you are able to do it.<br><br>If you don't fall inside the regular range, then receive yourself checked with other (omit) methods to figure out the amount of body fat. This usually provide a greater perspective and help you to do or keep up your ideal fat. |
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| The axioms were first discovered by Jacques Justin in 1989.<ref>Justin, Jacques, "Resolution par le pliage de l'equation du troisieme degre et applications geometriques", reprinted in ''Proceedings of the First International Meeting of Origami Science and Technology'', H. Huzita ed. (1989), pp. 251–261.</ref> Axioms 1 through 6 were rediscovered by [[Italy|Italian]]-[[Japan]]ese mathematician [[Humiaki Huzita]] and reported at ''the First International Conference on Origami in Education and Therapy'' in 1991. Axioms 1 though 5 were rediscovered by Auckly and Cleveland in 1995. Axiom 7 was rediscovered by Koshiro Hatori in 2001; [[Robert J. Lang]] also found axiom 7.
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| ==The seven axioms==
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| The first 6 axioms are known as Huzita's axioms. Axiom 7 was discovered by Koshiro Hatori. Jacques Justin and [[Robert J. Lang]] also found axiom 7. The axioms are as follows:
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| # Given two points ''p''<sub>1</sub> and ''p''<sub>2</sub>, there is a unique fold that passes through both of them.
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| # Given two points ''p''<sub>1</sub> and ''p''<sub>2</sub>, there is a unique fold that places ''p''<sub>1</sub> onto ''p''<sub>2</sub>.
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| # Given two lines ''l''<sub>1</sub> and ''l''<sub>2</sub>, there is a fold that places ''l''<sub>1</sub> onto ''l''<sub>2</sub>.
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| # Given a point ''p''<sub>1</sub> and a line ''l''<sub>1</sub>, there is a unique fold perpendicular to ''l''<sub>1</sub> that passes through point ''p''<sub>1</sub>.
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| # Given two points ''p''<sub>1</sub> and ''p''<sub>2</sub> and a line ''l''<sub>1</sub>, there is a fold that places ''p''<sub>1</sub> onto ''l''<sub>1</sub> and passes through ''p''<sub>2</sub>.
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| # Given two points ''p''<sub>1</sub> and ''p''<sub>2</sub> and two lines ''l''<sub>1</sub> and ''l''<sub>2</sub>, there is a fold that places ''p''<sub>1</sub> onto ''l''<sub>1</sub> and ''p''<sub>2</sub> onto ''l''<sub>2</sub>.
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| # Given one point ''p'' and two lines ''l''<sub>1</sub> and ''l''<sub>2</sub>, there is a fold that places ''p'' onto ''l''<sub>1</sub> and is perpendicular to ''l''<sub>2</sub>.
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| Axiom 5 may have 0, 1, or 2 solutions, while Axiom 6 may have 0, 1, 2, or 3 solutions. In this way, the resulting geometries of origami are stronger than the geometries of [[compass and straightedge]], where the maximum number of solutions an axiom has is 2. Thus compass and straightedge geometry solves second-degree equations, while origami geometry, or origametry, can solve third-degree equations, and solve problems such as [[angle trisection]] and [[doubling of the cube]]. However, in practice the construction of the fold guaranteed by Axiom 6 requires "sliding" the paper, or [[neusis]], which is not allowed in classical compass and straightedge constructions. Use of neusis together with a compass and straightedge does allow trisection of an arbitrary angle.
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| ==Details==
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| ===Axiom 1===
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| Given two points ''p''<sub>1</sub> and ''p''<sub>2</sub>, there is a unique fold that passes through both of them.
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| [[Image:Huzita axiom 1.png|Folding a line through two points]]
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| In parametric form, the equation for the line that passes through the two points is :
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| :<math>F(s)=p_1 +s(p_2 - p_1).</math>
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| ===Axiom 2===
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| Given two points ''p''<sub>1</sub> and ''p''<sub>2</sub>, there is a unique fold that places ''p''<sub>1</sub> onto ''p''<sub>2</sub>.
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| [[Image:Huzita axiom 2.png|Folding a line putting one point on another]]
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| This is equivalent to finding the perpendicular bisector of the line segment ''p''<sub>1</sub>''p''<sub>2</sub>. This can be done in four steps:
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| * Use '''Axiom 1''' to find the line through ''p''<sub>1</sub> and ''p''<sub>2</sub>, given by <math>P(s)=p_1+s(p_2-p_1)</math>
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| * Find the [[midpoint]] of ''p''<sub>mid</sub> of ''P''(''s'')
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| * Find the vector '''v'''<sup>perp</sup> perpendicular to ''P''(''s'')
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| * The [[parametric equation]] of the fold is then:
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| :<math>F(s)=p_\mathrm{mid} + s\cdot\mathbf{v}^{\mathrm{perp}}.</math>
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| ===Axiom 3===
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| Given two lines ''l''<sub>1</sub> and ''l''<sub>2</sub>, there is a fold that places ''l''<sub>1</sub> onto ''l''<sub>2</sub>.
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| [[Image:Huzita axiom 3.png|Folding a line putting one line on another]]
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| This is equivalent to finding a bisector of the angle between ''l''<sub>1</sub> and ''l''<sub>2</sub>. Let ''p''<sub>1</sub> and ''p''<sub>2</sub> be any two points on ''l''<sub>1</sub>, and let ''q''<sub>1</sub> and ''q''<sub>2</sub> be any two points on ''l''<sub>2</sub>. Also, let '''u''' and '''v''' be the unit direction vectors of ''l''<sub>1</sub> and ''l''<sub>2</sub>, respectively; that is:
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| :<math>\mathbf{u} = (p_2-p_1) / \left|(p_2-p_1)\right|</math>
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| :<math>\mathbf{v} = (q_2-q_1) / \left|(q_2-q_1)\right|.</math>
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| If the two lines are not parallel, their point of intersection is:
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| :<math>p_\mathrm{int} = p_1+s_\mathrm{int}\cdot\mathbf{u}</math>
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| where
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| :<math>s_{int} = -\frac{\mathbf{v}^{\perp} \cdot (p_1 - q_1)} {\mathbf{v}^{\perp} \cdot \mathbf{u}}.</math>
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| The direction of one of the bisectors is then:
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| :<math>\mathbf{w} = \frac{
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| \left|\mathbf{u}\right| \mathbf{v} +
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| \left|\mathbf{v}\right| \mathbf{u}}
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| {\left|\mathbf{u}\right| +
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| \left|\mathbf{v}\right|}.</math>
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| And the parametric equation of the fold is:
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| :<math>F(s) = p_\mathrm{int} + s\cdot\mathbf{w}.</math>
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| A second bisector also exists, perpendicular to the first and passing through ''p''<sub>int</sub>. Folding along this second bisector will also achieve the desired result of placing ''l''<sub>1</sub> onto ''l''<sub>2</sub>. It may not be possible to perform one or the other of these folds, depending on the location of the intersection point.
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| If the two lines are parallel, they have no point of intersection. The fold must be the line midway between ''l''<sub>1</sub> and ''l''<sub>2</sub> and parallel to them.
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| ===Axiom 4===
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| Given a point ''p''<sub>1</sub> and a line ''l''<sub>1</sub>, there is a unique fold perpendicular to ''l''<sub>1</sub> that passes through point ''p''<sub>1</sub>.
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| [[Image:Huzita axiom 4.png|Folding through a point perpendicular to a line]]
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| This is equivalent to finding a perpendicular to ''l''<sub>1</sub> that passes through ''p''<sub>1</sub>. If we find some vector '''v''' that is perpendicular to the line ''l''<sub>1</sub>, then the parametric equation of the fold is:
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| :<math>F(s) = p_1 + s\cdot\mathbf{v}.</math>
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| ===Axiom 5===
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| Given two points ''p''<sub>1</sub> and ''p''<sub>2</sub> and a line ''l''<sub>1</sub>, there is a fold that places ''p''<sub>1</sub> onto ''l''<sub>1</sub> and passes through ''p''<sub>2</sub>.
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| [[Image:Huzita axiom 5.png|Folding a point onto a line through another point]]
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| This axiom is equivalent to finding the intersection of a line with a circle, so it may have 0, 1, or 2 solutions. The line is defined by ''l''<sub>1</sub>, and the circle has its center at ''p''<sub>2</sub>, and a radius equal to the distance from ''p''<sub>2</sub> to ''p''<sub>1</sub>. If the line does not intersect the circle, there are no solutions. If the line is tangent to the circle, there is one solution, and if the line intersects the circle in two places, there are two solutions.
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| If we know two points on the line, (''x''<sub>1</sub>, ''y''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>), then the line can be expressed parametrically as:
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| :<math>x = x_1 + s(x_2 - x_1)\,</math>
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| :<math>y = y_1 + s(y_2 - y_1).\,</math>
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| Let the circle be defined by its center at ''p''<sub>2</sub>=(''x<sub>c</sub>'', ''y<sub>c</sub>''), with radius <math>r = \left|p_1 - p_2\right|</math>. Then the circle can be expressed as:
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| :<math>(x-x_c)^2 + (y-y_c)^2 = r^2.\,</math>
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| In order to determine the points of intersection of the line with the circle, we substitute the ''x'' and ''y'' components of the equations for the line into the equation for the circle, giving:
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| :<math>(x_1 + s(x_2-x_1) - x_c)^2 + (y_1 + s(y_2 - y_1) - y_c)^2 = r^2.\,</math>
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| Or, simplified:
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| :<math>as^2 + bs + c = 0\,</math>
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| where:
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| :<math>a = (x_2 - x_1)^2 + (y_2 - y_1)^2\,</math>
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| :<math>b = 2(x_2 - x_1)(x_1 - x_c) + 2(y_2 - y_1)(y_1 - y_c)\,</math>
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| :<math>c = x_c^2 + y_c^2 + x_1^2 + y_1^2 - 2(x_c x_1 + y_c y_1)-r^2.\,</math>
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| Then we simply solve the quadratic equation:
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| :<math>\frac{-b\pm\sqrt{b^2-4ac}}{2a}.</math>
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| If the discriminant ''b''<sup>2</sup> − 4''ac'' < 0, there are no solutions. The circle does not intersect or touch the line. If the discriminant is equal to 0, then there is a single solution, where the line is tangent to the circle. And if the discriminant is greater than 0, there are two solutions, representing the two points of intersection. Let us call the solutions ''d''<sub>1</sub> and ''d''<sub>2</sub>, if they exist. We have 0, 1, or 2 line segments:
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| :<math>m_1 = \overline{p_1 d_1} \, </math>
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| :<math>m_2 = \overline{p_1 d_2}. \, </math>
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| A fold ''F''<sub>1</sub>(''s'') perpendicular to ''m''<sub>1</sub> through its midpoint will place ''p''<sub>1</sub> on the line at location ''d''<sub>1</sub>. Similarly, a fold ''F''<sub>2</sub>(''s'') perpendicular to ''m''<sub>2</sub> through its midpoint will place ''p''<sub>1</sub> on the line at location ''d''<sub>2</sub>. The application of Axiom 2 easily accomplishes this. The parametric equations of the folds are thus:
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| :<math>
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| \begin{align}
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| F_1(s) & = p_1 +\frac{1}{2}(d_1-p_1)+s(d_1-p_1)^\perp \\[8pt]
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| F_2(s) & = p_1 +\frac{1}{2}(d_2-p_1)+s(d_2-p_1)^\perp.
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| \end{align}
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| </math>
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| ===Axiom 6===
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| Given two points ''p''<sub>1</sub> and ''p''<sub>2</sub> and two lines ''l''<sub>1</sub> and ''l''<sub>2</sub>, there is a fold that places ''p''<sub>1</sub> onto ''l''<sub>1</sub> and ''p''<sub>2</sub> onto ''l''<sub>2</sub>.
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| [[Image:Huzita axiom 6.png]]
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| This axiom is equivalent to finding a line simultaneously tangent to two parabolas, and can be considered equivalent to solving a third-degree equation as there are in general three solutions. The two parabolas have foci at ''p''<sub>1</sub> and ''p''<sub>2</sub>, respectively, with directrices defined by ''l''<sub>1</sub> and ''l''<sub>2</sub>, respectively.
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| This fold is called the Beloch fold after Margharita P. Beloch who in 1936 showed using it that origami can be used to solve general cubic equations.<ref>{{cite journal |title=Solving Cubics With Creases: The Work of Beloch and Lill |author=Thomas C. Hull |url=http://mars.wne.edu/~thull/papers/amer.math.monthly.118.04.307-hull.pdf |journal=American Mathematical Monthly |date=April 2011 |pages=307–315|doi=10.4169/amer.math.monthly.118.04.307}}</ref> | |
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| ===Axiom 7===
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| Given one point ''p'' and two lines ''l''<sub>1</sub> and ''l''<sub>2</sub>, there is a fold that places ''p'' onto ''l''<sub>1</sub> and is perpendicular to ''l''<sub>2</sub>.
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| [[Image:Huzita-Hatori axiom 7.png]]
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| This axiom was originally discovered by Jacques Justin in 1989 but was overlooked and was rediscovered by Koshiro Hatori in 2002.<ref>{{cite journal |title=One-, Two-, and Multi-Fold Origami Axioms |url=http://www.math.sjsu.edu/~alperin/AlperinLang.pdf |author1=Roger C. Alperin |author2=Robert J. Lang |journal=4OSME |publisher=A K Peters |year=2009}}</ref> [[Robert J. Lang]] has proven that this list of axioms completes the axioms of origami.
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| ==Constructibility==
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| Subsets of the axioms can be used to construct different sets of numbers. The first three can be used with three given points not on a line to do what Alpern calls Thalian constructions.<ref>{{cite journal |title=A Mathematical Theory of Origami Constructions and Numbers |first1=Roger C |last1=Alperin |journal=New York Journal of Mathematics |year=2000 |volume=6 |pages=119–133 |url=http://nyjm.albany.edu/j/2000/6-8.pdf}}</ref>
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| The first four axioms with two given points define a system weaker than [[compass and straightedge constructions]]: every shape that can be folded with those axioms can be constructed with compass and straightedge, but some things can be constructed by compass and straightedge that cannot be folded with those axioms.<ref>{{cite journal|authors=D. Auckly and J. Cleveland|title=Totally real origami and impossible paperfolding|journal=American Mathematical Monthly|issue=102|pages=pp. 215–226|arxiv=math/0407174 }}</ref> The numbers that can be constructed are called the origami or pythagorean numbers, if the distance between the two given points is 1 then the constructible points are all of the form <math>(\alpha,\beta)</math> where <math>\alpha</math> and <math>\beta</math> are Pythagorean numbers. The Pythagorean numbers are given by the smallest field containing the rational numbers and <math>\sqrt{1+\alpha^2}</math> whenever <math>\alpha</math> is such a number.
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| Adding the fifth axiom gives the [[Euclidean number]]s, that is the points constructible by [[straightedge and compass]] constructions.
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| Adding the [[Neusis construction|neusis]] axiom 6, the reverse becomes true: all compass-straightedge constructions, and more, can be made. In particular, the [[Constructible polygon|constructible]] regular polygons with these axioms are those with <math>2^a3^b\rho\ge3</math> sides, where <math>\rho</math> is a product of distinct [[Pierpont prime]]s. Compass-straightedge constructions allow only those with <math>2^a\phi\ge3</math> sides, where <math>\phi</math> is a product of distinct [[Fermat prime]]s. (Fermat primes are a [[subset]] of Pierpont primes.)
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| The seventh axiom does not allow construction of further point. The seven axioms give all the single fold constructions that can be done rather than being a minimal set of axioms.
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| ==References==
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| <references/>
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| ==External links==
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| {{Portal|Origami}}
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| * [http://mars.wnec.edu/~th297133/omfiles/geoconst.html Origami Geometric Constructions] by Thomas Hull
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| * [http://nyjm.albany.edu:8000/j/2000/6-8.html A Mathematical Theory of Origami Constructions and Numbers] by Roger C. Alperin
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| * {{cite journal
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| | author = [[Robert J. Lang|Lang, Robert J.]]
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| | title = Origami and Geometric Constructions
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| | publisher = Robert J. Lang
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| | year = 2003
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| | url = http://www.langorigami.com/science/math/hja/origami_constructions.pdf
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| | format = PDF
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| | accessdate = 2007-04-12
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| }}
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| {{DEFAULTSORT:Huzita-Hatori Axioms}}
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| [[Category:Geometry]]
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| [[Category:Mathematical axioms]]
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| [[Category:Paper folding]]
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| [[Category:Recreational mathematics]]
| |
, 72% of Americans have tried to get rid of weight at least when. However, various individuals that try to get rid of fat fail. Why is the fact that? Perhaps advertising got in the way. Though there are programs that promote healthy weight reduction, a few of the items advertisers want we to believe will really prevent healthy fat control. Here are 3 of the many usual lies advertisers need you to believe.
Over the last hundred years the average person has gotten taller and seems to carry more muscle mass. The result is the fact that BMI calculations tend to be a little bit off, many individuals will read high than they actually are. Nevertheless for most folks the results are nevertheless fairly accurate. If you are especially tall or you may be carrying a lot of muscle be prepared for the charts to tell we you are overweight.
As of 2010 the percentage of Americans with obesity is over 35% plus steadily growing. In 1985 less than 15% of Americans were fat. Obesity is defined has having a BMI (Body Mass Index) of 30 or high. For someone that is 5'9" that is 203 pounds or higher. For someone that is 5'5" which is regarding 180 pounds or high.
A bmi calculator is important tool to have specifically for females since they are more prone to get fat deposits than guys. Hormonal changes are the many significant reason. Women are moreover proven to emotionally devour food than guy. When ladies are depressed, happy or tired they tend to look for comfort foods. Women are structurally different than man to accommodate the growing baby inside the abdominal area. Plus a healthy woman (21-36%) would have twice more fat compared to a healthy man (8-25%). Since women start off with a better fat percentage than men, then it's not surprising which they are more liable to be overweight.
This really is the case with a individual i learn. She's 63, 5'9" and pretty close to 300 lbs. While she will walk, inside a limited way, her efforts to get rid of fat are really not functioning. She has other problems too like CHF plus osteoporosis. Her target weight is about 180, thus thats over 100 lbs she must loose. I devised this program for her!
This means which the question of the ideal weight range for athletic females in their 40's is a tough nut to crack. The body is the body, regardless what age you may be and regardless what amount of bodily activity you do. Ladies, if you need to have the body of a 21-year-old, you are able to do it.
If you don't fall inside the regular range, then receive yourself checked with other (omit) methods to figure out the amount of body fat. This usually provide a greater perspective and help you to do or keep up your ideal fat.