81. The Equiangular Spiral Spira Mirabilis Certainly, most will have witnessed the sublime geometry of a spiral in the vortex of The connection between the equiangular spiral, the golden mean, http://online.redwoods.cc.ca.us/instruct/darnold/CalcProj/Sp98/GabeP/Spiral.htm

Gabriel Peterson The Equiangular Spiral: Spira Mirabilis Gabriel Peterson Math 50C: Multivariable Calculus Abstract The purpose of this article is to develop the parametric equation for the plane curve of the equiangular spiral – also known as the logarithmic spiral or the logistique – from its geometric definition. Through analysis, it was found to be where a is an arbitrary constant and a is a constant angle made between the radial vector to any point on the curve and the tangent line at that point. Introduction Most readers will have had at least some informal introduction to the elegant beauty of the spiral prior to reading this. Certainly, most will have witnessed the sublime geometry of a spiral in the vortex of water rushing down a drain, the tempest of a hurricane, the hypnotic pattern of a magician’s spiral, the shape of a snail’s shell, the inscrutable movement of the arms of a spiral galaxy like our own... The formal mathematical study of spirals – as did so many other rich intellectual traditions present today – began with the ancient Greeks. Building upon the work of Conon of Samos, Archimedes of Syracuse wrote a definitive early work on the subject in 250 BC entitled On Spirals.

82. Links For Informal Geometry geometry through Art; Geometer s Sketchpad; golden mean, golden Ratio Etc . Fascinating Flat Facts about Phi (golden mean and golden section, http://euler.slu.edu/teachmaterial/hyperlinks_for_geometry.html

Faculty Undergraduate Math Undergraduate CS Graduate Program ... Teaching Resources Links to related topics in Informal Geometry (Courtesy Dr. R Freese) Contents: Area and Volume Circles Crystals Drawing ... Reference (NCTM) Area and Volume Experiment with Volume (simle worksheet, estimating volume) http://math.rice.edu/~lanius/Geom/cyls.html Volume Functions: (Worksheet)Calculating volumes http://math.rice.edu/~lanius/Geom/cyls2.html Geoboard Area : a worksheet exploring area. (Based on the geoboard) Circles Designs with Circles: construction of circles (rotation and reflection symmetry) http://forum.swarthmore.edu/alejandre/circles.html Percentage Circle http://forum.swarthmore.edu/~sarah/shapiro/gsp.percent.circle.gif Crystals Nets of crystals: paper construction of crystals, polyhedrahttp://forum.swarthmore.edu/alejandre/workshops/crystalnet.html Systems of Crystals (type of crystals, and physical examples) http://forum.swarthmore.edu/alejandre/workshops/chart.html Real World - Viewing Crystals (applications of Tesselations in the real world) http://forum.swarthmore.edu/alejandre/workshops/crystal.html Drawing Isometric Drawing http://forum.swarthmore.edu/workshops/sum98/participants/sanders/Isom.html

83. Feng Shui Seminars With Roger Green When this EI number gets to a multiple of golden mean (such as .62), we have found geometry puzzle which the golden mean Ratio (PhiLotaxes) solves. http://www.fengshuiseminars.com/articles/heartlink.htm

84. Nrich.maths.org::Mathematics Enrichment::Golden Mathematics If these ratios are equal then the point is called the golden mean and the ratiosare Moving from geometry to algebra the next challenge golden Powers http://nrich.maths.org/public/viewer.php?obj_id=2787&part=index&refpage=monthind

85. BuildPhiSpiral A Geometric Construction of the golden mean Spiral. First we will show how togenerate a golden mean Spiral geometrically. This is not the only way it can http://kjmaclean.com/Geometry/BuildPhiSpiral.html

A Geometric Construction of the Golden Mean Spiral First we will show how to generate a Golden Mean Spiral geometrically. This is not the only way it can be done, but it is an understandable way. It is also elegant, as the whole thing begins with a line AB of any length: Let's assume that the length of line AB is Unity, or 1. Now, bisect the line AB and transfer that distance to BC: BC is one-half AB, and is perpendicular to AB. Connect A to C with the straight-edge. Pin the compass at C and place the marking leg at B. Draw an arc to intersect AC at D. Now CD = CB. Now pin the compass at A and place the marking leg at D. Draw an arc to intersect the line AB at E. Now AE = AD. The line AB has now been divided in Mean and Extreme Ratio at E: The point of drawing this triangle is to construct a golden mean rectangle. From the golden mean rectangle we shall build the golden mean spiral. It's pretty cool the way it works out. Refer to the next drawing below: With the straight-edge, extend the line CB. Now, pin the compass at C and place the marking leg at A. Draw an arc to G and extend line CB to intersect the extended line CB, at G.

86. Why Befunge Is Better Than Perl The golden Ratio (also known as the divine proportion, the golden mean, You can use the golden Ratio in geometry, which is then called golden geometry. http://articles.mongueurs.net/spoofathon/befunge.html

Les Mongueurs de Perl Groupe de travail Articles Nous rejoindre CVS ... Planning Publications : LinuxMag Perles Comptes-Rendus Traductions ... Spoofathon Index Why Befunge is better than Perl A little mathematical reminder... The Golden Ratio The Golden Rectangle ... Conclusion Why Befunge is better than Perl Everyone knows Befunge, the famous programming language using a 2-dimensional cartesian Funge. It's obvious that Befunge, by its ease of use, and powerful syntax, is better than Perl. But very few people know that Befunge being the best language ever can be proven mathematically. A little mathematical reminder... The Golden Ratio Whoever has already done some mathematics already knows the Golden Ratio . The Golden Ratio (also known as the divine proportion , the golden mean , or the golden section ) is often encountered in the mathematics field. Supposedly discovered by Pythagoras, it is a Pisot-Vijayaraghavan constant, and has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers. But the Golden Ratio is in fact ubiquitous, and has some fascinating connections with a lot of theorems or well-known mathematical curiosities (such as the Fibonacci numbers).

87. Sacred Places: Sacred Sites, Places And Temples In India Sacred geometry and Sacred Space Spiritual Art, Music Dance indian treasures,the golden mean, pilgrimage tradition, pilgrimage sites, sacred space, http://www.experiencefestival.com/a/Sacred_Places/id/2328

var amazon_search = 'Sacred+Places'; Articles Archives Start page News Contact Community Experience Festival World University General Newsletter Contact information Site map Most recommended Search the site Archive Photo Archive Video Archive Articles Archive ... Site map Sacred Places Sacred Places: Sacred Sites, places and temples in India In India we find the oldest continually operating pilgrimage tradition in the entire world. The practice of pilgrimage in India is so deeply embedded in the cultural psyche and the number of pilgrimage sites is so large that the entire subcontinent may actually be regarded as one grand and continuous sacred space Martin Gray spent the last 20 years visiting more than thousand sacred sites around the world and he guides us to some of the sacred sites of India sacred sites holy place Sacred Places ... indian treasures , the golden mean pilgrimage tradition pilgrimage sites sacred space ... India Sacred Places: Sacred Sites, places and temples in India By Martin Gray India is a vast country , peopled with diverse and ancient civilizations, and its religious

88. THE FIBONACCI SEQUENCE, SPIRALS AND THE GOLDEN MEAN THE FIBONACCI SEQUENCE, SPIRALS AND THE golden mean Just in terms of puremathematics number theory, geometry and so on - the scope of his idea was so http://www.math.temple.edu/~reich/Fib/fibo.html

THE FIBONACCI SEQUENCE, SPIRALS AND THE GOLDEN MEAN The Fibonacci sequence exhibits a certain numerical pattern which originated as the answer to an exercise in the first ever high school algebra text. This pattern turned out to have an interest and importance far beyond what its creator imagined. It can be used to model or describe an amazing variety of phenomena, in mathematics and science, art and nature. The mathematical ideas the Fibonacci sequence leads to, such as the golden ratio, spirals and self- similar curves, have long been appreciated for their charm and beauty, but no one can really explain why they are echoed so clearly in the world of art and nature. The story began in Pisa, Italy in the year 1202. Leonardo Pisano Bigollo was a young man in his twenties, a member of an important trading family of Pisa. In his travels throughout the Middle East, he was captivated by the mathematical ideas that had come west from India through the Arabic countries. When he returned to Pisa he published these ideas in a book on mathematics called Liber Abaci , which became a landmark in Europe. Leonardo, who has since come to be known as

89. Loving Unified Field Geometry - Mountain Man's UseNet Archive fractal pattern compression, optimized by golden mean/phi Most of us recognizethat sound is a geometry of pressure . And those of us who have heard of http://www.mountainman.com.au/news96_2.html

Mountain Man's UseNet Archive Loving Unified Field Geometry for Softies by Dan Winter Web Publication by Mountain Man Graphics, Australia in the Southern Summer of 1995 Loving Unified Field Geometry for Softies by Dan Winter Date: 3 Jan 1996 From: Dan Winter To: sci.physics Subject: Quanta and Consciousness The Heart, Fractals and the Universe >> The quanta of consciousness is recursion.. >> as is the branching, the quantum collapse to center >> called gravity occurs only in the presence of >> fractal pattern compression, optimized by golden mean/phi >> (idealized translation vorticity).. >> fractality defines/limits spin density/ mass >> awareness-distributed as spin-onsets with recursion in harmonic >> system, dodeca(dna/earthgrid/zodiac) permits cross scalar >> info transfer in as above/so below, pressure fractal.. >> if time is spin measured and the time spiral is fractal(mayan), >> then when does it become self aware? "You Couldn't Get a Better Squeeze".. Loving Unified Field Geometry for Softies. by Dan Winter 12/24/95 On the side of the box containing a well designed crank operated Tomato Juicer, is printed the motto of today's inspiration: "You Couldn't Get a Better Squeeze". The news for preservers of sweet memories is juicy. Properly applied pressure, sorts waves into preserveable order. The key operating component of the Tomato juicer is a human operated turning spiral helix engraved on a cone vortex. What emerges from the apex of the cone in the shade of living blood, is a harvest of sweetness.

90. The 'Phinest' Source To The Golden Section, Golden Mean, Divine Proportion, Fibo The golden Section, Ratio or mean, Geometric construction of phi in the goldensection and rectangle. The Divine Proportion. The Fibonacci Series, 0, 1, 1, http://goldennumber.net/

GoldenNumber.net Home Contact Meet the Phi Guy "Phriends" in Phi Site Search ... News Dedicated to providing you with the phinest information on The Golden Section , Ratio or Mean The Divine Proportion The Fibonacci Series and the golden number, Phi ( NEW PhiMatrix Graphic Design Software Free Download Site Sections: Overview Design/Composition Life Mathematics ... Other Resources Welcome to GoldenNumber.Net - The Phi Source GoldenNumber.Net exists to share information on the pervasive appearance of Phi in life and the universe. Its goal is to present a broad sampling of phi related topics in an engaging and easy-to-understand format and to provide an online community (aka The Phi Nest), in which new findings about Phi can be shared. Select an area of interest below or START HERE . Enjoy the 'phi nomenon'! SITE CONTENTS Overview Design/Composition Life Mathematics ... Universe News: The Da Vinci Code Dan Brown's book "

91. Fibonacci Numbers And The Golden Section In Art, Architecture And Music golden sections in Violin construction; Did Mozart use the golden mean? and several geometric diagrams you will also have seen on other pages here. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html

Fibonacci Numbers and The Golden Section in Art, Architecture and Music This section introduces you to some of the occurrences of the Fibonacci series and the Golden Ratio in architecture, art and music. Contents of this page The icon means there is a Things to do investigation at the end of the section. The golden section in architecture The Parthenon and Greek Architecture Modern Architecture The Eden Project's new Education Building ... Links to Art sources including Contemporary Artists Fibonacci and Phi in fashioning Furniture Fibonacci in Films Fibonacci and Poetry ... More The Golden section in architecture The Parthenon and Greek Architecture The ancient Greeks knew of a rectangle whose sides are in the golden proportion (1 : 1.618 which is the same as 0.618 : 1). It occurs naturally in some of the proportions of the Five Plato nic Solids (as we have already seen) . A construction for the golden section point is found in Euclid 's Elements. The golden rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple, the Parthenon, in the Acropolis in

92. The Golden Rectangle And The Golden Ratio The golden ratio (and the golden triangle) shows up (as an exact fit) in I have since found the construction in geometry, by Harold R. Jacobs. http://www.jimloy.com/geometry/golden.htm

Return to my Mathematics pages Go to my home page The Golden Rectangle and the Golden Ratio click here for the alternative Golden Rectangle and Golden Ratio page This diagram shows a golden rectangle (roughly). I have divided the rectangle into a square and a smaller rectangle. In a golden rectangle, the smaller rectangle is the same shape as the larger rectangle, in other words, their sides are proportional. In further words, the two rectangles are similar. This can be used as the definition of a golden rectangle. The proportions give us: a/b = (a+b)/a This fraction, (a+b)/a, is called the golden ratio (or golden section or golden mean). Above I have defined the golden rectangle, and then said what the golden ratio is, in terms of the rectangle. Alternatively, I could have defined the golden ratio, using the above equation. And then a golden rectangle becomes any rectangle that exhibits this ratio. From our equation, we see that the ratio a/b=1/2+sqr(5)/2 -1/2+sqr(5)/2 or 0.61803398875 . . .) is called the golden ratio. Also, other mathematical quantities are called phi. The golden ratio is also called tau. Some people call the bigger one (1.61803398875 . . .) Phi (an uppercase phi) and the smaller one (0.61803398875 . . .) phi. By the way, a more accurage value is 1.6180339887498948482045868343656 . . .

93. The Archdome - Golden Mean Design golden mean The Archdome enfolds the golden spiral (also known as the using sacred geometric proportion, better known as the golden mean proportion. http://www.archdome.com/goldenmean.html

T he A rchdome™ • The Golden Mean “Space is naturally curved; people were never meant to be boxed in." Jocelyn Eastland, Graduate Architect Observe the gentle curves of nature. They are based on a mathematical proportion called the Golden Mean. The Golden Mean goes on forever and ever. The whole universe is based on that mathematical proportion . . . your body, the fish, the trees, the galaxy, tornadoes and the flow of wind and water... Sacred Geometry and the Archdome™ The Archdome™ is built using sacred geometric proportion, better known as the Golden Mean proportion. In mathematical terms, the Golden Mean is a ratio of the irrational number .618.... : 1 or 1 : 1.618..... This means that if you constructed a rectangle (or a wall) to Golden Mean proportions, its sides would be in a ratio of 1 (foot, yard, inch whatever) to 1.618... times that. For example: If a wall was 10 feet in height, to be in Golden Mean proportion its length would have to be either 16.18 or 6.18 feet. Now 1.618033988875.... is an irrational number which goes on forever and ever and ever. So what happens when you build a wall using this endless Golden Mean ratio of proportion? Well the wall is (supposedly) a finite structure. However, when it is built using this proportion, a dynamic is created, and the dynamic is this: The wall, as the physical representation of the Golden Mean ratio, is a continuum that approaches the Infinite, but never reaches it: It approaches perfect balance, the Ideal, but can never capture it. The dynamic essence of any structure using this proportion, is endless motion towards perfection.

94. Talk:Golden Ratio - Wikipedia, The Free Encyclopedia Work remains in this article to describe the golden mean better to the Conjugate is used with two geometric objects that have a reciprocal relation. http://en.wikipedia.org/wiki/Talk:Golden_ratio

Talk:Golden ratio From Wikipedia, the free encyclopedia. Contents why quadratic equation? Good page Errors in article Why? ... edit why quadratic equation? can anybody please suggest what should be the correct name for a quadratic equation since quad means four and is not approriately used in this context edit Good page Good page. Lots of info. Surely you don't need to keep the "Needs Attention" mention in place whilst you resolve final wordings about the validity of aesthetic claims. Take credit for the work done. edit Errors in article The following is in the article: I am not good at edits so could someone find a place to put in this the simplest equasion for "Phi is 5^.5 x .5 + .5" it is as universal as the fingers on your hands and the toes on your feet and uses only positive symbols thanks greg But more interestingly, it is usually found in natural shapes: Leaves length / width On faces, it's everywhere! Ratio mouth width / nose width, etc. More examples welcome! The first is obviously wrong - leaves come in all sorts of proportions. The second doesn't make much sense - mouth width divided by nose width may be close to the golden ratio for most people, but it's also close to π/2, 1.6, √ e , etc.

95. Abstract, Nexus 2000, Rachel Fletcher: Golden Proportions In A Great House: Pall from other geometric shapes are also present, the golden mean appears to ILLUSTRATION Room dimensions in Palladio s Villa Emo are in golden mean http://www.nexusjournal.com/conferences/N2000-Fletcher.html

Abstract. Rachel Fletcher examines Palladio's Villa Emo, reviewing the sources for harmonic proportions in architecture from Vitruvius to Alberti, and discussing both the rational proportions found in the plan of the villa published in the Quattro Libri as well as the incommensurable Golden Mean. Golden Proportions in a Great House: Palladio's Villa Emo Rachel Fletcher 113 Division St. Great Barrington, MA 01230 USA A mong the great houses of the Renaissance is Palladio's Villa Emo at Fanzolo in Northern Italy, built in the 1550s to realize Lunardo Emo's dream of a family seat in the country. Palladio's methods of harmonizing buildings organically with their physical surroundings borrow from rural traditions and local farm customs. The massive south-facing entry ramp in front, for example, serves as a threshing floor, where grain may be spread in the sun to dry. Palladio adopted a number of classical rules of building recorded in the ten books of De architectura by the Latin theorist Vitruvius, well known for applying mathematical proportion to a building's measures and equally adept at achieving harmony in more pragmatic and natural ways.

96. Intro Into The Golden Proportion When you here the golden mean or phi think proportion, ratio or relationship. The golden Proportion is a constant ratio derived from a geometric http://www.heartbeat2000.com/phi.htm

Here is a intro into the Golden Proportion, PHI continue here for a second page or see perfectly perfecting Numbers are symbols like letters or any other symbol or sign, they point to something other then itself. The numerical value or phi or the word points to something other. A dynamic relationship that is the only possible creative duality within the unity of the singularity. It is perfect, and perfects so many principles. T he first thing to understand when dealing with numbers is that there are two types that are generally discussed, Rational or whole numbers and Irrational Numbers, which Transcendental Numbers are the most curious. Irrational Numbers do not end in a whole number. Phi is a proportion , a number, a ratio and much more to those that have studied this most unusual of relationships. Below is from Robert Lawlors' 1982 book Sacred Geometry, pages 46-47 on the Golden Proportion. I quote Robert because it is the best, yet simplest explanation to the paradox of phi. This book is highly recommended as a workbook / primer to Sacred Geometry. First let me say that the Golden Proportion, the Golden Ratio, the Golden Mean or simple Phi all refer to the same magical proportion and for the discussions here are the same thing. It can be referred to in an mode or geometry, this because it is a relationship, from the line to the spiral or the base of a multidimensional holographic fractal. When you here the Golden Mean or phi think proportion, ratio or relationship.

97. Untitled Document Golden Ratio The golden mean. I believe that this geometric proportion served the Creator asan idea. when He introduced the continuous generation of similar ojects from http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/goldenrat

The Golden Mean I believe that this geometric proportion served the Creator as an idea when He introduced the continuous generation of similar ojects from similar objects. J. Kepler (1571-1630) We start with a problem in aesthetics. Consider a line segment. What is the most "pleasing" division of this line segment into two parts? Some people might say at the halfway point. Others might say at the one-quarters point. The "correct answer" is, however, none of these, and is found in Western art from the Ancient Greeks ownward (art theorists speak of it as the principle of "dynamic symmetry"): A geometric definition can be given in terms of the above figure. Let the ratio x = BC/AB. The Numerator and Denominator can then be taken as AB = a and BC = x without loss of generality. Now define the position of B by Plugging in gives or which can be solved using the Quadratic Equation to obtain This quantity known as the Golden Ratio, Divine Proportion, etc. The quantity where is the Golden Ratio. The golden ratio conjugate is sometimes also called the Silver Ratio. Click on the icon for a GSP scritp to divide a segment in golden ratio.

98. EMAT 6680 Class Page Forwrite-up5 Some golden geometry We can see there are lots of lines divided in the goldenratio. Such lines appear in the pentagon and the relationship between its http://jwilson.coe.uga.edu/EMT668/EMAT6680.2001/Youn/EMAT6690/Essay 1/Some golen

By Nami Youn Some Golden Geometry 1. Golden Rectangle A Golden Rectangle is a rectangle with proportions that are two consecutive numbers from the Fibonacci sequence. The Golden Rectangle has been said to be one of the most visually satisfying of all geometric forms. We can find many examples in art masterpieces such as in edifices of ancient Greece. GSP file 2. Golden Triangle If we rotate the shorter side through the base angle until it touches one of the legs, and then, from the endpoint, we draw a segment down to the opposite base vertex, the original isosceles triangle is split into two golden triangles. Aslo, we can find that the ratio of the area of the taller triangle to that of the smaller triangle is also 1.618Â . (=Phi) If the golden rectangle is split into two triangles, they are called golden triangles suing the Pythagorean theorem, we can find the hypotenuse of the triangle. 3. Golden Spiral The Golden Spiral above is created by making adjacent squares of Fibonacci dimensions and is based on the pattern of squares that can be constructed with the golden rectangle. If you take one point, and then a second point one-quarter of a turn away from it, the second point is Phi times farther from the center than the first point. The spiral increases by a factor of Phi.

99. Math And Art Students will be encouraged to use the golden mean. Ratio and Proportion They will break 3dimensional geometric objects down into their nets as Durer http://educ.queensu.ca/~egnatoff/digicurr/math _and_art/math_and_art.html

SchoolNet Digital Collections Curriculum Units Unit Summaries The Authors Math and Art Kim O'Halloran Lesson Overview Math Curriculum Topics Lesson 1 Students will be introduced to ratios by studying the proportions of the hand for the purpose of sketching lifelike features. Measurement Ratio and Proportion Data Collection and analysis Lesson 2 Students will practice using ratio and proportion by drawing and measuring the "nicest" rectangle they can visualize and comparing it to the golden rectangle. Students will try to verify golden rectangles in other works of art and devate whether or not the use of the golden rectangle is deliberate or not. Students will then come up with ideas on how the golden rectangle can be used to their advantage and create a pastel drawing involving the golden mean. Ratio and Proportion Measurement Data Analysis Lesson 3 Ratio and Proportion Measurement Estimation Data Collection and Analysis Lesson 4 Students will put their knowledge of ratio and proportion to the test as they design a 2-dimensional layout of a building that they will later draw and eventually model. Students will be encouraged to use the golden mean. Ratio and Proportion Area and Perimeter Measurement Drawing in 2-Dimensions Lesson 5 Students will draw prisms in 3-dimensions in preparation for a sketch of the building they designed in the previous lesson. They will break 3-dimensional geometric objects down into their nets as Durer had done, and they will construct 3-Dimensional objects from these nets.

100. Golden Mean The geometric construction that I have seen most often golden ratio uses a squareA, B, C, Find out more about constructions for the golden mean http://www.math.uwaterloo.ca/~ljdickey/geometry_corner/golden_mean/

The Golden Mean The Golden Mean (or golden ratio ) is a famous number that has been know from ancient times. For example some ancient buildings were built with this ratio used for its proportions. Here is an algebraic conditions that defines the golden ratio. It is positive and subtracting 1 from it gives a number that equals its reciprocal. So, it satisfies the equation x-1 = 1/x. It is not hard to show that the golden mean is one half of the sum of one and the square root of 5. Since the square root of 5 is about 2.236, and one plus the square root of 5 is 3.236, the golden ratio is approximately 1.618. The geometric construction that I have seen most often golden ratio uses a square A, B, C, D. Find the midpoint of the side AB and call it M. Make the circle with center M that passes through C (and D). Let E be the point where this circle meets the ray AB. Then the ratio AE / AB equals the ratio AB / BE. It is easy from this that that AE/AB is the golden ratio. Click here to see a picture (coming soon). A recent and surprising construction is one given by George Odom . George constructs an equilateral triangle ABC with vertices on a circle. Let M be the midpoint of AB and N be the midpoint of AC. Then the ray MN meets the circle at a point P. The ratio MP/MN is the golden ratio. Click here to see a picture (coming soon). Another pleasant surprise construction is given by Gabriel Bosia . Gabriel starts with the famous pythagorean triangle with sides 3, 4, and 5. Construct the circle of diameter 5 with center at the midpoint of the hypotenuse.

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z

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