41. The Geometry Junkyard: Pentagonal Geometry And The Golden Ratio The Geometry Junkyard. Pentagonal Geometry and the golden ratio. Fibonaccispirals, Ned May. The golden ratio in an equilateral triangle. http://www.ics.uci.edu/~eppstein/junkyard/pent.html

Pentagonal Geometry and the Golden Ratio This page includes geometric problems defined on regular pentagons, involving pentagonal angles, or based on the golden ratio (the ratio of diagonal to side length in a regular pentagon). A Brunnian link . Cutting any one of five links allows the remaining four to be disconnected from each other, so this is in some sense a generalization of the Borromean rings. However since each pair of links crosses four times, it can't be drawn with circles. Constructing a regular pentagon inscribed in a circle, by straightedge and compass. Scott Brodie. Also described by M. Gallant Cut-the-knot logo . With a proof of the origami-folklore that this folded-flat overhand knot forms a regular pentagon. Digital Diffraction , B. Hayes, Amer. Scientist 84(3), May-June 1996. What does the Fourier transform of a geometric figure such as a regular pentagon look like? The answer can reveal symmetries of interest to crystallographers. The downstairs half bath . Bob Jenkins decorated his bathroom with ceramic and painted pentagonal tiles. Equilateral pentagons . Jorge Luis Mireles Jasso investigates these polygons and dissects various polyominos into them. Fibonacci spirals , Ned May.

42. Pentagram & The Golden Ratio The Pentagram The golden ratio. Johann Kepler (1571-1630). The'ratio' has become known as the golden ratio or golden section. http://www.contracosta.cc.ca.us/math/pentagrm.htm

The Pentagram Golden Ratio Geometry has two great treasures: one the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel. Johann Kepler (1571-1630) The 'ratio' has become known as the golden ratio or golden section This ratio can be found in many places: in art, architecture, and mathematics. Consider the construction of the regular pentagon. If the side AB of a regular pentagon (see figure to the right) has unit length, then any diagonal, such as AC, has length and this is the golden ratio Notice also the diagonals of the pentagon form another regular pentagon in the center of the figure with, of course, the potential for additional diagonals to be drawn, thus generating the golden ratio again as well as another regular pentagon further inside the figure. Presumably this could continue indefinitely. The golden ratio also appears in comparing consecutive elements of certain kinds of sequences, most notably, the

43. Inter.View To George Cardas An interview with George Cardas, describing his use of the golden ratio in highend audio equipment cables.Category Science Math Recreations Specific Numbers phi...... by Lucio Cadeddu. A brief introduction to golden ratio. Let me writedown a brief survey on golden ratio and its amazing history. http://www.tnt-audio.com/intervis/cardase.html

TNT Who we are Inter.View to George Cardas - Cardas Cables by Lucio Cadeddu A brief introduction to Golden Ratio freely taken from "Golden sections and sequences in an unstable problem" by Lucio Cadeddu Golden Ratio is an easy concept of elementary geometry which has had, and still has, great relevance both in human designs and in Nature. Recently it has had wide application in HiFi Audio too. Let me write down a brief survey on Golden Ratio and its amazing history. Let us take a segment a of lenght 1. Another segment b is said to be the Golden Section of a if it solves the following equation: b + b - 1 = that is to say the two segments respect the following proportion: a : b = b : (a-b) . In simpler words, given the fact that a has lenght 1, b must be 0.618 approx. Historically the Golden Ratio was well known to the Egyptians who used it for building their pyramids but it achieved wider popularity thanks to the Greek geometers. We have to wait till 1496 in order to have that ratio called "Golden Ratio". Actually the mathematician (Friar) Pacioli wrote a paper called "De Divina Proportione" where he referred to that ratio as a God-given number one can find everywhere in Nature.

44. Spa And Salon Tables And Equipment By Golden Ratio Beauty golden ratio Beauty A division of golden ratio Woodworks 1 800345-1129 PO Box 2972896 Hwy. 89 South Emigrant, Montana 59027 406-333-4578 ph. golden ratio. http://www.goldenratiobeauty.com/

Golden Ratio Beauty A division of Golden Ratio Woodworks P.O. Box 297 2896 Hwy. 89 South Emigrant, Montana 59027 406-333-4578 ph. 406-333-4769 fax sales@goldenratiobeauty.com Our Spa Tables, Spa Chairs, Salon Tables and Wet Treatment Tables will perform reliably for all kinds of spa, skin care, faacials, manicures, pedicures, reflexology treatments and massage. Golden Ratio Bodyworks offers all the massage tools and massage accessories you'll need, including: Oils and Creams, Aromatherapy, Business Building Tools, Pillows, Power Massagers and Loungers, Hydrotherapy equipment, Charts, Decoders, Books, a great selection of training Videos and Music. Golden Ratio Sports. A line of top quality, tough massage tables built to perform in the sports massage arena. Models include The Master Trainer (heavy-duty portable massage table), The Athlete (solid portable massage table for the average athlete) and Stadium Flat Top (super strong stationary massage table), The Power Lift Flat Top (the ultimate electric-lift exam and treatment table for any size or weight of athlete) and the Field Chair (on-site massage chair built on a 15% larger frame than our QuickLite massage chair). See our catalog order page to order information.

45. 10000 Decimal Golden Ratio First 10,000 Digits of the golden ratio. This is the first publication of the GoldenRatio to 10,000 digits. If you know of an earlier one, please let me know. http://www.wwu.edu/~stephan/webstuff/ratio.digits.html

First 10,000 Digits of the Golden Ratio This is the first publication of the Golden Ratio to 10,000 digits. If you know of an earlier one, please let me know . How was this done? Here's how. There's more We're now listed as a Useless Page (search for "gold")! The Digit Warehouse gives the first million digits of the square root of five. I got the Golden Ratio by adding 0.5 to sqr(5) divided by 2. Most computers carry division out to a limited maximum number of decimal places. To divide the first 10,000 digits of sqr(5) by 2, I wrote the following Hypercard script - "long division" by 2. on mouseUp the first 10000 decimal digits of sqr(5) = 2.236067.... are in cd fld 1 when the program's done, add 0.5 to the result put empty into cd fld 2 repeat with i = 1 to 10000 put char i of cd fld 1 after holder if holder mod 2 = then put holder/2 after cd fld 2 put empty into holder else put trunc(holder/2) after cd fld 2 put 1 into holder end if end repeat end mouseUp

46. Proportion And The Golden Ratio - Mathematics And The Liberal Proportion and the golden ratio Mathematics and the Liberal Arts. Theauthor shows how the golden ratio occurs in music and art. http://math.truman.edu/~thammond/history/Proportion.html

Proportion and the Golden Ratio - Mathematics and the Liberal Arts To expand search, see Art . Laterally related topics: Symmetry Perspective Fractals in Art Weaving ... Origami , and Mazes The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews , published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet Comput. Math. Appl. Part B (1986), no. 1-2, 3962. SC: 92A27 (01A99 52-01), MR: 838 136. Certainly an unorthodox essay. It may be hard to understand the author's terms

47. WNYC - Reading Room WNYCmail Berlex FAQ. The golden ratio The Story of Phi, the World'sMost Astonishing Number By Mario Livio Random House Copyright http://www.wnyc.org/books/11057

On The Air Now FM 93.9 BBC World Update AM 820 BBC World Service Program Schedules Search the Site and Archives Search News Room Support WNYC On-Line Giving on-line is the most cost effective way to support WNYC. See the Thank You Gifts Email Updates Sign Up Now to receive WNYCmail The Golden Ratio: The Story of Phi, the World's Most Astonishing Number By Mario Livio Random House ISBN: 0-7679-0815-5 Available for purchase at Amazon.com Chapter One PRELUDE TO A NUMBER Numberless are the world's wonders.-Sophocles (495-405 b.c.) Less known than pi is another number, phi (f), which is in many respects even more fascinating. Suppose I ask you, for example: What do the delightful petal arrangement in a red rose, Salvador Dali's famous painting "Sacrament of the Last Supper," the magnificent spiral shells of mollusks, and the breeding of rabbits all have in common? Hard to believe, but these very disparate examples do have in common a certain number or geometrical proportion known since antiquity, a number that in the nineteenth century was given the honorifics "Golden Number," "Golden Ratio," and "Golden Section." A book published in Italy at the beginning of the sixteenth century went so far as to call this ratio the "Divine Proportion." In everyday life, we use the word "proportion" either for the comparative relation between parts of things with respect to size or quantity or when we want to describe a harmonious relationship between different parts. In mathematics, the term "proportion" is used to describe an equality of the type: nine is to three as six is to two. As we shall see, the Golden Ratio provides us with an intriguing mingling of the two definitions in that, while defined mathematically, it is claimed to have pleasingly harmonious qualities.

48. Golden Ratio Lesson They have established that the golden ratio can be found in the humanskeleton. Objectives. The student will find patterns in data. http://www.ruf.rice.edu/~winkler/goldenratio.html

This page is still under construction. Last updated August 6, 1996 Is Your Body in "Golden" Shape? Patricia Winkler DeBakey High School for Health Professions Houston, TX 77021 Overview This is a lesson intended to give students to opportunity to explore the claims of American researcher, Jay Hambridge, and others. They have established that the Golden Ratio can be found in the human skeleton. Objectives The student will: find patterns in data. use deductive reasoning to compare the data from all groups of students and draw general conclusions based on the data. determine sample size. discuss the effect of sample size on the conclusions drawn. develop statistical strategies for improved sample size. determine whether or not there is any data which will skew the results and why the data was inconsistent with the sample. Audience This lesson can be conducted with any age student who can read a tape measure with accuracy. Prerequisite Skills The student should be able to: accurately measure an object to the nearest millimeter or fraction of an inch. compute ratios of related body measures.

49. Golden Ratio, The Euclid defined this curious mathematical relationship, widely known as the GoldenRatio, more than 2,000 years ago because of its crucial role in the http://www.sciencenewsbooks.org/goldenratio.html

by Mario Livio The Golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras, who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as one of the greatest treasures of geometry; such medieval thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Debussy, Le Corbusier, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order, beauty, and eternal mystery will always coexist. from Broadway Broadway, 2002, 294 pages, 6 ¼" x 9 ½", hardcover More than 2,000 years ago, Euclid defined the number phi-1.6180339887-and linked it to the construction of the pentagram. Since then, this never-ending, never-repeating, irrational number has been called the Golden Number or the Golden Ratio and linked to patterns ranging from the petal arrangement of roses to the composition of the "Mona Lisa." Astrophysicist Livio takes an invigorating look at this ubiquitous number by examining the mathematical, aesthetic, and metaphysical qualities attributed to phi over the centuries. He also profiles the scientists and artists who have harnessed phi for a variety of purposes. He gives special attention to phi's appearance in natural patterns in the world. from Science News Golden Ratio, The

50. Golden Ratio A quick description of the golden ratio The golden ratio is often representedby Phi. Its approximate value it 1.61803 Links on the golden ratio http://www.math.uiuc.edu/~gfrancis/math306/math306web/GoldenRatioPeteWintermute.

What does it mean to be Golden Welcome to the wonderful world where everything is Golden!! presented to you by Pete Welcome! I am currently enrolled in a college course at the University of Illinois called The History of the Calculus and I thought for one of my course projects it would be very helpful for my fellow scholars to have a webpage that would help explain what it means to be "Golden" (that is in the mathematical sense). The topic of Golden is one of the more exciting topics in mathematics and one people should be familiar with. The goal of this annotated webpage is to list some very helpful websites that will give an overview of Golden with examples and some sites that contain some superb activities and lesson plans involving the concept of Golden for teachers. A quick description of the Golden Ratio: The Golden Ratio is often represented by Phi. Its approximate value it 1.61803... but more accurately is represented by (sqrt.of 5 + 1) / 2. As you notice Phi is an irrational number and has some very interesting properties and is often seen in the real world. To find out more about Phi please look at some of the intriguing webpages below. Links on the Golden Ratio: Fibonacci Numbers and the Golden Section - by Dr. Ron Knott. This is a very detailed webpage with a lot of information on the Golden section and Fibonacci Sequence but on numerous pages. Be ready to maneuver yourself through this page to find what you are looking for, but don't worry it is probably there. There is an overview on what Golden is; there are examples of the Golden section in nature, art, architecture, and music; Golden section occurring in Geometry and Trigonometry; constructions; you name it.

51. Golden Ratio Design Local doctor designs medical database applications for the Palm, as well as providing other Web design Category Regional North America Dansville Business and Economy...... LMRP Tool. The golden ratio or Golden Mean is a number revered since antiquity thatappears with suprising frequency in natural designs and great works of art. http://www.tonywitte.com/

Web and Palm OS Applications for Health Care LMRP Tool The Golden Ratio or Golden Mean is a number revered since antiquity that appears with suprising frequency in natural designs and great works of art. At Golden Ratio Design we believe well-designed computer applications and web tools are recognizable for their simplicity and elegance. We bring this concept to web-based and Palm OS applications for health care clinical desion making and productivity. The Palm OS is the dominant operating system for handheld computing, and for good reasons: simplicity, utility, dependability and a wide range of available software. As a physician dealing with volumes of critical information on a regular basis, the value of handheld computing has been very apparent to me. I have created a few applications for the Palm OS of use to the medical community, and am at work on other applications placing needed databases literally in the palm of the clinician's hand. LabCode NCD - Medical Necessity coding tool using latest Medicare medical necessity requirements included in the 23 lab test National Coverage Decisions. LabCode LMRP - Medical Necessity coding tool using latest local medical review policy medical necessity requirements.

52. IAHE - Shop Flash Cards, Gift Certificate, golden ratio Products, Gymnastic Balls, Herbal Pillows,Journals, golden ratio Products, From 1 to 10 of 95, Next 10 ». Name, Price, Qty. http://www.iahe.com/controller/IaheProductList?category=Golden Ratio Products

53. The Golden Ratio In Probability The golden ratio has been lurking in Probability. Results reach a perfectbalance when phi, f, the golden ratio, is the natural fulcrum. http://home.ozinet.aunz.com/~mervp/

In races, why does the most favoured starter win less than half the time? Why does it not win all the time? Just what % does it win? Not 0% either. What % then? What does nature say is the correct balance of success over failure? Not surprisingly, it is arithmetic and algebra and calculus that have been holding the answers to these questions all along. They just needed bringing out into the open for all to see. Results reach a perfect balance when phi, f , the Golden Ratio, is the natural fulcrum Mathematics predicts that favourites will win 38.2% of races, with an average of 38.2% of the people actually choosing that starter. In fact, the percentage of people who choose any starter is a very good indicator of its chances of success. This discovery is a major breakthrough in itself, but many things follow. You will find the Golden Ratio is also involved in election results. It is even at the footie, balancing how often the goal kicker will be successful. The theory was recently published. If you would like to know more, email

54. ARS Resources - Exclusive Educational Article What is the golden ratio? Its a little known marketing technique thatsay that, as long as the longest side on any oblong is 1.6 http://www.arsresources.com/articles/golden-ratio.shtml

Using Full Page Ad's Abuse Your Traffic The HTML Layout Typography Know How ... Whois What is the golden ratio? Its a little known marketing technique that say that, as long as the longest side on any oblong is 1.6 times the length of the shortest side of the same oblong people will be more likely to prefer that shape over any other shape. Why do they prefer this size? Because it occurs naturally and, subconsciously EVERY LIVING PERSON is attracted to this trait. An example of this is as follows: 1) Stand Up 2) Measure the distance from your head to your feet and write this measurement down 3) Measure the distance from your Navel (belly button to people like OTC :) write this figure down. The length of your ENTIRE body is 1.6 times longer than from your Navel to your feet!!

55. Golden Ratio golden ratio. see also golden ratio. Borissavliévitch, Miloutine. The Golden Number.Colman, Samuel. $61.95. Gardner, Martin. ``Phi The golden ratio.''Ch. http://www.ericweisstein.com/encyclopedias/books/GoldenRatio.html

Golden Ratio see also Golden Ratio The Golden Number. Colman, Samuel. Nature's Harmonic Unity: A Treatise on Its Relation to Proportional Form. Cook, Theodore Andrea. The Curves of Life, Being an Account of Spiral Formations and their Application to Growth in Nature, to Science, and to Art: With Special Reference to the Manuscripts of Leonardo da Vinci. New York: Dover, 1979. 479 p. $11.95. Coxeter, H.S.M. ``The Golden Section, Phyllotaxis.''Ch. 11 in Introduction to Geometry. New York: Wiley, 1989. $61.95. Gardner, Martin. ``Phi: The Golden Ratio.''Ch. 8 in The Second Scientific American Book of Mathematical Puzzles and Diversions: A New Selection. New York: Simon and Schuster, pp. 89-103, 1961. 253 p. 2nd Scientific American collection. $14.95. Ghyka, Matila Costiescu. The Geometry of Art and Life. New York: Dover, 1977. 174 p. $6.95. Herz-Fischler, R. A Mathematical History of the Golden Number. New York: Dover, 1998. 224 p. $14.95. Huntley, H.E. The Divine Proportion. New York: Dover, 1970. 186 p. $5.95. Pacioli, Luca.

56. Golden Ratio golden ratio. golden ratio and Fibonacci numbers. Phi Page Golden Section Ratio. Constructionby golden ratio. Egypt Cheops' pyramid. Alta Vista golden ratio. http://web.hep.uiuc.edu/home/karliner/golden.html

Golden Ratio Golden Ratio and Fibonacci numbers Phi Page Golden Section Ratio Number Games Construction by Golden ratio ... Lycos Golden Ratio Professor Sever Tipei sent this in response to a student who asked about the Golden Ratio in music: See also Erno Lendvai Bela Bartok : Analysis of His Music Dr. Sever Tipei, Professor of Music Manager, Computer Music Project of the University of Illinois Experimental Music Studios Urbana, Illinois 61801, USA send me mail

57. Nature's Golden Ratio, Alaska Science Forum May 20, 1985. Nature's golden ratio Article 716. Crosssection of nautilus shellshowing the growth pattern of chambers governed by the golden ratio. http://www.gi.alaska.edu/ScienceForum/ASF7/716.html

Alaska Science Forum May 20, 1985 Nature's Golden Ratio Article #716 by Larry Gedney This article is provided as a public service by the Geophysical Institute, University of Alaska Fairbanks, in cooperation with the UAF research community. Larry Gedney is a seismologist at the Institute. Cross-section of nautilus shell showing the growth pattern of chambers governed by the golden ratio. What do the chambers of a nautilus shell have in common with the Parthenon and playing cards? It turns out that their forms are examples of a standard proportion. There is a fundamental ratio found over and over again in nature that seems to please human perceptions. Geometrically, it can be defined as the ratio obtained if a line is divided so that the length of the shorter segment is in the same proportion to that of the longer segment as the length of the longer segment is to the entire line. Mathematically, these ratios are such that the longer segment is 1.618054 times the length of the shorter segment, while the shorter is 0.618054 times the longer. These are remarkable numbers. Not only are the figures after the decimal point identical in both, but each is the reciprocal of the other (that is, the number 1 divided by either yields the other). These are the only two numbers that demonstrate this property. Unlike pi, another fundamental constant in which the decimals extend to infinity (3.14159. . .), these factors are exact after the first six decimals.

58. Nature's Golden Ratio, Part II, Alaska Science Forum Alaska Science Forum. June 17, 1985. Nature's golden ratio, Part II Article 720. Theearlier column told only half the story of the golden ratio, however. http://www.gi.alaska.edu/ScienceForum/ASF7/720.html

Alaska Science Forum June 17, 1985 Nature's Golden Ratio, Part II Article #720 by Larry Gedney This article is provided as a public service by the Geophysical Institute, University of Alaska Fairbanks, in cooperation with the UAF research community. Larry Gedney is a seismologist at the Institute. Daisy head reveals two sets of opposing spirals formed by individual florets. The clockwise spiral contains 21 arms; the counter-clockwise spiral contains 34. These are two adjacent numbers in the Fibonacci series. Seldom has an article appearing in this space generated the volume of reader response as did last month's column on the Golden Ratio. The interest shown seems to justify a sequel. To recapitulate briefly, the Golden Ratio consists of the two numbers 1.618034 and 0.618034, each of which is the reciprocal of the other. Rectangles with sides proportioned 0.618034 to 1 (or 1 to 1.618034) are often the shape taken by such commonplace items as picture frames and playing cards. Thus, the shape seems to be subliminally pleasing to the human eye, as witnessed by the many ways in which it is used in art and in construction. It is also found in nature, reflected in essentially every spiral form from a snail shell to the arms of a galaxy. The earlier column told only half the story of the Golden Ratio, however. Historically, credit for recognition of the peculiar mathematical properties of this ratio must go to a 13th century Italian known as Fibonacci. The "Fi" part of his name meant "son of." The Bonacci part meant "simpleton."

59. Golden Ratio Prehistory drawing of phi rectangle and its construction. A brief pre history of the goldenratio. .gif version = xxx Kb. view vector version xxx Kb. info on vector viewer. http://www.recoveredscience.com/constgoldenprehistory.htm

recoveredscience .com We offer surprises about and numerals and their ancient religious uses in our e-book Ancient Creation Stories told by the Numbers by H. Peter Aleff Site Contents NUMERALS Numerals Introduction Horus Eye Fractions Creation by numerals ... Reader responses Visit our other Sections: Prime Patterns Board Games Astronomy Medicine Store Stuff Home Page Search this site FAQ about e-books Download free e-books ... email us Footnotes : Sir Thomas L. Heath: “Euclid: The Thirteen Books of The Elements”, Dover Publications, New York, 1956, Volume I, pages 46 and 47. Sir Thomas L. Heath: “Euclid: The Thirteen Books of The Elements”, Volume 2, edition cited above, page 99. H.E. Huntley: “The Divine Proportion: A Study in Mathematical Beauty”, Dover Publications, New York, 1970, page 30. The passage from Iamblichus which Huntley cites does not name or describe that sign of recognition, and the legend of the pentagram is hard to pin down. Sir Thomas Heath: “A Summary of Pythagorean Mathematical Discoveries”, pages 329 to 331 in Kenneth Sylvan Guthrie, compiler and translator: “The Pythagorean Sourcebook and Library”, Phanes Press, Grand Rapids, Michigan, 1987. Bruno Snell, ed.: “Heraklit - Fragmente”, Wissenschaftliche Buchgesellschaft, Darmstadt, 1995, sayings B 129 and B 81, pages 38 and 27.

60. Golden Ratio Properties The mathematics of Genesis 1. in the layout of the Jerusalem Temple. You areon page. Some properties of the golden ratio phi. Goldmea2.gif (24451 bytes). http://www.recoveredscience.com/const305goldenproperties.htm

recoveredscience .com We offer surprises about and numerals and their ancient religious uses in our e-book Ancient Creation Stories told by the Numbers by H. Peter Aleff Site Contents NUMERALS Numerals Introduction Horus Eye Fractions Creation by numerals ... Number perceptions Golden ratio properties Golden ratio prehistory Woman Wisdom Constant e ... Reader responses Visit our other Sections: Prime Patterns Board Games Astronomy Medicine Store Stuff Home Page Search this site FAQ about e-books Download free e-books ... email us Footnotes : Table adapted from H.E. Huntley: “The Divine Proportion a Study in Mathematical Beauty”, Dover Publications, New York, 1970,., page 40. O. Neugebauer: “The Exact Sciences in Antiquity”, 1957, edition consulted Dover, New York, 1969, see note 9, page 25. Simo Parpola: “The Assyrian Tree of Life: Tracing the Origins of Jewish Monotheism and Greek Philosophy”, Journal of Near Eastern Studies, Volume 52, July 1993, Number 3, pages 161-208, see note 103 on pages 188 and 189. Marshall Clagett: “Ancient Egyptian Science”, Volume 2: “Calendars, Clocks, and Astronomy”, American Philosophical Society, Philadelphia, 1995, page 49, citing Utterances 251 and 320 in which the word for “hours” is determined both times by three stars. R. Böker and F. Schmeidler: “Über Namen und Identifizierung der ägyptischen Dekane”, Centaurus 1984, Institute of History of Science, Aarhus, Denmark, Vol. 27, pp. 189 to 217.

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