7KH4XLQFXQ; Earth/matriX SCIENCE IN ANCIENT ARTWORK. Charles William Johnson

Similar documents
(1) Primary Trigonometric Ratios (SOH CAH TOA): Given a right triangle OPQ with acute angle, we have the following trig ratios: ADJ

REVIEW QUESTIONS TOPIC 5 TRIGONOMETRY I FLUENCY

Notes on Spherical Triangles

Section 6.1 Law of Sines. Notes. Oblique Triangles - triangles that have no right angles. A c. A is acute. A is obtuse

Geometric quantities for polar curves

SAMPLE. End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.

First Round Solutions Grades 4, 5, and 6

Polar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a

10.4 AREAS AND LENGTHS IN POLAR COORDINATES

SLOVAK UNIVERSITY OF TECHNOLOGY Faculty of Material Science and Technology in Trnava. ELECTRICAL ENGINEERING AND ELECTRONICS Laboratory exercises

Section 16.3 Double Integrals over General Regions

Section 10.2 Graphing Polar Equations

Triangles and parallelograms of equal area in an ellipse

by Kathy Brown of The Teacher s Pet

Spherical Geometry. This is an article from my home page:

REVIEW, pages

Domination and Independence on Square Chessboard

Patterns and Algebra

GLONASS PhaseRange biases in RTK processing

Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world

INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS

Probability and Statistics P(A) Mathletics Instant Workbooks. Copyright

ASY P.O. BOX 729 TERRELL, TEXAS / PAGE 1 OF 13 SAM

Vocabulary Check. Section 10.8 Graphs of Polar Equations not collinear The points are collinear.

c The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?

Theme: Don t get mad. Learn mod.

SOLVING TRIANGLES USING THE SINE AND COSINE RULES

Defining the Rational Numbers

Exercise 1-1. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION

Skills Practice Skills Practice for Lesson 4.1

9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations

Introduction 6 Basics 8 Projects 26. Stitching terms 94. About the author 95

Lecture 16. Double integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Comparing Fractions page 1 of 2 1 Color in the grid to show the fractions below. Each grid represents 1 whole. a 1 2 b 1. d 16

Student Book SERIES. Patterns and Algebra. Name

Macroscopic and Microscopic Springs Procedure

MATHEMATICS. Student Booklet

Multivariable integration. Multivariable integration. Iterated integration

Experiment 3: Non-Ideal Operational Amplifiers

Experiment 3: Non-Ideal Operational Amplifiers

Fitting & User Instructions

+ sin bsin. sin. tan

NONCLASSICAL CONSTRUCTIONS II

ISM-PRO SOFTWARE DIGITAL MICROSCOPE OPERATION MANUAL

8.1. The Sine Law. Investigate. Tools

Spiral Tilings with C-curves

METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin

Polar Coordinates. July 30, 2014

Samantha s Strategies page 1 of 2

CHAPTER 2 LITERATURE STUDY

Abacaba-Dabacaba! by Michael Naylor Western Washington University

Joanna Towler, Roading Engineer, Professional Services, NZTA National Office Dave Bates, Operations Manager, NZTA National Office

Seven Sisters. Visit for video tutorials

Lecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

& Y Connected resistors, Light emitting diode.

Analog Input Modules

THE STUDY OF INFLUENCE CORE MATERIALS ON TECHNOLOGICAL PROPERTIES OF UNIVERSAL BENTONITE MOULDING MATERIALS. Matej BEZNÁK, Vladimír HANZEN, Ján VRABEC

peyote stitch projects and patterns

SPECIAL EDITION. Spring 2012 Ezine. where crafty is contagious

Double Integrals over Rectangles

CAL. NX15 DUO-DISPLAY QUARTZ

So Many Possibilities page 1 of 2

Translate and Classify Conic Sections

Applications of a New Property of Conics to Architecture: An Alternative Design Project for Rio de Janeiro Metropolitan Cathedral

AGA56... Analog Input Modules. Siemens Building Technologies HVAC Products

Resistors, Current and Voltage measurements, Ohm s law, Kirchhoff s first and second law. Kirchhoff s first Objectives:

LINER MEASURING Made Easy

Available online at ScienceDirect. 6th CIRP International Conference on High Performance Cutting, HPC2014

Proposed Cable Tables for SAS2

The Math Learning Center PO Box 12929, Salem, Oregon Math Learning Center

ISSCC 2006 / SESSION 21 / ADVANCED CLOCKING, LOGIC AND SIGNALING TECHNIQUES / 21.5

EASY DISC Assessment

The UK Linguistics Olympiad 2017

WORKSHOP 15 PARASOLID MODELING

Series. Teacher. Numbers

Section 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17.

Solutions to exercise 1 in ETS052 Computer Communication

FP2 POLAR COORDINATES: PAST QUESTIONS

Unilateral and equitransitive tilings by squares of four sizes

Evaluating territories of Go positions with capturing races

Section Thyristor converter driven DC motor drive

Multi-beam antennas in a broadband wireless access system

Available online at ScienceDirect. Procedia Engineering 89 (2014 )

Glass Box Projection. Gives you 6 sides to view of an object. 10/2/14 2

5 I. T cu2. T use in modem computing systems, it is desirable to. A Comparison of Half-Bridge Resonant Converter Topologies

CHAPTER 3 AMPLIFIER DESIGN TECHNIQUES

Alternating-Current Circuits

PROGRAMMING MANUAL MTMA/01 MTMV/01 FB00329-EN

Introduction to Isometric Drawings

E GRAPHICS ISOMETRIC PROJECTIONS S.RAMANATHAN ASST PROF MVSREC PH: CONCEPTS.

Module 9. DC Machines. Version 2 EE IIT, Kharagpur

Redundancy Data Elimination Scheme Based on Stitching Technique in Image Senor Networks

MBR7030WTG. Switch Mode Power Rectifier SCHOTTKY BARRIER RECTIFIER 70 AMPERES, 30 VOLTS

MAYWOOD. Hospitality & ConferenCe tables FURNITURE CORP. GSA Approved! Contract #GS-28F-0050W. Established 1918

MONOCHRONICLE STRAIGHT

Yellowknife km Vancouver km NEL

Electronic Circuits I - Tutorial 03 Diode Applications I

Catty Corner. Side Lengths in Two and. Three Dimensions

ASSEMBLY INSTRUCTIONS

LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY

Transcription:

Erth/mtriX SCIENCE IN ANCIENT ARTWORK 7KH4XLQFXQ; Chrles Willim Johnson Erth/mtriX P.O. Box 231126, New Orlens, Louisin, 70183-1126 2001 Copyrighted y Chrles Willim Johnson www.erthmtrix.om www.the-periodi-tle.om ISBN 1-58616-213-6

Chrles Willim Johnson

Quinunx = five twelfths Etymologilly 7KHTXLQFXQ[LVDJHRPHWULFDOILJXUHRIILYHHOHPHQWV IRXQGLQWKHDUWZRUNRIPDQ\DQFLHQW0HVR$PHLFDQFXOWXUHV )RUH[DPSOHLWLVUHSUHVHQWHGLQDUHSHDW SDWWHUQRQWKH$]WHF&DOHQGDU Anient rtwork design. Signifies JADE Azte Clendr Design

7KH*ROGHQ0HDQ = 2 The quinunx my e derived from the proof of phi: 1.618033989 ::.6180339888

= ½( 5+1) = 1.618033989 = ½( 5-1) = 0.6180339888 B C A A line thus divided in extreme nd men rtio is often lled the Golden Setion. 1.618033989 ::.6180339888

7KH4XLQFXQ; The figure for the proof of phi is thus repeted nd rotted to fit in the mnner illustrted.

7KH4XLQFXQ; As we remove the smller irle employed in the proof, we now oserve the pperne of the Quinunx

7KH4XLQFXQ; Three figures of the proof of phi t right ngles to one nother.

7KH4XLQFXQ; Four figures of the proof of phi t right ngles to one nother.

7KH4XLQFXQ; Remove the smller irle of the proof.

Remove the smller irle of the proof.

As we remove more lines, we now my oserve the pperne of the Quinunx pled t the enter of the fourfold proof.

Now, we only need to d the entrl irle to omplete the Quinunx.

An revited proof of phi y fourfold figure. The entrl red lined figure is often ited s si design for the se of the Gret Pyrmid t Giz.

Let us employ the mesures orresponding to the Gret Pyrmid. We shll egin with the often-ited 756 feet mesure for the seline of the Gret Pyrmid. 756 x 756 = 571536 756 feet side mesure 756 feet side mesure

The Cross-Setion of the Gret Pyrmid: distint phi-like right-ngle tringultion of the se. We shll employ the sme proof of phi, sed on tringle whose se is one-hlf its height. 169 The 169, 338, 676 is n historilly reognized ount, from the Meso-Amerin Legend of the Fifth Sun or, the Legend of the Four Worlds 7KH4XLQFXQ; 338 756 side We hve inverted one hlf of the proof figure on its entrl xis: refleitve symmetry.

The Cross-Setion of the Gret Pyrmid: distint phi-like right-ngle tringultion of the se. When the se of the tringle is 756 s shown, then the other legs of the smller tringle re pproximte to 169 nd 338. Sine 169, 338, 676 is n historilly reognized ount, we my employ it s the invrile mesure nd djust the 378 mesurement. 7KH4XLQFXQ; 169 338 756 side 169 2 + 338 2 = 377.8954882 2 755.7909764 2

The Cross-Setion of the Gret Pyrmid: The Mesures 755.7909764 338 338 169 169 755.7909764 507 755.7909764 755.7909764 507 The Kemi knew tht if the frtions re on the outside, then the whole numers re on the inside of the figure. 7KH4XLQFXQ; 338 338 169 169 755.7909764

If we employ the numers of the Gret Pyrmid for the Quinunx design, then we hve the following vlues for the different spets of the design elements. 755.7909764 338 Bseline of the Gret Pyrmid of Giz 676 169 2 + 338 2 = 377.8954882 2 755.7909764 2

7KH*ROGHQ0HDQ [ FïDïEï SKL 755.7909764 = 2 676 338 B C A 417.7909764

7KH*ROGHQ0HDQ [ SKL 755.7909764 338 676 338 The phi reltion exists s 2phi 1.23067978 on the side of the hypotenuse 417.7909764

338 The rdius of eh irle would then e the Meso-Amein ount of 338.

755.7909764 2 338 Are of 755.7909764x Squre: 571220 Are of eh lrge irle: 358908.1111

Are of inside of the urviliner entrl ross without the entrl dot 212311.8889

Cirumferene of the lrge irle 2123.716634 Are of inside of urviliner entrl ross 212311.8889

3LDQG3KLLQ5HODWLRQWR2QH$QRWKHU 6TXDULQJWKH&LUFOH AREA OF SQUARE 571220 AREA OF CIRCLE 358908.1111 358908.1111 / 571220 =.6283185307 (2PI FRACTAL).3141592653 (PI is 3.141592654)

(1'),/( Erth/mtriX P.O. Box 231126, New Orlens, Louisin, 70183-1126 2001 Copyrighted y Chrles Willim Johnson www.erthmtrix.om www.the-periodi-tle.om ISBN 1-58616-213-6