THE WONDERS OF THE WORLD
CONTENTS CHAPTER 1 THE GOLDEN RATIO IN GEOMETRY 1 CHAPTER 2 UNEXPECTED CONCURRENCIES 53 CHAPTER 3 UNEXPECTED COLLINEARITIES 91 CHAPTER 4 SQUARES ON TRIANGLE SIDES 127 CHAPTER 5 SIMILAR TRIANGLES ON TRIANGLE SIDES 165 CHAPTER 6 DISCOVERING CONCYCLIC POINTS 189 CHAPTER 7 CIRCLE WONDERS 245 CHAPTER 8 POLYGONS AND POLYGRAMS 291 CHAPTER 9 GEOMETRIC SURPRISES 321 CHAPTER 10 GEOMETRIC FALLACIES 361 CHAPTER 11 HOMOTHETY SIMILARITY AND APPLICATIONS 401 CHAPTER 1 THE GOLDEN RATIO IN GEOMETRY FOR MANY CENTURIES ARCHITECTS ARTISTS MATHEMATICIANS SCIENTISTS AND EVERYDAY CITIZENS HAVE BEEN FASCINATED BY A RATIO THAT IS UBIQUI TOUS IN NATURE AND COMMONLY FOUND ACROSS MANY CULTURES IT WAS NAMED GOLDEN RATIO BECAUSE OF ITS PREVALENCE AS A DESIGN ELEMENT AND ITS SEEMINGLY UNIVERSAL AESTHETIC APPEAL THE FAMOUS GERMAN ASTRONO MER AND MATHEMATICIAN JOHANNES KEPLER 1571 1630 SUMMARIZED THIS RATIO S IMPORTANCE BY SAYING THAT GEOMETRY HARBORS TWO GREAT TREASURES ONE IS THE PYTHAGOREAN THEOREM AND THE OTHER IS THE GOLDEN RATIO THE FIRST CAN COMPARE WITH A HEAP OF GOLD AND THE SEC OND WE SIMPLY CALL A PRICELESS JEWEL THE HUMAN BODY POSSESSES THE GOLDEN RATIO AS DO THE HELIX STRUCTURE OF DNA AND THE PRINCIPLES OF GREEK ARCHITECTURE AS WELL AS MANY MODERN MASTERPIECES THE GOLDEN RATIO IS A UNIQUE PATTERN THAT NOT ONLY HAS WIDE RANGING AND ENDLESS APPLICATIONS BUT ALSO MANIFESTS ITSELF IN THE MOST UNEXPECTED PLACES PERHAPS THE MOST POPULAR APPEARANCE OF THE GOLDEN RATIO IS IN THE RECTANGLE WHOSE SIDES FORM THE GOLDEN RATIO THIS RECTANGLE IS REFERRED TO AS THE GOLDEN RECTANGLE AS WE BEGIN OUR JOURNEY INVESTIGATING THIS AMAZING GOLDEN RATIO MANIFESTED IN THE GOLDEN RECTANGLE AND BEYOND WE WILL FIRST DEFINE IT AND THEN MARVEL OVER ITS VARIOUS CONSTRUCTION POSSIBILITIES 1 DEFINING THE GOLDEN RATIO THE GOLDEN RATIO IS DEFINED AS THE RATIO OF THE LENGTHS OF THE TWO PARTS OF A LINE SEGMENT THAT ALLOWS US TO MAKE THE FOLLOWING EQUALITY OF TWO RATIOS NAMELY A PROPORTION THAT THE LONGER SEGMENT L IS TO THE SHORTER SEGMENT S AS THE ENTIRE ORIGINAL SEGMENT L S IS TO THE LON GER SEGMENT L SYMBOLICALLY THIS IS WRITTEN AS L S L S L AS SHOWN IN FIGURE 1 1 FIGURE 1 1 THIS IS CALLED THE GOLDEN RATIO OR THE GOLDEN SECTION THE LATTER TERM REFERS TO THE SECTIONING OR PARTITIONING OF A LINE SEGMENT THE TERMS GOLDEN RATIO AND GOLDEN SECTION WERE FIRST INTRODUCED DURING THE NINETEENTH CENTURY WE BELIEVE THAT THE FRANCISCAN FRIAR AND MATHEMATICIAN FRA LUCA PACIOLI CA 1445 1517 WAS THE FIRST TO USE THE TERM DE DIVINA PROPORTIONE DIVINE PROPORTION AS THE TITLE OF A BOOK IN 1509 WHILE THE GERMAN MATHEMATICIAN AND ASTRONOMER JOHANNES KEPLER 1571 1630 WAS THE FIRST TO USE THE TERM SECTIO DIVINA DIVINE SECTION MOREOVER THE GERMAN MATHEMATICIAN MARTIN OHM 1792 1872 IS CREDITED FOR HAVING FIRST USED THE TERM GOLDENER SCHNITT GOLDEN SECTION IN ENGLISH THE TERM GOLDEN SECTION WAS FIRST USED BY JAMES SULLY IN 1875 IN AN ARTICLE ON AESTHETICS IN THE 9 TH EDITION OF THE ENCYCLOPEDIA BRITANNICA YOU MAY BE WONDERING WHAT MAKES THIS RATIO SO OUTSTANDING THAT IT MERITS THE TITLE GOLDEN THIS DESIGNATION WHICH IT RICHLY DESERVES WILL BE MADE CLEAR THROUGHOUT THIS CHAPTER WE BEGIN BY CONSIDERING ITS NUMERICAL VALUE WHICH WILL BRING US TO ITS FIRST UNIQUE CHARACTERISTIC TO DETERMINE THE NUMERICAL VALUE OF THE GOLDEN RATIO L S WE WILL CHANGE THE EQUATION L S L S L L S L L S L OR TO AN EQUIVALENT WHERE X L S TO GET X X 1 1 THIS CAN BE REWRITTEN AS X 2 X 1 0 WE CAN NOW 2 THE WONDERS OF THE WORLD SOLVE THIS EQUATION FOR X BY USING THE QUADRATIC FORMULA 1 TO OBTAIN THE NUMERICAL VALUE OF THE GOLDEN RATIO THE NEGATIVE SOLUTION NEEDS NOT TO BE CONSIDERED L S X 1 5 2 THIS VALUE IS COMMONLY DENOTED BY THE GREEK LETTER PHI 2 THUS L S 1 5 2 1 61803398874989485 SO EVERY TIME ONE ENDS UP WITH THE EQUATION X 2 X 1 0 ONE CAN BE SURE WITHOUT THE QUADRATIC FORMULA THAT THE ONLY POSITIVE SOLUTION IS WHAT MAKES THIS RATIO L S MATHEMATICALLY UNIQUE IS ITS RECIPROCAL NAMELY S L 1 2 1 5 0 61803398874989485 NOW YOU SHOULD NOTICE A VERY UNUSUAL RELATIONSHIP THE VALUE OF AND 1 DIFFER BY EXACTLY 1 THAT IS 1 1 FROM THE NORMAL RELATIONSHIP OF RECIPROCALS THE PRODUCT OF AND 1 IS ALSO EQUAL TO 1 THAT IS 1 1 THEREFORE WE HAVE TWO NUMBERS AND 1 WHOSE DIFFERENCE AND PRODUCT ARE EQUAL TO 1 IT SHOULD BE NOTED THAT THESE ARE THE ONLY TWO NUMBERS FOR WHICH THIS IS TRUE BY THE WAY YOU MIGHT HAVE NOTICED THAT 1 5 SINCE 5 1 2 5 1 2 5 CONSTRUCTING THE GOLDEN SECTION HAVING NOW DEFINED THE GOLDEN RATIO NUMERICALLY WE SHALL NOW CON STRUCT IT GEOMETRICALLY THERE ARE SEVERAL WAYS TO CONSTRUCT THE GOLDEN SECTION OF A LINE SEGMENT YOU MAY NOTICE THAT WE APPEAR TO BE USING THE TERMS GOLDEN RATIO AND GOLDEN SECTION INTERCHANGEABLY TO AVOID CONFUSION WE WILL USE THE TERM GOLDEN RATIO TO REFER TO THE NUMERICAL 1?THE QUADRATIC FORMULA FOR THE GENERAL QUADRATIC EQUATION AX 2 BX C 0 YIELDS X B B AC A 2 4 2 2?THERE IS REASON TO BELIEVE THAT THE LETTER WAS USED BECAUSE IT IS THE FIRST LETTER OF THE NAME OF THE CELEBRATED GREEK SCULPTOR PHIDIAS CA 490 430 BCE IN GREEK PHEIDIAS FEIDIAS OR WHO PRODUCED THE FAMOUS STATUE OF ZEUS IN THE TEMPLE OF OLYMPIA AND SUPERVISED THE CONSTRUCTION OF THE PARTHENON IN ATHENS GREECE HIS FREQUENT USE OF THE GOLDEN RATIO IN THIS GLORIOUS BUILDING IS LIKELY THE REASON FOR THIS ATTRIBUTION IT MUST BE SAID THAT THERE IS NO DIRECT EVIDENCE THAT PHIDIAS CONSCIOUSLY USED THIS RATIO THE AMERICAN MATHEMATICIAN MARK BARR WAS THE FIRST USING THE LETTER IN ABOUT 1909 SEE THEODORE ANDREA COOK THE CURVES OF LIFE COURIER DOVER PUBLICATIONS 1914 P 420 NEW YORK DOVER PUBLICATIONS 1979 3 THE WONDERS OF THE WORLD