( Lucas(1000) has 209 digits!) Cycles in the Lucas numbers? On the The Mathematics of the Fibonacci Series we saw that the units digits of the Fibonacci numbers repeat in a cycle of length 60 (so that the units digits of F 60 = the units digits of F 0 , and so on for following digits). ● For the Lucas numbers, there is also a cycle of 60 - which is when the last two digits repeat in a cycle. There is a cycle of units digits in the Lucas numbers, which is much shorter. What is it? How long is it? Fibonacci - the man and His Times Fibonacci Home Page Fibonacci Forgeries! The Lucas Numbers WHERE TO NOW??? The Golden Section In Art, Architecture and Music Fibonacci, Phi and Lucas numbers Formulae Links and References Dr Ron Knott [email protected] Created: 18 October 1997 Updated: 27 November 1998 (5 of 5) [12/06/2001 17:23:04]
The Golden Section in Art, Architecture and Music 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 golden section in architecture ❍ The Parthenon and Greek Architecture ❍ Modern Architecture ❍ Architecture links ● The golden section and Art ❍ Leonardo's Art ❍ Links to Art sources including Contemporary Artists ● Fibonacci and Poetry ● Fibonacci and Music ❍ Golden sections in Violin construction ❍ Did Mozart use the Golden mean? ❍ Phi in Beethoven's Fifth ❍ Bartók, Debussy, Schubert, Bach and Satie ● A controversial issue ● References and Links on the golden section in Music and Art 1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 .. More .. The Golden section in architecture The Parthenon and Greek Architecture Even from the time of the Greeks, a rectangle whose sides are in the "golden proportion" (1 : 1.618 which is the same as 0.618 : 1) has been known since it occurs naturally in some of the proportions of the Five Platonic Solids (as we have already seen) . This rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple, the Parthenon, in the Acropolis in Athens , Greece. (There is a replica of the original building (accurate to one-eighth of an inch!) at Nashville which calls (1 of 11) [12/06/2001 17:23:19]
The Golden Section in Art, Architecture and Music itself "The Athens of South USA".) The Acropolis, in the centre of Athens, is an outcrop of rock that dominates this ancient city. Its most famous monument, now largely ruined, is the Parthenon, a temple to the goddess "Athena" built around 430 or 440 BC. Though no original plans of the temple exist, it appears that the temple was built on a square-root-of-5 rectangle, that is, it is 5 times as long as it is wide. These are also the dimensions of the longest side view of the temple. Also, the front elevation is built on a Golden Rectangle, that is, it is Phi times as wide as it is tall.