Description
The Golden Ratio is used to help design aesthetically pleasing shapes by artists and architects,
and also shows up in a surprising number of ways in nature. The Golden Ratio is also called the
“Divine Proportion.” It is an irrational number (it cannot be written as the ratio of two integers),
and is typically symbolized with the Greek letter φ (phi). As a continued fraction pattern, it is the
simplest continued fraction, as it consists of all 1’s:
φ = 1 +
1
1 + 1
1+ 1
1+ 1
···
≈ 1.618033988749895 . . .
The pattern in the denominator repeats infinitely. The Golden Ratio is not typically included as a
constant in part of a programming language’s package library, such as π = Math.PI in Java. One
possible application of φ is as some part of a procedural graphics program (artwork generated by
math formulae).
Write a Java program called GoldenRatio.java that does the following.
Create a variable of type double called phi.
Set phi = (1 + Math.sqrt(5)) / 2.
Estimate φ:
(a) create a variable of type double called approxPhi,
(b) set approxPhi to 1.0,
(c) set approxPhi to its reciprocal plus 1.0,
(d) repeat Step (c) four more times.
Then print the following statements:
The Golden Ratio is approximately
The Golden Ratio is
University of Victoria
CSC 110 Fundamentals of Programming I
8 May 2017
Compile your program and make sure there are no errors, and you have the above output. Your
approximation should agree with the actual value of φ in the first two digits.
Next, approximate φ again in your program:
(a) set approxPhi to 1.0,
(b) set approxPhi to its reciprocal plus 1.0,
(c) repeat Step (b) ninety-nine more times.
Then print the following statements:
Now for a more accurate approximation.
The Golden Ratio is approximately
The Golden Ratio is
The Golden Ratio can be used to generate a famous sequence of numbers called the Fibonacci
numbers. The first Fibonacci number is f1 = 1, and the second is f2 = 1. Any Fibonacci number
is the sum of the previous two: fn+2 = fn+1 + fn, for n ≥ 3. Observe the first nine Fibonacci
numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, . . .
After your previous code, calculate these Fibonacci numbers as follows. Declare and assign the
following variables:
• int f1 = 1;
• int f2 = 1;
• int fn = 0;
Write a for-loop with a loop-control variable i that iterates from 3 to 9, inclusive, with each iteration
performing the following steps:
(a) set fn equal to the sum of f1 and f2
(b) set f1 equal to f2
(c) set f2 equal to fn
(d) System.out.println(“f ” + i + ” = ” + fn);
(e) call checkFibonacci( approxPhi, i ) as described below,
(f) call starArt( f1, f2 ) as described below.
Now add the method checkFibonacci to your program:
public static void checkFibonacci( double approxPhi, int n ) {
double A = Math.pow(approxPhi, n);
double B = Math.pow(1 – approxPhi, n);
double Fibonacci = (A – B) / Math.sqrt(5);
long check fn = Math.round( Fibonacci );
System.out.println(“check f ” + n + ” = ” + check fn);
}
University of Victoria
CSC 110 Fundamentals of Programming I
Add another method starArt to your program that takes two integer parameters, a height and a
width. Inside starArt, use two nested loops to print as output a rectangle of star (*) characters,
with just less than 2.5*width characters per line of height number of lines. (The terminal/console
window typically has non-square character sizes, and so we need this adjustment in width to get
proper visual comparison of width against height here.)
Compile your program and make sure there are no errors. The output should give the 3rd to the
9th Fibonacci number, with each number followed by a rectangle of stars in an aspect ratio that is
said to be more pleasing to the eye than other aspect ratios.
The ratio of Fibonacci numbers fn+1/fn also approximates φ as n increases. The Golden Ratio
and the Fibonacci numbers can be used in many ways to hold the human eye’s attention. The
arrangement of branches of many trees loosely follow the sequence of Fibonacci numbers in efforts
to maximize the amount of light their leaves can photosynthesize. The Mona Lisa—arguably the
world’s most famous work of art—is among the notable applications of the Golden Ratio.
The Fibonacci numbers are named after an Italian mathematician born in the year 1170, but
appeared in history much earlier in the work of Indian mathematics. Fibonacci used this sequence
of numbers to describe the growth of a hypothetical rabbit population. As a tribute to this idea,
add one more method to your program called printRabbit that outputs the following ascii art
rabbit:
(\(\
( -.-)
o (“)(“)
This rabbit art uses the backslash, dash, period, double quote, underscore, and lowercase o characters, as well as left and right parenthesis. Make sure to call printRabbit at the end of your main
method.
Grading
Marks will be allocated out of 30 points:
• (0 points) You do not submit any program file.
• (5 points) Your program does not compile. The marker cannot fix the complier error easily,
or if they can, the output is severely mismatched with the assignment requirements.
• (10 points) Your program does not compile, but the marker can fix the error easily.
• (15 points) Your program compiles, but only outputs a few of the assignment requirements.
• (20 points) Your program compiles, outputs most of the assignment requirements, and is
missing documentation comments or has formatting issues.
• (25 points) Your program compiles, outputs everything required, but is missing documentation comments or has formatting issues.
• (30 points) Your program compiles, outputs everything required, and has proper documentation and formatting.
