[...]... Construction of Affine Fractals Using 3D Graphics Shapes 210 6.5 Construction of Affine Fractals Using 3D Parametric Curves 221 6.6 Construction of Affine Fractals Using 3D Parametric Surfaces 224 197 xi Contents Chapter 7 Julia and Mandelbrot Sets Constructed Using the Escape - Time Algorithm and Boundary Scanning Method 229 7.1 Julia Sets and Filled Julia Sets 230 7.2 Parameter Sets 267 7.3 Illustrations... 1) Find the Floor and IntegerPart of Sin[3.95 n], for n from 1 to 12 2) Find the Sign of Cos[3] • Pseudorandom Numbers We shall use the command Random quite often, in adding perturbations to graphic shapes and in coloring graphics, and in generating graphics which have a great deal of randomness ? Random Random[ ] gives a uniformly distributed pseudorandom Real in the range 0 to 1 Random[ type, range]... Booklet: Getting Started with Mathematica 1.1.1 Getting Started with Mathematica A booklet entitled 'Getting Started with Mathematica' , published by Wolfram Research, accompanies every copy of Mathematica In this Chapter, we start by showing you how to perform basic calculations, learn the basic syntax of Mathematica and some basic commands using this booklet and the Help in Mathematica Start by using... Master Index and inspect some of the sections of the Mathematica Book which apply to lists 1.7 Mathematical Functions 1.7.1 Standard Built-in Functions In 'The Mathematica Book1, 1.1.3, as we have seen, there is a list of the more common built-in functions Section 3.2 of the Mathematica Book contains a complete list of all standard mathematical functions used in Mathematica From Help - The Mathematica. .. to the Contraction Mapping Theorem forTYDR2] 168 5.4 Constructing Various Types of Fractals Using Roman Maeder's Commands 170 5.5 Construction of 2D Affine Fractals Using the Random Algorithm 193 Chapter 6 Constructing Non-affine and 3D Fractals Using the Deterministic and Random Algorithms 6.1 Construction of Julia Sets of Quadratic Functions as Attractors of Non-affine Iterated Function Systems 198... binomialExpand with argument n defines a procedure for expanding (l + x) n binomiaIExpand[n_] := Expand[(l +x)"]; Having entered the above command, we can use it calculate the expansion for different values of n: binomialExpand[8] Another example: Exercise: 1) Look up the command Apart in the Master Index and then define a procedure which for each n, resolves *" into partial fractions, and use it... should now be able to add two or more natural numbers with Mathematica 1.1.2 Your First Mathematica Calculations Go to this section of the booklet and try the first 2 examples given, noting the method of executing a Mathematica command Read the section on 'Some Mathematica conventions', noting the bracketing conventions, the capitalisation of built-in function names and the 3 ways of expressing the product... notes on the commands Expand and Factor, and then expand and re-factor the expression: (x - 2 y)3 (1 + y 2 ), using % 4) Look up Partial Fractions in the Master Index, and then: a) Find partial fractions for (x + l)/(x - I) 2 (x2 +1), and, using %, check your answer; b) Express as a single rational function: 1 / (x + 2)2 - 2 / (x2 + 1), and check your result From Help - The Mathematica Book 1.2.1: If... its effect on the variable x In Mathematica, we can use the above idea to great effect Mathematica has the following method of defining an 'anonymous' or 'pure' function The function we defined above can be defined in Mathematica as: #2 +#& Chapter 1 21 The symbol # stands for the argument of the function, and the ampersand, &, informs Mathematica that we are concerned with a 'pure' function Think of... used, so Mathematica printed a warning and evaluated the expression with normal code Exercise: Let q [z, n] = 2" =1 Sin[z], Use the command Compile to calculate, with maximum speed: Chapter 1 23 a ) q [ l + I , 3]; b) the numerical value of q [ j , 7]; c) the sequence {q[l, 1], q [ l , 2], q[l, 12]} 1.7.5 Functions as Procedures In 1.3, you were asked to look up the commands Expand and Factor, and apply . alt="" Graphics with Mathematica Fractals, Julia Sets, Patterns and Natural Forms This page is intentionally left blank Graphics with Mathematica Fractals, Julia Sets, Patterns and Natural Forms by Chonat. 224 Contents Chapter 7 Julia and Mandelbrot Sets Constructed Using the Escape - Time Algorithm and Boundary Scanning Method 7.1 Julia Sets and Filled Julia Sets 230 7.2 Parameter Sets 267 7.3 Illustrations. certain Julia sets, the Mandelbrot set and other parame- ter sets, using the ideas and constructions of Chapter 4. The main ideas needed for the construc- tion of the (filled) Julia sets of polynomials