Mathematica编程指南：入门与实践

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"Trott.M..The.Mathematica.Guidebook.for.Programming" 本书《The Mathematica Guidebook for Programming》由Michael Trott编写，是Mathematica编程的权威指南。书中内容涵盖广泛，旨在帮助读者深入理解和高效使用Mathematica进行程序设计。 0.1 概览 - 内容概述：书中的各个章节都提供了简洁的总结，便于读者快速了解主题。 - 四卷书的关系：这套指南由四卷组成，每卷分别针对Mathematica的不同方面，形成一个完整的知识体系。 - 章节结构：每个章节都按照逻辑结构编排，确保内容的连贯性。 - 代码展示风格：书中代码展示清晰，易于理解和实施。 0.2 要求 - 硬件和软件：读者需具备运行Mathematica的适当硬件和软件环境。 - 阅读前提：对计算机基础知识和基础编程概念有一定的了解。 0.3 指南的目的与非目的 - 计算数学：书中不仅教授Mathematica的使用，还涉及如何运用它进行复杂的数学计算。 - 编程范式：探讨了Mathematica支持的不同编程风格和策略。 0.4 练习与解答 - 练习：每章末尾提供练习题，以巩固学习内容。 - 解答：配套的解决方案帮助读者检查理解并解决遇到的问题。 0.5 书籍与电子组件 - Notebook工作方式：强调了使用Mathematica的Notebook界面进行交互式编程的优点。 - 结果可复制性：讨论了如何确保计算结果的一致性和可重复性。 - 笔记本早期版本：提醒读者不同版本的Mathematica可能存在的差异。 0.6 风格与设计元素 - 文本和代码格式：书中采用一致的格式标准，提高阅读体验。 - 参考文献：提供了丰富的引用资源，便于深入研究。 - 变量作用域、输入编号和警告信息：详细解释了这些编程关键概念。 - 图形：讲解了Mathematica在图形生成方面的功能。 - 符号和记号：介绍了Mathematica中的特殊符号和表示法。 - 单位：讨论了单位系统及其在计算中的应用。 - 封面图形：通过封面设计展示了Mathematica的图形能力。 0.7 生产历史：记录了书籍的创作和修订过程。 0.8 四个一般性建议：作者给出了提升学习效果和使用体验的建议。第1章“Introduction to Mathematica”则开始介绍Mathematica作为编程语言的基础知识，包括其背景和基本概念。这本指南对于希望掌握Mathematica编程技术的读者来说是一份宝贵的资源，它将引导读者深入探索这个强大的计算平台，并学习如何利用它进行各种复杂任务的处理。通过书中的实例和练习，读者可以逐步提升自己的编程技能，从而更好地利用Mathematica进行科学计算、数据分析以及可视化等任务。

1.6 Coloring Closed Curves

Coloring Plots † Finding Curve Intersections † Sorting 2D Line

Segments

† Loop Construction † Constructing the Clusters †

Checkerboard Coloring

† Some Examples † Checking if Polygons

are Disjoint

@ @ Overview

Exercises

Game of Life † Langton’s Ant † Brillouin Zones † Maxwell–Helmholtz

Color Triangle

† Conformal Maps † Cornet Isogons † Jarník

Polygons

† Light Ray Reflections in a Water Drop † Warped

Patterns † Moiré Patterns † Triptych Fractal † Multiple Reflected

Pentagons

† Random Lissajous Figures † Walsh Function † Sorting

Game † Ball Moves † Rectangle Packings † Smoothed L-Systems †

Polygonal Billiards

† Random Walk on a Sierpinski Fractal † Voronoi

Tessellations

† Lévy Flights † Random Supersymmetric Potential †

Common Plotting Problems

† Nomogram for Quadratic Equation †

Clusters on Square Grids

† Aperiodic Triangle Tilings

@ @ Solutions

Random Cluster Generation † Leath Clusters † Midsector Lines †

nalyzing Mathematica

Code † Visualizing Piecewise Linear

pproximations † Cartesian Ray † Kepler Cubes † Modulated Sin-

Curves

† Superimposed Lattices † Triptych Fractals † Two

Superimposed Bumps Forming Three Bumps † Repeatedly Mirrored

Decagons

† Smoothly Connected Curves † Randomly Deformed

Graphics † Random Expressions

@ @ References

CHAPTER

Three–Dimensional Graphics

2.0 Remarks

2.1 Fundamentals

2.1.1 Graphics Primitives

Points, Lines, and Polygons † Cuboids † Projecting a Hypercube into

† Nonplanar and Nonconvex Polygons † Translating 3D Shapes †

Escher’s Cube World

2.1.2 Directives for Three-Dimensional Graphics Primitives

bsolute and Relative Sizes of Points and Lines † Constructing an

Icosahedron from Quadrilaterals

† Coloring Polygons in the

Presence of Light Sources † Diffuse and Specular Reflection †

Edges and Faces of Polygons

† Rotating 3D Shapes † Random

Rotations † Stacked Tubes † Text in 3D Graphics

2.1.3 Options for 3D Graphics

The 34 Options of 3D Graphics † Relative and Absolute Coordinate

Systems

† Space Curves versus Space Tubes

2.1.4 The Structure of Three-Dimensional Graphics

Resolving Automatic Option Settings † Nested Primitives and

Directives † Converting 3D Graphics to 2D Graphics

Printed from THE MATHEMATICA GUIDEBOOKS

2.1.5 Discussion of Selected Options

Platonic Solids † Choosing the Viewpoint † Simple 3D Shapes † Light

Sources and Colored Polygons

† Cluster of Dodecahedra † Views on

an Octant Filled with Cubes † Restricting the Plot Range † The 3D

Graphics Enclosing Box

† View Direction † Sizing Identical Graphics

Independently of the Viewpoint † Rendering All versus Rendering

Only Visible Polygons

† Intersecting Polygons † Colliding Platonic

Solids

† A Scale with Platonic Solids † Diamond Faces † Rolled

Checkered Paper † Woven Tubes † Smooth Dodecahedron–

Icosahedron Transition

† Platonic Solid Metamorphosis † Slicing a

Cube

2.2 Display of Functions

2.2.1 Functions Given Analytically

Graphing Functions of Two Variables † Special Plotting Options †

Wireframes † Showing Multiple Plots † Parametrized Vector

Functions

† Cubed Torus † Klein Bottle † Parametrized Surfaces

Samples † Using Symmetries to Construct Graphics † Constructing a

Candelabra

† Surfaces of Revolution † Emission of an Accelerated

Point Charge † Borromaen Rings † Spiraling Spiral † Constructing a

Birthday Bow

2.2.2 Functions Given at Data Points

Visualizing 2D Arrays of Data † Visualizing Computation Timings †

Time Evolution on a Torus

† 3D Bar Charts † Randomized Geode

2.3 Some More Complicated Three-Dimensional Graphics

2.3.0 Remarks

2.3.1 3D Graphics of Iterative Mappings

Rauzy Fractal From a 3D Projection † 3D Sierpinski Sponge †

Exercising a Sierpinski Sponge † Kepler Tiling † 3D Iterated Function

System

† Random Clusters of Tetrahedra † Quaquaversal Tiling †

3D Truchet Graphics

† 3D Space Fillers

2.3.2 Tubular Curves

Frenet Frame † Tangents, Normals, and Binormals of Space

Curves

† Tubes around Space Curves † Knots † Mapping Textures

to Knots

† Tubes around Piecewise Straight Curves † Biased

Random Walk

† Osculating Circles of Curves

2.3.3 Recursively Colored Easter Eggs

Recursively Subdividing Surfaces † Deformed Spheres † Mapping

Patterns to Spheres

† Rough Surfaces

2.3.4 Klein Bottles

Making Surfaces by Gluing the Edges of a Square † Spine Curves †

Cross Sections of Klein Bottles † Slicing and Coloring Klein Bottles †

Deformed Klein Bottles

† Cubistic Klein Bottles

2.3.5 A Hypocycloidal Torus

Triangulating Quadrilaterals † Rotating Curves to Sweep out

Surfaces † Triangulations † Surfaces with Holes

2.3.6 The Penrose Tribar

Constructing a Tribar † Coordinate System Transformations †

Choosing the Right View Point

† Calculating the Optimal Viewpoint †

n Impossible Crate

2.3.7 Riemann Surfaces of Simple Functions

Plotting Multivalued Functions † Riemann Surfaces of Algebraic

Functions

† Cutting Surfaces along Branch Cuts † Surfaces

Subdivided Using Tilings † A Family of Polynomial Riemann

Surfaces

† Implicit Parametrizations † Riemann Surfaces of Nested

Logarithms

† Riemann Surfaces over the Riemann Sphere

THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS

2.3.8 Interwoven Polygonal Frames

Planes Intersecting Convex Bodies † Calculating All Intersections †

Creating Frames

† Interweaving Frames † Examples of Interwoven

Frames

2.3.9 Selfintersecting Origami and 4D Hilbert Curves

Paper Folding Models † Goffinet Kite † Folding Animation † Hilbert

Curves in Higher Dimensions

2.3.10 The Cover Image: Hyperbolic Platonic Bodies

Triangulating Platonic Solids † Symmetry Considerations † Compact

Code † Evolution of the Cover Graphics from Version 2 to Version

† Nonplanar Contraction and Expansion of Polyhedra

2.4 Brillouin Zones of Cubic Lattices

Higher Degree Voronoi Regions † Simple Cubic Lattice † Bisector

Planes

† Intersection of Planes † Symmetry of a Cube † Forming

Brillouin Zones from Polygons † Gluing Polygons Together † Body-

Centered Lattice

† Face

Centered

Lattice

Overview

@ @ Exercises

3D Surface Sampler † Warped, Twisted, and Interlocked Tori †

Dodecahedra Iteratively Reflected on its Faces

† Snail † Trinomial

Theorem Visualization † Ball Blending Method † Loop Subdivision †

3 -Subdivision Algorithm † Averaging Closed Curves † Projective

Plane Model

† Counting Surfaces for a Given Genus † Lattice

Pyramids † Fractal Mountains † Random Walk on a Sphere †

Projecting onto Polyhedra

† Alexander’s Horned Sphere † Polyhedral

Caustic

† Sliced Möbius Strip † Perspective Modeling † Displaying

Hidden Edges † Generating Platonic Solid Clusters † A 4D Platonic

Solid

—

The

120

Cell

† Folding a Dodecahedron † Continuously

Changing Polyhedra † Inscribing Five Cubes in a Dodecahedron †

Interwoven Bands around a Dodecahedron

† Knot Made from

Knots † Knot with Escher Tiling † Gear Chain Animation † 3D Peano

Polygon

† Tetraview Riemann Surface Animation † Riemann Surface

of Kepler Equation

† Sierpinski Plant

@ @ Solutions

Cayley Cusp † Boy Surface † Möbius Strip † Steiner’s Cross Cap †

Henneberg Surface

† Flying Saucer Construction † Random

Parametrized Surfaces † Dodecahedral Flowers † Extruded Platonic

Solids

† Smoothing through Graph Plotting † Staggered Trefoil

Knots

† Field Lines of Two Charged Spheres † Random Symmetric

Polyhedra † Graphics of a Screw † Arranging Worn Stones Tightly †

Random Cones

† Broken Tube † Weaving a Torus † Constructing

Double and Triple Tori from Torus Pieces † Massive Wireframes of

Platonic Solids

† Smoothing a Cube Wireframe † Smoothing a

Stellated Icosahedron

† Pyramids on Lattices † Closed Random

Walks

† Slicing and Coloring a Möbius Strip † Coordinate System

Transformations

† Kochen

–

Specker

Theorem

† Smooth Random

Functions

† Subdividing Concave Polygons

Printed from THE MATHEMATICA GUIDEBOOKS

@ @ References

CHAPTER 3

Contour and Density Plots

3.0 Remarks

3.1 Contour Plots

Contour Graphics † Converting Contour Graphics † Options of

Contour Graphics † Cassini Curve † Various Sample Contour Plots †

Functions Varying Strongly

† Homogeneous Contour Line Density †

Coloring Contour Plots † Contour Graphics in Nonrectangular

Domains

† Speckles and Scarlets from Superimposing 2D Waves †

Smoothing Contour Lines

† Superimposed 2D Waves in Symmetric

Directions † Comparing Options and Option Settings of Plotting

Functions

† Algebraic Description of Polygons † Blaschke Products †

Charged Goffinet Dragon † Square Well-Scattering Amplitude

3.2 Density Plots

Density Graphics † Converting Density Graphics † Arrays of Gray or

Color Values † Lifting Color Value Arrays to 3D † Earth Graphics †

rra

Plots † Gauss Sums † Visualizing Difference Equation

Solutions † Visualizing Matrices † Saunders Pictures † Making

Photomosaics from Density Plots

3.3 Plots of Equipotential Surfaces

Visualizing Scalar Functions of Three Variables † Marching Cubes †

Plots of Implicitly Defined Algebraic Surfaces

† Implicit Descriptions

of Riemann Surfaces † Gluing Implicitly Defined Surfaces Smoothly

Together

† Using Reflection and Rotation Symmetries to Visualize

lgebrai

Surfaces † Examples of Surfaces from Spheres, Tubes,

and Tori Glued Together † An Algebraic Candelabra † Joining Three

Cylinders Smoothly

† Zero

Velocity

Surfaces

† Implicit Form of an

Oloid † Isosurfaces of Data

@ @ Overview

Exercises

Clusters of Irreducible Fractions † Chladny Tone Figures in

Rectangles and Triangles

† Helmholtz Operator Eigenfunctions of a

Tetrahedron

† Liénard

–

Wiechert

Potential of a Rotating Point

Charge

† Shallit–Stolfi–Barbé Plots † Random Fractals † Functions

with the Symmetry of Cubes and Icosahedra

† Icosahedron

Equation † Belye Functions † Branch Cuts of Hyperelliptic Curves †

Equipotential Plots of Charged Letters

† Charged Random Polygon †

Gauss

–

Bonnet

Theorem

† Interlocked Double and Triple Tori †

Inverse Elliptic Nome † Contour Plots of Functions with Boundaries

of Analyticity

† Isophotes on a Supersphere † Structured Knots †

Textures on a Double Torus

THE MATHEMATICA GUIDEBOOKS to PROGRAMMING—GRAPHICS—NUMERICS—SYMBOLICS

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