利用3D打印探索科学：课堂、科学展与家庭实践

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■ IntroduCtIon

The models span a variety of topics, and we tried to cover as many disciplines as

possible. Briefly, here is what you can look forward to:

Chapter 1 gives you a few options to print many different

types of mathematical surface. This ability underlies some of

the other models.

Chapter 2 creates models of waves to allow you to explore

what happens when various waves overlap and interact. You

can print yourself a model of Young’s famous double-slit

experiment to see how light from two slits can interfere.

Chapter 3 takes us back to Newton and Kepler to learn about

planets and stars and how they speed up and slow down in

their orbits.

Chapter 4 allows you to create wings with classic airfoil shapes

from the early days of flight. You will be able to make yourself

a very simplistic test stand that you can use to measure the lift

from the wing with just a fan and a postal scale.

Chapter 5 lets you create basic models of all the “simple

machines”—wedge, inclined plane, lever, pulleys, and screws.

Chapter 6 allows you to model plants and their ecosystems,

and to design plants for different environments. Maybe you

can create a garden for another planet (or for Earth after

another few hundred years of climate change).

Chapter 7 lets you begin to explore carbon atoms, and how

water molecules come together to make two different types of

ice crystals.

Chapter 8 explores 2D and 3D trusses and how you can use

them in various explorations.

As we noted earlier, Appendix A reviews how to 3D print, and

Appendix B aggregates all the links in the book.

Finally, we are making the 3D-printable models used in this book

(although not the book itself!) open source, licensed under a Creative

Commons Attribution-NonCommerical-ShareAlike 4.0 International License

(https://creativecommons.org/licenses/by-nc-sa/4.0/). That means you can use

them for noncommercial purposes (in a classroom, for instance) and alter and remix

them as long as you credit us. Appendix A has some notes about where to find the

repositories if you would like to add to these models. We hope these models are the

beginning of a set of models that students everywhere can play with and learn from.

CHAPTER 1 ■ 3D MATH FUNCTIONS

■ Tip This book presumes you are generally familiar with 3D printing practices. If not,

you can learn how to use a printer from Joan’s previous book, Mastering 3D Printing

(Apress, 2014) or our book 3D Printing With MatterControl (Apress, 2015). Unless we

specifically note otherwise, prints in this book were created on a Deezmaker Bukito in

polylactic acid (PLA) plastic, using the MatterSlice engine in MatterControl (although we

could have used any software compatible with an open source printer).

Math Background

This chapter presumes you know what a function is—a relationship among a number of

variables. In this case, we are dealing with functions using three variables, which we will call

x, y, and z. Function notation looks like this: z = f (x,y). All that means is that our variable, z,

can be computed for any given pair of values for the x and y variables. Having three variables

means we can define shapes in three dimensions, with one variable corresponding to each

dimension. Normally these three-dimensional shapes would be shown on a page with

two-dimensional projections. Often, this is fine and you can see what is going on.

Sometimes, however, it really helps to hold a 3D model in your hand and turn it this way

and that. This chapter will give you the ability to do that for many types of functions.

■ Note 3D printing convention holds that x and y are in the plane of the platform that your

model is being built up on, and z is vertical height above that. In other words, the bottom

of the surfaces generated in this chapter is always the z = 0 plane. In this convention, you

always have to transform what you are printing to have z greater than or equal to zero, since

you cannot build under the platform. In other words, if you know that z would be negative for

some values of x and y that you want to use, you may have to add an offset to your equation

so that z is always greater than zero and remember that the offset is there when you think

about what your model represents.

We will get you started with a model entirely in OpenSCAD that creates surfaces of

functions z = f (x,y), where x and y are the plane of the 3D printer’s build platform, and z is

the height of the surface above that plane. First we will show you how the basic 3D math

model Rich has written and included here works, and what kind of functions you can

print. Then we will show you a simple model that creates surfaces that might be a starting

point for your own projects in OpenSCAD.

Alternatively you may have code you developed that produces a surface you would

like to 3D print. It may not be practical to port that code to an OpenSCAD model. We will

also show you an example in which we wrote a separate Python script that produces a file,

which is then read into OpenSCAD and made into a surface. Finally we will give you some

ideas about how you might use these tools as a teacher or as part of a student project.

CHAPTER 1 ■ 3D MATH FUNCTIONS

The values of x and y go from 0 to xmax and ymax respectively. A maximum of 100

points in each dimension (xmax = 99 and ymax = 99) is allowed. The model will step in

units of 1, which cannot be changed. If you want your model to step in smaller or larger

increments, you will need to scale the variable in the equation.

For example, if in your original equation you had a function you wanted to increment

by 0.02 in each step, replace the x in your original equation by (0.02 * x) to accomplish

the same thing when you increment by 1. Or if you want to step from –500 to –400 in the

original function, replace the x in your original function with (x – 500) everywhere to

accomplish the same thing. Be careful if variables are raised to a power, or are inside a

function like cosine, that you do this scaling correctly and consistently.

■ Caution Because we are creating a flat bottom, the equation being represented here is

actually 0 <= z <= f (x, y ). As a result, z = f (x, y ) must be greater than zero everywhere for

the flat-bottomed (thick = 0) version. A model will still be produced if there are z values

that are less than zero, but it will be an invalid model, and even if you manage to repair it,

it won’t print easily.

■ Note The OpenSCAD models in this book are written by Rich Cameron and licensed

under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International

License—see http://creativecommons.org/licenses/by-nc-sa/4.0/ for details.

Listing 1-1. The Basic OpenSCAD Model to Create a 3D Print of a Surface

// OpenSCAD model to print out an arbitrary surface defined as z = f(x,y)

// Either prints the surface as two sided and variable thick = thickness

// Or if thick = 0, prints a top surface with a flat bottom

// File surfaceprint.scad

function f(x, y) = ((x - 50) * (y - 50)) / 100 + 30; // Saddle point

//z height, in mm

thick = 0; //set to 0 for flat bottom. else mm thickness of surface

xmax = 99; //Number of points in x direction - 99 is the max

ymax = 99; // Number of points in y direction - 99 is the max

// If you want a rough surface (to make it more tactile) set blocky=true.

// Otherwise surface will be smoothed

blocky = false; //if true, xmax and ymax must be less than 100.

//number of points that will be plotted

toppoints = (xmax + 1) * (ymax + 1);

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