Table of Contents
Preface x
Who this book is for x
What you'll need x
Teaching and Learning with this book xi
Alignment to Standards xii
Acknowledgements xii
About the authors xiii
Chapter 1 What Is Geometry 2
Euclid 3
Beyond Euclid 3
Think Like a Mathematician 4
Structure of this Book 4
Summary and Learning More 6
Chapter 2 Openscad and Math Modeling 8
The Models 9
Some Models Have Small Parts 9
3D Printing 10
Materials 12
Downloading the Models: Github 13
Openscad 14
Openscad Workflow 15
Navigating on the Screen 15
Commenting Out or Editing Parts of a Model 16
Creating Basic Shapes 17
Combining Basic Shapes 18
Coordinate Systems in Openscad 19
Programming Tips for Openscad 21
No True Variables 21
Openscad is Case Sensitive 23
If and Else 23
Vectors and Ranges 23
Loops 25
Functions 25
Openscad Variants 26
Tinkercad 26
Summary and Learning More 28
Chapter 3 Simulating Geometry 30
3D Printable Models Used In This Chapter 31
Preliminaries 32
Circles and PI 32
Angles and Degrees 33
Parallel and Perpendicular 33
Code Versus Math Conventions 34
Equal Sign 34
Defaults 34
Polygons and Polyhedrons 35
Regular Polygons 35
Polyhedrons 38
Platonic Solids 38
Platonic Solids Models 38
Euler's Characteristic 40
Translation 41
Openscad Order of Operations 41
Rotation and Mirroring 42
Translating or Rotating Multiple Objects 44
Mirroring 44
Scaling 45
Math Functions 46
Extrusion 46
Extruding Linearly 46
Extruding While Rotating 49
Convex Hull 49
Make a Castle 50
The Walls 51
The Tower 53
Making Four Walls 55
Fancier Castles 56
Tinkercad 58
Summary and Learning More 59
Chapter 4 Constructions 62
Constructions 63
3D Printable Models Used in This Chapter 63
How to Use a Compass 64
Bisecting an Angle 65
Perpendicular Bisector 66
Equilateral Triangle 69
Other Constructions to Try 69
Reuleaux Triangles 69
3D Printable Model 70
Rolling Reuleaux 71
Other Reuleaux Polygons 72
Summary and Learning More 73
Answers 73
Equilateral Triangle Construction 73
Chapter 5 The Triangle Bestiary 74
What Is a Triangle? 75
Angles of a Triangle 75
3D Printable Models Used in This Chapter 76
Prove Triangle Angles Add to 180 Degrees 77
How to Use a Protractor 78
Specifying Triangles 78
Congruent Triangles 78
The Triangle Family Tree 79
Making Triangles with Openscad 81
Drawing and Labeling Triangles 82
Similar Triangles 83
Area of a Triangle 85
Triangle Area Models 85
Right Triangle Model 87
Acute Triangle Model 88
Obtuse Triangle Model 89
Paper Version of Triangle Area Model 93
Why Triangles Make Things Strong 95
3D Printing Meshes 96
Viewing A 3D Print's Mesh 98
Summary and Learning More 99
Answers 100
Triangle-Type Table 100
Chapter 6 Pythagoras and a Little Trigonometry 102
3D Printable Models Used In This Chapter 103
Squares and Square Roots 104
Pythagorean Theorem 104
Pythagoras' Lego 104
Making a Spiral of Theodorus 105
A Little Trigonometry 108
Sine, Cosine, Tangent 108
Hypotenuse Model 109
Using a Triangle as an Angle Template 111
Complementary Angles 111
Other Ratios 111
Degrees, Radians, and PI 112
Finding the Length of a Side 113
Calculating with Sine and Cosine 113
Angles Not Between 0° and 90° 114
Angles Between 90° and 180° 115
Angles Between 180° and 270° 116
Angles Between 270° and 360° (Or -90° and 0°) 116
Arcsin, Arccos, Arctan 118
Summary and Learning More 119
Answers 120
Hypotenuse Model 120
Calculating with Sine and Cosine 120
Angles Greater Than 90° 120
Chapter 7 Circles 122
Area of a Circle 123
3D Printable Models Used in this Chapter 123
Inscribed and Circumscribed Polygons 124
Modifying the Models 125
Bracketing PI 127
Area of Inscribed Polygons 127
Area of Circumscribed Polygons 129
Leonardo's Puzzle 130
Measuring PI with String 132
Latitude and Longitude 133
Finding Your Latitude 134
Elevation of the Sun at Local Noon 137
Solar Declination 140
Solar Declination Example 141
Calculating Latitude 142
Arctic and Tropics 143
Finding Your Longitude 143
Historical Efforts to Find Longitude 144
Approximating Longitude and the Equation of Time 144
Longitude Calculation Example 145
Summary and Learning More 147
Answers 148
Arctic and Tropics 148
Chapter 8 Volume, Density, and Displacement 150
3D Printable Models Used in this Chapter 152
Volume 153
Volume of a Rectangular Solid 153
Volume of Prisms 154
Modifying the Models 156
Volume of a Cylinder 157
Volume of a Cone 157
Volume of a Sphere 158
Volume Special Ratios 158
Volume of Pyramids 161
Comparing Volumes 163
If You Don't Have a 3D Printer 163
Cavalieri's Puzzle 164
Density 165
Mass V. Weight 165
Measuring Density 166
Displacement 167
Measuring Volume with Displacement 168
The Questionable Crown-Maker 169
Finding Density from Displacement 170
Archimedes' Principle 171
Buoyancy Force 172
Buoyancy Experiment 173
Summary and Learning More 174
Chapter 9 Surface Area and Nets 176
Nets 177
3D Printable Models Used in This Chapter 177
3D Printing the Nets 178
Printing the Nets on Paper 179
3D Printing the Nets on Fabric 180
Checking Answers 180
Platonic Solids 181
Modifying the Model 181
Equilateral Triangle Faces 181
Square Faces 182
Pentagonal Faces 182
Checking Surface Area of Platonic Solids 184
Right Prisms 185
Modifying the Model 185
Pyramid 186
Cone 188
Cylinder 190
Sphere 191
Gluing or Printing a Net Onto Fabric 191
Gluing Nets onto Fabric 191
Fabric Printing Process 192
Pausing a 3D Print 193
Fabric to Use 194
Attaching Fabric to the Platform 194
Keeping a Wary Eye 195
Summary and Learning More 195
Answers 196
Cone 196
Cylinder 196
Chapter 10 Slicing and Sections 198
Slicing Prisms 199
Modifying the Model 199
3D Printable Models Used in This Chapter 199
Slicing Cubes 200
Rectangular (4-Sided) Cross-Section 201
Triangular (3-Sided) Cross-Section 201
Pentagonal (5-Sided) Cross-Section 202
Hexagonal (6-Sided) Cross-Section 203
Comparing Cases 203
Slicing Other Prisms 203
Slicing a Cylinder 205
Slicing a Sphere 206
Conic Sections 206
Slant Angle and Slant Height 207
Prelude: Flashlight Experiment 208
Modifying the Models 209
Circle Cross-Section 210
Ellipse Cross-Section 210
Parabola Cross-Section 211
Hyperbola Cross-Section 213
Conic Sections and Angles 216
Slices Intersecting the Vertex of a Cone 216
Summary and Where to Learn More 216
Chapter 11 Constructing Conics 218
Ellipses and Circles 219
3D Printable Models Used in This Chapter 219
Foci of an Ellipse and Internal Reflection 221
Whispering Galleries 222
Circumference and Area of an Ellipse 223
Orbits of Planets and Kepler's Laws 223
Parabola 225
Directrix and Generatrix 225
Drawing a Parabola 226
Parabolic Mirrors and Antennas 228
Kicking a Ball into the Air 229
Hyperbola 231
Constructing a Hyperbola 231
Hyperbola Applications 234
Tying the Curves Together 234
Summary and Learning More 236
Answers 236
Finding the Location of the Foci of an Ellipse 236
Chapter 12 Geometry, Space and Time 238
The Equation of Time 239
3D Printable Models Used in This Chapter 239
The Analemma 241
The Geometry of the Analemma 241
Measuring the Analemma 243
Analemmas on Other Planets 246
Paths in the Sky of Other Objects 247
Making a Sundial 247
Creating the Gnomon 247
Using Your Sundial 251
Clockwise 253
Other Sundial Designs 253
Summary and Learning More 253
Chapter 13 The Geometry Museum 254
Arches 255
3D Printable Models Used in this Chapter 256
Circular and Gothic Arches 257
Making the Model 258
Testing Your Arches 259
Paper (Or Cut-Out) Alternatives 261
Renovating the Castle 261
Gothic Tracery 264
Making the Model 265
The Trefoil 265
Printing Caveats 265
Trusses 266
Preparation 267
Tetrahedron 268
Cube 270
Combine Cube and Tetrahedron 272
The Gift Shop 273
The Tetrahedron and Cube Puzzle 273
Reuleaux and Constant-Width Shapes in 3D 274
Revolved Reuleaux Polygons 274
Reuleaux Tetrahedron 275
Meissner Tetrahedrons 276
Printing These Models 276
Bounding Box 277
Dual Polyhedrons 277
The Golden Ratio, PHI 279
The Golden Angle 280
Relation to Fibonacci Series 281
More Than 3D: Polytopes 281
Summary and Learning More 281
Answers 282
Gothic vs. Circular Arch 282
Appendix: Resources and Standards 284
Companion Resources 284
Other Sources 284
Index 288
