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E-raamat: Make: Geometry

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  • Formaat: 312 pages
  • Ilmumisaeg: 28-Jun-2021
  • Kirjastus: Make Community, LLC
  • Keel: eng
  • ISBN-13: 9781680456691
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Make: GeometrySuurem pilt
  • Formaat: 312 pages
  • Ilmumisaeg: 28-Jun-2021
  • Kirjastus: Make Community, LLC
  • Keel: eng
  • ISBN-13: 9781680456691
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Geometry, of all the branches of mathematics, is the one that is most easily visualized by making something. However, it is all too easy to reduce it to reams of formulas to memorize and proofs to replicate. This book aims to take geometry back to its practical roots with 3D printed models and puzzles as well as demonstrations with household objects like flashlights and paper towel tubes.

This is not a traditional geometry textbook, but rather builds up understanding of geometry concepts encountered primarily in middle school while also bringing in elements of concepts normally learned much later. Some of the models are counterintuitive, and figuring out how and why they work will both entertain and give insights. Two final chapters suggesting open-ended projects in astronomy and physics, and art and architecture, allow for deeper understanding and integration of the learning in the rest of the book.

Preface x
Who This Book Is For
x
What You'll Need
x
Teaching And Learning With This Book
xi
Alignment To Standards
xii
Acknowledgments
xii
About The Authors
xiii
Chapter 1 What Is Geometry 2(6)
Euclid
3(1)
Beyond Euclid
3(1)
Think Like A Mathematician
4(1)
Structure Of This Book
4(2)
Summary And Learning More
6(2)
Chapter 2 OpenSCAD And Math Modeling 8(22)
The Models
9(5)
Some Models Have Small Parts
9(1)
3D Printing
10(2)
Materials
12(1)
Downloading The Models: Github
13(1)
OpenSCAD
14(12)
OpenSCAD Workflow
15(1)
Navigating On The Screen
15(1)
Commenting Out Or Editing Parts Of A Model
16(1)
Creating Basic Shapes
17(1)
Combining Basic Shapes
18(1)
Coordinate Systems In OpenSCAD
19(2)
Programming Tips For OpenSCAD
21(5)
No True Variables
21(2)
OpenSCAD Is Case Sensitive
23(1)
If And Else
23(1)
Vectors And Ranges
23(2)
Loops
25(1)
Functions
25(1)
OpenSCAD Variants
26(1)
Tinkercad
26(2)
Summary And Learning More
28(2)
Chapter 3 Simulating Geometry 30(32)
3D Printable Models Used In This
Chapter
31(1)
Preliminaries
32(3)
Circles And Pi
32(1)
Angles And Degrees
33(1)
Parallel And Perpendicular
33(1)
Code Versus Math Conventions
34(1)
Equal Sign
34(1)
Defaults
34(1)
Polygons And Polyhedrons
35(6)
Regular Polygons
35(3)
Polyhedrons
38(8)
Platonic Solids
38(1)
Platonic Solids Models
38(2)
Euler's Characteristic
40(1)
Translation
41(1)
OpenSCAD Order Of Operations
41(1)
Rotation And Mirroring
42(3)
Translating Or Rotating Multiple Objects
44(1)
Mirroring
44(1)
Scaling
45(1)
Math Functions
46(1)
Extrusion
46(3)
Extruding Linearly
46(3)
Extruding While Rotating
49(1)
Convex Hull
49(1)
Make A Castle
50(8)
The Walls
51(2)
The Tower
53(2)
Making Four Walls
55(1)
Fancier Castles
56(2)
Tinkercad
58(1)
Summary And Learning More
59(3)
Chapter 4 Constructions 62(12)
Constructions
63(1)
3D Printable Models Used In This
Chapter
63(10)
How To Use A Compass
64(1)
Bisecting An Angle
65(1)
Perpendicular Bisector
66(3)
Equilateral Triangle
69(1)
Other Constructions To Try
69(1)
Reuleaux Triangles
69(1)
3D Printable Model
70(1)
Rolling Reuleaux
71(1)
Other Reuleaux Polygons
72(1)
Summary And Learning More
73(1)
Answers
73(1)
Equilateral Triangle Construction
73(1)
Chapter 5 The Triangle Bestiary 74(28)
What Is A Triangle?
75(1)
Angles Of A Triangle
75(1)
3D Printable Models Used In This
Chapter
76(2)
Prove Triangle Angles Add To 180 Degrees
77(1)
How To Use A Protractor
78(1)
Specifying Triangles
78(1)
Congruent Triangles
78(5)
The Triangle Family Tree
79(2)
Making Triangles With OpenSCAD
81(1)
Drawing And Labeling Triangles
82(1)
Similar Triangles
83(2)
Area Of A Triangle
85(10)
Triangle Area Models
85(8)
Right Triangle Model
87(1)
Acute Triangle Model
88(1)
Obtuse Triangle Model
89(4)
Paper Version Of Triangle Area Model
93(2)
Why Triangles Make Things Strong
95(1)
3D Printing Meshes
96(3)
Viewing A 3D Print's Mesh
98(1)
Summary And Learning More
99(1)
Answers
100(2)
Triangle-Type Table
100(2)
Chapter 6 Pythagoras And A Little Trigonometry 102(20)
3D Printable Models Used In This
Chapter
103(1)
Squares And Square Roots
104(1)
Pythagorean Theorem
104(4)
Pythagoras' Lego
104(1)
Making A Spiral Of Theodorus
105(3)
A Little Trigonometry
108(11)
Sine, Cosine, Tangent
108(1)
Hypotenuse Model
109(2)
Using A Triangle As An Angle Template
111(1)
Complementary Angles
111(1)
Other Ratios
111(1)
Degrees, Radians, And Pi
112(1)
Finding The Length Of A Side
113(1)
Calculating With Sine And Cosine
113(1)
Angles Not Between 0° And 90°
114(4)
Angles Between 90 °and 180°
115(1)
Angles Between 180 °and 270°
116(1)
Angles Between 270° And 360°(Or -90° And 0°)
116(2)
Arcsin, Arccos, Arctan
118(1)
Summary And Learning More
119(1)
Answers
120(2)
Hypotenuse Model
120(1)
Calculating With Sine And Cosine
120(1)
Angles Greater Than 90°
120(2)
Chapter 7 Circles 122(28)
Area Of A Circle
123(1)
3D Printable Models Used In This
Chapter
123(10)
Inscribed And Circumscribed Polygons
124(1)
Modifying The Models
125(2)
Bracketing Pi
127(3)
Area Of Inscribed Polygons
127(2)
Area Of Circumscribed Polygons
129(1)
Leonardo's Puzzle
130(2)
Measuring Pi With String
132(1)
Latitude And Longitude
133(1)
Finding Your Latitude
134(9)
Elevation Of The Sun At Local Noon
137(3)
Solar Declination
140(1)
Solar Declination Example
141(1)
Calculating Latitude
142(1)
Arctic And Tropics
143(1)
Finding Your Longitude
143(4)
Historical Efforts To Find Longitude
144(1)
Approximating Longitude And The Equation Of Time
144(1)
Longitude Calculation Example
145(2)
Summary And Learning More
147(1)
Answers
148(2)
Arctic And Tropics
148(2)
Chapter 8 Volume, Density, And Displacement 150(26)
3D Printable Models Used In This
Chapter
152(1)
Volume
153(11)
Volume Of A Rectangular Solid
153(1)
Volume Of Prisms
154(2)
Modifying The Models
156(1)
Volume Of A Cylinder
157(1)
Volume Of A Cone
157(1)
Volume Of A Sphere
158(1)
Volume Special Ratios
158(3)
Volume Of Pyramids
161(2)
Comparing Volumes
163(2)
If You Don't Have A 3D Printer
163(1)
Cavalieri's Puzzle
164(1)
Density
165(2)
Mass V. Weight
165(1)
Measuring Density
166(1)
Displacement
167(7)
Measuring Volume With Displacement
168(1)
The Questionable Crown-Maker
169(1)
Finding Density From Displacement
170(1)
Archimedes' Principle
171(1)
Buoyancy Force
172(1)
Buoyancy Experiment
173(1)
Summary And Learning More
174(2)
Chapter 9 Surface Area And Nets 176(22)
Nets
177(1)
3D Printable Models Used In This
Chapter
177(4)
3D Printing The Nets
178(1)
Printing The Nets On Paper
179(1)
3D Printing The Nets On Fabric
180(1)
Checking Answers
180(1)
Platonic Solids
181(4)
Modifying The Model
181(1)
Equilateral Triangle Faces
181(1)
Square Faces
182(1)
Pentagonal Faces
182(3)
Checking Surface Area Of Platonic Solids
184(1)
Right Prisms
185(1)
Modifying The Model
185(1)
Pyramid
186(2)
Cone
188(2)
Cylinder
190(1)
Sphere
191(4)
Gluing Or Printing A Net Onto Fabric
191(1)
Gluing Nets Onto Fabric
191(1)
Fabric Printing Process
192(1)
Pausing A 3D Print
193(1)
Fabric To Use
194(1)
Attaching Fabric To The Platform
194(1)
Keeping A Wary Eye
195(1)
Summary And Learning More
195(1)
Answers
196(2)
Cone
196(1)
Cylinder
196(2)
Chapter 10 Slicing And Sections 198(20)
Slicing Prisms
199(1)
Modifying The Model
199(1)
3D Printable Models Used In This
Chapter
199(1)
Slicing Cubes
200(3)
Rectangular (4-Sided) Cross-Section
201(1)
Triangular (3-Sided) Cross-Section
201(1)
Pentagonal (5-Sided) Cross-Section
202(1)
Hexagonal (6-Sided) Cross-Section
203(1)
Comparing Cases
203(1)
Slicing Other Prisms
203(2)
Slicing A Cylinder
205(1)
Slicing A Sphere
206(1)
Conic Sections
206(10)
Slant Angle And Slant Height
207(1)
Prelude: Flashlight Experiment
208(1)
Modifying The Models
209(1)
Circle Cross-Section
210(1)
Ellipse Cross-Section
210(1)
Parabola Cross-Section
211(2)
Hyperbola Cross-Section
213(3)
Conic Sections And Angles
216(1)
Slices Intersecting The Vertex Of A Cone
216(1)
Summary And Where To Learn More
216(2)
Chapter 11 Constructing Conics 218(20)
Ellipses And Circles
219(1)
3D Printable Models Used In This
Chapter
219(6)
Foci Of An Ellipse And Internal Reflection
221(1)
Whispering Galleries
222(1)
Circumference And Area Of An Ellipse
223(1)
Orbits Of Planets And Kepler's Laws
223(2)
Parabola
225(6)
Directrix And Generatrix
225(1)
Drawing A Parabola
226(2)
Parabolic Mirrors And Antennas
228(1)
Kicking A Ball Into The Air
229(2)
Hyperbola
231(3)
Constructing A Hyperbola
231(3)
Hyperbola Applications
234(1)
Tying The Curves Together
234(2)
Summary And Learning More
236(1)
Answers
236(2)
Finding The Location Of The Foci Of An Ellipse
236(2)
Chapter 12 Geometry, Space And Time 238(16)
The Equation Of Time
239(1)
3D Printable Models Used In This
Chapter
239(2)
The Analemma
241(6)
The Geometry Of The Analemma
241(2)
Measuring The Analemma
243(3)
Analemmas On Other Planets
246(1)
Paths In The Sky Of Other Objects
247(1)
Making A Sundial
247(6)
Creating The Gnomon
247(4)
Using Your Sundial
251(2)
Clockwise
253(1)
Other Sundial Designs
253(1)
Summary And Learning More
253(1)
Chapter 13 The Geometry Museum 254(30)
Arches
255(1)
3D Printable Models Used In This
Chapter
256(10)
Circular And Gothic Arches
257(4)
Making The Model
258(1)
Testing Your Arches
259(2)
Paper (Or Cut-Out) Alternatives
261(1)
Renovating The Castle
261(3)
Gothic Tracery
264(2)
Making The Model
265(1)
The Trefoil
265(1)
Printing Caveats
265(1)
Trusses
266(7)
Preparation
267(1)
Tetrahedron
268(2)
Cube
270(2)
Combine Cube And Tetrahedron
272(1)
The Gift Shop
273(8)
The Tetrahedron And Cube Puzzle
273(1)
Reuleaux And Constant-Width Shapes In 3D
274(3)
Revolved Reuleaux Polygons
274(1)
Reuleaux Tetrahedron
275(1)
Meissner Tetrahedrons
276(1)
Printing These Models
276(1)
Bounding Box
277(1)
Dual Polyhedrons
277(2)
The Golden Ratio, Phi
279(2)
The Golden Angle
280(1)
Relation To Fibonacci Series
281(1)
More Than 3D: Polytopes
281(1)
Summary And Learning More
281(1)
Answers
282(2)
Gothic vs. Circular Arch
282(2)
Appendix: Resources And Standards 284
Companion Resources
284(1)
Other Sources
284
Index 28
Joan Horvath is an MIT alumna, a recovering rocket scientist, and educator. She is a cofounder of Nonscriptum LLC. Rich Cameron is an open-source 3D printer hacker who designed the RepRapWallace and Bukito printers. He is a cofounder of Nonscriptum LLC.
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