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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
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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
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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 CUTOUT) ALTERNATIVES ..............................................................................261
RENOVATING THE CASTLE ..................................................................................................261
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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
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