ssrs barcode font free 17.1B Regular Polyhedra in Objective-C

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17.1B Regular Polyhedra
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Regular polyhedra are solids having faces that are regular polygons, with the same number of faces meeting at each vertex. There are only five such solids, shown in Fig. 17-11. Note that their faces are equilateral triangles, squares, or regular pentagons.
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CHAPTER 17 Extending Plane Geometry into Solid Geometry
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Regular Tetrahedron (4 faces)
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Regular Hexahedron (6 faces)
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Regular Octahedron (8 faces)
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Regular Dodecahedron (12 faces)
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Regular Icosahedron (20 faces)
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Fig. 17-11
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Regular hexagons cannot be faces of a regular polyhedron. If three regular hexagons had a common vertex (Fig. 17-12), the sum of the measures of the three interior angles at that vertex would be 3(120 ) or 360 . As a result, the three regular hexagons would lie in the same plane, so they could not form a solid.
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Fig. 17-12
17.2 Extensions to Solid Geometry
17.2A Extension of Plane Geometry Principles to Space Geometry Principles
Sphere in space geometry corresponds to circle in plane geometry. Similarly, plane corresponds to straight line. By interchanging circle and sphere, or straight line and plane, each of the following dual statements can be interchanged. When you do so, many plane geometry principles with which you are acquainted become space geometry principles. Related Dual Statements 1. Every point on a circle is at a distance of one radius from its center. 2. Two intersecting straight lines determine a plane.
l2 l1
1. Every point on a sphere is at a distance of one radius from its center. 2. Two intersecting planes determine a straight line.
l p2 p1
3. Two lines perpendicular to the same plane are parallel.
3. Two planes perpendicular to the same line are parallel.
p1 p2 l
p l1 l2
CHAPTER 17 Extending Plane Geometry into Solid Geometry
Be sure, in obtaining dual statements, that there is a complete interchange of terms. If the interchange is incomplete, as in the following pair of statements, there is no duality. 4. The locus of a point at a given distance from a given line is a pair of lines parallel to the given line and at the given distance from it. 4. The locus of a point at a given distance from a given line is a circular cylindrical surface having the given line as axis and the given distance as radius.
Axis
Unlike a cylinder, a cylindrical surface is not limited in extent; nor does it have bases. Similarly, a conical surface is unlimited in the extent and has no base. Extension of Distance Principles Here are some dual statements involving distance in a plane and in space. Distance in a Plane Distance in Space
1. The distance from a point to a line is the length 1. The distance from a point to a plane is the length of the perpendicular from the point to the line. of the perpendicular from the point to the plane.
2. The distance between two parallel lines is the length of a perpendicular between them.
2. The distance between two parallel planes is the length of a perpendicular between them.
3. The distance from a point to a circle is the external segment of the secant from the point through the center of the circle.
3. The distance from a point to a sphere is the external segment of the secant from the point through the center of the sphere.
A B O
CHAPTER 17 Extending Plane Geometry into Solid Geometry
Distance in a Plane 4. Parallel lines are everywhere equidistant.
Distance in Space 4. Parallel planes are everywhere equidistant.
5. Any point on the line which is the perpendicular bisector of a segment is equidistant from the ends of the segment.
5. Any point on the plane which is the perpendicular bisector of a segment is equidistant from the ends of the segment.
6. Any point on the line which is the bisector of the angle between two lines is equidistant from the sides of the angle.
6. Any point on the plane which is the bisector of the dihedral angle between two planes is equidistant from the sides of the angle.
Extension of Locus Principles The following dual statements involve the locus of points in a plane and in space. Locus in a Plane 1. The locus of points at a given distance from a given point is a circle having the given point as center and the given distance as radius. Locus in Space 1. The locus of points at a given distance from a given point is a sphere having the given point as center and the given distance as radius.
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