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Convex Polyhedra with Regular Polygonal Faces

Convex Polyhedra with Regular Polygonal Faces. David McKillop. Making Math Matter Inc. Visualization and Logical Thinking. Close your eyes and visualize a regular octahedron Visualize its faces: How many? What shapes?

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Convex Polyhedra with Regular Polygonal Faces

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  1. Convex Polyhedra with Regular Polygonal Faces David McKillop Making Math Matter Inc.

  2. Visualization and Logical Thinking • Close your eyes and visualize a regular octahedron • Visualize its faces: How many? What shapes? • Visualize its vertices: Where are they located? How many? Is there vertex regularity? • Visualize its edges: Where are they located? How many? • Visualize one of its nets: What do you see? Making Math Matter Inc.

  3. Visualization and Logical Thinking • Close your eyes and visualize how you constructed a regular icosahedron • Visualize its faces: How many? What shapes? • Visualize its vertices: Where are they located? How many? Is there vertex regularity? • Visualize its edges: Where are they located? How many? • Visualize one of its nets: What do you see? Making Math Matter Inc.

  4. Regular Polyhedra • There are only 5 of these 3-D shapes: regular tetrahedron, cube, regular octahedron, regular dodecahedron, regular icosahedron • Each shape has only one type of regular polygon for its faces • They have vertex regularity • All angles formed by two faces (dihedral angles) are equal Making Math Matter Inc.

  5. Visualization and Logical Thinking • Close your eyes and visualize a uniform decagon-based prism • Visualize its faces: How many? What shapes? • Visualize its vertices: Where are they located? How many? Is there vertex regularity? • Visualize its edges: Where are they located? How many? • Visualize one of its nets: What do you see? Making Math Matter Inc.

  6. Uniform Prisms • Except for the uniform square prism (cube), there are two regular polygons of one type as bases (on parallel planes) and the rest of the faces are squares • They have vertex regularity, usually {4,4,n} but uniform triangular prism is {3,4,4} • A net of a uniform n-gonal prism is easily visualized as a regular n-gon with a square attached to each side and another n-gon attached to the opposite side of one of the squares, OR as a belt of n squares with an n-gon attached on opposite sides of the belt. Making Math Matter Inc.

  7. Visualization and Logical Thinking • Close your eyes and visualize how you would construct a uniform hexagonal antiprism • Visualize its faces: How many? What shapes? • Visualize its vertices: Where are they located? How many? Is there vertex regularity? • Visualize its edges: Where are they located? How many? • Visualize one of its nets: What do you see? Making Math Matter Inc.

  8. Uniform Antiprisms • Except for the uniform triangular antiprism (regular octahedron), there are two regular polygons of one type as bases (on parallel planes) and the rest of the faces are equilateral triangles • They have vertex regularity, usually {3,3,3,n} • A net of a uniform n-gonal antiprism is easily visualized as two regular n-gons with an equilateral triangle attached to each side and these two configurations joined, OR as a belt of 2n equilateral triangles with an n-gon attached on opposite sides of the belt. Making Math Matter Inc.

  9. How are these sets of polyhedra alike? Different? 1 1 Making Math Matter Inc.

  10. Deltahedra • Any 3-D shape constructed using only equilateral triangles is called a deltahedron • There are an infinite number of deltahedra; however, there is a finite number of convex deltahedra. Making Math Matter Inc.

  11. dipyramids The Convex Deltahedra Making Math Matter Inc.

  12. The Convex Deltahedra • All faces are equilateral triangles • They all have an even number of faces • There are only 8 of them • Only 3 of them have vertex regularity: the regular tetrahedron, octahedron, and icosahedron • 3 of them are dipyramids (6, 8, and 10 faces) Making Math Matter Inc.

  13. How are these sets of polyhedra alike? Different? 5 2 1 1 1 Making Math Matter Inc.

  14. The Archimedean Solids • Two or three different regular polygons as faces • Always 4 or more of any regular polygon • There are only 13 of these solids • They have vertex regularity • They are very symmetrical, looking the same when rotated in many directions Why are uniform prisms and uniform antiprisms NOT Archimedean solids? Making Math Matter Inc.

  15. How are these sets of polyhedra alike? Different? 2 1 1 1 Making Math Matter Inc.

  16. Johnson Solids • Have only regular polygons as faces (1 or more different types) • They do NOT have vertex regularity • There are only 92 of them (5 of them are convex deltahedra) Making Math Matter Inc. Making Math Matter Inc.

  17. Convex Polyhedra With Regular Polygonal Faces 87 5 13 Making Math Matter Inc.

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