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Star and Convex Regular Polyhedra by Origami. Build Polyhedra by Origami.] Star and convex regular polyhedra by Origami. Build polyhedra by Origami.] Marcel Morales Alice Morales 2009 E D I T I O N M ORALES Polyhedron by Origami I) Table of convex regular Polyhedra ................................................................................................. 4 II) Table of star regular Polyhedra (non convex) .............................................................................. 5 III) Some Polyhedra that should be named regular done with equilateral triangles .......................... 6 IV) Some Polyhedra that should be named regular done with rectangle triangles ............................ 7 V) Some geometry of polygons ........................................................................................................ 9 VI) The convex regular polygons ..................................................................................................... 12 VII) The star regular polygons .......................................................................................................... 12 VIII) Regular convex polygons by folding paper ............................................................................. 14 1) The square .......................................................................................................................................... 14 2) The equilateral triangle ....................................................................................................................... 14 3) The regular pentagon .......................................................................................................................... 18 4) A few of vocabulary about Polyhedra .................................................................................................. 21 5) Making regular Polyhedra by folding paper ......................................................................................... 23 IX) Making a cube by folding paper ................................................................................................ 25 1) Making a basic square module ............................................................................................................ 25 2) Assembly the cube .............................................................................................................................. 31 X) Making regular Polyhedra with equilateral triangles as faces ................................................... 36 1) Folding the basic triangle module ........................................................................................................ 36 2) Folding the symmetric basic triangle module ....................................................................................... 40 3) Assembly the Tetrahedron .................................................................................................................. 43 4) Assembly the Octahedron ................................................................................................................... 45 5) Assembly the Icosahedron ................................................................................................................... 49 XI) Assembly the Dodecahedron ..................................................................................................... 54 XII) Assembly of the great triangular star dodecahedron ................................................................ 59 XIII) Assembly the small triangular star dodecahedron .................................................................. 67 XIV) Assembly the great Icosahedron (regular) .............................................................................. 74 Here is a photo of the great Icosahedron. ............................................................................................. 74 XV) Assembly of the regular great dodecahedron ............................................................................ 79 XVI) Assembly the inverse star great dodecahedron also called third stellation of the icosahedron 82 XVII) Making Polyhedra which faces are isosceles triangles rectangles ........................................... 87 1) Folding the special square module ...................................................................................................... 87 2) Assembly the Triakis Icosahedron (small star dodecahedron rectangle) ............................................... 90 3) Assembly of the great dodecahedron rectangle ................................................................................... 97 4) Assembly of the great icosahedron rectangle .................................................................................... 105 5) The rectangular great icosahedron .................................................................................................... 108 XV) Assembly other Polyhedra ...................................................................................................... 109 Marcel Morales, Alice Morales 1) The rectangular final Stellation of the Icosahedron............................................................................ 109 2) The Epcot’s ball ................................................................................................................................. 111 3) Making a dipyramide (or hexaedra) with triangular base ................................................................... 114 4) The polyhedron Stella Octangula ....................................................................................................... 116 5) The Stella Octangula-bis .................................................................................................................... 119 6) The star Cube-Octahedron ................................................................................................................. 122 7) The Trihedron or framework ............................................................................................................. 125 8) The Heptahedron .............................................................................................................................. 129 XVI) Star Polyhedra. Stellation and Excavation. ........................................................................... 132 XVII) A few of geometry on the sphere of radius one .................................................................... 136 XVIII) Euler’s formula ..................................................................................................................... 139 3 Polyhedron by Origami I) Table of convex regular Polyhedra Cube Tetrahedron 6 square pieces of the same type, 2 triangular pieces of distinct 6 faces, 12 edges, 8 vertices type (A+B) 4 faces, 6 edges, 4 vertices Octahedron Dodecahedron 4 triangular pieces of the same 12 pentagons type, 8 faces, 12 edges, 6 12 faces, 30 edges, 20 vertices vertices Marcel Morales, Alice Morales Icosahedron 10 triangular pieces (5A+5B) 20 faces, 30 edges, 12 vertices II) Table of star regular Polyhedra (non convex) The great star dodecahedron 30 pieces of the same type: The small star dodecahedron 12 faces (star pentagons called 30 pieces of the same type: pentagrams), 30 edges, 20 12 faces (star pentagons called vertices pentagrams), 30 edges, 20 vertices 5 Polyhedron by Origami The great dodecahedron 30 pieces of the same type: The great icosahedron 12 faces (star pentagons called 12 pieces of the same type, pentagrams), 30 edges, 20 vertices 20 faces(equilateral triangles), 30 edges, 12 vertices III) Some Polyhedra that should be named regular done with equilateral triangles The triangular small star The triangular great star dodecahedron dodecahedron 30 pieces of the same type: 30 pieces of the same type: 60 faces, 90 edges, 32 vertices 60 faces, 90 edges, 32 vertices Marcel Morales, Alice Morales The triangular great dodecahedron The triangular great 30 pieces of the same type: icosahedron 60 faces, 90 edges, 32 vertices 30 pieces of the same type: The great star dodecahedron inversed or the third stellation of the icosahedron 30 pieces of the same type: 60 faces, 90 edges, 32 vertices IV) Some Polyhedra that should be named regular done with rectangle triangles 7 Polyhedron by Origami The great star The small star Le great dodecahedron inversed dodecahedron dodecahedron or the third stellation Dodecahedron rectangle of the icosahedron rectangle 30 square pieces of 30 pieces of the same 30 square pieces of the same type: type: the same type: 60 faces, 90 edges, 32 60 faces, 90 edges, 32 60 faces, 90 edges, 32 vertices vertices vertices The rectangular great Rectangular Final The Epcot’s ball icosahedron Stellation of the 270 square pieces of 120 square pieces of the icosahedron the same type. same type: 90 square pieces of 180 faces, 270 edges, 92 the same type. vertices Marcel Morales, Alice Morales V) Some geometry of polygons A polygon is a figure in the plane bounded by a sequence of segments of lines, called edges or sides, a point common to the extremities of two edges is a corner. We give some examples: We can see that we can distinguish such polygons, first by the number of vertices, then by the number of sides, then by the shape. Let me be more precise: The number of sides is very important: Triangle or Polygon with three sides, Quadrilateral or Polygon with four sides, Pentagon or Polygon with five sides, 9 Polyhedron by Origami Hexagon or Polygon with six sides, Heptagon or Polygon with seven sides, Octagon or Polygon with eight sides, Enneagon or Polygon with nine
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