logging in or signing up Math bkdimegurl Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 599 Category: Education License: All Rights Reserved Like it (0) Dislike it (0) Added: December 08, 2008 This Presentation is Public Favorites: 0 Presentation Description No description available. Comments Posting comment... Premium member Presentation Transcript The Platonic Solids : The Platonic Solids Regular polygons : Regular polygons Have all sides equal Have all angles equal Example: Equilateral triangle Sum of degrees in the three angles: 180 Each angle = 180/3 = 60 Quadrilateral (four sides) : Quadrilateral (four sides) The Square Sum of angles = sum of two triangles = 2 (180) = 360 Each angle = 360/4 = 90 More sides : More sides Sum of angles = 3 (180) = 540 Each angle = 540/ 5 = 108 Pentagon Hexagon Sum of angles = 4 (180) = 720 Each angle = 720/6 = 120 For a regular polygon with n sides and n angles : For a regular polygon with n sides and n angles Sum of the interior angles = (n-2) 180 Each individual angle = Sum / n We can have as many sides as we like All regular polygons can be inscribed in circles : More sides bigger inside angles Is there a “biggest shape” we are heading to? The circle! All regular polygons can be inscribed in circles Sides=3 Sides=4 Sides=5 Sides=6 Sides=8 Bigger regular polygons : Bigger regular polygons The ultimate “end” of this process is a polygon with an infinite number of sides Basically …a circle We can make a regular polygon with as many sides as we like. The interior angles just keep getting bigger. Can we do this in three dimensions? : Can we do this in three dimensions? A polyhedron is a three dimensional solid. Face Edge Vertex Regular polyhedron : Regular polyhedron A regular polyhedron would have these properties: Its faces are identical regular polygons The number of edges coming from each vertex is the same Each “corner” is identical to every other corner (vertex) Building regular polyhedra : Building regular polyhedra Let’s start with the simplest shape, the equilateral triangle. How many do you need to make a corner? What will the bottom look like? The four triangles together give us our first regular polyhedron…. Three will work. Note that 3x60 = 180, and that 60 divides 360 equally The tetrahedron : The tetrahedron Tetrahedron means four faces More equilateral triangles : More equilateral triangles Make a corner from 4 triangles. Note that 4 x 60 = 240, and that 60 divides 360 equally. What will this look like folded up? Place two square pyramids together to get….. The octahedron : The octahedron Octahedron means eight faces Now try 5 triangles : Now try 5 triangles Now fold up 5 triangles. Note that 5x60 = 300 and 60 divides 360 equally. Twenty such triangles together make…. The icosahedron : The icosahedron Icosahedron means 20 faces Now what? : Now what? Could we fold a corner from 6 triangles? More than six triangles? No more regular solids with equilateral triangle faces are possible. Let’s move to… Square faces : Square faces Can we make a corner from two squares? From three? Note that 3x90=270 and 90 divides 360 evenly From more than three? Only one regular polyhedron with square faces is possible. We have…. The hexahedron : The hexahedron Otherwise known as – the cube Hexahedron means six faces. Cubos in Greek means a knucklebone. Let’s try pentagons as faces : Let’s try pentagons as faces Can we fold a corner with 3 regular pentagons? Notice that 3x 108 = 324 Can we use more than 3 pentagons? No, because 4x108 is larger than 360. So only one regular polyhedron with pentagonal faces is possible. We have…. The dodecahedron : The dodecahedron Dodecahedron means 12 faces. Could we use hexagons as sides? : Could we use hexagons as sides? We can’t fold a corner here. Where three hexagons come together, we get three angles of 120. And 3x120 is 360. So no regular solid with six sides is possible. Since regular polygons with more than six sides have even bigger interior angles, no more regular solids are possible. There are only five regular three dimensional solids. : There are only five regular three dimensional solids. History : History These rocks carved as the solids are from a Neolithic village site in Scotland. They date to about 1300 BC and are at least a thousand years older than Plato’s time. Pythagoras : Pythagoras It may have been the Pythagoreans who first studied the solids in the Greek world. They may have known only the tetrahedron, cube and dodecahedron. Plato : Plato A contemporary of Plato’s, named Theaetetus was said to have first written about all five solids. He may also have been the first to prove there were only five. Plato wrote about the solids in his book Timaeus. He associated each of the solids with one of the classical elements. Plato : Plato The pointy tetrahedron represented fire, which feels sharp and stabbing Plato : Plato The octahedron, represents the whirling element of air Plato : Plato Water which flows freely out of one’s hands, is represented by the icosahedron. Water fills much of the earth, and the icosahedron has the largest interior of the Platonic solids. Plato : Plato The cube, solid and steady, represented the earth itself. Plato : Plato Plato obscurely remarked about the dodecahedron that “the god used this to arrange the constellations of heaven.” Other people have noticed that the twelve signs of the zodiac fit nicely on the 12 faces. Euclid : Euclid The solids were the crowning glory of Euclid's Elements, featured in Book XIII, the final part of this great work. Johannes Kepler : Johannes Kepler In 1596 Kepler tried to associate the solids with the five planets known at that time. His model of the solar system nested the five solids inside each other separated by circumscribed and inscribed spheres. Euler’s Formula : Euler’s Formula V + F = E + 2 Examples: Tetrahedron 4 + 4 = 6 + 2 Cube 8 + 6 = 12 + 2 Duality : Duality Start with a cube. Put a dot in the center of each face. Connect the dots Duality : Duality Interchanging faces and vertexes Octahedron in a cube Cube and octahedron duality Duality : Duality The octahedron is the dual of the cube Faces Vertexes Cube 6 8 Octahedron 8 6 Duality : Duality Duality : Duality What is the dual of the dodecahedron? Duality : Duality The dodecahedron and icosahedron are duals Faces Vertexes Dodecahedron 12 20 Icosahedron 20 12 Duality : Duality Dodecahedron and icosahedron are duals Duality : Duality What is the terahedron’s dual? The tetrahedron is self-dual The Stella Octangula : The Stella Octangula Tetrahedron is its own dual The Golden Rectangle : The Golden Rectangle Take three Golden rectangles intersected with a common central point. Join the vertices of the rectangles, producing twenty equilateral triangles – the icosahedron. The Golden Rectangle : The Golden Rectangle An icosahedron can be nested in an octahedron. This is not a dual. The icosahedron does not touch the middle of the edge of the octahedron. It cuts the edge into two pieces which make a golden ratio. You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.
Math bkdimegurl Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 599 Category: Education License: All Rights Reserved Like it (0) Dislike it (0) Added: December 08, 2008 This Presentation is Public Favorites: 0 Presentation Description No description available. Comments Posting comment... Premium member Presentation Transcript The Platonic Solids : The Platonic Solids Regular polygons : Regular polygons Have all sides equal Have all angles equal Example: Equilateral triangle Sum of degrees in the three angles: 180 Each angle = 180/3 = 60 Quadrilateral (four sides) : Quadrilateral (four sides) The Square Sum of angles = sum of two triangles = 2 (180) = 360 Each angle = 360/4 = 90 More sides : More sides Sum of angles = 3 (180) = 540 Each angle = 540/ 5 = 108 Pentagon Hexagon Sum of angles = 4 (180) = 720 Each angle = 720/6 = 120 For a regular polygon with n sides and n angles : For a regular polygon with n sides and n angles Sum of the interior angles = (n-2) 180 Each individual angle = Sum / n We can have as many sides as we like All regular polygons can be inscribed in circles : More sides bigger inside angles Is there a “biggest shape” we are heading to? The circle! All regular polygons can be inscribed in circles Sides=3 Sides=4 Sides=5 Sides=6 Sides=8 Bigger regular polygons : Bigger regular polygons The ultimate “end” of this process is a polygon with an infinite number of sides Basically …a circle We can make a regular polygon with as many sides as we like. The interior angles just keep getting bigger. Can we do this in three dimensions? : Can we do this in three dimensions? A polyhedron is a three dimensional solid. Face Edge Vertex Regular polyhedron : Regular polyhedron A regular polyhedron would have these properties: Its faces are identical regular polygons The number of edges coming from each vertex is the same Each “corner” is identical to every other corner (vertex) Building regular polyhedra : Building regular polyhedra Let’s start with the simplest shape, the equilateral triangle. How many do you need to make a corner? What will the bottom look like? The four triangles together give us our first regular polyhedron…. Three will work. Note that 3x60 = 180, and that 60 divides 360 equally The tetrahedron : The tetrahedron Tetrahedron means four faces More equilateral triangles : More equilateral triangles Make a corner from 4 triangles. Note that 4 x 60 = 240, and that 60 divides 360 equally. What will this look like folded up? Place two square pyramids together to get….. The octahedron : The octahedron Octahedron means eight faces Now try 5 triangles : Now try 5 triangles Now fold up 5 triangles. Note that 5x60 = 300 and 60 divides 360 equally. Twenty such triangles together make…. The icosahedron : The icosahedron Icosahedron means 20 faces Now what? : Now what? Could we fold a corner from 6 triangles? More than six triangles? No more regular solids with equilateral triangle faces are possible. Let’s move to… Square faces : Square faces Can we make a corner from two squares? From three? Note that 3x90=270 and 90 divides 360 evenly From more than three? Only one regular polyhedron with square faces is possible. We have…. The hexahedron : The hexahedron Otherwise known as – the cube Hexahedron means six faces. Cubos in Greek means a knucklebone. Let’s try pentagons as faces : Let’s try pentagons as faces Can we fold a corner with 3 regular pentagons? Notice that 3x 108 = 324 Can we use more than 3 pentagons? No, because 4x108 is larger than 360. So only one regular polyhedron with pentagonal faces is possible. We have…. The dodecahedron : The dodecahedron Dodecahedron means 12 faces. Could we use hexagons as sides? : Could we use hexagons as sides? We can’t fold a corner here. Where three hexagons come together, we get three angles of 120. And 3x120 is 360. So no regular solid with six sides is possible. Since regular polygons with more than six sides have even bigger interior angles, no more regular solids are possible. There are only five regular three dimensional solids. : There are only five regular three dimensional solids. History : History These rocks carved as the solids are from a Neolithic village site in Scotland. They date to about 1300 BC and are at least a thousand years older than Plato’s time. Pythagoras : Pythagoras It may have been the Pythagoreans who first studied the solids in the Greek world. They may have known only the tetrahedron, cube and dodecahedron. Plato : Plato A contemporary of Plato’s, named Theaetetus was said to have first written about all five solids. He may also have been the first to prove there were only five. Plato wrote about the solids in his book Timaeus. He associated each of the solids with one of the classical elements. Plato : Plato The pointy tetrahedron represented fire, which feels sharp and stabbing Plato : Plato The octahedron, represents the whirling element of air Plato : Plato Water which flows freely out of one’s hands, is represented by the icosahedron. Water fills much of the earth, and the icosahedron has the largest interior of the Platonic solids. Plato : Plato The cube, solid and steady, represented the earth itself. Plato : Plato Plato obscurely remarked about the dodecahedron that “the god used this to arrange the constellations of heaven.” Other people have noticed that the twelve signs of the zodiac fit nicely on the 12 faces. Euclid : Euclid The solids were the crowning glory of Euclid's Elements, featured in Book XIII, the final part of this great work. Johannes Kepler : Johannes Kepler In 1596 Kepler tried to associate the solids with the five planets known at that time. His model of the solar system nested the five solids inside each other separated by circumscribed and inscribed spheres. Euler’s Formula : Euler’s Formula V + F = E + 2 Examples: Tetrahedron 4 + 4 = 6 + 2 Cube 8 + 6 = 12 + 2 Duality : Duality Start with a cube. Put a dot in the center of each face. Connect the dots Duality : Duality Interchanging faces and vertexes Octahedron in a cube Cube and octahedron duality Duality : Duality The octahedron is the dual of the cube Faces Vertexes Cube 6 8 Octahedron 8 6 Duality : Duality Duality : Duality What is the dual of the dodecahedron? Duality : Duality The dodecahedron and icosahedron are duals Faces Vertexes Dodecahedron 12 20 Icosahedron 20 12 Duality : Duality Dodecahedron and icosahedron are duals Duality : Duality What is the terahedron’s dual? The tetrahedron is self-dual The Stella Octangula : The Stella Octangula Tetrahedron is its own dual The Golden Rectangle : The Golden Rectangle Take three Golden rectangles intersected with a common central point. Join the vertices of the rectangles, producing twenty equilateral triangles – the icosahedron. The Golden Rectangle : The Golden Rectangle An icosahedron can be nested in an octahedron. This is not a dual. The icosahedron does not touch the middle of the edge of the octahedron. It cuts the edge into two pieces which make a golden ratio.