PLATONIC SOLIDS BY PRABHNOOR KAUR AND OLIVIA COHEN

PLATONIC SOLIDS BY: PRABHNOOR KAUR AND OLIVIA COHEN

WHAT IS A PLATONIC SOLID? v A 3 -D solid figure with faces made of the same regular polygon v This 3 -D shape is created because of the interior angle sum at a vertex in each platonic sold the interior angle of certain polygons multiplied by the number of faces at each vertex is less than 360 degrees creating a 3 D structure. v In a platonic solid, the angles sum of all vertices has to be less than 360 degrees which is why only a tetrahedron, cube, octahedron, dodecahedron, icosahedron are the only platonic solids able to be formed because of the polygons they use for their faces. v The relationship between the faces, edges and vertices of a platonic solid is F+V-2=E

THE HISTORY OF PLATONIC SOLIDS v Based off of the writings of Plato who was a Greek philosopher (430 B. C)—he associated each of the solids with the five different atoms of the universe which were fire, air, water, earth, and cosmos (the universe). v Plato associated the tetrahedron with fire, cube with the earth, octahedron with air, icosahedron with water, and the dodecahedron with cosmos

TETRAHEDRON v v v Fire element: symbol of spiritual balance because of its flat sitting Regular polygon used: triangle There are 4 faces, 6 edges, and 4 vertices There are 3 faces and 3 edges at a vertex The interior angles at a vertex is 60 degrees -The sum of the interior angles at a vertex is 180 degrees

CUBE v v v Earth element: symbol of connection and stability because it stands grounded Regular polygon used: the square There are 6 faces, 12 edges, and 8 vertices There are 3 faces at a vertex and 3 edges at a vertex The interior angles at a vertex is 90 degrees -The sum of the interior angles at a vertex is 270 degrees

OCTAHEDRON v Air element: symbol of reflection in life because of its selfreflecting shape v Regular polygon used: triangle v There are 8 faces, 12 edges, and 6 vertices v There are 4 faces and 4 edges at a vertex v The interior angles at a vertex is 60 degrees -The sum of the interior angles at a vertex is 240 degrees

DODECAHEDRON v Universe element: symbolized a higher connection with the universe v Regular polygon used: pentagon v There are 12 faces, 30 edges, and 20 vertices v There are 3 faces and 3 edges at a vertex v The interior angles at a vertex is 108 degrees -The sum of the interior angles at a vertex is 324 degrees

ICOSAHEDRON v Water element: symbolized the importance of transformation and movement v Regular polygon used: triangle v There are 20 faces, 30 edges, and 12 vertices v There are 5 faces and 5 edges at a vertex v The interior angles at a vertex is 60 degrees -The sum of the interior angles at a vertex is 300 degrees

EACH PLATONIC SOLID HAS A DUAL v Every platonic solid has a dual pair v A dual pair has a pattern between the vertices and faces v A platonic solid with the same number of vertices as another platonic solid’s sides, fits into that platonic solid v A platonic solid’s dual pair can be made by putting a dot on each face of the platonic solid and then connecting them all v Duals have the same amount of edges

PATTERN OF DUALS Platonic Solid # Faces # Edges # Vertices Tetrahedron 4 6 4 Cube 6 12 8 Octahedron 8 12 6 Dodecahedron 12 30 20 Icosahedron 20 30 12

IMAGES OF DUALS Icosahedron and Dodecahedron Octahedron and Cube Tetrahedron

BIBLIOGRAPHY https: //www. asiact. org/archive/numerology/10 sequence. asp http: //www. latroika. com/mathoman/pix/ http: //www. thingiverse. com/thing: 693660 http: //www. shlomifish. org/Math. Ventures/dodeca. gif https: //encryptedtbn 3. gstatic. com/images? q=tbn: ANd 9 Gc. TGJuj. UE 3 _w. Ol. N 2 Fp. Xu. A 13 v 2 FUnfz. ZCSJYonj 7 OIyy. LXae 8 Sux http: //nlvm. usu. edu/en/nav/frames_asid_131_g_4_t_3. html? o pen=instructions&from=category_g_4_t_3. html