POLYHEDRA. SPHERE. EARTH GLOBE We can classify three-dimensional shapes in two big groups: polyhedra and bodies with curved surface. Also they can be studied according to other properties: prisms - cylinders, pyramids cones and other polyhedra-sphere (and so we wil do it in this case ) Polyhedron is a part of space bounded by polygons which are called faces. Other elements of a polyhedron are: edge, vertex and polyhedron angle. Its surface is developable. Below, one of them is not a polyhedron. Which one? CONVEX AND CONCAVE POLYHEDRA EULER’S POLYHEDRA FORMULA VERTICES + FACES = EDGES + K(constant) • When the polyhedron is simple (without hole) K = 2 V + F = E + 2 (the classic formula) If the polyhedron has a hole k = 0 and V + F = E A little of Geography and History (1) Syracuse, where Archimedes was born. Mine of hauerita. (2)Crotona in Magna Greece. The Pythagorean school. Mine of pyrite. (8) Athens, Plato and his students’ town. REGULAR POLYHEDRA. PLATONIC SOLIDS Polyhedra can be combined into pairs called duals Tetrahedron is self-dual. Cube an octahedron are dual. Dodecahedron and icosahedron are dual. Cubes painted by Ibarrola (Basque painter) View of Peace Camp in Barcelona Forum Cutting cubes SEMIREGULAR POLYHEDRA. ARCHIMEDEAN SOLIDS Kepler’s drawings to “Harmonices Mundi” 1619 CUBOCTAHEDRON : cutting a cube by the middle points of its edges Truncated octahedron (Kelvin solid): truncation of an octahedron to one third of its edges. More Kelvin solids Truncated icosahedron: by truncation to one third of the edges SPHERE AND SPHERICAL SURFACE Locus , surface of revolution and limit polyhedron. This surface is not developable (it can’t be flattened onto a plane without distortion). Volume V = (4 π r3 )/3 Surface area A = 4 π r2 TORUS (SPHERICAL RING) EARTH GLOBE The Earth globe on a truncated icosahedron which has been inflated like a sphere.