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 dualsTetrahedron 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 SOLIDSKepler’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 SURFACELocus , 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.