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.