II)
The regular Polyhedra
1) 1) The
Platon's Solids
.
|
|
Tetrahedral 4
faces (equilateral triangles) 4
vertex 6
edges |
|
|
|
Cube 6
faces (squares) 8
vertex 12 edges |
|
|
|
Octahedral 8
faces (equilateral triangles) 6
vertex 12 edges |
|
|
|
Dodecahedra 12
faces (regular pentagons) 20
vertex 30 edges |
|
|
|
Icosahedra 20
faces (equilateral triangles) 12
vertex 30 edges |
|
III) The
non convexes regular polyhedrons
Johannes
Kepler, discovers in 1619 two others regular polyhedrons non convexes: the
small star dodecahedra and the big star dodecahedra (Kepler's star). Two
centuries later, in 1809 Louis Poinsot discovers, two more regular polyhedrons
non convexes: the star dodecahedra and the star icosahedra.
.
|
small star dodecahedra |
big
star dodecahedra |
|
the star dodecahedra |
big star icosahedra |
IV) Realization of regular polyhedrons by
paper folding (ORIGAMI)
1)
1)
The dodecahedra
In order to construct the dodecahedra, we need 12 pieces similar to the elementary piece
(2).
We remark that each piece is composed of a regular
pentagon and two languet's.
|
(1) Make a knot with a strip of paper
of dimensions 41,7 cm by 5,5 cm. |
(2) |
2)
2)
The regular polyhedrons with
triangular faces
In order to construct these solids, we will use
the elementary piece (A) or its symmetric (B). We remark that each piece
is composed by two equilateral triangles in the middle of the piece and two
others languet's. All these languet's must be used in the construction
|
|
|
|
Pièce A |
Pièce B |
Production of the piece A
|
(1) Take
a shit of paper, of rectangular shape and
dimensions L and |
(2) the
vertex on the left down must go over the vertex on the right top. Fold it |
|
(3) Open
and fold from the bottom following the line of the previous folding (2). |
(4) fold
from the top following the line of the previous folding (2). |
|
(5) Fold
the two corners and hide them. |
(6) Turn
the piece. |
|
(7) Fold
it in order to have four equilateral triangles. |
(8) Here
is the elementary piece A. |
Réalisation de la pièce B
|
Take
a shit of paper, of rectangular shape and
dimensions L and |
(2) the
vertex on the right down must go over the vertex on the left top. Fold it |
|
(3) Open
and fold from the bottom following the line of the previous folding (2). |
(4) fold
from the top following the line of the previous folding (2). |
|
(5) Fold
the two corners and hide them.. |
(6) Turn
the piece. |
|
(7) Fold
it in order to have four equilateral triangles. |
(8) Here
is the elementary piece B. |
3)
3)
The cube
In
order to construct the cube, we need six elementary pieces identicals, starting
with a square. Remark that this piece
is composed by a square and two triangles. These two triangles will be used
during the construction of the cube
|
(1) fold
the square by the mediane line |
(2) folding
again in the middle each half part of the piece (1) . |
|
(3) fold
the opposite small triangles
rectangles. |
(4) fold
the piece. |
|
(5) fold
two triangles in order to have a parallelogram |
(6) put
the two triangles obtained in (5) in
the interior of the parallelogram |
|
|
Turn the piece. Fold two triangles rectangles, in order to get a square and two triangles
rectangles |
Take care in the step (3) , since you can obtains a
symmetrical piece (7’) the cube need 6 pieces identicals.
(3’) (7’)
4)
4)
Number of pieces needed to construct the regular polyhedra with
triangular faces
|
Tetrahedron |
2 symmetrical pieces (A+B) |
|
Octahedron |
4 pieces identical |
|
Icosahedra |
10 pieces (5A+5B) |
|
Small star dodecahedron |
30 pieces identical |
|
Big star dodecahedron |
30 pieces identical |
|
Star Dodecahedron |
35 pieces identical |
|
Star Icosahedra |
120 pieces identical |