II)
The regular Polyhedra by Origami
The text, the models by Origami of the convex regular polyhedra
and the Delta big
Stellated Dodecahedron are well known.
All the other Origami models presented here are due to Marcel Morales,
who realized them by Origami for the first time in the world.
The photos are original by Marcel Morales
1) 1) The
Platonic Solids
Two Thousand years ago, greek mathematiciens founded that there are only five regular convex polyhedra,
called Platonic Solids.They are realized by paper models.
To realize them by Origami is simpler, quicker and nicer.
Euler's formula applies to the Platonic Solids.

Tetrahedron 4
faces (equilateral triangles) 4
vertex 6
edges 


Cube 6
faces (squares) 8
vertex 12 edges 


Octahedron 8
faces (equilateral triangles) 6
vertex 12 edges 


Dodecahedron 12
faces (regular pentagons) 20
vertex 30 edges 


Icosahedron 20
faces (equilateral triangles) 12
vertex 30 edges 

III) The
non convex regular polyhedra
Johannes
Kepler, discovers in 1619 two regular polyhedrons non convex:
the
small Stellated Dodecahedra and the big Stellated Dodecahedra (Kepler's stellation).
Two
centuries later, in 1809 Louis Poinsot discovers,
two more regular polyhedrons
non convex: the great Dodecahedra and the great Icosahedra.
They are called regular because every visible face
(convex or Stellated regular pentagon, or equilateral triangle) is in a plane.
The small Stellated Dodecahedra has 12 faces (star pentagons called pentagrams), 30 edges, 20 vertices.
The great Stellated dodecahedron has
12 faces (star pentagons called pentagrams), 30 edges, 20 vertices.
The great dodecahedron has
12 faces (pentagons), 30 edges, 20 vertices.
The great icosahedron has
20 faces (equilateral triangles), 30 edges, 12 vertices.
.
small Stellated Dodecahedron 
big
Stellated Dodecahedron 
the great Dodecahedron 
the great Icosahedron 
The following nice polyhedra, are not considered to be regular,
they look like the Kepler Poinsot regular Polyhedra,
their faces are regular triangles.
They are in the family of Delta Polyhedra or Deltahedron.
Delta small Stellated Dodecahedron 
Delta big
Stellated Dodecahedron 
Delta great Dodecahedron 
Delta great Icosahedra 
IV) Realization of Delta regular polyhedra by
paper folding (ORIGAMI)
1)
1)
The Dodecahedron
In order to construct the Dodecahedron, we need 12 similar pieces to the elementary piece
(2).
We remark that each piece is composed of a regular
pentagon and two tabs.
(1) Make a knot with a strip of paper
of lengths 41,7 cm by 5,5 cm. 
(2) 
2)
2)
The Delta regular polyhedra with
equilateral 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 tabs. All these tabs 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_{}L. 
(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_{}L 
(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 
(7)=> 
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 Delta regular polyhedra with
equilateral triangular faces
Tetrahedron 
2 symmetrical pieces (A+B) 
Octahedron 
4 pieces identical 
Icosahedra 
10 pieces (5A+5B) 
Delta Small Stellated Dodecahedron 
30 identical pieces 
Delta Big Stellated Dodecahedron 
30 identical pieces 
Delta Great Dodecahedron 
30 identical pieces 
Delta Great Icosahedra 
120 identical pieces 