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 andL.	(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 andL	(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