The five convex regular polyhedra (or
Platonic solids), divided into polyhedra with cubic symmetry (cube, tetrahedron
and octahedron)
and icosahedral symmetry (icosahedron and dodecahedron), can be correlated in different ways:
Each is inscribable in another, according, for istance, to the following
sequence:
icosahedron  octahedron  tetrahedron  cube  dodecahedron

From left to right:
icosahedron inscribed in an octahedron
octahedron inscribed in a tetrahedron
tetrahedron inscribed in a cube
cube inscribed in a dodecahedron

Also, the polyhedra belonging to the dual couples cubeoctahedron and dodecahedronicosahedron are
reciprocally inscribable, This is a consequence of their duality; the fifth
Platonic solid, the tetrahedron,
is inscribable in another tetrahedron, being autodual.
Concerning the other combinations of polyhedra, it is worth mentioning both the
octahedron inscribed in dodecahedron and the
icosahedron inscribed in
cube.
Using the pleasing notation
proposed by John Conway, tetrahedron is linked by the operators named join, ambo, gyro and snub
to the other four Platonic solids (cube, octahedron, dodecahedron and
icosahedron, respectively).

Conway operators linking the tetrahedron to the other four Platonic solids. 
A further relation between the two platonic solids having the
least and the maximum number of faces is given from the possibily to generate
an icosahedron by the intersection of five tetrahedra having their faces
orientated like the icosahedral ones.

Animated gif showing the orientation of the five tetrahedra with rispect to icosahedron and, step by step,
the result of their addition leading to a 5colored icosahedron.

The procedure leading from tetrahedron to icosahedron (already
described
here) can be summarized as follows.
The starting point is a tetrahedron whose faces have the Miller indices:
(111), (
11
1),
(
1 1 1),
(
1 1 1).
The choice, as second tetrahedron, of the one whose faces are characterized by the indices:
(
1 1 1),
(1
11), (
111), (11
1),
hence placed centrosymmetrically with respect to the faces of the first tetrahedron, would
lead to a compound polyhedron named
stella octangula by Kepler. It can
be regarded as the stellation of the octahedron resulting from the intersection of the two previous tetrahedra.
(clicking on the following images one can visualize the corresponding VRML files)
Such choice of the second tetrahedron is not suitable to reconstruct an
icosahedron from tetrahedra, since the remaining
twelve faces of the icosahedron could not be distributed among the three other
tetrahedra with faces orientated as the icosahedral ones. Nevertheless, each of the four
other tetrahedra that are fit to reconstruct the icosahedron can be derived
just from this "forbidden" tetrahedron by a rotation of 44.48° around the direction
perpendicular to one of its faces: this implies that the other three faces of each
tetrahedron, making an angle of 109.46° with the first face, undergo a rotation of 41.81° (corresponding to the angle
between the perpendiculars to every pair of contiguous icosahedral faces).
In the following image one can see, in
their own orientations, the four tetrahedra obtained, by the described
procedure, starting from the "forbidden" grey tetrahedron (in the centre);
their faces, and the faces of the red tetrahedron, result to be orientated like the twenty icosahedral faces.

Four coloured tetrahedra, in their own orientation, obtained from the "forbidden" one (in grey, at the centre of the image):
with the addition of the red tetrahedron, their intersection corresponds to an icosahedron.

The next view of the icosahedron along the [111] direction evidences the yellow
faces involved in the rotation of the "forbidden" tetrahedron.
(clicking on the following image one can visualize the corresponding VRML file)
Animated view of an icosahedron along the [111] axis (perpendicular to a face
deriving from the red tetrahedron). The darkgrey and yellow icosahedral faces are involved in the procedure,
described in the text, that has been applied in order to orientate properly the second tetrahedron;
each icosahedral face derived from the yellow tetrahedron can be obtained also by a rotation of 41.81°
of the contiguous darkgrey face, derived from the "forbidden" tetrahedron,
around the edge shared by the two faces.

There is an alternative method that permits to obtain
a second tetrahedra starting from the red one. The stereographic projection of the rotation axes
of icosahedron, described in a previous work concerning
icosahedral polyhedra, lets one ascertain that there are three 2fold axes
perpendicular to every 3fold axis present in icosahedral forms. In particular, the three 2fold axes
perpendicular to the [111] axis (and therefore parallel to the [111] face of the
red tetrahedron) have the irrational indices: [1
1/τ τ], [1/τ τ 1],
[τ 1 1/τ ] where τ = (√5+1)/2
=1.61803... denotes the golden ratio: by rotating the faces of red tetrahedron around each of the 2fold axes, one
directly obtains the faces of the yellow tetrahedron.
It must be pointed out that these 2fold axes are not symmetry operators of tetrahedron, just as it happens during the process of crystal twinning.


a) Stereographic projection of the icosahedral
rotation axes, including the drawing of the zones of 3fold axes: three 2fold axes
belong to each zone.
b) view along [001] of the polyhedron resulting from the intersection of {111}
red and {111} yellow tetrahedra,
reporting also the three 2fold axes perpendicular to the 3fold axis directed along [111].
The relative VRML image is visible clicking
here.  
In the next table one can see that each 2fold axis of
rotation correlates four different pairs of faces made of a red and a yellow face.
Directions of 2fold axes correlating pairs of faces belonging to
red and yellow tetrahedra 

(1
1 1) 
(1 1
1) 
(1 1
1) 
(1 1 1) 
(τ 1/τ 0) 
[1/τ τ 1] 
[1
1/τ τ] 
[τ 1
1/τ] 
 
(0
τ 1/τ) 
[τ 1
1/τ] 
[1/τ τ 1] 
[1
1/τ τ] 
 
(1/τ 0 τ ) 
[1
1/τ τ] 
[τ
1 1/τ] 
[1/τ τ 1] 
 
(1
1
1) 
 
 
 
[1/τ τ 1]
[1
1/τ τ]
[τ 1
1/τ]

Views, along different directions, both of the compound
polyhedron made of red and yellow tetrahedra (make a comparison with stella octangula) and the polyhedron
resulting from the intersection of the two tetrahedra. 
The following tables show the single steps leading from tetrahedron to icosahedron
through intermediate polyhedra deriving from the intersection of two, three and four polyhedra.
In the first table of the series one can see the ten polyhedra
deriving from all the possible pairs of tetrahedra: also the indices of all the tetrahedral faces are reported.
(clicking on the images one can visualize the corresponding VRML files)
Congruent polyhedra corresponding to the intersection of pairs of tetrahedra
that originate from the decomposition of an icosahedron into five tetrahedra 


1^{st
}tetrahedron 

Indices of each tetrahedral face 
(1 1 1)
( τ 1/τ 0)
(0 τ 1/τ)
(1/τ 0 τ)

( 1 1 1)
(τ 1/τ 0)
(0 τ 1/τ)
(1/τ 0 τ ) 
(1 1 1)
( τ
1/τ 0)
(0 τ 1/τ )
(1/τ 0 τ )

( 1
1 1)
(τ 1/τ 0)
(0 τ 1/τ)
(1/τ 0 τ) 
(1 1 1)
(1 1 1)
(1 1
1)
(1
1 1) 
2^{nd
}
t
e
t
r
a
h
e
d
r
o
n

(1 1 1)
(1 1 1)
(1 1
1)
(1
1 1) 





( 1 1 1 )
(τ 1/τ 0)
(0 τ 1/τ)
(1/τ 0 τ) 




(1 1 1)
( τ
1/τ 0)
(0 τ 1/τ )
(1/τ 0 τ )




( 1 1 1)
(τ 1/τ 0)
(0 τ 1/τ)
(1/τ 0 τ ) 


(1 1 1)
( τ 1/τ 0)
(0 τ 1/τ)
(1/τ 0 τ)


It is not difficult to realize that the previous polyhedra, made of pair of tetrahedra,
are all congruent (regardless of colouring),
as visualized by the animated gif images contained in the following table relative to compound polyhedra,
deriving from the combination of the red tetrahedron with the four others,
and their intersections: starting from every initial orientation of the different couples of tetrahedra,
by appropriate rotations a final common orientation is obtained in every case.
In the same way, ten congruent sets of three tetrahedra can be obtained adding a third tetrahedron
to the previous pairs of tetrahedra.
(clicking on the images one can visualize the corresponding VRML files)
Congruent intersections of three tetrahedra obtained by the addition
of a third tetrahedron to each pair of tetrahedra 
The next step shows how the five congruent
sets of four tetrahedra can be obtained from the
ten sets of three tetrahedra by the addition of a fourth tetrahedron.
(clicking on the images one can visualize the corresponding VRML
files)
Congruent intersections of four tetrahedra obtained by the addition
of a fourth tetrahedron to each set of three tetrahedra 

The last step consists in the addition of a fifth tetrahedron
to each set of four tetrahedra, giving as result a 5coloured icosahedron.
(clicking on the images one can visualize the corresponding VRML
files)
Addition of a fifth tetrahedron to each set of four tetrahedra leading to icosahedron


To summarize, in the following table both the compound
polyhedra and the result of the intersections of tetrahedra are shown,
starting from the red tetrahedron with the successive addition, step by step, of
the yellow, green, paleblue and violet tetrahedra.(clicking on the images, one can visualize the corresponding VRLM files)
Pairs of images relative to the compound polyhedra, made
of 2345 tetrahedra (on the left), and their intersections (on the right),
leading from tetrahedron to icosahedron; viewing direction, scale and
orientation of the polyhedra are the same for all the images.


Compound polyhedron made of 2
tetrahedra and solid generated by their intersection
The following animated
GIF, consisting of four frames, shows sequentially the images
along different directions, included 2fold and 3fold rotation axes, relative to the pairs of:
chiral compound polyhedra made of red and yellow tetrahedra (first row)
chiral solids generated by the intersection of the two tetrahedra (second row)
The polyhedra in the second column have been properly orientated to evidence their mirror symmetry
(concerning both shape and colouring) with respect to the polyhedra in the first column.

Animated GIF of chiral compounds of two tetrahedra and their intersections, seen
by ortographic viewing and along 2fold and 3fold axes. 

Starting from the initial position a), the solid generated by the intersection of red and yellow tetrahedra
is shown in b) after an appropriate rotation, in order to evidence the presence of a vertical 3fold axis and three 2fold axes
perpendicular to the 3fold axis, characterizing the 32 point group.
In c) the same colour has been given to all the faces belonging to each of the two crystallographic forms:
trigonal trapezohedron (6 brown faces) and pinacoid (2 fuchsia faces). 

Left: view, along the [001] direction, of the faces of the polyhedron corresponding to the
intersection of two tetrahedra and relative stereographic projection; in conformity with the 32 point group,
the forms are a trigonal trapezohedron and a pinacoid
(here and in all the following stereographic projections, circles indicate the positions of
faces belonging to the North hemisphere, crosses indicate faces belonging to the South hemisphere)
Right: orthographic view of the only trigonal trapezohedron.


Three alternative colourings of the faces of the
polyhedron resulting from the intersection of two tetrahedra. The first two colourings refer to:
the faces deriving from red and yellow tetrahedra
the trigonal trapezohedron and the pinacoid constituting the polyhedron
In order to evidence the single faces, the sharing of edges by faces having
equal colouring should be avoided; since the triangular faces share their edges
with three pentagonal faces, at least four colours need to be used (third
image): one for the two triangular faces and three other colours for the
two sets of three pentagonal faces related by the 3fold axis. Moreover each
pentagonal face of a set of three faces is related by two 2fold axes to
two pentagonal faces, coloured by the other two colours, belonging
to the second set of three faces. Therefore, in case
of this 4coloured polyhedon, the relative symmetry results to be 3'2',
where 3' and 2' indicates chromatic 3fold and 2fold axes, respectively.

The geometric features of the pentagonal faces are visible
clicking
here: the side lengths are relative to an unit distance
of the face from the centre of the polyhedron.
4coloured planar nets of the chiral polyhedra deriving from the intersection of two tetrahedra: the alternative nets, in which
one colour corresponds to each crystallographic form (trapezohedron or pinacoid), can be seen clicking
here; a further pair of chiral nets,
coloured accordingly to the red and yellow intersecting tetrahedra,
is shown here.

Compound polyhedron made of 3 tetrahedra and solid generated by their intersection
The following animated
GIF, consisting of four frames, shows sequentially the images
along different directions, included 2fold and 3fold rotation axes, relative to the pairs of:
chiral compound polyhedra made of red, yellow and green tetrahedra
(first row)
chiral solids generated by the intersection of the three tetrahedra
(second row)
The polyhedra in the second column have been properly orientated to evidence their mirror symmetry
(concerning both shape and colouring) with respect to the polyhedra in the first column.

Animated GIF of chiral compounds of three tetrahedra and their intersections, seen
by ortographic viewing and along 2fold and 3fold axes.


Starting from the initial position a), the solid generated by the intersection of red, yellow
and green tetrahedra is shown in b) after an appropriate rotation, in order to evidence the presence
of both a vertical 3fold axis and three 2fold axes, perpendicular to the 3fold axis,
characterizing the 32 point group.
In c) the same colour has been assigned to all the faces belonging to each of the two crystallographic forms:
trigonal trapezohedron (6 brown faces) and rhombohedron (6 fuchsia faces).


View along the [001] direction of the faces of the polyhedron resulting from the intersection of
three tetrahedra and relative stereographic projection: in conformity with the 32 point group, the forms present are
a trigonal trapezohedron (6 brown faces) and a rhombohedron (6 fuchsia faces).


Orthographic views of trigonal trapezohedron (left)
and rhombohedron (right) constituting the polyhedron that derives from the intersection of three tetrahedra.

Clicking
here, one can see the geometry of the quadrilateral and pentagonal faces characterizing the polyhedron
that derives from the intersection of three tetrahedra; the side lengths are
relative to a unit distance of the faces from the centre of the polyhedron.

Planar nets of the chiral polyhedra deriving from the intersection of red, yellow
and green tetrahedra: clicking here, one can see alternative
nets, in which the same colour is assigned to all the faces belonging to to each
crystallographic form (trapezohedron or rhombohedron).

Compound polyhedron made of 4 tetrahedra and solid generated by their intersection
The following animated
GIF, consisting of three frames, shows sequentially the images along different directions, included 2fold and 3fold rotation axes, relative to the pairs of:
chiral compound polyhedra made of red, yellow, green and paleblue tetrahedra (first row)
chiral solids generated by the intersection of the four tetrahedra (second row)
The polyhedra in the second column have been properly orientated to evidence their mirror symmetry (concerning both shape and
colouring) with respect to the polyhedra of the first column.

Animated GIF of chiral compounds of four tetrahedra and their intersections seen
by orthographic viewing and along 3fold and 2fold axes. 

Starting from the initial position a), the solid generated by the intersection of
four tetrahedra is shown in b) after an appropriate rotation, in order to evidence the presence
of four threefold axis and three perpendicular twofold axes, characterizing the 23 point group.
In c) the same colouring has been assigned to all the faces belonging to each of the two crystallographic forms:
pentagondodecahedron (12 brown faces) and tetrahedron (4 fuchsia faces).


View along the [001] direction and stereographic projection of the faces of the polyhedron
resulting from the intersection of four tetrahedra. 
A
s well as by the intersection of four tetrahedra, the same polyhedron can be obtained,
of course, eliminating from an icosahedron four faces, orientated like the faces of a tetrahedron
(the one with {111} indices, for
example). This may be achieved by extending the three triangular faces contiguous
to each face of the chosen tetrahedron, until they meet in a new vertex:
consequently the shape of twelve faces (out of twenty) changes from triangular to deltoidal (or
kiteshaped quadrilateral), whereas the remaining four faces, centrosymmmetric with respect to the
four faces eliminated, retain their triangular shape.

Animated GIF showing the stellation process relative to four out of twenty faces of the icosahedron,
leading to the tetrahedrally stellated icosahedron. 
Four 3fold axis are parallel to the directions connecting each new vertex with the centre
of the opposite triangular face, whereas three 2fold axes are perpendicular to the midpoint of the long sides
of the deltoids not shared with the triangular faces: the overall resulting symmetry corresponds
to the crystallographic 23 point group.
Consequently, the sixteen faces can be grouped in two forms: a
{τ 1/τ 0} pentagondodecahedron and a
{1 1
1} tetrahedron.
(Due to the lack of both inversion centre and any mirror, also an enantiomorphic polyhedron exists, made of the same
{τ 1/τ 0}
pentagondodecahedron and a {111} tetrahedron).

Orthographic views of the forms in which the solid generated by the intersection of four tetrahedra
can be decomposed: a {τ 1/τ
0} pentagondodecahedron and a {
1 1
1} tetrahedron. 
Taking into account the procedure followed to get the polyhedron, George W.
Hart proposed for it the name: tetrahedrally stellated icosahedron.
To summarize, the polyhedron consists of sixteen faces and sixteen vertices: the twelve
original vertices of icosahedron
plus the four new vertices (of course, the edges are thirty, from the known rule
E = F+V2, valid for convex polyhedra). This circumstance implies that also
its dual has the same number of faces, vertices and edges; therefore the two
polyhedra are topologically identical, but not geometrically: in fact
the faces of the dual consist of four equilateral triangles and twelve isosceles trapezoids (or trapezia).
The faces of the dual can be grouped according to two
crystallographic forms, coexisting in the 23 point group: a {111}
tetrahedron and a {1τ 0} pentagondodecahedron, geometrically identical
to a regular dodecahedron. Consequently, the dual of a tetrahedrally stellated
icosahedron can be obtained truncating a dodecahedron by a tetrahedron, till
the twelve new vertices of the truncated dodecahedron coincide with
twelve of the original twenty vertices of the dodecahedron (in total, due to
the truncation process, the vertices from 20 become
at first (20  4) + (3 x 4) = 28 and, at the end,
20  4 = 16).
Tetrahedrally truncated dodecahedron is the
appropriate name, obviously correlated with the name of its dual, given from George W. Hart to this polyhedron.

Animated GIF showing the truncation of four vertices of a dodecahedron,
leading to the tetrahedrally truncated dodecahedron. 

View, along the [001] direction, of the faces of
the tetrahedrally truncated dodecahedron and relative stereographic projection. 

First row: chiral couple made of tetrahedrally stellated icosahedra
Second row: chiral couple made of tetrahedrally truncated dodecahedra
First column: dual couple made of tetrahedrally stellated icosahedra and tetrahedrally truncated dodecahedra
Second column: dual couple made of chiral tetrahedrally stellated icosahedra and
chiral tetrahedrally truncated dodecahedra
Every polyhedron includes 16 faces, 16 vertices and 30 edges and belongs to 23 point group,
being characterized by three 2fold axes and four 3fold axes.

A further interesting feature concerning the
canonicalization of the previous dual couple of polyhedra has been pointed out by George W. Hart:
the canonical forms of tetrahedrally stellated icosahedron and tetrahedrally truncated dodecahedron are geometrically selfdual.
The kiteshaped faces of the canonical form have edges in the ratio 1: 0.803
and the angle between the shorter sides is 111.15°, whereas the other 3 angles
are equal (82.95°).

Geometry of kiteshaped faces of the tetrahedrally stellated icosahedron (left),
trapezoidal faces of the tetrahedrally truncated dodecahedron (centre),
kiteshaped faces of the related canonical form (right). 
(clicking on the following images one can visualize the VRML file
relative to the canonical form)


View, along a 2fold axis (left) and a 3fold axes (right), of a polyhedron approximating rather
closely the canonical form of both the tetrahedrally stellated icosahedron and
the chiral tetrahedrally truncated dodecahedron.


Planar nets of chiral tetrahedrally stellated icosahedra: their four colours remind
one that the polyhedra derive from the intersection of red, yellow, green and paleblue tetrahedra; alternative nets, in which one colour corresponds to each crystallographic form (pentagondodecahedron or tetrahedron),
can be seen clicking here.


Nets of a chiral couple made of tetrahedrally truncated
dodecahedra, duals of the chiral couple made of tetrahedrally stellated icosahedra.
The isosceles trapezia have the following remarkable features: the length of the shorter
parallel side is equal to the two not parallel sides and the value of the ratio
between the parallel sides corresponds to the golden ratio
τ = (√5+1)/2
=1.61803... 


Nets of the chiral canonical forms of
tetrahedrally stellated icosahedron and tetrahedrally truncated dodecahedron. 
Compound polyhedron made of 5 tetrahedra and icosahedron
generated by their intersection
The following animated
GIF, consisting of three frames, shows sequentially the images along different directions, included 5fold and 3fold rotation axes, relative to the pairs of:
chiral compound polyhedra made of red, yellow, green, paleblue and violet tetrahedra (first row)
chiral colouring of the solids generated by the intersection of the five tetrahedra (second row)
The polyhedra in the second column have been properly orientated to evidence their mirror colouring relatively to
the polyhedra of the first column; in the instance of the compound polyhedra, the mirror symmetry concerns also their shape.

Animated GIF of chiral compounds of five tetrahedra and their intersections
seen by orthographic viewing and along 5fold and 3fold axes.


Views, along the [001] direction, of the chiral colouring of the icosahedron,
whereas the stereographic projection is unique; in fact, the presence of both fifteen mirrors and inversion centre
rules out the possibility of enantiomorphism concerning the geometrical shape of icosahedron itself. 
Views, along all the directions corresponding to 5fold axes, of the two chiral compound polyhedra made of five tetrahedra.
In every row there are three pairs of views, along [uvw] and [u
v w] centrosymmetric directions, showing that both colour pattern and geometric shape of chiral compound polyhedra are symmetric with respect to an
intermediate vertical mirror (compare the images of 1st and 4th, 2nd and 5th, 3rd and 6th columns).


First row:
chiral compound polyhedra made of five tetrahedra, whose vertices
coincide with the vertices of a dodecahedron;
their enantiomorphism concerns both colouring and shape of the polyhedra.
Second row: chiral icosahedra resulting from the intersection of
the previous compound polyhedra, positioned with respect to the outer dodecahedron (also the
position of the vertices of each visible coloured tetrahedron has been reported);
they are enantiomorphic only concerning the colouring of the triangular faces.

As shown in the following image, an interesting dissection of the icosahedron generates two pentagonal pyramids
and a pentagonal antiprism (or parabidiminished icosahedron,
according to Johnson's systematic nomenclature): it suggests, among the 43380
possible different nets of the icosahedron (Buekenhout, F. and Parker, M., 1998),
the choice of a net including a chain of ten sidesharing triangles; two other sets of five
vertexsharing triangles, placed on the opposite sides of the chain, are connected to it.

Dissection of an icosahedron into two pentagonal pyramids and a pentagonal antiprism.


Chiral 5coloured nets of icosahedron.

Many thanks are due to Maria Rosa Ardigo' and Riccardo Basso for the fruitful discussions and to Fabio Somenzi for his precious help concerning the translation of the text.
References
H. Martyn Cundy and A. P. Rollet
"Mathematical Models", Oxford University Press,
1954
Peter R. Cromwell
"Polyhedra", Cambridge University Press, 1997
F. Buekenhout and M. Parker
"The Number of Nets of the Regular Convex Polytopes in
Dimension Less or Equal to 4", Disc. Math., 186, 6994, 1998
Norman W.
Johnson
"Convex Polyhedra with Regular Faces", Canadian Journal of Mathematics, 18,
169200, 1966.
Links