Polyhedra deriving from the progressive truncation by cube of the Archimedean truncated tetrahedron and relative duals

The completely analogous sequence relative to the truncation of the icosi-dodecahedron by a rhomb-triacontahedron is shown in Fig.7b. The face transitive duals of the series of vertex-transitive polyhedra go from the rhomb-triacontahedron to the deltoid-hexecontahedron {τ10}, passing through a continuous series of intermediate hexakis-icosahedra (also called disdyakis-triacontahedra).

Sequence of vertex-truncations of the icosi-dodecahedron by a rhomb-triacontahedron (RT) (d_dodecahedron= 1/(√1+(1/τ²), d_icosahedron= τ/√3 and 1/(1+1/τ²) ≤ d_RT ≤ 1)

d_RT = 1 The faces of rhomb-triacontahedron are tangent to the thirty vertices of the icosi-dodecahedron	d_RT = 0.97	d_RT = (4τ+3)/10 = 0.9472 The faces of dodecahedron assume the shape of a regular decagon

d_RT = (τ+4)/6 = 0.9363 The faces of icosahedron have the shape of a regular hexagon	d_RT = 0.92	d_RT = (τ²+1)/4 = 0.9045 Each vertex is shared among four faces

d_RT = 0.89	d_RT = τ²/3 = 0.8727 Vanishing of the faces of icosahedron	d_RT = 1/(1+1/τ²)= 0.7236 The only faces of rhomb-triacontahedron survive
FIG.7b- Sequence of the complete vertex-truncation of the icosi-dodecahedron by a rhomb-triacontahedron: only the forms reported in the first two rows are vertex-transitive.

The two names truncated cuboctahedron and truncated icosi-dodecahedron attributed to as many Archimedean polyhedra are somehow misleading (alternatively, the respective synonyms "great rhombicuboctahedron" and "great rhombicosidodecahedron" [3] are sometimes found in literature), since they cannot be obtained after truncating the cuboctahedron by means of a rhomb-dodecahedron or the icosi-dodecahedron by means of a rhomb-triacontahedron [4].
The edge-truncations leading to these two other Archimedean polyhedra start instead from a non-Archimedean truncated octahedron and a non-Archimedean truncated icosahedron, as shown in Fig.8a and Fig8b; the ratios d_octahedron/d_cube or d_dodecahedron/d_icosahedron remain constant along each of all these truncation processes.

d_octahedron /d_cube= √3/(3-√2) = 1.0922	d_octahedron /d_cube = 1.0922 d_cube/d_{rhomb-dodecahedron} = 0.8673 d_octahedron/d_{rhomb-dodecahedron} = 0.9473

FIG. 8a - The edge-truncation by a rhomb-dodecahedron of a non Archimedean truncated octahedron leads to the Archimedean truncated cuboctahedron when the ratio d_octahedron/ d_cube= √3/(3-√2).

d_dodecahedron/d_icosahedron = √(3-τ)³/3 = 0.9380	d_dodecahedron /d_icosahedron = 0.9380 d_icosahedron /d_{rhomb-triacontahedron} = 0.9819 d_dodecahedron /d_{rhomb-triacontahedron} = 0.9210

FIG.8b - The edge-truncation of a non Archimedean truncated icosahedron by a rhomb-triacontahedron leads to the Archimedean truncated icosidodecahedron when the ratio d_dodecahedron/ d_icosahedron= √(3-τ)³/3.

Analogously, in Fig. 9a and Fig. 9b one can see how the Archimedean rhomb-cuboctahedron and rhomb-icosidodecahedron derive from the edge-truncation of two other non-Archimedean polyhedra, a truncated octahedron and a truncated icosahedron, both slightly different from the previous ones shown in Fig. 8a and Fig. 8b.

d_octahedron/d_cube = (2√2 -1)/√3 = 1.0556	d_octahedron /d_cube = 1.0556 d_cube/d_{rhomb-dodecahedron} = 1.0 d_octahedron/d_{rhomb-dodecahedron} = 1.0556

FIG. 9a - Edge-truncation by a rhomb-dodecahedron of a non-Archimedean truncated octahedron, slightly different from the previous one (ratio d_octahedron/ d_cube = 1.0556), leading to the rhomb-cuboctahedron.

d_dodecahedron/d_icosahedron = 3√3 / [(3+τ)√3-τ] = 0.9571	d_dodecahedron/d_icosahedron= 0.9571 d_icosahedron/d_{rhomb-triacontahedron}= 1.0184 d_dodecahedron/d_{rhomb-triacontahedron}= 0.9748

FIG. 9b - Edge-truncation by a rhomb-triacontahedron of a non-Archimedean truncated icosahedron, slightly different from the previous one (ratio d_dodecahedron/d_icosahedron = 0.9571), leading to the rhomb-icosidodecahedron.

A further truncation of the Archimedean truncated cuboctahedron and truncated icosidodecahedron (left column of Fig.10) can be obtained by diminishing the central distance of rhomb-dodecahedron and rhomb-triacontahedron respectively: the resulting vertex-transitive polyhedra are shown in the central column.
They both can be compared with the very similar rhomb-cuboctahedron and rhomb-icosidodecahedron reported in the right column, which are the archetypes also of the solids, shown in Fig.5, resulting from the vertex- and edge-truncation of the other Archimedean solids.


Archimedean truncated cuboctahedron	truncation by rhomb-dodecahedron of the Archimedean truncated cuboctahedron	Archimedean rhomb-cuboctahedron

Archimedean truncated icosidodecahedron	truncation by rhomb-triacontahedron of the Archimedean truncated icosidodecahedron	Archimedean rhomb-icosidodecahedron
FIG. 10- The further truncation of both Archimedean truncated cuboctahedron and truncated icosidodecahedron (on the left) leads to the vertex-transitive polyhedra shown in the central images, which can be compared with the isomorphic rhomb-cuboctahedron and rhomb-icosidodecahedron (on the right).

All the truncation processes previously described are summarized in the animated sequences reported in Fig.11 and Fig.12.

tC	CO	tO	tCO	RCO

FIG. 11 - Sequences of vertex-transitive polyhedra deriving from the truncation by a rhomb-dodecahedron of: three Archimedean solids, the truncated cube (tC), the cuboctahedron (CO) and the truncated octahedron (tO) two specified (non-Archimedean) truncated octahedra, leading to the truncated cuboctahedron (tCO) and the rhomb-cuboctahedron (RCO) respectively A further decrease of the central distance of the intersecting rhomb-dodecahedron transforms the Archimedean truncated cuboctahedron in a vertex transitive polyhedron whose dual, analogously to the other four cases, is a face-transitive deltoid-icositetrahedron.

tD	ID	tI	tID	RID

FIG. 12 - Sequences of vertex-transitive polyhedra deriving from the truncation by a rhomb-triacontahedron of: three Archimedean solids, the truncated dodecahedron (tD), the icosidodecahedron (ID) and the truncated icosahedron (tI) two specified (non-Archimedean) truncated icosahedra, leading to the Archimedean truncated icosidodecahedron (tID) and the rhomb-icosidodecahedron (RID) respectively A further decrease of the central distance of the intersecting rhomb-triacontahedron transforms the truncated icosidodecahedron in a vertex transitive polyhedron whose dual, analogously to the other four cases, is a face-transitive deltoid-hexecontahedron.

The face-transitive duals of the polyhedra obtained in the early stages of the truncation processes consist in a continuous series of hexakis-octahedra and hexakis-icosahedra, whose Catalan archetypes are shown in Fig.13.

CATALAN HEXAKIS-OCTAHEDRON	CATALAN HEXAKIS-ICOSAHEDRON

FIG.13 - Face-transitive hexakis-octahedron (left) and hexakis-icosahedron (right), Catalan archetypes of the duals of the series of polyhedra deriving from the truncation processes.

In turn, the Catalan deltoid-icositetrahedron and deltoid-hexecontahedron shown in Fig.14 are the archetypes of the face-transitive duals of the ultimate vertex-transitive polyhedra resulting from each truncation process.

CATALAN DELTOID-ICOSITETRAHEDRON	CATALAN DELTOID-HEXECONTAHEDRON

FIG.14 - Catalan deltoid-icositetrahedron (left) and deltoid-hexecontahedron (right), archetypes of the duals of the vertex-transitive polyhedra being the final result of each truncation process.

FROM THE TETRAHEDRON TO THE ARCHIMEDEAN TRUNCATED TETRAHEDRON

The tetrahedron is the fifth Platonic solid; it plays a singular role among the regular convex polyhedra, because of some peculiar features:

it is the Platonic solid having fewest faces, vertices and edges: its edges are six, as many as the faces of the cube and the vertices of the octahedron, whereas from the fact that both its faces and vertices are four it follows that the tetrahedron is self-dual. The dual of the tetrahedron {111} is a second tetrahedron, having {111} Miller's indices, congruent to the first by each of three 90° rotations around the orthogonal directions [001], [010] and [100], passing through the midpoints of opposite edges of each tetrahedron (right image of Fig.15).
the elements of symmetry relative to the tetrahedron are different in respect to the other Platonic solids, included cube and octahedron which are characterized by the 4/m 3 2/m cubic point group symmetry [5]: infact the three 4-fold rotation axes of cube and octahedron in tetrahedron become three 4-fold rotoinversion axes (passing through the midpoint of opposite edges) and, concerning the other elements of symmetry present in cube and octahedron, the only remaining in tetrahedron are the four 3-fold rotation axes (through each face and the opposite vertex) and six of the nine mirror planes (each mirror passes through an edge and the midpoint of the opposite edge). The maximum point group symmetry of tetrahedron is therefore 43m (left image of Fig.15), even if it can be described also by the 23 point group symmetry, in which the mirrors are absent and the 4 rotoinversion axes reduce to 2-fold rotation axes (right image).


FIG.15- (left) view of a tetrahedron with the elements of symmetry relative to the 43m point group; (right) couple of self-dual (and congruent) tetrahedra, showing the lower symmetry peculiar to the 23 point group. Clicking on this image, one can see the compound solid, made of the two self-dual tetrahedra, named "stella octangula" by Kepler.

as already pointed out in [6], the tetrahedron can be linked to all the other four Platonic solids by the operators named: join, ambo, gyro and snub, according to the notation [7] proposed by John Conway (Fig.16).

FIG.16 - Conway operators linking tetrahedron to the other four Platonic solids.

Just as the truncation of each other Platonic solid by its dual can lead to an Archimedean solid, the truncation of a tetrahedron, because of its self-duality, involves a second tetrahedron.
In Fig.17 one can see the frames of both sequences relative to the reciprocal truncation of the self-dual tetrahedra, leading to a pseudo-octahedron; the value of the ratio between the distances, relative to the barycentre, of the faces belonging to the two intersecting tetrahedra decreases from three, when an only tetrahedron is present, to one, when the two tetrahedra are equidistant and their intersection gives rise to the pseudo-octahedron.


FIG. 17 - Sequences of the reciprocal truncation of {111} and {111} tetrahedra, leading to the same final form, geometrically identical to an octahedron.

In both sequences, the most interesting frame corresponds to the situation in which the ratio of the distances of the two tetrahedra from the center is equal to 5/3: the solid resulting is the Archimedean truncated tetrahedron. Infact it is a semi-regular polyhedron, since the faces of the truncating tetrahedron, namely the one more distant from the centre of the solid, have the shape of equilateral triangles, whereas the faces of the truncated tetrahedron, the one nearer to the centre, are regular hexagons (Fig. 18); in addition, it is also vertex-transitive, being all its vertices, shared by a triangular and two hexagonal faces of the two tetrahedra, all equivalent by the action of the elements of symmetry characterizing it.


FIG.18 - Couple of views of the Archimedean truncated tetrahedron, obtained when the value of the ratio d_{111} /d_{111} between the distances of the tetrahedra {111} and {111} from the center of the solid is 5/3 (left) or 3/5 (right).

It is important to point out that the two truncated tetrahedra shown in Fig.18 are not chiral, being congruent by a 90° rotation, just as the couple of dual tetrahedra; in practice, they are two alternative views of an unique Archimedean truncated tetrahedron.
The same holds for the couple of triakis-tetrahedra, shown in Fig.19, being the {311} (on the left) or the {311} (on the right) triakis-tetrahedron. They are two alternative views of the Catalan triakis-tetrahedron, dual of the Archimedean truncated tetrahedron.


FIG.19 - Couple of congruent Catalan triakis-tetrahedra {311} and {311}, duals of the Archimedean truncated tetrahedron, set in two alternative orientations, 90 degrees apart. In each Catalan polyhedron all the dihedral angles between couples of contiguous faces have a constant value: in case of the triakis-tetrahedron such value is 50.48°.

Even the intersection between the Archimedean truncated tetrahedron and its dual triakis-tetrahedron does not lead to a face transitive polyhedron, as one can see in Fig.20 (left): in correspondence to each vertex of the resulting polyhedron, two faces coming from the triakis-tetrahedron are shared either with two different faces of the truncated tetrahedron (triangular the smaller face, hexagonal the larger one) or with two hexagonal faces.
Even if not equivalent, all the vertices are placed at the same central distance, and therefore the same happens with the faces of its dual (shown on the right of Fig.20), consisting in the intersection of two forms: the cube {100} and the deltoid-dodecahedron {221}.


FIG.20 - (left) Composite form, resulting from the intersection of the Archimedean truncated tetrahedron with the Catalan triakis-tetrahedra {311}, and (right) the relative dual form, made of a cube and the deltoid-dodecahedron {221}.

INTERSECTION BETWEEN THE ARCHIMEDEAN TRUNCATED TETRAHEDRON AND THE CUBE
In general, the intersection of truncated tetrahedra with another polyhedron can lead to its further truncation.
As it will be illustrated in a next paper, a series of vertex-transitive polyhedra can be obtained, in case of the 43m point group symmetry, by the intersection of each truncated tetrahedron with a cube of progressively reduced dimensions: in particular, the following animated sequence (Fig. 21) describes the edge truncation by a cube of the Archimedean truncated tetrahedron.

FIG. 21 - Animated sequence describing the progressive edge-truncation process deriving from the intersection between the Archimedean truncated tetrahedron and a cube having decreasing dimensions.

The choice of selected frames from the sequence allows to highlight the more interesting steps of the edge-truncation process (left column of Fig. 22) and the relative duals are shown in the right column. The lower row of each frame reports also the corresponding stereographic projection between the views, along the vertical [001] direction, of the couple of dual polyhedra.

Intermediate polyhedra (and relative duals) resulting from the progressive truncation of the Archimedean truncated tetrahedron by a cube

a) d_{111}= √3/3; d_{111}= 5√3/9 Archimedean truncated tetrahedron (the faces of the cube are tangent to six edges of the truncated tetrahedron when d_{100}= 1.0)	a') catalan triakis-tetrahedron {311}



b) d_{111}= √3/3; d_{111}= 5√3/9; d_{100}= 5/6	b') hexakis-tetrahedron {523}



c) d_{111}= √3/3; d_{111}= 5√3/9; d_{100}= 7/9	c') hexakis-tetrahedron {735}



d) d_{111}= √3/3; d_{111}= 5√3/9; d_{100}= 3/4	d') hexakis-tetrahedron {947}



e) d_{111}= √3/3; d_{111}= 5√3/9; d_{100}= 2/3	e') deltoid-dodecahedron {212}



f) d_{111}= √3/3; d_{111}= 5√3/9; d_{100}= 3/5	f') {979} and {313} deltoid-dodecahedra



g) d_{111}= √3/3; d_{111}= 5√3/9; d_{100}= 5/9 (the decreased central distance of the cube causes the disappearance of the more distant {111} tetrahedron)	g') tetrahedron {111} and deltoid-dodecahedron {515}



h) d_{111}= √3/3; d_{111}= 5√3/9; d_{100}= 1/2	h') tetrahedron {111} and rhomb-dodecahedron {101}



i) d_{111}= √3/3; d_{111}= 5√3/9; d_{100}= 2/5	i') tetrahedron {111} and deltoid-dodecahedron {212}



j) d_{111}= √3/3; d_{111}= 5√3/9; d_{100}= 1/3 (due to the intersecting cube, also the remaining tetrahedron has disappeared)	j') {111} and {111} tetrahedra

FIG: 22 - Selected frames of the animated sequence reported in Fig.20, illustrating the more significant steps of the progressive truncation by a cube of the Archimedean truncated tetrahedron. The polyhedron obtained at each step is shown on the left of the upper row, whereas the relative dual is shown on the right; the stereographic projection of the faces of the dual polyhedron are reported in the lower row, between the views along the vertical [001] direction of the couple of dual polyhedra.

A detailed description of the forms obtained at each stage of the truncation process (and also of the relative duals) is given in the following table.

First row d_{100} = 1	When d_{100}= 1.0 the faces of the cube are just tangent to six edges of the Archimedean truncated tetrahedron and do not intersecate it: therefore the solid in (a) consists in the Archimedean truncated tetrahedron, made of triangular and hexagonal faces, both regular, and the solid in (a') consists in its dual, the {311} Catalan triakis-tetrahedron, having twelve faces.
Second row d_{100} = 5/6	As a consequence of the beginning of the truncation by the cube, the faces coming from both tetrahedra assume the shape of a non-regular hexagon, whereas the faces of the cube are rectangular; since each vertex (shared by three faces, one of the cube and two of the different tetrahedra) is equivalent by symmetry to all the other 23 vertices, the form in (b) is vertex-transitive (or isogonal) and its dual shown in (b') is the face-transitive (or isohedral) hexakis-tetrahedron {523}, having twenty-four faces.
Third row d_{100} = 7/9	In (c) the smaller hexagonal faces become regular in consequence of the increased truncation; consequently, in the dual {735} hexakis-tetrahedron shown in (c'), the dihedral angles between each couple of contiguous faces included in the set of six faces sharing a vertex along the direction [111] are all equal (17.860°).
Fourth row d_{100} = 3/4	In (d) the faces derived from both tetrahedra have again the shape of non-regular hexagons; the dual isohedral form, shown in (d'), is the {947} hexakis-tetrahedron.
Fifth row d_{100} = 2/3	In (e) the truncation originates a form, having only twelve vertices, which is still vertex-transitive: the faces derived from both tetrahedra have the shape of equilateral triangles (having different sizes) and each vertex is shared by four faces: a couple of rectangular faces coming from the cube, the two others from the couple of tetrahedra. The relative dual, shown in (e'), is the deltoid-dodecahedron {212}, consisting of twelve faces.
Sixth row d_{100} = 3/5	Since two are the different vertices present in (f), the result of the truncation is a form that is not vertex-transitive: each vertex is shared by three faces, two coming from the cube and the third, alternatively, from each of the two tetrahedra. Consequently also the dual shown in (f') is not face-transitive, and consists in the intersection of two deltoid-dodecahedra, having the indices {313} and {979}.
Seventh row d_{100} = 5/9	This is another noteworthy stage of the truncation process, since (g) marks the complete disappearance of one of the tetrahedra. The number of vertices decreases to sixteen: twelve are still shared by two faces of cube and a face of the remaining tetrahedron, whereas the other four are shared by three faces belonging to the only cube. Then the dual shown in (g') consists in the intersection of two forms: the deltoid-dodecahedron {515} and the tetrahedron {111}.
Eighth row d_{100} = 1/2	The number of vertices is sixteen even in (h), but the twelve vertices shared by two faces of cube and a face of tetrahedron are now localized along [110] and equivalent directions. Therefore the dual in (h') consists in the intersection of the rhomb-dodecahedron {110}, made of twelve faces, and the tetrahedron {111}.
Ninth row d_{100} = 2/5	The decreased central distance of the faces of the cube causes in ( i ) the progressive reduction of the size of the faces belonging to the remaining tetrahedron. In (i') the dual consists in the intersection between the tetrahedron {111} and the deltoid-dodecahedron {212} (note the negative value of the second index), made of the twelve faces related, by duality, to the vertices shared in ( i ) by a face of tetrahedron and two faces of cube.
Tenth row d_{100} = 1/3	In (j) even the second tetrahedron disappears, lefting the only cube; in consequence of the 43m point group symmetry, its dual shown in (j') is a pseudo-octahedron, actually resulting from the intersection of the tetrahedron {111} and the tetrahedron {111}.

In conclusion, in Fig.23 one can see the animated sequence relative to the duals of the forms reported in Figure 21.

FIG. 23 - Animated sequence including the duals of the forms obtained by the intersection between the Archimedean truncated tetrahedron and a cube with decreasing size (to be compared with Figure 21).

LINKS AND REFERENCES

Cromwell P.
Polyhedra
Cambridge University Press, 1997, ISBN 9-521-55432-2

Zefiro L., Ardigo' M.R.
Platonic and Catalan polyhedra as archetypes of forms belonging to the cubic and icosahedral systems
VisMath, volume 11, No. 2, 2009

Hart G.
Archimedean Polyhedra
www.georgehart.com/virtual-polyhedra
Zefiro L., Ardigo' M.R.
Description of the Forms Belonging to the 235 and m35 Icosahedral Point Groups Starting from the Pairs of Dual Polyhedra: Icosahedron-Dodecahedron and Archimedean Polyhedra-Catalan Polyhedra
VisMath, volume 9, No. 4, 2007
International Union of Crystallography
International Tables for Crystallography, Vol. A
Theo Hahn Editor, Kluwer Academic Publisher, 1989
Zefiro L.
Generation of an icosahedron by the intersection of five tetrahedra: geometrical and crystallographic features of the intermediate polyhedra
VisMath, volume 10, No. 3, 2008
Hart G.
Conway Notation for Polyhedra
www.georgehart.com/virtual-polyhedra

ARCHIMEDEAN TRUNCATED CUBE	ARCHIMEDEAN TRUNCATED DODECAHEDRON

d_octahedron /d_cube= (√2+1)/√3 = 1.3938	d_dodecahedron/d_icosahedron = √3(τ²+1)/(3τ-1) = 0.8548

CUBOCTAHEDRON	ICOSI-DODECAHEDRON

d_octahedron /d_cube= 2/√3 = 1.1547	d_dodecahedron/d_icosahedron = √3/√τ² +1 = 0.9106

ARCHIMEDEAN TRUNCATED OCTAHEDRON	ARCHIMEDEAN TRUNCATED ICOSAHEDRON

d_octahedron /d_cube= √3/2 = 0.8660	d_dodecahedron/d_icosahedron = (3+1/τ²)/√3(τ²+1) = 1.0265


edge-truncation by rhomb-dodecahedron of the Archimedean truncated cube	vertex-truncation by rhomb-dodecahedron of the Archimedean cuboctahedron	edge-truncation by rhomb-dodecahedron of the Archimedean truncated octahedron

edge-truncation by rhomb-triacontahedron of the Archimedean truncated dodecahedron	vertex-truncation by rhomb-triacontahedron of the Archimedean icosi-dodecahedron	edge-truncation by rhomb-triacontahedron of the Archimedean truncated icosahedron
FIG.5- Vertex-transitive polyhedra, duals of deltoid-icositetrahedra or deltoid-hexecontahedra, resulting from the intersection of cubic or icosahedral Archimedean polyhedra with the rhomb-dodecahedron (upper row) or the rhomb-triacontahedron (lower row).

Sequence of vertex-truncations of the cuboctahedron by a rhomb-dodecahedron (d_cube= √2/2, d_octahedron= √6/3 and 1/2 ≤ d_{rhomb-dodecahedron} ≤ 1)

d_{rhomb-dodecahedron} = 1 The faces of the rhomb-dodecahedron are tangent to the twelve vertices of the cuboctahedron	d_{rhomb-dodecahedron} = 0.9	d_{rhomb-dodecahedron} = 0.8536 = (√2+2)/4 The faces deriving from the cube have the shape of a regular octagon

d_{rhomb-dodecahedron} = 0.8333 = 5/6 The faces deriving from the octahedron have the shape of a regular hexagon	d_{rhomb-dodecahedron} = 0.8	d_{rhomb-dodecahedron} = 3/4 Each vertex is shared among four faces

d_{rhomb-dodecahedron} = 0.72	d_{rhomb-dodecahedron} = 0.6 Vanishing of the faces of octahedron	d_{rhomb-dodecahedron} = 1/2 The only faces of rhomb-dodecahedron survive
FIG.7a- Sequence of the complete vertex-truncation by rhomb-dodecahedron of cuboctahedron: only the forms of the first two rows are vertex-transitive.

RHOMB-DODECAHEDRON	RHOMB-TRIACONTAHEDRON

FIG.3- Rhomb-dodecahedron (left) and rhomb-triacontahedron (right), Catalan duals of the Archimedean cuboctahedron and icosi-dodecahedron respectively

Edge-truncation of cube and octahedron by a rhomb-dodecahedron

FIG.4a- Sequences of the edge-truncation by a rhomb-dodecahedron of a cube (left) and an octahedron (right)

Edge-truncation of dodecahedron and icosahedron by a rhomb-triacontahedron

FIG.4b- Sequences of the edge-truncation by a rhomb-triacontahedron of a dodecahedron (left) and an icosahedron (right)

TRUNCATED CUBE and TRIAKIS-OCTAHEDRON	TRUNCATED DODECAHEDRON and TRIAKIS-ICOSAHEDRON

FIG.6a- Intersections between couples of dual Archimedean and Catalan polyhedra consisting in: a truncated cube and a triakis-octahedron (left) a truncated dodecahedron and a triakis-icosahedron (right)

TRUNCATED OCTAHEDRON and TETRAKIS-CUBE	TRUNCATED ICOSAHEDRON and PENTAKIS-DODECAHEDRON

FIG.6b- Intersections between couples of dual Archimedean and Catalan polyhedra consisting in: a truncated octahedron and a tetrakis-cube (left) a truncated icosahedron and a pentakis-dodecahedron (right)