Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra

Livio Zefiro*
*Dip.Te.Ris, Universita' di Genova, Italy

 Notes Tested with Internet Explorer 8.0, Mozilla Firefox 5.0, Opera 11.11, Google Chrome 11.0 and Safari 5 at 1024x768 and 1280x1024 pixels All the images were created by SHAPE 7.0, a software program for drawing the morphology of crystals, distributed by Shape Software

INTRODUCTION
The five regular convex polyhedra called Platonic solids are universally known (Fig. 1). Leaving aside tetrahedron, described in detail successively, the four others consist in two couples of dual solids, which interchange the number and position of faces and vertices; they are:
• cube and octahedron, having 4/m 3 2/m cubic symmetry (or Oh according to the notation adopted by Schoenflies)
• dodecahedron and icosahedron, having 2/m 3 5 (or Ih) icosahedral symmetry
The Platonic solids are the only convex polyhedra that are vertex-transitive (or isogonal), face-transitive (or isohedral) and edge-transitive (or isotoxal) at the same time. It means that each vertex, face or edge can be carried out to any other by a symmetry operation [1].

• TRUNCATION OF PLATONIC SOLIDS

Each Platonic solid can be vertex-truncated by its dual: when the ratios doctahedron /dcube and ddodecahedron /dicosahedron, relative to the distances of the faces from the center of the solid, get appropriate values [2], their intersection gives rise to the following Archimedean solids (shown in Fig.2):
• truncated cube, truncated octahedron and cuboctahedron, derived from the intersection of cube and octahedron
• truncated dodecahedron, truncated icosahedron and icosi-dodecahedron, from dodecahedron and icosahedron
These Archimedean solids are vertex-transitive (and their duals are face-transitive Catalan solids), but vertex-transitive are also all the non-Archimedean solids, obtained from the vertex-truncation of Platonic solids when the ratio of the distances gets any other value included in the intervals:
1/√3 < doctahedron /dcube < √3   and  τ2 /√3(τ2+1) < ddodecahedron /dicosahedron < 3(τ2+1)2
where τ corresponds to the value of the golden ratio (√
5+1)/2 = 1.61803...

•  ARCHIMEDEAN TRUNCATED CUBE ARCHIMEDEAN TRUNCATED DODECAHEDRON doctahedron /dcube=  (√2+1)/√3 = 1.3938 ddodecahedron /dicosahedron = √3(τ2+1)/(3τ-1) = 0.8548

 CUBOCTAHEDRON ICOSI-DODECAHEDRON doctahedron /dcube= 2/√3 = 1.1547 ddodecahedron /dicosahedron  = √3/√τ2 +1 = 0.9106

 ARCHIMEDEAN TRUNCATED OCTAHEDRON ARCHIMEDEAN TRUNCATED ICOSAHEDRON doctahedron /dcube=  √3/2 = 0.8660 ddodecahedron /dicosahedron = (3+1/τ2)/√3(τ2+1) = 1.0265

 FIG.2- Archimedean polyhedra resulting from the intersection between the two couples of dual solids consisting of cube and octahedron (left column) and dodecahedron and icosahedron (right column).

The edge-truncation of the previous four Platonic solids can instead be performed by the rhomb-dodecahedron or the rhomb-triacontahedron, depending on whether the polyhedron to be truncated has a cubic or an icosahedral symmetry.
It is noteworthy to point out that the two edge-truncating polyhedra, rhomb-dodecahedron and rhomb-triacontahedron (Fig.3), are the Catalan duals of the Archimedean cuboctahedron and icosi-dodecahedron respectively, in turn obtained (as shown previously) by the vertex truncation of a Platonic solid.

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

The edge truncations of cube and octahedron by a rhomb-dodecahedron are shown in Fig.4a, whereas in Fig.4b one can see the edge truncations of dodecahedron and icosahedron by a rhomb-triacontahedron.
At any stage of these edge-truncation processes it is not possible to achieve a vertex-transitive polyhedron.

 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)

TRUNCATION OF ARCHIMEDEAN SOLIDS
The two sets of three Archimedean polyhedra shown in Fig.2, which were obtained by the intersection of dual Platonic solids, can be furtherly truncated by a rhomb-dodecahedron or a rhomb-triacontahedron, depending on the symmetry of the Archimedean polyhedron to be truncated.
The most significant result of each truncation process consists in the vertex-transitive solids shown in Fig.5, in which all the vertices are shared by four faces: two rectangular faces derived from the Catalan polyhedron and two other faces, having different shapes, derived from the Archimedean polyhedron.

 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).

Conversely, no vertex-transitive polyhedron would result from the truncation process, even if the solid truncating each of the other Archimedean polyhedra listed in Fig.2 were the respective Catalan dual (Fig.6); however, although not equivalent, the vertices of each resulting polyhedron are all equidistant from its centre.

 COMPOSITE POLYHEDRA RESULTING FROM THE VERTEX-TRUNCATION OF AN ARCHIMEDEAN SOLID BY ITS CATALAN DUAL

 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)

However, even when the truncating polyhedron is a rhomb-dodecahedron or a rhomb-triacontahedron, the vertex-transitivity of the resulting solid is preserved only in the initial stage of each truncation process, till to the solid shown in Fig.5: afterwards, the vertices of the polyhedra resulting from the truncations are no longer equivalent by symmetry.
For instance, in Fig.7a one can see the sequence relative to the complete truncation of the cuboctahedron by a rhomb-dodecahedron: all the solids of the first two rows are vertex-transitive, whereas non-equivalent vertices of two different kinds characterize the solids of the third row.
The face transitive duals of the series of vertex-transitive polyhedra go from the rhomb-dodecahedron {110} to the deltoid-icositetrahedron {211}, passing through a continuous series of intermediate hexakis-octahedra (also called disdyakis-dodecahedra).

 Sequence of vertex-truncations of the cuboctahedron by a rhomb-dodecahedron  (dcube= √2/2,  doctahedron= √6/3  and  1/2 ≤ drhomb-dodecahedron ≤ 1) drhomb-dodecahedron = 1 The faces of the rhomb-dodecahedron are tangent to the twelve vertices of the cuboctahedron drhomb-dodecahedron = 0.9 drhomb-dodecahedron = 0.8536 = (√2+2)/4 The faces deriving from the cube have the shape of a regular octagon drhomb-dodecahedron = 0.8333 = 5/6 The faces deriving from the octahedron have the shape of a regular hexagon drhomb-dodecahedron = 0.8 drhomb-dodecahedron = 3/4 Each vertex is shared among four faces drhomb-dodecahedron = 0.72 drhomb-dodecahedron = 0.6   Vanishing of the faces of octahedron drhomb-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.

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) (ddodecahedron= 1/(√1+(1/τ2),  dicosahedron= τ/√3  and  1/(1+1/τ2) ≤ dRT ≤ 1) dRT = 1 The faces of  rhomb-triacontahedron are tangent to the thirty vertices of the icosi-dodecahedron dRT = 0.97 dRT = (4τ+3)/10 = 0.9472 The faces of dodecahedron assume the shape of a regular decagon dRT = (τ+4)/6 = 0.9363 The faces of icosahedron have the shape of a regular hexagon dRT = 0.92 dRT = (τ2+1)/4 = 0.9045 Each vertex is shared among four faces dRT = 0.89 dRT = τ2/3 = 0.8727 Vanishing of the faces of icosahedron dRT = 1/(1+1/τ2)= 0.7236The 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 /dcube or ddodecahedron /dicosahedron remain constant along each of all these truncation processes.

 doctahedron /dcube= √3/(3-√2) = 1.0922 doctahedron /dcube = 1.0922  dcube /drhomb-dodecahedron = 0.8673  doctahedron /drhomb-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 doctahedron / dcube= √3/(3-√2).

 ddodecahedron /dicosahedron = √(3-τ)3/3 = 0.9380 ddodecahedron /dicosahedron = 0.9380  dicosahedron /drhomb-triacontahedron = 0.9819  ddodecahedron /drhomb-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 ddodecahedron / dicosahedron = √(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.

 doctahedron /dcube = (2√2 -1)/√3 = 1.0556 doctahedron /dcube = 1.0556  dcube /drhomb-dodecahedron = 1.0  doctahedron /drhomb-dodecahedron = 1.0556 FIG. 9a - Edge-truncation by a rhomb-dodecahedron of a non-Archimedean truncated octahedron, slightly different from the previous one (ratio doctahedron / dcube = 1.0556), leading to the rhomb-cuboctahedron.

 ddodecahedron/dicosahedron = 3√3 / [(3+τ)√3-τ] = 0.9571 ddodecahedron /dicosahedron= 0.9571  dicosahedron /drhomb-triacontahedron= 1.0184  ddodecahedron /drhomb-triacontahedron= 0.9748 FIG. 9b - Edge-truncation by a rhomb-triacontahedron of a non-Archimedean truncated icosahedron, slightly different from the previous one (ratio ddodecahedron /dicosahedron = 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).

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