Vertex and edgetruncation of the Platonic and Archimedean solids leading to vertextransitive polyhedra
Livio Zefiro*
*Dip.Te.Ris, Universita' di Genova, Italy
(Email address:
livio.zefiro@fastwebnet.it)
Notes

ARCHIMEDEAN TRUNCATED CUBE  ARCHIMEDEAN TRUNCATED DODECAHEDRON 


d_{octahedron} /d_{cube}= (√2+1)/√3 = 1.3938 
d_{dodecahedron }/d_{icosahedron} = √3(τ^{2}+1)/(3τ1) = 0.8548 
CUBOCTAHEDRON 
ICOSIDODECAHEDRON 


d_{octahedron} /d_{cube}= 2/√3 = 1.1547 
d_{dodecahedron }/d_{icosahedron} = √3/√τ^{2} +1 = 0.9106 
ARCHIMEDEAN TRUNCATED OCTAHEDRON  ARCHIMEDEAN TRUNCATED ICOSAHEDRON 
d_{octahedron} /d_{cube}= √3/2 = 0.8660 
d_{dodecahedron }/d_{icosahedron} = (3+1/τ^{2})/√3(τ^{2}+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). 
RHOMBDODECAHEDRON  RHOMBTRIACONTAHEDRON 
FIG.3 Rhombdodecahedron (left) and rhombtriacontahedron (right), Catalan duals of the Archimedean cuboctahedron and icosidodecahedron respectively 
Edgetruncation of cube and octahedron by a rhombdodecahedron 

FIG.4a Sequences of the edgetruncation by a rhombdodecahedron of a cube (left) and an octahedron (right) 
Edgetruncation of dodecahedron and icosahedron by a rhombtriacontahedron 

FIG.4b Sequences of the edgetruncation by a rhombtriacontahedron of a dodecahedron (left) and an icosahedron (right) 
edgetruncation 
vertextruncation

edgetruncation

edgetruncation 
vertextruncation 
edgetruncation 
FIG.5 Vertextransitive polyhedra, duals of deltoidicositetrahedra or deltoidhexecontahedra, resulting from the intersection of cubic or icosahedral Archimedean polyhedra with the rhombdodecahedron (upper row) or the rhombtriacontahedron (lower row). 
COMPOSITE POLYHEDRA RESULTING FROM THE VERTEXTRUNCATION OF AN ARCHIMEDEAN SOLID BY ITS CATALAN DUAL 
TRUNCATED CUBE and TRIAKISOCTAHEDRON  TRUNCATED DODECAHEDRON and TRIAKISICOSAHEDRON 
FIG.6a
Intersections between couples of dual Archimedean and Catalan
polyhedra consisting in:

TRUNCATED OCTAHEDRON and TETRAKISCUBE 
TRUNCATED
ICOSAHEDRON and PENTAKISDODECAHEDRON 
FIG.6b
Intersections between couples of
dual Archimedean and Catalan polyhedra consisting in: 
Sequence of vertextruncations of the cuboctahedron
by a rhombdodecahedron 

The faces of the rhombdodecahedron are tangent to the twelve vertices of the cuboctahedron 
d_{rhombdodecahedron} = 0.9 
The faces deriving from the cube have the shape of a regular octagon 
The faces deriving from the octahedron have the shape of a regular hexagon 
d_{rhombdodecahedron} = 3/4 Each vertex is shared among four faces 

d_{rhombdodecahedron} = 0.72 
d_{rhombdodecahedron} = 0.6
Vanishing of the faces of octahedron 
d_{rhombdodecahedron} = 1/2
The only faces of rhombdodecahedron survive

FIG.7a Sequence of the complete vertextruncation by rhombdodecahedron of cuboctahedron: only the forms of the first two rows are vertextransitive. 
The completely analogous sequence relative to the
truncation of the icosidodecahedron by a rhombtriacontahedron is shown in Fig.7b.
The face transitive duals of the series of vertextransitive polyhedra go from
the rhombtriacontahedron to the deltoidhexecontahedron {τ10}, passing
through a continuous series of intermediate hexakisicosahedra (also called disdyakistriacontahedra).
Sequence of vertextruncations of the icosidodecahedron by
a rhombtriacontahedron (RT) 

The faces of rhombtriacontahedron are tangent to the thirty vertices of
the icosidodecahedron 
d_{RT} = 0.97 
The faces of dodecahedron assume the shape of a regular decagon 
The faces of icosahedron have the shape of a regular hexagon 
d_{RT}
= (τ^{2}+1)/4 = 0.9045


d_{RT} = 0.89 
d_{RT} = τ^{2}/3 = 0.8727 
d_{RT} = 1/(1+1/τ^{2})= 0.7236 The only faces of rhombtriacontahedron 
FIG.7b Sequence of the complete vertextruncation of the icosidodecahedron by a rhombtriacontahedron: only the forms reported in the first two rows are vertextransitive. 
d_{octahedron} /d_{cube}= √3/(3√2) = 1.0922 
d_{octahedron} /d_{cube}
= 1.0922 d_{cube }/d_{rhombdodecahedron} = 0.8673 d_{octahedron }/d_{rhombdodecahedron} = 0.9473 
FIG. 8a  The edgetruncation by a rhombdodecahedron 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} = √(3τ)^{3}/3 = 0.9380 
d_{dodecahedron
}
/d_{icosahedron
}
= 0.9380 
FIG.8b  The edgetruncation of a non Archimedean truncated icosahedron by a rhombtriacontahedron leads to the Archimedean truncated icosidodecahedron when the ratio d_{dodecahedron }/ d_{icosahedron }= √(3τ)^{3}/3. 
Analogously, in Fig. 9a and Fig. 9b one can see how the Archimedean
rhombcuboctahedron and
rhombicosidodecahedron derive from the
edgetruncation of two other nonArchimedean 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_{rhombdodecahedron} = 1.0 d_{octahedron }/d_{rhombdodecahedron} = 1.0556 
FIG. 9a  Edgetruncation by a rhombdodecahedron of a nonArchimedean truncated octahedron, slightly different from the previous one (ratio d_{octahedron }/ d_{cube} = 1.0556), leading to the rhombcuboctahedron. 
d_{dodecahedron}/d_{icosahedron} = 3√3 / [(3+τ)√3τ] = 0.9571 
d_{dodecahedron }/d_{icosahedron}= 0.9571 
FIG. 9b  Edgetruncation by a rhombtriacontahedron of a nonArchimedean truncated icosahedron, slightly different from the previous one (ratio d_{dodecahedron }/d_{icosahedron} = 0.9571), leading to the rhombicosidodecahedron. 
Archimedean

truncation
by rhombdodecahedron 
Archimedean rhombcuboctahedron 
Archimedean 
truncation by
rhombtriacontahedron 
Archimedean 
FIG. 10 The further truncation of both Archimedean truncated cuboctahedron and truncated icosidodecahedron (on the left) leads to the vertextransitive polyhedra shown in the central images, which can be compared with the isomorphic rhombcuboctahedron and rhombicosidodecahedron (on the right). 
tC  CO  tO  tCO  RCO 
FIG. 11 
Sequences of vertextransitive polyhedra deriving from the truncation by a rhombdodecahedron of:
A further decrease of the central distance of the intersecting rhombdodecahedron transforms the Archimedean truncated cuboctahedron in a vertex transitive polyhedron whose dual, analogously to the other four cases, is a facetransitive deltoidicositetrahedron. 
tD  ID  tI  tID  RID 
FIG. 12 
Sequences of vertextransitive polyhedra deriving from the truncation by a rhombtriacontahedron of:
A further decrease of the central distance of the intersecting rhombtriacontahedron transforms the truncated icosidodecahedron in a vertex transitive polyhedron whose dual, analogously to the other four cases, is a facetransitive deltoidhexecontahedron. 
CATALAN HEXAKISOCTAHEDRON  CATALAN HEXAKISICOSAHEDRON 
FIG.13  Facetransitive hexakisoctahedron (left) and hexakisicosahedron (right), Catalan archetypes of the duals of the series of polyhedra deriving from the truncation processes. 
CATALAN DELTOIDICOSITETRAHEDRON  CATALAN DELTOIDHEXECONTAHEDRON 
FIG.14  Catalan deltoidicositetrahedron (left) and deltoidhexecontahedron (right), archetypes of the duals of the vertextransitive 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:
FIG.16  Conway operators linking tetrahedron to the other
four Platonic solids. 
FIG. 17  Sequences of the reciprocal truncation of {111} and {111} tetrahedra, leading to the same final form, geometrically identical to an octahedron. 
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). 
FIG.19  Couple of congruent Catalan triakistetrahedra {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 triakistetrahedron such value is 50.48°. 
FIG.20  (left) Composite form, resulting from the intersection of the Archimedean truncated tetrahedron with the Catalan triakistetrahedra {311}, and (right) the relative dual form, made of a cube and the deltoiddodecahedron {221}. 
FIG. 21  Animated sequence describing the progressive edgetruncation 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 edgetruncation 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 triakistetrahedron {311} 
b) d_{{111}}= √3/3; d_{{111}}= 5√3/9; d_{{100}}= 5/6 
b') hexakistetrahedron {523} 
c) d_{{111}}= √3/3; d_{{111}}= 5√3/9; d_{{100}}= 7/9  c') hexakistetrahedron {735} 
d) d_{{111}}= √3/3; d_{{111}}= 5√3/9; d_{{100}}= 3/4  d') hexakistetrahedron {947} 
e) d_{{111}}= √3/3; d_{{111}}= 5√3/9; d_{{100}}= 2/3  e') deltoiddodecahedron {212} 
f) d_{{111}}= √3/3; d_{{111}}= 5√3/9; d_{{100}}= 3/5  f') {979} and {313} deltoiddodecahedra 
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 
h) d_{{111}}= √3/3; d_{{111}}= 5√3/9; d_{{100}}= 1/2 
h') tetrahedron {111} and rhombdodecahedron {101} 
i) d_{{111}}= √3/3; d_{{111}}= 5√3/9; d_{{100}}= 2/5 
i') tetrahedron {111} and 
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. 
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 triakistetrahedron, 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 nonregular
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 vertextransitive (or isogonal) and its dual shown in
(b') is the facetransitive (or isohedral)
hexakistetrahedron {523}, having twentyfour 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} hexakistetrahedron 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 nonregular hexagons; the dual isohedral form, shown in (d'), is the {947} hexakistetrahedron. 
Fifth row d_{{100}} = 2/3 
In (e) the
truncation originates a form, having only twelve
vertices, which is still vertextransitive: 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. 
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 vertextransitive: each vertex is shared by three faces,
two coming from the cube and the third, alternatively, from each of the two tetrahedra. 
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. 
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. 
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. 
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 pseudooctahedron, actually resulting from the intersection of the tetrahedron {111} and the tetrahedron {111}. 
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