Abstract
Cube and its dual Octahedron exist in any multi-dimensional space; as shapes, they compose multi-shell clusters of octahedral symmetry (resulted by operations on maps) and crystal networks. A particular attention was given to clusters decorated with octahedra and dodecahedra, respectively.
Cube is the only Platonic solid that can tessellate the 3D space. This chapter was focused on two space fillers: the cube C and the rhombic dodecahedron, Rh12 (i.e. d(mC).14, or dual of cuboctahedron) and to their networks, derived by map operations, like dual, medial, truncation or leapfrog. The clusters and networks were characterized by figure count, ring signature and centrality index. An atlas section illustrates the discussed multi-shell polyhedral clusters and triple periodic structures, respectively.
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Chapter 7 Atlas: Octahedral Clusters
Chapter 7 Atlas: Octahedral Clusters
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C.8 | O.6 | OP.7 |
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O.6 | C.8 | CP.9 |
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C.8 | A4.8 | CP.9 |
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mA4.16 | mCP.20 | d(mCP).56_4 |
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mA4.16 | mCP.20 | m(mCP).60_4 |
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TC.24 | mCP.20 | t(mCP).120_4 |
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C.8 | TO.24 | CP.9 |
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O.6 | C.8 | CP.9 |
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C.8 | CO.12 | O@CO.18 |
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C8 | CO.12 | O@CO.18 |
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O.6 | TO.24 | O@CO.18=mOP.18 |
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O@3O.15 | dCO@CO.26 | COP@dCO.27 |
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O@3O.15 | CO@dCO.26 | m(dCO@CO).96_4 |
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CO.12 | O@3O.15 | CO@dCO.26 |
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O@3O.15 | dCO@COP.27 | CO@dCO.26 |
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COP@dCO.27 P12@CO@dCO.27 | C108X.60 | m(CO@dCO27).96 |
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t sel (P4(C)).44 | s 2(C).56 | t sel (P4(C))@(8D).100 |
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D.20 | t sel (P4(C)).44 | t sel (P4(C))@(8D).100 |
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t sel (P4(C))@(8D).100_2 | t sel (P4(C))@(8D).100_3 | t sel (P4(C))@(8D).100_4 |
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d(C100).84_2 | d(C100).84_3 | t sel (P4(C))@(8D).100_4 |
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m(C100).180_2 | m(C100).180_3 | t sel (P4(C))@(8D).100 |
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dm(C100).244_2 | dm(C100).244_3 | Rh30.32 |
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t(C100).360_2 | t(C100).360_3 | t(C100).360_4 |
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l(C100).420_2 | l(C100).420_3 | l(C100).420_4 |
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C16 (8(4.5.5).8(5.5.5)) | C.20 | t sel (P4(C))@(8D).100_4 |
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t(C108)X.184 | t(C100).360 | C108 |
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C108 | C400 | C184 |
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l(C108).528X.204 | l(C100).420_4 | l(C108).528_4 |
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m(Q 4)X.20 | m(Q 4)X.20 | m(Q 4).32 |
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dm(Q 4)X.50a (r) Rh12@(12mP3;8T).50 C2 × S4; Order 48; |{6};{8};{12};{24}| | dm(Q 4)X.50a (b) Rh12@(12mP3;8T).50 C2 × S4; Order 48; |{6};{8};{12};{24}| | dm(Q 4).88_3 |
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d(m(Q 4)88)X.50 C50a | (d(mQ 4)88)X50.222.310 C50a.222.310 | d(m(C24)84)166X50.222.310 C50b.222.310 |
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O@14O.24 C2 × S4 Classes: |2{6};{12}| | d(m(C24)84).166X.50 Rh12@12mP3.50 | d(m(C24)84).166 |
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Rh12@12mP3.50 C50b | d(m(C24)84)166X50.222.310 C50b.222.310 | (d(mQ 4)88)X50.222.310 C50a.222.310 |
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Rh24@24mP3.98 C50b co-net | Rh12@12mP3.50_3 | Rh12@12mP3.50 C50b net |
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Rh12@12mP3.50 | Rh24@24mP3.98_3 | Rh24@24mP3.98 |
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C.8 | O.6 | C.222.27 |
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C.222.27 | O.6 | CO.12 |
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C.222.27 | O.6 | TC.24 |
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l(C.222).144 | O.6 | TO.24 |
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C.8 | CO.12 | RCO.24= mmC.24 |
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RCO.24= mmC.24 | CO.12 | C.8 |
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TCO.48 | TO.24 | C.8 |
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TCO.48 | TO.24 | C.8 |
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DSCO=d(stCO).48 | TC.24 | CP.9 |
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DSCO=d(stCO).48 | TC.24 | CP.222.35 |
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Rh12@12Rh12.94 | Rh12.14 = d(CO).14 [4^12] | l((Rh12@12Rh12).480 |
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RCO.24= mmC.24 | C.8 | T.4 |
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RCO.24= mmC.24 | C.8 | T.4 |
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Rh12.14 = p 4(T).14 d(mC).14 = d(CO).14 | TCO.48 | l((DCO).333.324 co-net |
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TCO.48 [4^12.6^8.8^6] | TC.24 [3^8.8^6] | TT.12 [3^4.6^4] |
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Diudea, M.V. (2018). Clusters of Octahedral Symmetry. In: Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-64123-2_7
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