Skip to main content
SpringerLink
Log in

Small Icosahedral Clusters

  • Chapter
  • First Online:
Multi-shell Polyhedral Clusters

Part of the book series: Carbon Materials: Chemistry and Physics ((CMCP,volume 10))

  • 561 Accesses

Abstract

In geometry, any polyhedron with 12 faces is named a dodecahedron, among which only one is the regular dodecahedron (i.e. the Platonic solid), composed of 12 regular pentagonal faces, 3 of which meeting at each vertex; it has the Schläfli symbol {5,3} and icosahedral (point group) symmetry, I h . The dual of a dodecahedron is an icosahedron, referring to shapes, if one disregards the angles and bond length, rather than to regular polyhedra. The fifth chapter shows the transforming, by map operations, of small seeds, like “point centered polyhedra” and “cell-in-cell”, into more complex multi-shell clusters, of rank 4 or 5. Among the transformed polyhedra, a special attention was given to rhombic polyhedra, obtained by the sequence d(m(P)) (i.e. dual of medial polyhedra). An atlas section illustrates the discussed multi-shell polyhedral clusters.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Coxeter HSM (1973) Regular polytopes, London, 1948. Dover, New York, NY

    Google Scholar 

  • Diudea MV, Nagy CL (2007) Periodic nanostructures. Springer, Dordrecht

    Book  Google Scholar 

  • Diudea MV, Ştefu M, John PE, Graovac A (2006) Generalized operations on maps. Croat Chem Acta 79:355–362

    CAS  Google Scholar 

  • Euler L (1752–1753) Elementa doctrinae solidorum. Novi Comment Acad Sci I Petropolitanae 4:109–160

    Google Scholar 

  • Johnson NW (1966) Convex solids with regular faces. Can J Math 18:169–200

    Article  Google Scholar 

  • Nagy CL, Diudea MV (2009) Nano Studio. Univ Babes–Bolyai, Cluj

    Google Scholar 

  • Nagy CL, Diudea MV (2017) Ring signature index. MATCH Commun Math Comput Chem 77(2):479–492

    Google Scholar 

  • Papacostea C (1930–1935) P1aton. Opere, 1–11 (Roum. translation), Casa Şcoalelor, Bucureşti

    Google Scholar 

  • Parvan-Moldovan A, Diudea MV (2015) Cell@cell higher dimensional structures. Stud Univ Babes-Bolyai Chem 60(2):379–388

    Google Scholar 

  • Pisanski T, Randić M (2000) Bridges between geometry and graph theory, geometry at work. MAA Notes 53:174–194

    Google Scholar 

  • Schläfli L (1901) Theorie der vielfachen Kontinuität Zürcher und Furrer, Zürich (Reprinted in: Ludwig Schläfli, 1814–1895, Gesammelte Mathematische Abhandlungen, Band 1, 167–387, Verlag Birkhäuser, Basel, 1950)

    Google Scholar 

  • Schulte E (1985) Regular incidence-polytopes with Euclidean or toroidal faces and vertex-figures. J Combin Theory Ser A 40(2):305–330

    Article  Google Scholar 

  • Schulte E (2014) Polyhedra, complexes, nets and symmetry. Acta Cryst A 70:203–216

    Article  CAS  Google Scholar 

  • Taylor AE (1928) A commentary on Plato’s Timaeus. Clarendon, Oxford

    Google Scholar 

  • Wikipedia (2016) https://en.wikipedia.org/wiki/Pentagonal_gyrobicupola. Accessed 12 Dec 2016

  • Wolfram Research, Inc. (2016) Mathematica, Version 10.4. Champaign, IL

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemistry, Faculty of Chemistry and Chemical Engineering, Babes-Bolyai University, Cluj-Napoca, Romania

    Mircea Vasile Diudea

Authors
  1. Mircea Vasile Diudea
    View author publications

    You can also search for this author in PubMed Google Scholar

Chapter 5 Atlas: Small Icosahedral Clusters

Chapter 5 Atlas: Small Icosahedral Clusters

figure a

I.20

Py5.6

P@D.21

figure e

D.20

T.4

IP.13

P@T20@I.13

figure i

I@D.32_2

I@D.32_3

I@D.32_5

figure m

IP@D.33_2

IP@D.33_3

I@D.32_5

figure q

I.12

ID.30 = mI.30

mIP.42_5

figure u

IP.13

ID.30

I@ID.42

figure y

D@ID.50_2

D@ID.50_3

ID@D.50_5

figure ac

D.20

I.12

mD.30 = ID.30 = d(Rh30).30

figure ag

D.20

tPy5.20

DP.21

figure ak

I.12

tPy3.12 = TT.12

IP.13

figure ao

D@ID.50

ID@C80.110

C50@C110.130_5

d(C84).130

figure as

cd(C84).21&C84_2

cd(C84).21&C84_3

cd(C84).21&C84_5

figure aw

ID@C80.110_2

ID@C80.110_3

T@3T.7

figure ba

l(IP).150_2

l(IP).150_3

IP.13

P@T20@I.13

figure be

D@tD.80

D@(12hmA5;20T)@tD.80

C60@tD.120

C60@(12hmA5;20hCO;30T)@tD.120

d(I@D32).140_5

figure bi

A5.10

ID.30

I@D.32

figure bm

I.12

ID.30

m(I@D32).120

figure bq

A5.10

TT.12

I@D.32

figure bu

TT@3TT.39

TT.12 = tT.12

I@D.32

figure by

I@ID.42

m(IaD32).120

P12@I@D.33

figure cc

I.12

m(IP@D33)132

ID@20CO.150

figure cg

I.12

m(IP@D33)132

ID@20CO.150

figure ck

I.12

m(IP@D33)132

ID@20CO.150

figure co

d(510).870_2

d(510).870_3

dCO.14 = Rh12.14

figure cs

t(IP@D33).264

TT.12

I@D.33

figure cw

TT@3TT.39

TT.12 = tT.12

IP@D.33

figure da

C

mP5.15

I@ID.42

figure de

ID.30

CO.12

I@ID.42

figure di

ID.30

CO.12

I@ID.42

figure dm
figure dn

m2(C).24

I@ID.42

m(I@ID42).150

figure dr

P5.10

TO.24

mIP.42

figure dv

P5.10

C24 = TO

C60

figure dz

d(IP@ID).160_2

d(IP@ID).160_3

IP@ID.43

figure ed

ID@20CO.150

I@ID.42

IP@ID.43

figure eh

C84

C300

IP@ID.43

figure el

C114

I@(12(Rh10);20mP3).114

C122

(Rh30)@(20T;30mP3).122

d(D@ID50).134_5

figure ep

m(D@ID50).150_5a

m(D@ID50).150_5b

mA5

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Cite this chapter

Diudea, M.V. (2018). Small Icosahedral Clusters. In: Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-64123-2_5

Download citation

Publish with us

Policies and ethics

Search

Navigation