The Non-Euclidean Symmetry of Escher's Picture 'Circle Limit III'

Leonardo ◽  
1979 ◽  
Vol 12 (1) ◽  
pp. 19 ◽  
Author(s):  
H. S. M. Coxeter
2003 ◽  
pp. 166-174
Author(s):  
Helaman Ferguson ◽  
Claire Ferguson
Keyword(s):  

2014 ◽  
Vol 70 (2) ◽  
pp. 168-180 ◽  
Author(s):  
A. Janner

The fullerenes of the C60series (C60, C240, C540, C960, C1500, C2160etc.) form onion-like shells with icosahedralIhsymmetry. Up to C2160, their geometry has been optimized by Dunlap & Zope from computations according to the analytic density-functional theory and shown by Wardman to obey structural constraints derived from an affine-extendedIhgroup. In this paper, these approaches are compared with models based on crystallographic scaling transformations. To start with, it is shown that the 56 symmetry-inequivalent computed carbon positions, approximated by the corresponding ones in the models, are mutually related by crystallographic scalings. This result is consistent with Wardman's remark that the affine-extension approach simultaneously models different shells of a carbon onion. From the regularities observed in the fullerene models derived from scaling, an icosahedral infinite C60onion molecule is defined, with shells consisting of all successive fullerenes of the C60series. The structural relations between the C60onion and graphite lead to a one-parameter model with the same Euclidean symmetryP63mcas graphite and having ac/a= τ2ratio, where τ = 1.618… is the golden number. This ratio approximates (up to a 4% discrepancy) the value observed in graphite. A number of tables and figures illustrate successive steps of the present investigation.


1970 ◽  
Vol 11 (5) ◽  
pp. 1655-1668 ◽  
Author(s):  
Gérard G. Emch ◽  
Hubert J. F. Knops ◽  
Edward J. Verboven
Keyword(s):  

1984 ◽  
Vol 91 (4) ◽  
pp. 225
Author(s):  
J. B. Wilker
Keyword(s):  

Sign in / Sign up

Export Citation Format

Share Document