Breaking of Euclidean Symmetry with an Application to the Theory of Crystallization

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

Alternative approaches to onion-like icosahedral fullerenes

2014 ◽  
Vol 70 (2) ◽  
pp. 168-180 ◽  
Author(s):  
A. Janner
Keyword(s):  
Density Functional Theory ◽  
Density Functional ◽  
Structural Constraints ◽  
Functional Theory ◽  
Carbon Onion ◽  
Golden Number ◽  
Affine Extension ◽  
Alternative Approaches ◽  
Euclidean Symmetry ◽  
Structural Relations

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.

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

Leonardo ◽  
1979 ◽  
Vol 12 (1) ◽  
pp. 19 ◽  
Author(s):  
H. S. M. Coxeter
Keyword(s):  
Euclidean Symmetry ◽  
Circle Limit

Mobius Equivalence and Euclidean Symmetry

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

scholarly journals Affine-Distance symmetry sets

2003 ◽  
Vol 93 (2) ◽  
pp. 247
Author(s):  
P. J. Giblin ◽  
P. A. Holtom
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
Convex Curve ◽  
Euclidean Case ◽  
Generic Structure ◽  
Symmetry Set ◽  
Convex Curves ◽  
Euclidean Symmetry

The affine distance symmetry set (ADSS) of a plane curve is an affinely invariant analogue of the euclidean symmetry set (SS) [7], [6]. We list all transitions on the ADSS for generic 1-parameter families of plane curves. We show that for generic convex curves the possible transitions coincide with those for the SS but for generic non-convex curves, further transitions occur which are generic in 1-parameter families of bifurcation sets, but are impossible in the euclidean case. For a non-convex curve there are also additional local forms and transitions which do not fit into the generic structure of bifurcation sets at all. We give computational and experimental details of these.

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