Affine isomorphism - Revision history

BrianMcMahon: Added Spanish translation (U. Mueller)

2017-11-08T17:50:32Z

Added Spanish translation (U. Mueller)

BrianMcMahon: Added German translation (U. Mueller)

2017-11-08T17:48:06Z

Added German translation (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-12T11:10:07Z

Style edits to align with printed edition

MassimoNespolo: /* See also */

2007-04-23T14:02:17Z

‎See also

MassimoNespolo at 15:50, 22 April 2007

2007-04-22T15:50:26Z

New page

Each symmetry operation of crystallographic group in ''E''3 may be represented by a 3×3 matrix '''W''' (the ''linear part'') and a vector '''w'''. Two crystallographic groups ''G''1 = {('''W'''1''i'','''w'''1''i'')} and ''G''2 = {('''W'''2''i'','''w'''2''i'')} are called '''affine isomorphic''' is there exists a non-singular 3×3 matrix '''A''' and a vector '''a''' such that:
<div align="center">
''G''2 = {('''A''','''a''')('''W'''1''i'','''w'''1''i'')('''A''','''a''')-1}
</div>
Two crystallographic groups are affine isomorphic if and only if their arrangement of symmetry elements may be mapped onto each other by an [[affine mapping]] of ''E''3. Two affine isomorphic groups are always [[Group isomorphism|isomorphic]].

==See also==
[[Group isomorphism]]

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 17:50, 8 November 2017
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−	<font color="red">Affiner Isomorphismus</font> (''Ge'').	+	<font color="red">Affiner Isomorphismus</font> (''Ge''). <font color="green">Isomorfismo afín</font> (''Sp'').

	Each symmetry operation of a crystallographic group in ''E''<sup>3</sup> may be represented by a 3×3 matrix '''W''' (the ''linear part'') and a vector '''w'''. Two crystallographic groups ''G''<sub>1</sub> = {('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)} and ''G''<sub>2</sub> = {('''W'''<sub>2''i''</sub>,'''w'''<sub>2''i''</sub>)} are called '''affine isomorphic''' if there exists a non-singular 3×3 matrix '''A''' and a vector '''a''' such that:		Each symmetry operation of a crystallographic group in ''E''<sup>3</sup> may be represented by a 3×3 matrix '''W''' (the ''linear part'') and a vector '''w'''. Two crystallographic groups ''G''<sub>1</sub> = {('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)} and ''G''<sub>2</sub> = {('''W'''<sub>2''i''</sub>,'''w'''<sub>2''i''</sub>)} are called '''affine isomorphic''' if there exists a non-singular 3×3 matrix '''A''' and a vector '''a''' such that:

← Older revision		Revision as of 17:48, 8 November 2017
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		+	<font color="red">Affiner Isomorphismus</font> (''Ge'').
		+
	Each symmetry operation of a crystallographic group in ''E''<sup>3</sup> may be represented by a 3×3 matrix '''W''' (the ''linear part'') and a vector '''w'''. Two crystallographic groups ''G''<sub>1</sub> = {('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)} and ''G''<sub>2</sub> = {('''W'''<sub>2''i''</sub>,'''w'''<sub>2''i''</sub>)} are called '''affine isomorphic''' if there exists a non-singular 3×3 matrix '''A''' and a vector '''a''' such that:		Each symmetry operation of a crystallographic group in ''E''<sup>3</sup> may be represented by a 3×3 matrix '''W''' (the ''linear part'') and a vector '''w'''. Two crystallographic groups ''G''<sub>1</sub> = {('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)} and ''G''<sub>2</sub> = {('''W'''<sub>2''i''</sub>,'''w'''<sub>2''i''</sub>)} are called '''affine isomorphic''' if there exists a non-singular 3×3 matrix '''A''' and a vector '''a''' such that:
	<div align="center">		<div align="center">

@@ Line 1: / Line 1: @@
-Each symmetry operation of crystallographic group in ''E''<sup>3</sup> may be represented by a 3&times;3 matrix '''W''' (the ''linear part'') and a vector '''w'''. Two crystallographic groups ''G''<sub>1</sub>  = {('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)} and ''G''<sub>2</sub>  = {('''W'''<sub>2''i''</sub>,'''w'''<sub>2''i''</sub>)} are called '''affine isomorphic''' is there exists a non-singular 3&times;3 matrix '''A''' and a vector '''a''' such that:
+Each symmetry operation of a crystallographic group in ''E''<sup>3</sup> may be represented by a 3&times;3 matrix '''W''' (the ''linear part'') and a vector '''w'''. Two crystallographic groups ''G''<sub>1</sub>  = {('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)} and ''G''<sub>2</sub>  = {('''W'''<sub>2''i''</sub>,'''w'''<sub>2''i''</sub>)} are called '''affine isomorphic''' if there exists a non-singular 3&times;3 matrix '''A''' and a vector '''a''' such that:
 <div align="center">
-''G''<sub>2</sub> = {('''A''','''a''')('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)('''A''','''a''')<sup>-1</sup>}
+''G''<sub>2</sub> = {('''A''','''a''')('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)('''A''','''a''')<sup>-1</sup>}.
 </div>
 Two crystallographic groups are affine isomorphic if and only if their arrangement of symmetry elements may be mapped onto each other by an [[affine mapping]] of ''E''<sup>3</sup>. Two affine isomorphic groups are always [[Group isomorphism|isomorphic]].

@@ Line 6: / Line 6: @@
 ==See also==
-[[Group isomorphism]]
+*[[Automorphism]]
 [[Category:Fundamental crystallography]]