Jesu juva
Interactive 3D Space Group Visualization with CLUCalc based on Geometric Algebra www.spacegroup.info AGACSE 3, Grimma, Leipzig, Germany 19 August 2008
Christian Perwass (Bosch/Germany) Eckhard Hitzer (Fukui/Japan) vectors Æ geometric product Æpointgroup Æ spacegroups Æ CLUCalc Æ SGV = crystal class
Acknowledgements z C. Perwass, Univ. Kiel / Bosch z D. Hestenes & J. Holt z Students: D. Ichikawa, M. Sakai, K. Yamamoto, Univ. of Fukui, GA group z M. Aroyo, Univ. of Bilbao, N. Ashcroft, IUCr Chester, H. Wondratschek, Univ. of Karlsruhe z Organizers of AGACSE 3!
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Google Maps
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Acknowledgements But where can wisdom be found? Where does understanding dwell? Man does not comprehend its worth; it cannot be found in the land of the living. … Neither gold nor crystal can compare with it, nor can it be had for jewels of gold.
Background
Job 28:12-17, Bible, New International Version
< I do thank my family < E. Hitzer
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Contents z Introduction z Geometric Algebra Approach z Point Groups z 3D Space Groups {Geometry {Example z Implementation z Conclusions z More Information 19 Aug. 2008
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Geometric Approach to Symmetry - Reflections & Rotations -
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Reflection at plane reverse component perp. to plane
x’
X||
−X||
x
x a
normal to plane
x’=−a xa −1 a = a/a 2 −1
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2 Reflections Æ 1 Rotation x’’
x’
Angles: α x,x’’ = 2 α a,b
x b
a
x’ =− a−1 x a −1 −1 −1 x’’ = b a x ab = (ab) x ab 19 Aug. 2008
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Rotary-Reflection −1
x’ = − (abc) x abc
Inversion i −1
-1
x’ = − (abc) x abc = − i x i = − x a b c a, e.g. e1 , e2 , e3
Rotary-Inversion …
ÎAll transformations of 2D and 3D crystal cell point groups E. Cartan 1920ies, Coxeter 1934, Coxeter & Moser 1957, Hestenes 2002 GA representation 19 Aug. 2008
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10 2D Point Groups
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2D Point Groups
Image: SnowCrystals.com
regular polygons (n=2,3,4,6)
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select
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2 vectors from each reg. polygon
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2D Point Groups
regular polygons (n=2,3,4,6)
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2 vectors from each reg. polygon generate its 2D point groups 11
2D Point Group Generation n=6 zReflections
a,b,aR2,bR2,aR4,bR4 z60º Rotations
R=ab,R2,R3,R4,R5,R6 = −1
Representations: Tables 1,2 of ICNAAM 2005 proc., pp. 939,940 19 Aug. 2008
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32 3D Point groups
- Crystal classes -
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Seven types of space lattices with elementary cells
zTriclinic zMonoclinic zOrthorhombic zTetragonal zTrigonal zHexagonal zCubic 19 Aug. 2008
select
3 vectors from each cell generate all 32 point groups! www.spacegroup.info E. Hitzer, C. Perwass
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3 vectors a, b, c selected from each crystal cell:
triclinic monoclinic
orthorhombic
tetragonal
hexagonal 19 Aug. 2008
trigonal
www.spacegroup.info E. Hitzer, C. Perwass
15 cubic (isometric)
Hestenes and Holt’s Geometric Notation Isomorphic to Table 2 of Coxeter & Moser, 4th ed. 1980
Generators
a a,b ab a,b,c ab , c a , bc ab , bc abc
(ab)p = (bc)q = (ac)2 = −1 ; p,q ∈{1,2,3,4,6}
reflections rotation
c π/q b
)
Point Group Symbol p=1 p≠1 p pq pq pq pq pq
roto reflection = roto inversion
π/2
)π/p a
sufficient
+ Bravais prefix + glide & screw indexes Æ GA space group symbols 19 Aug. 2008
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Crystal Classes = Point Groups
* * *
* * *
* *
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*
polar (pyroelectric)
*
* www.spacegroup.info E. Hitzer, C. Perwass
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7 Holohedric Point Groups - highest order -
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Holohedry
Hexagonal Point Group 62 geometric z 6 reflections a, b, aR2, bR2, aR4, bR4 z 6 rots. (60º,120º,180º,240º,300º,360º) R=ab, R2 , R3 , R4 , R5 ,1 z 6 rotary-reflections c, cR, cR2 , cR3 = i, cR4, cR5 z six 180º rotations ac, bc, acR2, bcR2,acR4, bcR4 Total: 24 symmetry transformations 19 Aug. 2008
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Beryl
SchoolofGemology.com 19
スリーディー
230 3D Spacegroups
- Crystallographic Symmetry -
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What is a Spacegroup ? Reflections (Rotations, …) a, b, c and Translations Tka, Tlb, Tmc, k,l,m ∈ Q fractions
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Conformal Model of Euclidean space (double projective) in Cl(4,1) Ref.: Wachter (Horosphere), Lounesto 1978, Angles 1980, Hestenes,Li,Rockwood 2001, Hitzer 2004,2005 for including translations origin
e0 = 0 2
e0
5D
X = x + 1/2 x 2 e∞ +e0 projective
infinity
e∞ e∞ 2 = 0 19 Aug. 2008
{e1 , e2 , e3}⊥ e0,e∞
R3 X 2 = 0, X ・ Y = − (x − y)2 / 2
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Conformal 5D model of Euclidean space in Cl(4,1) Vectors represent points X,P,Q and (mid) planes m,r mr … generates general rotations & translations Q
P
m
r t
X'
X
mid plane X・(P-Q)=0
m
gen. rotation
t /2
Tt = mr
gen. translation
Conformal Conformalgroup groupC(3) C(3)== Orthogonal Orthogonalgroup groupO(4,1) O(4,1) Eucl. Eucl.group groupE(3) E(3)==subgroup subgroupofofO(4,1) O(4,1)leaving leavinginfinity infinityee∞∞invariant. invariant. Use UseofofVersor Versor(Clifford, (Clifford,Lipschitz) Lipschitz)group grouprepresentation. representation. 19 Aug. 2008
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3D Spacegroups - Notation zTranslator (by t) X T -1 X T Tt = exp(te∞/2) = 1 + te∞/2 zAlways: Ta, Tb, Tc zEx: Glide reflection aTb/2 , a丄b
z Bravais: P, I, H, F, R (A,C,B side cent.) Æ means extra translations z Locating symmetries: T-t generator Tt 19 Aug. 2008
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P221 multivector generator products Æ symmetries n sio r e inv
screw axis
i= a∧bc cTc/2 a∧bTc/2
reflection plane example:
inversion center general 19 Aug. 2008 element
Kinoite http://webmineral.com/data/Kinoite.shtml
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Interactive Visualization - software -
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Interactive Software Implementation zClifford Geometric Algebra zSymmetry definitions: XML zVisual multivector software CLUCalc (OpenGL graphics) www.clucalc.info zSpace Group Visualizer Demo 2.0 free 8 group demo www.spacegroup.info zIUCr Expert Review version (18 Aug. 2008) 19 Aug. 2008
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Space Group Visualizer Quick Guide
User Interface
Space SpaceGroup Group Select ion Selection
Menu Menu
Toolbar Toolbar
Visualized Visualized Space SpaceGroup Group Name Name
Rotation Rotation Center Center Selector Selector
Symmetry Symmetry Select ion Selection
Visualization Visualization Window Window
Locus Locus Selector Selector Drag Dragborder border to tochange change window windowsize size
Info Info Window Window
Context Context Help Help
Experience Nature's Symmetries Powered by CLUCalc 5
Developed by Dr. Ch. Perwass & Prof. Dr. E. Hitzer
Space Group Visualizer Quick Guide The Toolbar Click Click to to show/hide show/hide loci loci
Vary Vary size size of of loci loci
Click Click to to show/hide show/hide cell cell grid grid
Vary Vary angles angles between between basis basis vectors vectors
Click Click to to show/hide show/hide generator generator basis basis vectors vectors
Vary Vary number number of of displayed displayed cells cells in in three three space space directions directions
Vary Vary length length of of three basis three basis vectors vectors
Vary Vary size size of of reflection reflection and and glide-reflection glide-reflection planes planes
Depending on the displayed space group, basis vector lengths and/or angles may not be changed. The corresponding toolbar elements are displayed in gray in this case.
Experience Nature's Symmetries Powered by CLUCalc 5
Developed by Dr. Ch. Perwass & Prof. Dr. E. Hitzer
Space Group Visualizer Quick Guide Selecting a Space Group
Just as with an HTML browser, all blue text elements are links. That is, you can click on them to obtain the named result.
Click Click here here Click Click here here on '28 on '28'' Click Click here here on ' Cubic' on 'Cubic'
Select Crystal System
Select Point Group
Click Click here here on on '195 '195''
Select Space Group
Space Group Selected
Experience Nature's Symmetries Powered by CLUCalc 5
Developed by Dr. Ch. Perwass & Prof. Dr. E. Hitzer
Space Group Visualizer Quick Guide Selecting Symmetries
Click Click here here on on 'only 'only''
Below 'Symmetries' all symmetries present in the current group are listed. The symmetries that are to be displayed can be selected by their properties. If a number of properties are selected, only those symmetries that satisfy all of them are displayed. Click Click here here on on 'hide' 'hide' to to hide hide available available angle angle Click Click here here Click properties Click here here on on properties again again on on 'select 'select'' '120°' to toggle '120°' to toggle display display of of 120° 120° rotation rotation symmetries symmetries
The meaning of the various commands is the same at all levels of the symmetry selection menu.
Click Click here here on on 'only' 'only' to to only only show show 120° 120° rotation rotation symmetries symmetries
Click Click here here on on 'other' 'other' to show all rotation to show all rotation symmetries symmetries apart apart from 120° rotations from 120° rotations
Experience Nature's Symmetries Powered by CLUCalc 5
Developed by Dr. Ch. Perwass & Prof. Dr. E. Hitzer
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ITA online Ù SGV interaction
Group Selection online and in SGV simultaneously
Tables vs. Computer Graphics
Conclusions
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Conclusions z z z
z z z z z
Geometric Algebra with Conformal Model of R3 represents 2D,3D Pointgroups and Spacegroups Generation only by physical vectors of cell (lattice) Interactive software Space Group Visualizer Future. Combination with International Tables of Crystallography, A? Extraordinary spacegroup orbits Subperiodic/magnetic spacegroups Molecule/Ion group toolbox Æ Physical crystal + full symmetry Combination with protein visualizer? 19 Aug. 2008
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More Information - internet, literature -
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More Information z Geometric Calculus Fukui http://sinai.mech.fukui-u.ac.jp/gcj/gcjportal.html
z Cognitive Systems Group (Kiel) http://www.ks.informatik.uni-kiel.de/
z Visualization software
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3D Interactive Point Group Visualization free download from
www. spacegroup.info
Generate all 32 point groups by clicking successive reflections!
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Literature - general z D. Hestenes, Point Groups and Space Groups in GA in L. Dorst, C. Doran, J. Lasenby (eds.), Appl. of GA in Comp. Sc. and Eng., Birkhaeuser, 2002. z D. Hestenes, J. Holt, The Cyrstallographic Space Groups in GA, JMP 48, 023514, 2007. z J.D.M. Gutierrez, Operaciones de simitria mediante algebra geometrica aplicadas a grupos cristalograficos, Tesis, UNAM, Mexico 1996. z T. Hahn, Int. Tables of Crystallography, Vol. A Springer, 2005.
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Literature – E. Hitzer / C. Perwass Geometric Algebra background Crystal Cell and Space Lattice Symmetries in Clifford Geometric Algebra, in TE. Simos et. al. (eds.), ICNAAM 2005, Wiley-VCH, Weinheim (2005), pp. 937-941. Full Geometric Description of All Symmetry Elements of Crystal Space Groups by the Suitable Choice of Only Three Vectors for Each Bravais Cell or Crystal Family, Proc. of ISAME 2005, pp. 19-25. Bulletin of the Society for Science on Form, Vol. 20(1) (2005), pp. 105,106. Bulletin of the Society for Science on Form, Vol. 21(1) (2006), pp. 55,56. Point Group Visualizer (theory and use) Crystal Cells in Geometric Algebra, Proc. of ISAME 2004, (2004), pp. 290-295. Bulletin of the Society for Science on Form, Vol. 20(1) (2005), p. 128. Space Group Visualizer (theory and use) Crystallographic space groups: representation and interactive visualization by geometric algebra, submitted to Proc. of 26th ICGTMP, S. Catto, ed., New York, June 2006. Interactive Visualization of Full Geometric Description of Cyrstal Space Groups, Proc. of ISAME 2005, pp. 276-282. Bulletin of the Society for Science on Form, Vol. 21(1) (2006), pp. 38,39. The hidden Beauty of Gold, ISMAPE 2007, pp. 157-167. Sym. of orthor. materials & interact. 3D visualization in GA, ISMAPE 2007, pp. 302-312 19 Aug. 2008
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Online available from: http://sinai.mech.fukui-u.ac.jp/gcj/pubs.html
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Literature on Conformal Model z P. Lounesto, E. Latvamaa, Conformal Transformations and Clifford Algebras, Proc. of The AMS, Vol. 79, No. 4, pp. 533-538 (1980). z P. Angles, Construction de revetements du groupe conforme d'un espace vectoriel muni d'une metrique de type (p,q), Ann. de l'I.H.P., Sect. A, Vol. 33, No. 1, pp. 33-51 (1980). z A.W.M. Dress, T.F. Havel, Distance geometry and GA, Found. Phys., Vol. 23, pp. 1357-1374. z D. Hestenes, H. Li, A. Rockwood, New Algebraic Tools for Classical Geometry, in G. Sommer (ed.), Geometric Computing with Clifford Algebras, Springer, Berlin, pp. 4-26 (2001). z E. Hitzer, Euclidean Geometric Objects in the Clifford Geometric Algebra of {Origin, 3Space, Infinity}, Bulletin of the Belgian Mathematical Society – Simon Stevin, Vol. 11, No. 5, pp. 653-662 (2004). z E. Hitzer, Conic Sections and Meet Intersections in Geometric Algebra, Computer Algebra and Geometric Algebra with Applications, GIAE 2004, Revised Selected Papers, Springer, Lecture Notes in Computer Science, 3519, pp. 350-362 (2005). z Hongbo Li, Invariant Algebras and Geometric Reasoning, ISBN 978-981-270-808-3, Price: US$ 98, 532 pp., (Mar 2008), Homepage: http://www.worldscibooks.com/mathematics/6514.html
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GA-Net, GA-Net Updates zGA-Net email newsletter http://sinai.mech.fukui-u.ac.jp/GA-Net/ zGA-Net Updates (blog) http://gaupdate.wordpress.com/ 19 Aug. 2008
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Soli Deo Gloria
end - fin
www.spacegroup.info To God alone be the glory E. Hitzer, C. Perwass
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J.S. Bach, Leipzig
Image source: Google Images
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