4LetterWalksInDiamondLattice.nb

4 Letter Walks in Diamond Lattice
Based on Context Independent L–Systems
25 Feb 2004

Erik Jensen
University of California, Santa Cruz

Veikko Keränen
Rovaniemi Polytechnic, School of Technology
http://south.rotol.ramk.fi
http://south.rotol.ramk.fi/IMS_IAS_Events.html
veikko.keranen@ramk.fi

2003 Vectorgram concept by Erik Jensen
2004 Tetrahedron concept by Veikko Keränen
The support of pLAB at Rovaniemi Polytechnic is gratefully acknowledged.

You may interactively rotate and zoom the following LiveGraphics3D representations.
Just use CTRL, SHIFT, and MOUSE. Try to see loops (in case of equally many a, b, c, d letters),
symmetrices, semipalindromes, three letter projections, or even 2D walks.
The rest of the page is Mathematica code and still graphics. Enjoy!

LiveGraphics3D representations follow:

     tetrahedron     zeroSum

     g85aPath     g85aStart34     reducedg85aPath     g85to2ofa

     g98aPath     g98to2ofa

abacWalkWithForbiddenPoints

abacabaWalkWithForbiddenPoints

From here you may download the full Mathematica notebook as a zip file:

4LetterWalksInDiamondLattice.zip (zip: 1010 kB, nb: 7212 kB)

L-system basics

2D Walks

3D Walks

Starting algorithms for 3D walks

$verticesOfTetrahedronDisplacementsForabcd =  {{0, 0, 1}, {0, (2 2^(1/2))/3, -1/3}, {-2/3^(1/2), -2^(1/2)/3, -1/3}, {2/3^(1/2), -2^(1/2)/3, -1/3}} ;$

{aD, bD, cD, dD} = (verticesOfTetrahedronDisplacementsForabcd[[#1]] &)/@{1, 2, 3, 4}

${{0, 0, 1}, {0, (2 2^(1/2))/3, -1/3}, {-2/3^(1/2), -2^(1/2)/3, -1/3}, {2/3^(1/2), -2^(1/2)/3, -1/3}}$

Old graphics

New graphics (see code below)

[Graphics:HTMLFiles/4LetterWalksInDiamondLattice_104.gif]

Code

Matrices for coordinate rotations about a coordinate axis
(here the coordinate system is being rotated)

Combined rotations of a point about any axis
(component rotations are about a coordinate axis)

Designing of the heads for arrows

Arrows, our3DdisplacementForm, our3DShow, and newShow

Code for RealTime3D (newRTShow)

Code for reduced RealTime3D (reducedRTShow)

Code for producing LiveGraphics3D Java 1.1 applet

3D Walks from a-2-free strings over four letters

visualShuffleAndCodingToFourLetters

cyclicPermutation = {"a""b", "b""c", & ... ; g85c = StringReplace[g85b, cyclicPermutation] ; g85d = StringReplace[g85c, cyclicPermutation] ;

ga = "abcacdcbcdcadbdcbdbabcbdcacbabdbabcabdadcdadbdcbdbabdbcbacbcdbabdcdbdcacdbcbacbcdca ... ion] ; gd = StringReplace[gc, cyclicPermutation] ; {g98a = ga, g98b = gd, g98c = gc, g98d = gb} ;

$productions85 = {"a"g85a, "b"g85b, "c"g8 ... 8[x_] := StringReplace[x, productions98] g98iter[n_] := Characters[Nest[g98, "a", n]] ;$

abcacdcbcdcadcdbdabacabadbabcbdbcbacbcdcacbabdabacadcbcdcacdbcbacbcdcacdcbdcdadbdcbca

[Graphics:HTMLFiles/4LetterWalksInDiamondLattice_194.gif]

$SetDirectory["C:\_IMS2004\JavaApplet"] ; (* Change this *)$

-Graphics3D-

[Graphics:HTMLFiles/4LetterWalksInDiamondLattice_204.gif]

-Graphics3D-

g85 (a) = abc acdcbcdcadcdbdabacabadbabcbd bcbacbc dcacbabd abac adcb cdc acdb cbacbcdcacdcbdcd adbdcbca

RowBox[{newShow, [, RowBox[{"abc"<>"acdcbcdcadcdbdabacabadbabcbd"<>"bcb", ,, .005, ,, RowBox[{{, RowBox[{RowBox[{-, 2.7}], ,, -3, ,, 1.7}], }}]}], ]}]

[Graphics:HTMLFiles/4LetterWalksInDiamondLattice_214.gif]

RowBox[{g85aStart34, =, RowBox[{newRTShow, [, RowBox[{"abc"<>"acdcbcdcadc ... gt;"bcb", ,, .005, ,, RowBox[{{, RowBox[{RowBox[{-, 2.7}], ,, -3, ,, 1.7}], }}]}], ]}]}]

[Graphics:HTMLFiles/4LetterWalksInDiamondLattice_218.gif]

g98a = "abc acdcbcdca db dcbdbabcbdc acbabdbabca bda <br />dcdadbdcbdbabdbcbacbcdbabdc d bdcacdbcbacbcdcacdcbdcdadbd cbca"

RowBox[{newShow, [, RowBox[{"dcdadbdcbdbabdbcbacbcdbabdcdbdcacdbcbacbcdcacdcbdcdadbd", ,, 0.005, ,, 3dD}], ]}]

[Graphics:HTMLFiles/4LetterWalksInDiamondLattice_224.gif]

RowBox[{newRTShow, [, RowBox[{"dcdadbdcbdbabdbcbacbcdbabdcdbdcacdbcbacbcdcacdcbdcdadbd", ,, 0.005, ,, {2, 0, 0}}], ]}]

-Graphics3D-

The cycles in the 3D graphics below come from the grey areas
(in which letters a, b, c, d occur equally often) of

g85(a) = abc acdcbcdcadcdbdabacabadbabcbd
bcbacbc dcacbabd abac adcb cdc acdb
cbacbcdcacdcbdcd adbdcbca

g85(a) = abc acdcbcdcadcdbdabacabadbabcbd
bcbacbcdcacba bdabacadcbcd c acdb
cbacbcdcacdcbdcd adbdcbca

RowBox[{RowBox[{newRTShow, [, RowBox[{g85["a"], ,, 0.005, ,, {0, 0, 1}}], ]}], ;}] R ... 371;, RowBox[{newShow, [, RowBox[{g85["d"], ,, 0.005, ,, {0, 0, 1}}], ]}], ;}], , *)}]

-Graphics3D-

RowBox[{RowBox[{RowBox[{RowBox[{newRTShow, [, RowBox[{g98["a"], ,, 0.005, ,, {0, 0, ... RowBox[{newShow, [, RowBox[{g98["d"], ,, 0.005, ,, {0, 0, 1}}], ]}], ;}], , *)}]}], }]

-Graphics3D-

RowBox[{RowBox[{RowBox[{RowBox[{g98aPath, =, RowBox[{newShow, [, RowBox[{g98["a"], , ... RowBox[{newShow, [, RowBox[{g98["d"], ,, 0.005, ,, {0, 0, 1}}], ]}], ;}], , *)}]}], }]

[Graphics:HTMLFiles/4LetterWalksInDiamondLattice_238.gif]

-Graphics3D-

$SetDirectory["C:\_IMS2004\JavaApplet"] ; (* Change this *)$

RowBox[{g85to2ofa, =, RowBox[{RowBox[{reducedRTShow, [, RowBox[{g85iter[2], ,, RowBox[{ViewPoi ... {RowBox[{-, 0.223}], ,, , RowBox[{-, 3.376}], ,, , RowBox[{-, 0.026}]}], }}]}]}], ]}], //, N}]}]

-Graphics3D-

Please edit the last lines of "g85to2ofa.m": Change PlotLabel ->, to PlotLabel ->"", .
Sorry, I had no time to check why this went wrong in the first place.

RowBox[{g98to2ofa, =, RowBox[{RowBox[{reducedRTShow, [, RowBox[{g98iter[2], ,, RowBox[{ViewPoi ... {RowBox[{-, 0.223}], ,, , RowBox[{-, 3.376}], ,, , RowBox[{-, 0.026}]}], }}]}]}], ]}], //, N}]}]

-Graphics3D-

Once again: Change PlotLabel ->, to PlotLabel ->"", .

Computing the final 2D point reached by extremely long walks for g85 and g98

DNA sequences

Acknowledgements

Created by Mathematica (February 18, 2004)