(4,4) Tiling
From Ultrafractal Wiki
(4,4) Tiling is a Transformation in sam.ufx.
Contents |
What is it?
This transformation produce tilings based on the regular tessellation of the plane by squares.
The standard square tiling is rather uninteresting. This transform can replace each sides of the tiles with a broken line. This broken line is mirrored or flipped in some definite way and quite nice tilings appear. It is exactly this idea Escher used in his well-known tessellations. All the periodic tilings consist of blocks of four tiles repeating regularly. In one block, the tiles may differ from one another. I think I exhausted all the tilings of this type (I found 25 of them). A "quasi-periodic" mode has also been added. Some other features are available.
How to Start
Load "Pixel" in mt.ufm. As a coloring, choose a gradient-like or fbm-like one. Load the transform. Choose "Mode" : "Drop Points". You should see a square. This is the base tile. You can now replace its sides by broken lines. To do it, select "Point 1", take the eyedropper and click near the bottom side of the square (between the two corners). To avoid problems, you shouldn't put the point too high or too low (its imaginary coordinate should be between -0.2 and 0.2. The shape should have changed. The bottom side (0,0)-(1,0) has been transformed into (0,0)-Point 1-(1,0) and the other sides have been modified accordingly. You can add more points by choosing "Point 2", then "Point 3", ... You're not obliged to use all the points (usually, two or three should be sufficient). Take care not to put them too far from the bottom side of the tile. If you want to "erase" the latest point you put, just reset it to (1,0). Once you're satisfied, go back to the "Tiling" mode, zoom out a bit and play with "Longitudinal Mode" and "Transverse Mode" to see the different tilings. You may want to use a mask (Checkerboard, for instance) in order to see the tiling better. Note that you won't get anything interesting from this transform if you leave all the points to (1,0).
Parameters
Mode
- Tiling : The usual mode.
- Drop Points : Use it to define the parameters "Point 1", "Point 2", ... , "Point 10".
Longitudinal Mode
Describes on which sides of the tiles the broken line will be mirrored. Use it to produce different tilings. The "quasi-periodic" choose this randomly and produce an quasi-periodic tiling. The tiling obtained is not periodic, but consists of a finite number of different tiles.
Transverse Mode
Describes on which sides of the tiles the broken line will be flipped. Works the same as "Longitudinal Mode". The combination of the five periodic modes give you 25 different tesselations.
Rotation Mode
Tells how the image mapped in each shape will be rotated.
Mask
Enables you to mask some tiles, i.e. assign them solid color. It's useful to give them another color or texture, for instance. There are two "strange" masks : "Selected Tiles" and "Unselected Tiles". To use them, choose the tiles you want to be set solid (or not solid) with the "Selected Tile" parameters (there are four of them). Use the eyedropper and just click or the tile you want to select.
Mapping Center\Rotation\Magnification
Center\Rotation\Magnification of the image that will be mapped on the tiles.
Shape Mask
A shape made of tiles will be assigned solid color. The shape is defined by "Shape Mask p1\p2". See how the "Pattern Rotation" parameter will affect the shape...
Shape Mask p1\p2
To choose the size and the location of the shape mask. Try them with the eyedropper.
Example
Copyright 2000 Samuel Monnier
recttiling {
; Copyrights 2000 S. Monnier
; http://www.crosswinds.net/~s31415/index/index.htm
fractal:
title="recttiling" width=500 height=500 author="Samuel"
created="June 24, 2000" numlayers=5
layer:
mapalpha=0 method=multipass caption="New Layer 4" opacity=100 visible=yes
alpha=yes mergemode=multiply
mapping:
center=-2.46982576519457088/5.51365078195329408 magn=0.402184517497348928
angle=45 numtransforms=1
transform:
filename="sam.uxf" entry="Tiling11" p_chmode="Tiling"
p_lmode="Quasiperiodic" p_tmode="Quasiperiodic" p_rmode="Quasiperiodic"
p_mask="Checkerboard In" p_size=1 p_rot=0 p_center=0/0 p_mrot=0 p_mag=1
p_mask2="Square Out" p_smaskp1=-4.38036517051726976/7.44763238856829568
p_smaskp2=-0.406912051471902464/3.45073706823062528
p_p1=0.367041305856666752/0.0729485968423333376
p_p2=0.435023165267097536/0.326145875753897984
p_p3=0.533707972523333376/0.079615263509
p_p4=0.420374639190000064/-0.213718069824333344 p_p5=1/0 p_p6=1/0
p_p7=1/0 p_p8=1/0 p_p9=1/0 p_p10=1/0 p_stile1=1E20/0 p_stile2=1E20/0
p_stile3=1E20/0 p_stile4=1E20/0 p_ffact=0.3 p_seed=123456234234 f_f1=exp
formula:
filename="sam.ufm" entry="pixeldis" maxiter=100 percheck=normal
inside:
transfer=none repeat=yes
outside:
filename="sam.ucl" entry="SFBMpix" transfer=linear repeat=yes
p_mode="Perlin's" p_niter=20 p_string=yes p_power1=0.1
p_scaledis="1/sqrt(scale)" p_size=1 p_magn=1.2 p_rot=28 p_power=1
p_sin="Raw" p_perturb=no p_seed=123094 p_randmode="Division"
p_mod="Use sqrt(i)" f_f1=exp
gradient:
smooth=no numnodes=2 index=34 color=65957 index=263 color=6080490
layer:
mapalpha=0 method=multipass caption="New Layer 2" opacity=100 visible=yes
alpha=yes
mapping:
center=-2.46982576519457088/5.51365078195329408 magn=0.402184517497348928
angle=45 numtransforms=1
transform:
filename="sam.uxf" entry="Tiling11" p_chmode="Tiling"
p_lmode="Quasiperiodic" p_tmode="Quasiperiodic" p_rmode="Quasiperiodic"
p_mask="Checkerboard In" p_size=1 p_rot=0 p_center=0/0 p_mrot=0 p_mag=1
p_mask2="Square Out" p_smaskp1=-4.38036517051726976/7.44763238856829568
p_smaskp2=-0.406912051471902464/3.45073706823062528
p_p1=0.367041305856666752/0.0729485968423333376
p_p2=0.435023165267097536/0.326145875753897984
p_p3=0.533707972523333376/0.079615263509
p_p4=0.420374639190000064/-0.213718069824333344 p_p5=1/0 p_p6=1/0
p_p7=1/0 p_p8=1/0 p_p9=1/0 p_p10=1/0 p_stile1=1E20/0 p_stile2=1E20/0
p_stile3=1E20/0 p_stile4=1E20/0 p_ffact=0.3 p_seed=123456234234 f_f1=exp
formula:
filename="sam.ufm" entry="pixeldis" maxiter=100 percheck=normal
inside:
transfer=none repeat=yes
outside:
filename="sam.ucl" entry="SFBMpix" transfer=linear repeat=yes
p_mode="Raw... (Sam's)" p_niter=20 p_string=yes p_power1=0.1
p_scaledis="1/sqrt(scale)" p_size=0.4 p_magn=1.2 p_rot=28 p_power=1
p_sin="Raw" p_perturb=no p_seed=123094 p_randmode="Division"
p_mod="Use sqrt(i)" f_f1=exp
gradient:
smooth=no numnodes=2 index=163 color=3586533 index=334 color=65957
layer:
mapalpha=0 method=multipass caption="New Layer 5" opacity=25 visible=yes
alpha=yes mergemode=softlight
mapping:
center=-2.46982576519457088/5.51365078195329408 magn=0.402184517497348928
angle=45 numtransforms=1
transform:
filename="sam.uxf" entry="Tiling11" p_chmode="Tiling"
p_lmode="Quasiperiodic" p_tmode="Quasiperiodic" p_rmode="Quasiperiodic"
p_mask="None" p_size=1 p_rot=0 p_center=0.6/0.6 p_mrot=0 p_mag=1
p_mask2="None" p_smaskp1=-4.38036517051726976/7.44763238856829568
p_smaskp2=-0.406912051471902464/3.45073706823062528
p_p1=0.367041305856666752/0.0729485968423333376
p_p2=0.435023165267097536/0.326145875753897984
p_p3=0.533707972523333376/0.079615263509
p_p4=0.420374639190000064/-0.213718069824333344 p_p5=1/0 p_p6=1/0
p_p7=1/0 p_p8=1/0 p_p9=1/0 p_p10=1/0 p_stile1=1E20/0 p_stile2=1E20/0
p_stile3=1E20/0 p_stile4=1E20/0 p_ffact=0.3 p_seed=123456234234 f_f1=exp
formula:
filename="sam.ufm" entry="pixeldis" maxiter=100 percheck=normal
inside:
transfer=none repeat=yes
outside:
filename="sg.ucl" entry="SG_Gradient" density=40 transfer=linear
repeat=yes p_type="Angle" p_center=0/0
gradient:
smooth=no numnodes=2 index=232 color=16777215 index=233 color=0
layer:
mapalpha=0 method=multipass caption="New Layer 1" opacity=100 visible=yes
alpha=yes
mapping:
center=-2.48640190469429856/5.49707464245356608 magn=0.402184517497348928
angle=45 numtransforms=1
transform:
filename="sam.uxf" entry="Tiling11" p_chmode="Tiling"
p_lmode="Quasiperiodic" p_tmode="Quasiperiodic" p_rmode="Quasiperiodic"
p_mask="Checkerboard Out" p_size=1 p_rot=0 p_center=0/0 p_mrot=0 p_mag=1
p_mask2="None" p_smaskp1=-4.38036517051726976/7.44763238856829568
p_smaskp2=-0.406912051471902464/3.45073706823062528
p_p1=0.367041305856666752/0.0729485968423333376
p_p2=0.435023165267097536/0.326145875753897984
p_p3=0.533707972523333376/0.079615263509
p_p4=0.420374639190000064/-0.213718069824333344 p_p5=1/0 p_p6=1/0
p_p7=1/0 p_p8=1/0 p_p9=1/0 p_p10=1/0 p_stile1=1E20/0 p_stile2=1E20/0
p_stile3=1E20/0 p_stile4=1E20/0 p_ffact=0.3 p_seed=123456234234 f_f1=exp
formula:
filename="sam.ufm" entry="pixeldis" maxiter=100 percheck=normal
inside:
transfer=none repeat=yes
outside:
filename="sam.ucl" entry="SFBMpix" transfer=linear repeat=yes
p_mode="Raw... (Sam's)" p_niter=20 p_string=yes p_power1=0.1
p_scaledis="1/sqrt(scale)" p_size=0.4 p_magn=1.2 p_rot=28 p_power=1
p_sin="Raw" p_perturb=no p_seed=123094 p_randmode="Division"
p_mod="Use sqrt(i)" f_f1=exp
gradient:
smooth=no numnodes=2 index=121 color=15131360 index=258 color=6109704
layer:
mapalpha=0 method=multipass caption="New Layer 3" opacity=100 visible=yes
alpha=yes
mapping:
center=-2.46982576519457088/5.51365078195329408 magn=0.402184517497348928
angle=45 numtransforms=1
transform:
filename="sam.uxf" entry="Tiling11" p_chmode="Tiling"
p_lmode="Quasiperiodic" p_tmode="Quasiperiodic" p_rmode="Quasiperiodic"
p_mask="Checkerboard In" p_size=1 p_rot=0 p_center=0/0 p_mrot=0 p_mag=1
p_mask2="None" p_smaskp1=2/2 p_smaskp2=-2/-2
p_p1=0.367041305856666752/0.0729485968423333376
p_p2=0.435023165267097536/0.326145875753897984
p_p3=0.533707972523333376/0.079615263509
p_p4=0.420374639190000064/-0.213718069824333344 p_p5=1/0 p_p6=1/0
p_p7=1/0 p_p8=1/0 p_p9=1/0 p_p10=1/0 p_stile1=1E20/0 p_stile2=1E20/0
p_stile3=1E20/0 p_stile4=1E20/0 p_ffact=0.3 p_seed=123456234234 f_f1=exp
formula:
filename="sam.ufm" entry="pixeldis" maxiter=100 percheck=normal
inside:
transfer=none repeat=yes
outside:
filename="sam.ucl" entry="SFBMpix" transfer=linear repeat=yes
p_mode="Raw... (Sam's)" p_niter=20 p_string=yes p_power1=0.1
p_scaledis="1/sqrt(scale)" p_size=0.4 p_magn=1.2 p_rot=28 p_power=1
p_sin="Raw" p_perturb=no p_seed=123094 p_randmode="Division"
p_mod="Use sqrt(i)" f_f1=exp
gradient:
smooth=no numnodes=2 index=159 color=16777215 index=273 color=5992839
}
