Ossa Sepia

While I can ~never be absolutely certain the implementation is a bug-free translation of the spec, I can certainly check again and check better and take a break from that code before checking then yet again with fresh eyes and even concrete illustrations to try as best as possible to either find the trouble or otherwise set the matter to rest and set own mind at rest that there isn’t anything to that niggling thought that somewhere kept insisting that hm, *some* of those hopefuls in the 4th parade are not quite as expected, somehow.

So I left the generator code alone for a few days and did instead all the writing of textures and then some more writing and different code and new textures and even various other code tidy ups before I sat down again to recalculate all the formulae (it’s not the first time either), to re-read the whole code and to check – with pictures if need be! – that it does as expected. The formulae turned out the same and correct, everything supposedly was fine except… a typo, of course, what fucking else. I fixed it and then tracked back to see where it got in exactly: at the moment when I switched from mirroring only original points to mirroring all in the 2nd plane, so it affects only generation 4 (and only the hopefuls with sym2), not any of the previous generations nor the sym1 hopefuls of gen4. So thank you for giving voice to that doubt and asking that question, apologies for the mess up and I’ll try to take that needed break in the future before it’s overdue rather than after.

With the fix in, I ran again the whole testing step by step and took screenshots to provide a clearer illustration of this whole important symmetry part, as I think it might help even to spot if there’s anything else unexpected/undesired in there. To help make the points clear, I ran each step twice: once with normal fit of the mesh on the full bone’s length and once with fitting the mesh only on the end point of the bone (and a fixed, small size, 0.02, so that something is visible). Step 1 is just the set of pseudo-randomly chosen points (7 inside the volume, 17 on the surface of the unit sphere), without any mirroring (nor collapsing) at all, first normal fit of the mesh, then end-of-bone fit:

Step 2 uses the same seed as above but adds mirroring in one and then in both planes *without* any collapsing – so you can see visually the symmetry introduced:

Worth noting here since I’m going in detail through it anyway: the 2nd plane of symmetry is obtained by taking first the 2 points that are furthest apart and then a 3rd point that is closest to those 2 but *also* not collinear to them (since collinear wouldn’t help to define a plane); at the moment I’m not checking anywhere specifically so that all points are distinct. In principle this can introduce ~duplicate points – either by pure chance or by the weird effects of working with floating point numbers on the computer, since the distance of a point to the plane that contains it may still end up not-exactly-zero so that the mirroring can literally double the points, nevermind that 3 of them are in principle their own mirroring. Even if introduced, those duplicate points are the first ones to go at collapsing time but it’s true that they would in principle affect that calculated average distance so let me know if this is a problem and you’d rather I add checks to prevent it entirely.

Finally, step 3 uses the same seed and does mirrorings as above but adds afterwards the collapsing too:

Let me know if there’s anything else looking dubious in the above and/or if there’s something else to change regarding this mirroring and collapsing of points.

Update with the required data dump for seed 99. Note that the dump rounded to 2 decimals above while the script works with the values not rounded (hence also the mirroring at distance -0.00 – the check is too strict and so there is this, as I mentioned in the text):
Initial points:
0.28 -0.55 0.38
-0.01 -0.02 -0.01
-0.01 -0.01 -0.18
-0.04 -0.02 -0.26
0.23 -0.01 0.51
-0.00 0.00 -0.01
-0.01 -0.25 -0.01
-0.72 0.70 -0.72
-0.87 0.03 -1.00
0.84 -0.40 0.90
-0.72 0.55 -0.79
0.85 -0.14 0.99
-0.91 -0.14 -0.99
0.68 0.46 0.83
0.33 -0.79 0.39
0.23 0.93 0.24
-0.70 -0.25 -0.94
-0.21 0.77 -0.26
0.31 -0.71 0.40
0.11 -0.21 0.47
0.17 0.94 0.18
-0.59 -0.76 -0.61
0.11 0.08 0.83
-0.79 0.49 -0.85
Symmetry plane 1 determined by the 2 furthest away points 11 and 12, together with 3rd closest (and non-collinear), 6:
0.85 -0.14 0.99
-0.91 -0.14 -0.99
-0.01 -0.25 -0.01
Equation symplane1: -0.21 * x + (-0.06) * y + (0.19) * z + (-0.01) = 0
Reflected points in symplane1:
Point 24 reflection of 0 at distance -0.10 from plane: 0.32 -0.54 0.34
Point 25 reflection of 1 at distance 0.05 from plane: -0.03 -0.02 0.01
Point 26 reflection of 2 at distance 0.15 from plane: -0.08 -0.02 -0.12
Point 27 reflection of 3 at distance 0.19 from plane: -0.12 -0.04 -0.19
Point 28 reflection of 4 at distance -0.12 from plane: 0.28 -0.00 0.47
Point 29 reflection of 5 at distance 0.05 from plane: -0.02 -0.00 0.01
Point 30 reflection of 6 at distance -0.00 from plane: -0.01 -0.25 -0.01
Point 31 reflection of 7 at distance 0.13 from plane: -0.77 0.68 -0.67
Point 32 reflection of 8 at distance 0.07 from plane: -0.90 0.02 -0.97
Point 33 reflection of 9 at distance -0.00 from plane: 0.84 -0.40 0.90
Point 34 reflection of 10 at distance 0.15 from plane: -0.78 0.54 -0.74
Point 35 reflection of 12 at distance -0.00 from plane: -0.91 -0.14 -0.99
Point 36 reflection of 13 at distance 0.09 from plane: 0.64 0.45 0.87
Point 37 reflection of 14 at distance -0.12 from plane: 0.38 -0.78 0.34
Point 38 reflection of 15 at distance 0.24 from plane: 0.13 0.91 0.33
Point 39 reflection of 16 at distance 0.10 from plane: -0.74 -0.26 -0.90
Point 40 reflection of 17 at distance 0.22 from plane: -0.30 0.74 -0.18
Point 41 reflection of 18 at distance -0.12 from plane: 0.37 -0.69 0.36
Point 42 reflection of 19 at distance -0.22 from plane: 0.20 -0.19 0.39
Point 43 reflection of 20 at distance 0.24 from plane: 0.07 0.92 0.27
Point 44 reflection of 21 at distance -0.13 from plane: -0.54 -0.75 -0.66
Point 45 reflection of 22 at distance -0.39 from plane: 0.28 0.12 0.68
Point 46 reflection of 23 at distance 0.12 from plane: -0.84 0.48 -0.81

Symmetry plane 2 based on normal to plane1 & going through 2 furthest away points:
-0.11 * x + (0.75) * y + (0.10) * z + (0.10) = 0

Point 47 reflection of 0 at distance 0.52 from plane: 0.17 0.23 0.48
Point 48 reflection of 1 at distance -0.15 from plane: 0.03 -0.25 -0.04
Point 49 reflection of 2 at distance -0.14 from plane: 0.02 -0.22 -0.20
Point 50 reflection of 3 at distance -0.11 from plane: -0.01 -0.19 -0.28
Point 51 reflection of 4 at distance -0.20 from plane: 0.28 -0.31 0.47
Point 52 reflection of 5 at distance -0.18 from plane: 0.04 -0.27 -0.04
Point 53 reflection of 6 at distance 0.14 from plane: -0.04 -0.03 0.02
Point 54 reflection of 7 at distance -1.08 from plane: -0.48 -0.93 -0.93
Point 55 reflection of 8 at distance -0.21 from plane: -0.82 -0.29 -1.04
Point 56 reflection of 9 at distance 0.35 from plane: 0.77 0.12 0.97
Point 57 reflection of 10 at distance -0.89 from plane: -0.52 -0.78 -0.97
Point 58 reflection of 11 at distance -0.00 from plane: 0.85 -0.14 0.99
Point 59 reflection of 12 at distance -0.00 from plane: -0.91 -0.14 -0.99
Point 60 reflection of 13 at distance -0.77 from plane: 0.85 -0.70 0.68
Point 61 reflection of 14 at distance 0.84 from plane: 0.15 0.46 0.55
Point 62 reflection of 15 at distance -1.37 from plane: 0.53 -1.12 -0.03
Point 63 reflection of 16 at distance 0.17 from plane: -0.74 0.01 -0.91
Point 64 reflection of 17 at distance -1.15 from plane: 0.05 -0.96 -0.49
Point 65 reflection of 18 at distance 0.73 from plane: 0.15 0.38 0.55
Point 66 reflection of 19 at distance 0.04 from plane: 0.10 -0.16 0.48
Point 67 reflection of 20 at distance -1.39 from plane: 0.48 -1.13 -0.10
Point 68 reflection of 21 at distance 0.80 from plane: -0.77 0.43 -0.46
Point 69 reflection of 22 at distance -0.39 from plane: 0.20 -0.51 0.75
Point 70 reflection of 23 at distance -0.81 from plane: -0.61 -0.73 -1.01
Point 71 reflection of 24 at distance 0.52 from plane: 0.21 0.24 0.44
Point 72 reflection of 25 at distance -0.15 from plane: 0.01 -0.25 -0.02
Point 73 reflection of 26 at distance -0.14 from plane: -0.05 -0.23 -0.15
Point 74 reflection of 27 at distance -0.11 from plane: -0.09 -0.21 -0.21
Point 75 reflection of 28 at distance -0.20 from plane: 0.33 -0.30 0.43
Point 76 reflection of 29 at distance -0.18 from plane: 0.02 -0.27 -0.02
Point 77 reflection of 30 at distance 0.14 from plane: -0.04 -0.03 0.02
Point 78 reflection of 31 at distance -1.08 from plane: -0.53 -0.94 -0.88
Point 79 reflection of 32 at distance -0.21 from plane: -0.85 -0.29 -1.02
Point 80 reflection of 33 at distance 0.35 from plane: 0.77 0.12 0.97
Point 81 reflection of 34 at distance -0.89 from plane: -0.58 -0.79 -0.91
Point 82 reflection of 35 at distance -0.00 from plane: -0.91 -0.14 -0.99
Point 83 reflection of 36 at distance -0.77 from plane: 0.81 -0.71 0.71
Point 84 reflection of 37 at distance 0.84 from plane: 0.20 0.48 0.51
Point 85 reflection of 38 at distance -1.37 from plane: 0.43 -1.15 0.06
Point 86 reflection of 39 at distance 0.17 from plane: -0.78 -0.00 -0.87
Point 87 reflection of 40 at distance -1.15 from plane: -0.04 -0.99 -0.41
Point 88 reflection of 41 at distance 0.73 from plane: 0.20 0.39 0.50
Point 89 reflection of 42 at distance 0.04 from plane: 0.20 -0.13 0.40
Point 90 reflection of 43 at distance -1.39 from plane: 0.38 -1.16 -0.01
Point 91 reflection of 44 at distance 0.80 from plane: -0.71 0.44 -0.51
Point 92 reflection of 45 at distance -0.39 from plane: 0.37 -0.47 0.60
Point 93 reflection of 46 at distance -0.81 from plane: -0.66 -0.74 -0.97

Fwiw I calculated yet again the plane equations and I still don’t see anything wrong with them. Perhaps there’s something going wrong at reflection itself, so I’ll re-calc that too, though tomorrow with a fresher mind.

Updated 22 May 2020: *fixed* the incorrect use of size of normal instead of squared size of normal when calculating the lambda for the first plane. Updated data dump with full decimals too:
Seed 99
Initial points:
0.281813 -0.548861 0.380002
-0.005405 -0.017001 -0.009531
-0.012594 -0.005508 -0.177122
-0.038420 -0.020412 -0.262836
0.230660 -0.013599 0.512881
-0.003318 0.003075 -0.005014
-0.005029 -0.247561 -0.005563
-0.717715 0.696048 -0.717859
-0.870788 0.032036 -0.999324
0.842434 -0.403239 0.901994
-0.717069 0.552947 -0.791900
0.851877 -0.140657 0.986641
-0.910115 -0.140719 -0.988257
0.682939 0.455476 0.831947
0.331493 -0.788936 0.387371
0.228824 0.932114 0.238411
-0.698641 -0.248287 -0.942265
-0.208097 0.765032 -0.262473
0.312717 -0.708648 0.403725
0.111998 -0.209795 0.470940
0.170545 0.944810 0.177637
-0.592100 -0.761696 -0.613727
0.112007 0.076307 0.826439
-0.791548 0.490804 -0.849881
Symmetry plane 1
2 furthest away points 11 and 12:
0.851877 -0.140657 0.986641
-0.910115 -0.140719 -0.988257
3rd point closest to the above, 6: -0.005029 -0.247561 -0.005563
Normal symplane1: -0.211063 -0.055953 0.188303
d=-0.013857
Point 24 reflection of 0 with lambda -0.347961: 0.428696 -0.509922 0.248958
Point 25 reflection of 1 with lambda 0.163105: -0.074256 -0.035254 0.051896
Point 26 reflection of 2 with lambda 0.532182: -0.237242 -0.065063 0.023300
Point 27 reflection of 3 with lambda 0.650727: -0.313109 -0.093233 -0.017769
Point 28 reflection of 4 with lambda -0.418546: 0.407339 0.033239 0.355254
Point 29 reflection of 5 with lambda 0.171685: -0.075791 -0.016138 0.059644
Point 30 reflection of 6 with lambda -0.000101: -0.004987 -0.247550 -0.005601
Point 31 reflection of 7 with lambda 0.438981: -0.903020 0.646923 -0.552537
Point 32 reflection of 8 with lambda 0.240978: -0.972511 0.005069 -0.908570
Point 33 reflection of 9 with lambda -0.008977: 0.846223 -0.402235 0.898613
Point 34 reflection of 10 with lambda 0.512010: -0.933202 0.495649 -0.599075
Point 35 reflection of 12 with lambda -0.000201: -0.910030 -0.140697 -0.988333
Point 36 reflection of 13 with lambda 0.322703: 0.546718 0.419363 0.953479
Point 37 reflection of 14 with lambda -0.400105: 0.500388 -0.744161 0.236689
Point 38 reflection of 15 with lambda 0.834958: -0.123634 0.838677 0.552861
Point 39 reflection of 16 with lambda 0.360113: -0.850654 -0.288586 -0.806645
Point 40 reflection of 17 with lambda 0.747765: -0.523748 0.681352 0.019139
Point 41 reflection of 18 with lambda -0.430777: 0.494559 -0.660441 0.241492
Point 42 reflection of 19 with lambda -0.756851: 0.431485 -0.125098 0.185906
Point 43 reflection of 20 with lambda 0.833199: -0.181170 0.851569 0.491424
Point 44 reflection of 21 with lambda -0.459078: -0.398311 -0.710322 -0.786619
Point 45 reflection of 22 with lambda -1.369474: 0.690098 0.229560 0.310687
Point 46 reflection of 23 with lambda 0.412428: -0.965645 0.444651 -0.694559
Symmetry plane 2 based on normal to plane1 & going through 2 furthest away points:
-0.110514 * x + (0.748616) * y + (0.098576) * z + (0.102183) = 0
Point 47 reflection of 0 with lambda 0.519249: 0.167044 0.228576 0.482373
Point 48 reflection of 1 with lambda -0.153022: 0.028417 -0.246110 -0.039699
Point 49 reflection of 2 with lambda -0.140792: 0.018525 -0.216306 -0.204880
Point 50 reflection of 3 with lambda -0.112025: -0.013659 -0.188139 -0.284922
Point 51 reflection of 4 with lambda -0.201026: 0.275092 -0.314582 0.473248
Point 52 reflection of 5 with lambda -0.179198: 0.036290 -0.265226 -0.040343
Point 53 reflection of 6 with lambda 0.142761: -0.036584 -0.033814 0.022582
Point 54 reflection of 7 with lambda -1.084918: -0.477917 -0.928326 -0.931754
Point 55 reflection of 8 with lambda -0.212739: -0.823766 -0.286484 -1.041266
Point 56 reflection of 9 with lambda 0.350085: 0.765055 0.120919 0.971014
Point 57 reflection of 10 with lambda -0.888306: -0.520729 -0.777054 -0.967032
Point 58 reflection of 11 with lambda -0.000000: 0.851877 -0.140657 0.986641
Point 59 reflection of 12 with lambda -0.000000: -0.910115 -0.140719 -0.988257
Point 60 reflection of 13 with lambda -0.772199: 0.853617 -0.700686 0.679706
Point 61 reflection of 14 with lambda 0.836044: 0.146703 0.462817 0.552200
Point 62 reflection of 15 with lambda -1.370623: 0.531771 -1.120028 -0.031811
Point 63 reflection of 16 with lambda 0.170624: -0.736354 0.007177 -0.908626
Point 64 reflection of 17 with lambda -1.153969: 0.046963 -0.962729 -0.489982
Point 65 reflection of 18 with lambda 0.726503: 0.152139 0.379097 0.546957
Point 66 reflection of 19 with lambda 0.035763: 0.104093 -0.156249 0.477991
Point 67 reflection of 20 with lambda -1.387716: 0.477270 -1.132924 -0.095955
Point 68 reflection of 21 with lambda 0.795214: -0.767865 0.428925 -0.456949
Point 69 reflection of 22 with lambda -0.392193: 0.198692 -0.510897 0.749117
Point 70 reflection of 23 with lambda -0.812742: -0.611909 -0.726060 -1.010116
Point 71 reflection of 24 with lambda 0.519249: 0.313928 0.267515 0.351329
Point 72 reflection of 25 with lambda -0.153022: -0.040434 -0.264363 0.021727
Point 73 reflection of 26 with lambda -0.140792: -0.206123 -0.275861 -0.004457
Point 74 reflection of 27 with lambda -0.112025: -0.288348 -0.260960 -0.039855
Point 75 reflection of 28 with lambda -0.201026: 0.451772 -0.267744 0.315621
Point 76 reflection of 29 with lambda -0.179198: -0.036183 -0.284439 0.024314
Point 77 reflection of 30 with lambda 0.142761: -0.036541 -0.033803 0.022544
Point 78 reflection of 31 with lambda -1.084918: -0.663223 -0.977451 -0.766431
Point 79 reflection of 32 with lambda -0.212739: -0.925490 -0.313451 -0.950512
Point 80 reflection of 33 with lambda 0.350085: 0.768844 0.121923 0.967633
Point 81 reflection of 34 with lambda -0.888306: -0.736862 -0.834352 -0.774207
Point 82 reflection of 35 with lambda -0.000000: -0.910030 -0.140697 -0.988333
Point 83 reflection of 36 with lambda -0.772199: 0.717395 -0.736798 0.801238
Point 84 reflection of 37 with lambda 0.836044: 0.315599 0.507592 0.401518
Point 85 reflection of 38 with lambda -1.370623: 0.179312 -1.213465 0.282639
Point 86 reflection of 39 with lambda 0.170624: -0.888367 -0.033122 -0.773006
Point 87 reflection of 40 with lambda -1.153969: -0.268688 -1.046409 -0.208369
Point 88 reflection of 41 with lambda 0.726503: 0.333982 0.427304 0.384724
Point 89 reflection of 42 with lambda 0.035763: 0.423580 -0.071553 0.192956
Point 90 reflection of 43 with lambda -1.387716: 0.125554 -1.226164 0.217832
Point 91 reflection of 44 with lambda 0.795214: -0.574076 0.480299 -0.629840
Point 92 reflection of 45 with lambda -0.392193: 0.776783 -0.357644 0.233366
Point 93 reflection of 46 with lambda -0.812742: -0.786006 -0.772213 -0.854793

Having given more time and more background thinking to the question in the title, I now have 3 answers of yet to be decided length (instead of the original ~500 words one) and I still like none! So I’m writing it all down already, since it turns out to be still at the expansion and exploration stage rather than the contraction and synthesis stage. First of all though and for any innocent readers that might happen one day, here’s the question in its larger context:

Diana Coman: I was in fact looking at maybe trying out generating at least a few *types* of meshes as opposed to this single-mesh-for-everything approach. But it’s a long road to explore and I’m not sure if it’s burning /first in line given how many other things are to do – arguably and as a barely-formed/fuzzy idea for instance heads would be more reasonable starting with an egg-like shape (rather than a rounded cylinder) and deforming that based on/around some feature points. Not that it wasn’t fun to get a proper implicit equation for eggs!

Mircea Popescu: I think it should start with the following single step : explain to me how they’d be different from one another ?

The first and easy answer to the above would be the shortest too: seeing how the whole business of this fractal deformation thing means precisely fuzzying clarity itself, fracturing previously whole dimensions and otherwise in general and in plain language making an unrecognizable mess out of any initial given thing, there can’t possibly be any way in which the results of such activity depend on the shape that one starts with! After all, if you twist and deform something long and hard enough, what you get is still a pile of rubbish, regardless of what you started with. In which case the whole problem is solved, there’s nothing to do here, we can move on and work on something else already. Wouldn’t that be tempting, too?

The trouble with the above first and easy answer is that it’s pure (and even sligthly bent) theory pushed to the extreme and not much to do with the practical matter at hand: first, the goal is precisely to make a *recognizable* mess of a mesh rather than a totally random mess (of a mesh) – even if, perhaps, barely recognizable rather than strongly so and even if recognizable by the implicit system that human perception uses rather than by an explicit set of rules; second, fractals may result in messy, seemingly random shapes indeed, but their randomness is just an illusion and an artefact of either limited space/time available or of our own limited perception of everything that the fractal can pack in such a tiny space. So the second and unhelpful answer is that in theory they need not be any different but in practice they will be (as the starting shape gives the space in which the fractal deformation works)!

Finally, the third and very long answer says at core that they’d be different in style (in the way for instance swirly textures make a clearly distinct group from Escher-type textures and both are still different from the naive/surreal set), as the initial shape acts as a sort of starting point – while one can arguably get anywhere from any starting point, the sort of things one meets first are going to be different around one starting point as opposed to around a different starting point and moreover, the distance to travel and the difficulty in finding/following the road to the sort of thing desired will also depend as well on where one starts from. As illustration to what “different in style” might mean specifically, here’s a few concrete pictures anyway as they can’t hurt – although they are just a random pick really and they can’t really say all that much of the whole story either (for all the popular pretense otherwise). For a simple thing that is perhaps not that obvious at first: note that the deformation of the cylinder shape resulted in stray parts – unconnected to the main, as a result of having to crank the noise up (and for an even clearer illustration of cranking the noise way too much up, see the first attempts at mesh deformation) quite a lot to get from a straight cylinder to something else, basically as an effect of that longer distance to travel; the egg-shape has no such trouble, as there is way less deformation applied (and even limited as to affected area) though with significant and visible effect:

On a superficial level (that can be easily shot down with “that’s an artefact of the implementation/approach, damn it!”), the reason for there being a difference between starting with egg-shape vs starting with cyllinder shape in the first place is that the fractal mesh deformation effectively acts on a fuzzy region around the initial shape and is called upon to do at least 2 types of things: at the higher level, to morph the overall shape into one that is close enough to be recognizable as part of one specific group (e.g. “heads” or “limbs” – those do tend to have different proportions and overall shape); at the lower level, to create detail that fits that specific group’s features (e.g. “eyes and mouth”). Having one single deformation do reasonably well both is theoretically possible, practically more likely a nightmare as it’s quite unlikely that deformations making plausible eyes for instance are *also* helpful to inflate a straight cylinder into a plausible head-like shape. Certainly, there’s also the route of applying several distinct deformations, basically one for the higher level shape and one for the lower-level features – the question in this case is whether it’s worth the effort to do both. For one possible direction, the high level shape might even be better addressed at setting-on-bone time: perhaps the mesh should really remain just a surface and thus concerned with feature-level details, leaving the rest to be decided by the underlying bone on which the mesh is to be stretched (this would require that the bone has therefore at least 2, possibly 3 dimensions). Nevertheless, what having all those ultimately confusing rather than clear/clarifying options means is that so far the approach is not general enough to be fully independent of the starting shape – and perhaps that is an issue worth discussing first and in more detail anyway.

Before attempting though a discussion of the very approach itself and of choices regarding the mix of chaos and structure in mesh or texture generation, there are a few basic terms that I’ll surely end up needing to remember in some years from now if I ever re-read this:

1. Chaotic behaviour is the result of errors/noise overpowering the input/main signal. Any tame and otherwise very well behaved system that reacts deterministically and predictably to a known input signal can in principle become chaotic: all it takes is for errors/noise to accumulate at a pace greater than the system’s own capacity to self-correct and/or identify the original signal as distinct; as soon as the original signal is drowned by errors/noise, the behaviour of the system becomes chaotic, with little prediction possible *in the short term*. The short term part is quite important here because it’s really that the unpredictability, as to what the system might do immediately next, as it depends so much on just what random bit it interprets as signal, after all. Long-term though, at the limit (of infinite time), chaos itself is the tamest and most predictable of things, really (see below).
2. Fractals are feedback systems and as such, their limits (aka what they tend to, given infinite time to run) are called attractors. The feedback nature of fractals means that they can accumulate and/or magnify errors very quickly indeed, basically turning into chaotic systems as per the definition above. And at the intersection of attractors with chaotic systems, there are some of the most interesting of attractors indeed, namely the unique attractors:
3. Unique attractors define the shapes towards which chaotic systems reliably tend to, in the long term. In other words, regardless of the unpredictability of a chaotic system’s *next* step, the final result of *all* those steps is itself entirely and reliably determined & predictable upfront (whether it’s also *correctly* predicted or not depends on the knowledge of the one attempting to predict, not on the chaotic system that is already walking that shape anyway). All it takes for that determined outcome to be fully achieved is time – possibly an infinite amount of it, too (it’s the result at the limit, what the fractal tends to).

Considering the goals outlined earlier (recognizable but diverse volumes/shapes/images), one way to look at mesh/texture generation is as an exercise in finding the right balance between – and even place for! – chaos (which brings the desired diversity) and structure (or order, if you prefer; at any rate, that part which ensures the result is still recognizable as part of one given group as opposed to just random noise). The exact way in which order is encoded as well as the concrete way in which chaos is brought in and harnessed make together for a significant decision that goes even more directly to the core of the matter than does the question in the title and trying to distill the answer to the latter kept pushing me towards grappling with this underlying decision and making it more clearly explicit. While I’m not quite sure I am fully there with it, the whole exercise certainly helped in various ways, so I’ll go through what clarity I’ve got out of it, such as it is.

Contemplating this how-to-mix-chaos-with-order-for-pleasing-effect issue, one can be tempted to say that fractals already contain both chaos and structure, being thus in and by themselves the perfect and full solution: as chaotic systems, fractals provide short-term unpredictability (hence the desired diversity) AND long-term structure (aka the shapes defined by unique attractors, the limits towards which the fractal will tend). The trouble is that the practice is more limited than the theory: the structure as defined by the unique attractor is fixed indeed but this doesn’t mean that it’s either a recognizable structure or in any way known/easy to figure out upfront; moreover, banking on the structure that is the limit may mean that one needs to run the generator for a very long time indeed to get to the part where the chaos-order balance turns just right for the purpose at hand and stopping any earlier than that turns out rubbish and nothing more. Add also to this the fact that the whole theory assumes of course perfectly random noise as opposed to the fake sort that one can get on a computer – and funnily enough, the punishment for trying to sneak in more structure through non-fully-random noise is a postponement of reaching that desired limit of determinism and structure otherwise. In other words, there are a lot of potentially infinite roads to travel and very few signs (if any at all) to help you avoid at least those that take a long and unknown time to get in the end nowhere you have any interest to go to!

To avoid therefore the above death-like temptation of pure theory, there are only 2 main avenues left to still get both chaos and structure in without having to wait a lifetime to even find out if one got it wrong or possibly almost-right: one can use the short-term unpredictability of the fractal’s chaotic nature to provide the desired variety, eschewing the potentially-infinite wait for having the fractal’s structure manifest too – in which case the core structure has to come (or at the very least be augmented) from a different part that is external to the fractal itself; alternatively, one can use the long-term deterministic nature of the fractal’s chaotic nature to provide the core structure – to the extent that this is known upfront and/or can be determined easily enough. Separate from this, one can of course further use fractals simply as a sort of efficient packers, basically as non-chaotic feedback systems – and I’d say this is by far the commonest type of use, too!

Almost all the attempts I made so far at both mesh and texture generation ¹ use mainly the first option above: it’s the fractal’s short-term unpredictability that is called upon, as a relatively cheap -and controllable to a reasonable extent- way to provide pleasing variety. And the sort of error introduction into the fractal’s feedback loop is quite carefully limited so as to give a sort of push towards chaotic system but not too much of it – as previously noted, too much of that and the result quickly turns to …noise, unsurprisingly. Structure is injected through all sorts, from prng to the structure of noise in use (lattice for Perlin’s, cells for Worley’s) and to the functions used for domain warping or combination of fractal transformations. As noticed with the trig textures, the external structure thus provided can be perhaps dry but surprisingly natural-looking otherwise, if well chosen. The trouble is at that “well chosen” – basically this route has the pitfall of ending up forever tweaking various parameters in the search of the “well chosen” for a specific purpose. And it seems to me that this is how one ends up in the “sea of parameters” I mentioned previously (and that I tried to escape with the tidy up of my texture generator, only to find I merely got a bit of a better grip on it while still nevertheless fully submerged otherwise).

Does the above conclusion mean that one should therefore go instead for that second option of relying on the deterministic (at limit) behaviour of chaotic systems to provide the required structure (thus avoiding the sea of parameters and eternal tweaking within it)? It certainly makes for an appealing route (at least while submerged otherwise and fighting to keep track of the 1001 parameters of the other approach) but what would it even mean, right? Well, for textures at least, I had to get the answer and that meant not only an entirely new set of textures but… a new generator, too! At any rate, before going through the technical details, here’s the first sample of outputs, a most basic sketch of what this approach might mean – it reminds me of caveman drawing style, for better (seems quite realistic really) and for worse (some are such awful scribbles and scratches!):

The above sample are but a few textures from what is essentially an entirely new set made with an entirely new generator too. The switch of the approach goes deep here, it’s literally changing from evaluating the value of each pixel to drawing the image with repeated pixel-sized brush strokes that accumulate to different extents in different areas. A chaotic feedback system (an IFS – iterated functions system) is created pseudo-randomly (aka MT is used to pick all the required parameters within some carefully constrained intervals to ensure that there IS a unique attractor) and then iterated 90 million times ² in the 2D space of the image. Each position records the number of times it was hit and then the colour of the corresponding pixel is simply calculated for now by multiplying the number of hits with 16 (I didn’t experiment much with it, this made the pattern visible and so it was good enough for now, for this totally new thing). On the most attractive side, it’s indeed, way, way simpler to generate this way as many textures as one might possibly want, pretty much in the way the skeleton generation goes – it’s out of that sea of parameters to tweak with and into the more interesting place of finding the constraints that filter the sort of thing one is after. On the troublesome part, it’s unclear to me if this approach can indeed produce *all* the different types of textures one might want. On the even more troublesome part for the topic of this article, it’s even less clear to me if the same approach makes sense (or can even work directly with the current polygonizer, sheesh) for the mesh generation!

At the end of those 3k words, I think that far from answering the question all that clearly, I have added to it a bigger question as to the very core of the approach to explore in the first place. Ignoring that though and coming back to cylinder vs something-else, the best I can extract at the moment a sort of answer from all this exploration is that egg-shapes would have potentially better feature-like deformation (ie for making passable heads). More generally, starting with a shape that is close enough to that desired for a group of meshes (ie assuming that one *does* intend to create different meshes for types of body parts) allows the deformation to focus more closely on smaller details without having to become noisy just to achieve that too.

For an ending, here’s one of the textures from the new set, used as a night-sky because why-not:

1. The exception would be the pure Mandelbrot textures as that’s exactly what the depiction there is – the shape defined by the unique attractor.[↩]
2. In principle the number of required iterations can be reduced through a better choice of some of the parameters. Specifically, the IFS is made of a set of transformations with associated probabilities so that at each iteration, one transformation is picked with a biased die from the set and applied; the IFS that I generate contain a preset number of transformations (in the set illustrated here 4), where each transformation contains one rotation around the origin, one translation and one contraction in the 2D space; the contraction is the part that ensures the existence of a unique attractor; a better choice of probabilities can in principle help the IFS converge faster to the shape defined by its attractor. For my generator, the seed used for MT serves essentially to identify each unique attractor since all the other parameters are generated from the corresponding MT sequence – as it can be easily noticed even in this selected sample of images, some attractors have more interesting shapes than others, heh. For the record, there are ways to find the IFS that has as unique attractor a given shape – basically by going at it backwards, using rotated/translated/contracted versions of the shape to extract the corresponding parameters for the transformations and then picking probabilities to reduce the time needed for convergence. My point being exploration though rather than fitting to some given shape, I didn’t implement any of that finding-specific-shape part.[↩]

Since this is anyway the second article for today and one coming after going out into the space, no less, let us start with the side for once – aka the turbulence: this is yet another use of the perlin noise, in principle more specifically geared towards gases and other potential sky-textures. The naked effect tends to be quite strong – see the first two images below that adjust the considered pixel size (aka indirectly the number of iterations that apply the noise). Nevertheless, it can perhaps qualify as a different type than previous sets of textures, though to my eye it does tend in any case towards the effect obtained otherwise with several iterations of a multifractal. Comparing it with some of the original pictures and descriptions of this turbulence use that supposedly provides images of perfect-gas (to the extent of even perfect-steam!), I tend to think it takes quite the futzing and the precise fitting to some exact colour scheme to get it to really look like gas/steam. In any case, I could tame down the colours using my own colour mappings that are basically biased towards white (3rd image) and blue (4th image), respectively:

The above side aside, the fun part for me was playing with the trig functions as distorters of the domain essentially. And I do wonder now just why on Earth don’t they illustrate the 4 quadrants better in all those Maths books, not like it’s not possible, have a look (in order, below, the images are obtained with the original colouring scheme, the same scheme only all colours lighetened up a bit; the rgb unit sphere colouring scheme):

The above obviously pack the sin,cos and tan, all in one so it’s clear that one *can* separate them too, if so desired – though at the cost of losing that cool reflection-effect, heh. Nevertheless, separating the circle part creates basically a vortex that can further get pastel colours this time by use of worley noise instead of perlin noise (ha, did you think the rgb sphere was always good or the only option for pastel colours? nope, it’s not):

Separating instead the vertical and horizontal threads gets the very illustration of weaving:

And since we are weaving cloth of sorts and trying it all out, why not make it a rectangular proper patch and get some truly regular patterns on it, too:

Messing about with the initial domain mapping means that one can turn inside out the four quadrants, of course:

Finally, for all the nostalgy of sine waves, here’s some quite plain and bold sinus let loose:

Using two of those turbulence textures as sky this time, the fashion parade wore trigonometry proudly on its skins (and bones):

And with this I am *finally* caught up on the texture sets and it’s good too as I’m clearly starting to shut down the word volume, it’s like I wrote way too much those past few days or something. Anyways, having now all of this in the open should help with getting out that explanation re shapes as well so there’s that at least still pending.

The texture planets, skies and outer space seem to be easiest to find at the intersection of the layered approach ¹ with various mappings for texture and colour domain. The stereo mapping of the texture domain provides as usual the way to zoom in and out, so that you can admire the newly found texture planet and its surrounding galaxy, all obtained through 2 rounds of perlin-based fbm (with incremental addition of results) and using otherwise the most basic of colour mappings (simply interpreting the x,y,z values as percentages of red, green and blue, respectively):

With a triangle-based domain mapping (z is considered the hypothenuse of a right angled triangle with given x and y as sides) and turning the knobs on the layered approach and/or the noisiness level of the fractal used, the results make rather atmospheric skies of sorts:

To tone down a bit all that clash of bright colours, an initial approach was to actually combine the final values for red, green and blue since that brings them basically again into a closer area and “closer” in the r,g,b space means closer colours & shades:

Given the success with the public of that unitsphere mapping in the r,g,b space, I applied it today to a few of those textures as well and it turned out perhaps not all that bad, as a sort of milky-rivers in the sky for instance:

Keeping this short but using the pics for all they can give, I went ahead and set two of them as skies as well when taking the shots at the hopefuls’ latest fashion parade – you’ll surely notice the milky one, though both are quite the departure from those tame clouds that are currently stuck to the euloran sky. The thing with making the sky more interesting seems to be that now one *really* wants those things in the sky to also …move about!

1. Calculate the new x,y,z coordinates in turn as the result of fractal brownian motion with perlin noise at the given point, always adding the previously calculated value as offset to the corresponding coordinate; possibly iterate this a couple of times.[↩]

This time I’ll start with 3 illustrations since I found that I’d rather paint of sorts than write these days ¹ and supposedly the pictures are anyway worth lots and lots of words. So here’s in order my texture generator’s take on Impressionism, Naive and Surrealism painting styles and feel free to disagree most strongly for it totally doesn’t care about your disagreement:

If you compare the above with the previous styles featured in this series (such as the Escher-like or the swirly river), you might get perhaps some idea that there are some specific choices made so as to obtain one style or another. Those choices focus currently on 3 main aspects that are worth setting out in clear and perhaps discussing in a bit more detail, too, in the order in which they come into play at texture generation time:

1. Texture domain mapping.
2. Pattern (where the whole fractal fun is going on atm).
3. Colour domain mapping.

The texture domain mapping refers to the 3D shape that the generator assumes essentially to be the target of this texture – in other words, the resulting texture is meant to be a 2D picture of the surface of that shape, if one were to fully peel and set it flat. Unsurprisingly perhaps, the choice of the specific 3D shape *and* of the way in which the mapping to 2D is done has quite significant effects on the image obtained, even if *everything else* is kept the same. In principle though, provided that the exact inverse mapping is used when rendering the texture on the corresponding 3D shape, the two “different” images should end up as similar “skins” – though so far I haven’t really spent the time to verify this fully in practice. In any case and for illustration, have a look at 2 textures that are generated using the very same pattern and colour mapping, only assuming first a hyperboloid 3D shape and then an egg 3D shape (just in case you needed to dress up your eggs and hyperboloids):

In addition to the shape itself, it’s worth noting also that any 3D to 2D mapping will make further choices, out of necessity. Even for the very simple case of a sphere, there are several different mappings to a 2D representation and the choice depends mostly on what specific property of the sphere’s surface interests you most (meaning: you’d rather have preserved in the 2D representation). Without going so far into all that detail, as I don’t think it’s all that crucial right now, I still had fun with adjusting the “zoom” factor of a stereographic mapping, hence the part of the sphere that is captured in the limited space of my 512×512 pixels texture. Unlike the earlier diagonal travel by means of (x,y,z) offsets through the river of eyes, this is literally a travel towards or away from whatever pattern you are setting otherwise. Besides the fun factor, this sort of movement in the texture space gives both different patterns (by focusing on one spot or another, as they can be quite different looking anyway) and a neat way to control otherwise where the detail ends up (or not) on the rendered mesh. At any rate, for the style of texture that is the focus of this article, the zoom in turns out for instance like this:

With respect to the pattern used for a texture, the choices are trickier to fully describe since in principle anything can go in here, from a predefined fixed, traditional pattern to any combination of fractals, multi-fractals, prng, trng, noise, interpolations and whatever else you can think of. Such choice doesn’t necessarily mean that there is no overlap ie that you can’t perhaps obtain the same thing through different means – in fact I’d rather say you surely can. What it means however is that it’s SO very easy to get lost and play for hours with it all, even stumble upon some interesting stuff and then go nuts trying to …find it again! So what can I say, like in any other labyrinth whose design is wider than you fully comprehend just yet, take your steps one at a time, keep detailed notes and be aware that even that might not be always quite enough.

The above warning aside (and surely immediately forgotten too), for this set of textures I wanted to experiment with introducing some prng into the mix. Essentially the whole game of generating textures and shapes consists at core in finding just the right mix of structure (not enough structure means that the result will look simply like noise/nothing/random rubbish) and chaos (not enough chaos and you get stunted, boring, “un-natural”/nothing/rubbish). In other words, the structure has to be there for the meaning, while the chaos is desired in there for the fun, variety and anti-boredom property. In principle, fractals themselves are already neatly tuned combinations of both structure and chaos but tipping this balance and then re-setting it at a different point is exactly how you get different *styles* of pictures to start with – the previous swirls and river of eyes is what you get for instance with fixed offsets added to all three coordinates, basically translating the texture’s domain with slightly different amounts on each of the axes. So if translation got me swirls, my next curiosity was to see: what does rotation do? And since I like all rotations just the same, so that I don’t want to choose any in particular, I went all out and picked at *each* pixel, 3 pseudo-random numbers from an MT prng, to serve as the angles of rotation around each of the 3 axes. Note that using the MT prng is effectively introducing some tamed, approximate form of chaos really, given the pseudo-part. At any rate, the fbm with perlin noise is then used to calculate a value for the (x,y,z) triplets that are obtained in turn by rotating around the x axis, then around the y axis, then around the z axis. The result of it all raises some interesting sharp “ridges” and looks for instance like this:

The above is only the beginning though. For once I got that, it was a given I’d want to further mess about with it. So next step was to apply on top of it, an effect I had previously discovered to give a sort of layered or transparent look to the whole thing – it’s yet another application of the fractal brownian motion (fbm) with perlin noise, but applying different translations for x, y and z coordinates. Because a direct full “layered” application was clearly way too much coming after the rotation and one round of fractal transformations, I’ve further introduced in between a clamp on the coordinates so that the whole rotation+translation stays within some closer bounds. Interestingly enough, whether this clamp allows negative values or not makes in practice a *lot* of difference – basically straddling the origin and being comfortable with negative numbers gets one all the way from the naive to the impressionist:

Further keeping the same but changing basically the “noisiness” of the fractal deformation itself yields a busier pattern both when using only the rotation + one fractal application and when using rotation+fractal+translation+fractal:

Having made it so far and therefore having set both the initial 3D shape (domain mapping) and the pattern (that rotation and possibly added translation with 2 sets of fractal brownian motions in the mix), I had to try out various combinations of the two, of course. Keeping all that rotation and translation with clamping and everything else but simply changing the domain mapping from the unit sphere ² changes of course quite radically the result but I admit that I like that little face kind of forming on the right:

Finally, the last big choice to make is the colour domain mapping – basically deciding how to interpret as colours whatever bunch of numbers come out of all the previous calculations. Generally I kept this part really straightforward and very simple so far, mainly because I found it way more interesting to play with other bits: the values obtained are translated to [0,1] interval and then interpreted as % of the r,g,b (x for r, y for g, z for b). But since I tidied up and set nicely all those three different areas of play and whatnot, the unbalance of options was unsufferable really, so I did first this half-hearted what-if of considering instead a unit sphere in the R,G,B space – and I mean half-hearted as I didn’t bother too much with either more interesting options or at the very least moving about that sphere in the space to spice it up a bit. Later on and being quite annoyed with it all, I did implement a proper colour spline too but so far I didn’t get around to really try it out so it doesn’t count. Anyways, at least for the above impressionist-style texture, the R,G,B unit sphere kind of switched the colours to pastels:

Finally and for completeness rather than for anything else, here’s this article’s fashion parade with yet another set of hopefuls wearing the above textures:

1. Could it have *anything* to do with the fact that I have a *mountain* of stuff to write up? No, of course not, nothing at all, none, not possible.[↩]
2. To… ahem, the complex plane: x and y are considered the re and im, while z is calculated as arctan(y/x), obviously; don’t ask me why I had implemented this particular mapping, I just… had it implemented there from earlier attempts and so I used it now.[↩]

The swirly texture style clearly held some promise of both types of materials (e.g. metalic) and recognizable shapes of faces and eyes. So it was one of the first directions I explored further with my newly tidied up texture generation scripts that turned out in short order a whole… river of crazy eyes effectively flowing diagonally as you advance from one image to the next. The “flow” was rather expected, since the images are generated with a given offset that is applied to all three coordinates so that they are indeed pushed at the same time. The multitude of eyes and half-baked faces forming or half-forming and then dissolving again as the river flows was less expected but rather fun to watch. Here’s a short sequence to illustrate how this works:

Given the obvious abundance of eyes and suggestive faces even in that river of textures, I was quite curious to see if one can indeed bring out some that may be even hidden under the surface. And it turns out – surprised? – that one actually can do even better – under the obvious face contour that is quite obvious at the very first glance, there is in fact quite an entirely different guy all sad for his lack of visibility – can you even spot him all hidden by the layers upon layers of fast-flowing features in this first picture?

Once he gets space to be seen all on his own, he can even be further nudged to smile at you

For all the coolness factor of that river of eyes from where this whole set of textures can be fished out and for all the fun factor of the textures themselves when seen as images on their own, their usefulness as skin covers for my hopefuls is rather dubious. Basically evaluating the fit of a texture has to take into account its match with the shape on which it will be painted and that’s not an easy task: for one thing the shape so far is quite difficult to describe anyway, seeing how it’s all fractally-deformed into all sorts; for another thing, the mapping of the texture to the shape makes it even less straightforward to directly figure out how the result will look like just by looking at the image shown plainly like this. In principle, the mapping currently in use wraps the texture around the shape – as the shape is not regular though, this round wrapping creates all sorts of effects and/or hides or distorts all sorts of features that are otherwise visible in the texture when seen as an image. As a result, whatever eyes or faces may be obvious in the texture itself, whether they show as such on a mesh or not is less clear upfront, as it can be seen in those shots of hopefuls wearing some textures from this set:

There is some little help that I can offer to someone trying to figure out this link between a texture image and the actual result when rendered on a hopeful: the whole trouble (and therefore potential solution too) stems from the repeated transformations between 2D and 3D domains. Starting from the texture generation itself, there’s first a 2D to 3D transformation of the texture’s own domain: the texture being a 2D image, it has only 2 dimensions that are called by convention u,v; as one iterates through the whole image to decide on a colour for each pixel, one has to make a first choice: is the texture to be considered 3D or 2D? If one sticks to 2D, things are more straightforward but there are of course more limitations as to the result itself. If one chooses 3D ¹, one way to look at it is that the 2D texture itself is generated as if it were carved out of the corresponding 3D material – through whichever path/way this first mapping of the texture domain itself defines. Once this is done, there’s a 2nd mapping in any case, this time to the 3D domain of colours. Finally, when used, there’s a 3rd mapping that goes the opposite way, from the 2D domain of the texture to the 3D domain of the shape on which the texture is applied. (And then there are of course further mappings as the whole thing is rendered on a 2D screen, after all.)

The above is the basic thing behind my previous generation of stereographically mapped textures (aka their 2D representation is the mapping of a sphere in the 3D “material” domain) with the interesting part in the centre so that it maps then in turn neatly onto the “north pole” of the mesh too. In a way, this does mean though that the fit between a texture and a given mesh would naturally be best if and only if the 3D domain of the texture at generation time is taken to be exactly the mesh itself with all its irregularities and weird shape. Perhaps it’s a direction worth exploring though at the moment I really can’t tell. The downside to it would be that fitting them *that* closely literally means one has to generate a texture for *each* mesh. On the other hand, if meshes are anyway reused to stand in for many parts of a creature and if the whole generation is automated anyway, it might still be worth a try. Nevertheless, this doesn’t solve otherwise the trouble of actually getting a pattern that one *wants* to see, anyway. That’s a different matter entirely and a different exploration all in itself.

1. For the record, all the textures I generated so far are in fact 3D textures, mainly because my implementation of noises and fractal transformations are 3D. I suppose I should at some point implement some 2D ones too, at least to be able to say that I explored that option too.[↩]

As the previous parade of hopefuls is essentially still ongoing, I focused for a bit on painting with numbers, functions and scripts to automate it all. This is the write-up, with pics further down, as usual, playing catch-up (but not yet winning) with what came out of it so far.

Initially I wanted simply to tidy up all those loose scripts for texture generation and tie them together into a more easily controllable tool for exploring the R,G,B space where all possible textures await discovery. The whole of it didn’t take all that long – in one day and a half (with assorted C/CPP related misery) over the weekend, all the relevant scripts were neatly separated into relevant categories, ready for mix-and-match or add-new-ones: image domain mapping, pattern generation, colour mapping. Then the full list of what was available got some additions as I started implemented all sorts of ideas gotten from my previous reading and then I started to give it a spin or two and then of course the whole carefully set up overall scripting went overboard because new things require new knobs to explore and there’s no way to set it all upfront for *all* of it. And when I finally sat down today to “write it up”, I looked in mild surprise at the accumulated *sets* of textures that have each a different style really because of the different underlying approach and various knobs and all that. While I could perhaps cram them all in a single article and be done with it, I think it makes more sense to take at least the write-up (and images per page) at a more reasonable pace and make a series out of it so that’s it a bit easier to load and to digest too. This way I might even remember to pace the rest too, next time! ¹

In a rather amusing inversion of terms, from a fractal point of view it turns out that the M.C. Escher style is really just about the most boringly classical you could make – all it takes is a perlin-based typical multifractal ² + the most basic of colouring mappings (such as interpreting the result as % of each colour) and you’re all set. Fiddle with the multifractal’s classical knobs controlling noise and amplitude and you’ll fit, as expected, increasingly more of it in each tiny patch until it all quickly turns to noise essentially – indistinguishable from noise, at least. Fiddle further with the additional offset knob and you can get some variation in pattern – as long as you keep the amplitude (hence number of octaves knob) quite low, up to a maximum of about 4 really. Worth noting here that the offset in this case simply acts as a fixed amount of *additional* amplitude at each iteration so basically making the fractal pack increasingly more/less in each shape already determined by the rest of knobs. So in this sense, the offset’s effect is limited really, at least compared to what one can truly do with offsets that are not tied up so tightly with the multifractal’s own parameters.

Without spending the time to fit any specific colour mappings to work just-so (or any other similar overfittings essentially), the pictures obtained are rather… boring, I should say, with all my heartfelt apologies going to Maurits Cornelis who did it all before there were computers to unleash the fractals just like that at the push of a button (after a lot of calculations and some code writing). Variations of pattern can be obtained to some degree with that limited knob controlling the offset but the essential trouble with this typical multifractal is that there’s only one offset and no way to further play around with it during the iterations themselves. Nevertheless, hopefuls were found to want to give a go at getting under those escher-like skins, so here’s the list of shots, starting with the textures themselves and then some hopefuls showing off the improved rotations by different angles on x,y and z axes as well as their newly found, skin-deep love of M.C. Escher:

In the next articles of this series, I shall leave classical Escher behind and illustrate for instance an exploration of the eyes-river, a travel through texture-galaxy and a stare into the trigonometry vortex that surely ate many other hopefuls before. Stay tuned or get out of tune, for the pictures are generated already and so I’ll have to write them up anyway!

1. I can imagine it. Just about… imagine it.[↩]
2. The very term multifractal is rather slippery since basically any combination of “fractal transformations” fits the notion of multifractal. Nevertheless, the simplest of those that I could find is this one used here: an fbm based on perlin noise, having in addition to the usual H and octaves, a fixed offset that gets applied (multiplication) at each iteration.[↩]

This week’s parade of hopefuls features a heartfelt attempt at zombiefication as well as several commendable efforts at horsification, frogification, carnivorousplantification and bring-your-own-imagination-ication. The show’s 4th and improved edition provides all participants with an updated set of skins (that come each with their own specific pose, too!) including the best ones from the latest fashion, a doubling up on that 2nd plane of symmetry (of sorts), an abundance of bones going at times all over the place and bigger screenshots for all the bigger monitors out there. There’s still a list of various things I want to try whenever I get a chance, including both texture and mesh attempts. For now though and for future reference, here’s again the list of changes implemented this week:

1. first plane is defined by the 2 points furthest apart from one another + the one point that is closest to those 2, where “closest to those 2” means that the “sum of distances from candidate to point 1 and to point 2 is minimum”;
2. ALL points (original as well as those reflected in the 1st plane) are reflected in the 2nd plane (so total ends up 4 times the original number but then the collapsing drops up to 3/4 of this total and usually it seems to drop indeed that many);
3. took out textures 8 and 9 (mandelbrot grays & mandelbrot blue+yellow);
4. added textures 1,4,13,14 from new set so the total textures now is 13;
5. replaced fixed pose with a rotation by a prng-number of degrees (around all three axes) using texture’s number as seed (so that each texture has its own); this is shown in the file’s name too, for easy ref;
6. client’s window is made bigger so that the screenshot is 1280px wide to start with, so the detail is there;
7. the fix to remove that seam from the texture mapping is in as well.

Here’s the absolute winner of this show’s edition as far as I’m concerned:

In other commendable entries that will therefore remain here even after the full dump ¹ gets wiped out, there’s Horsey and his gang of hopefuls:

Until next time, the past, present and future hopefuls, all want to wish you a well-jointed weekend with rounded symmetries, interesting skins and most pleasing rotations!

1. 50 skeleton seeds * 2 symmetry planes * 8 combination of points * 13 textures = 10400 screenshots. I’ve made them .jpg though as there isn’t any visible loss really and the size is half of png so that the whole thing does not weigh 5 Gigabytes or more, sheesh. Fwiw, it took me about 2 hours to just go through all of them this time.[↩]

Those past two days I finally got around to explore a few more directions I had in mind. My focus this time was on textures – mainly because it’s a faster way to see what various changes do to the result. Nevertheless, what I’m looking for otherwise is still focused more on the meshes – while the skeleton might be getting there, I think the mesh(es) still needs quite some improvement before there is more hope for the hopefuls. At any rate, I find I’d rather unload already the new stuff so read further for the list or scroll to the end for the pretty pictures, as usual.

The fun had yesterday and today yielded:

1. Cellular noise.I’ve finally implemented a new noise to play with alongside Perlin’s – it’s Worley’s and the main thing is that it’s quite on purpose “cellular” aka it’s based on linear combinations of the shortest n distances to a set of feature points that are probabilistically (Poisson) distributed in the 3D space. The main reason why I really wanted to see this in action is that a set of feature points – although possibly not Poisson distributed – might make perfect sense to consider when deforming shapes with the aim of getting something looking more like creatures. As usual, things are not all that clear to start with, mainly because there are so many ways to use each and every thing that at the end of the day ~anything is potentially possible – at least until one finally understands the whole thing enough to be able to figure out the reasons why it might not be possible! In any case, for starters I aimed to keep as close as possible to Worley’s original paper describing this cellular noise so I have an implementation that generates indeed -when used …plainly, let’s say- textures with “cells”. Scroll further down for the pics too.
2. Metallic, swirly, sparkly and other new textures. Trying out Worley’s noise plain was not half bad but trying it out with fractals turned interesting as I found out how easily it can all tend to …noise, heh. This had in turn the effect that I went back to playing with Perlin’s noise and multifractals and between the two, the result is that I’ve got what I can only describe as “metallic” textures, sparkly textures, swirly textures and ~any combination of those (swirly metallic texture being quite a thing for instance). By this stage I wish I could set some of those on fabric and start a fashion line as it would be way more interesting than many rags they fill the shops with. Come to think of it, it would be quite funny -although not all that surprising anymore- if Euloran characters end up better dressed than various “influencers” that are supposedly more than just a bunch of pixels on a screen.
3. Tweaked texture mapping. Another unexpected but productive side effect of this round of play is that I finally got thoroughly annoyed with that line/seam artefact that the stereo mapping of texture used so far always seems to create. And so I went and had another look at all my formulae there, with an eye to adjusting them so as to figure out if I can *also* have a version of it that eliminates that seam ¹. Like many other things, it turns out of course that all it takes is to know what to look for. Both coordinates use the arctan which returns results between -pi and pi, while the valid range for texture coordinates is between 0 and 1. So there is yet another place where options are available – and make a visible difference – depending on how this particular adjustment/mapping of intervals is made. At any rate, changing it to the more direct “just add pi and then divide all by 2*pi” turns out I’d say a better mapping in terms of how the full area of the texture gets shown on the mesh (though yes, if you want that seam, it’s not there anymore). One can further fiddle with it and keep the old adjustment for *one* of the coordinates while using the new for the other coordinate – yet another look and all that. Basically that’s the core trouble with looks – you can have so many of them for so little that it’s extremely easy to spend all the time just trying out nonsense in all its shapes and forms. At times even pretty nonsense, indeed.

As to the pretty pictures, there are some new textures and then a few screenshots to see them with the old and with the updated mapping as well. To start with, Worley’s noise is by definition quite “structured” and this shows immediately if simply plugged into the fractal brownian motion function as such. This is not to say that resulting images are not interesting of sorts – it’s simply that I think they end up rather tiresome on the eye and moreover, essentially too regular to serve that well as cover for any creature. In any case and for the memory of it if for nothing else, here are a few of them (some simple fractals, some multi-fractal giving them that layered look mainly):

Plain Worley’s noise without fractal messing about creates some cell structures as expected but that can be as simple as one big cell or as intricate as one desires – depending on the mapping of the original domain and otherwise on how the closest n distances are combined to give the final result. The textures that look “round” are by design exactly so – they use a stereographic mapping of the image’s own domain that is passed on to the Worley function, with the difference between the versions below being in how big (or small) the domain is considered. The ones with a tiny “centre” where everything happens can be quite useful if one wants to pop out some detail on a creature while simply colouring the rest to match but still remain more tame:

The additional exploration of multifractals using Perlin’s noise got this time all sorts, from sparkly texture to pronounced swirls – and I’ve specifically got a series of one set so that the effect of added detail is quite clearly visible (if one insists, after a point it all quite goes into noise/sparkle but most are really more noise than sparkle):

And here’s the fashion hopefuls-parade, experimental edition:

1. I’m saying a version of it because otherwise in some cases that seam might come in handy – it tends to create… noses perhaps, heh; at least a clearly separating midline that is not always such a bad thing.[↩]