Menger Sponge as an Isosurface?

 
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nextstep
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Joined: 06 Jan 2007
Posts: 158
PostPosted: Tue Sep 11, 2007 7:53 am    Post subject: Menger Sponge as an Isosurface? Reply with quote
Hi all,
This is an attempt to draw the Menger Sponge cube at a fourth level (M4) with K3DSurf Cool .
This example shows that the resolution of this model should be correct with a grid >150 but the complete definition of this model (the Isosurface formula) should take some hundred of pages Sad .
Maybe there is a better way to define it...It should be great to have a super compact formulas to draw it Razz
Code:
Name: MengerSponge
F(): if((x^2 >0.109 | y^2>.109) &(z^2 >0.109 | y^2>.109) & (x^2 >0.109 | z^2>.109)  &((x-2/3)^2 >0.0123456789 | (y-2/3)^2> 0.0123456789)&((x-2/3)^2 >0.0123456789 | (z-2/3)^2> 0.0123456789)&((y-2/3)^2 >0.0123456789 | (z-2/3)^2> 0.0123456789)&((x-2/3-2/9)^2 >0.0013717 | (y-2/3-2/9)^2> 0.0013717)&((x-2/3-2/9)^2 >0.0013717 | (z-2/3-2/9)^2> 0.0013717)&((y-2/3-2/9)^2 >0.0013717 | (z-2/3-2/9)^2>0.0013717 )&((x-2/3-2/9-2/27)^2 >.00015241383936 | (y-2/3-2/9-2/27)^2>.00015241383936 )&((x-2/3-2/9-2/27)^2 >.00015241383936 | (z-2/3-2/9-2/27)^2> .00015241383936)&((y-2/3-2/9-2/27)^2 >.00015241383936 | (z-2/3-2/9-2/27)^2>.00015241383936 ), x^100 + y^100 + z^100 -1, 1)
[x]: -1.01 , 1.01
[y]: -1.01 , 1.01
[z]: -1.01 , 1.01
;



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Taha
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jotero



Joined: 27 Jan 2007
Posts: 63
Location: Germany Hannover
PostPosted: Tue Sep 11, 2007 9:40 am    Post subject: Reply with quote
hello all Smile

WOW Shocked very good idea taha Smile

the finished model looks however differently Cool

ciao
torolf
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nextstep
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Joined: 06 Jan 2007
Posts: 158
PostPosted: Tue Sep 11, 2007 9:53 pm    Post subject: Reply with quote
Hi all,
Thanks Torolf and here is the Menger Sponge at level 2 .
I think I've found a way to make it up to the level 4 Razz .
wait and see Cool
Quote:
Name: MengerSponge_L2
F(): if(( abs(x) >1/3 | abs(y) > 1/3 ) &( abs(z) >1/3 | abs(y) > 1/3 ) &( abs(x) >1/3 | abs(z) > 1/3 ) & ( (( ( (abs(x-2/3) <1/9) + (abs(x+2/3) <1/9) + (abs(x) <1/9) ) * ((abs(y-2/3) < 1/9) +(abs(y) < 1/9) +(abs(y+2/3) <1/9) + (abs(z-2/3) < 1/9) +(abs(z) < 1/9) + (abs(z+2/3) < 1/9))) +( ( (abs(z-2/3) <1/9) + (abs(z+2/3) <1/9) + (abs(z) <1/9) ) * ( (abs(y-2/3) < 1/9) + (abs(y) < 1/9) + (abs(y+2/3) < 1/9))) ) =0), x^100 + y^100 + z^100 -1,1)
[x]: -1.001 , 1.001
[y]: -1.001 , 1.001
[z]: -1.001 , 1.001;



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nextstep
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PostPosted: Wed Sep 12, 2007 4:03 am    Post subject: Reply with quote
Hi all,
Welcome to the Menger Sponge Level 3 Razz :
Quote:
Name: MS3
F(): if(( abs(x) >1/3 | abs(y) > 1/3 ) &( abs(z) >1/3 | abs(y) > 1/3 ) &( abs(x) >1/3 | abs(z) > 1/3 ) & ( (( ( (abs(x-2/3) <1/9) + (abs(x+2/3) <1/9) + (abs(x) <1/9) ) * ((abs(y-2/3) < 1/9) +(abs(y) < 1/9) +(abs(y+2/3) <1/9) + (abs(z-2/3) < 1/9) +(abs(z) < 1/9) + (abs(z+2/3) < 1/9))) +( ( (abs(z-2/3) <1/9) + (abs(z+2/3) <1/9) + (abs(z) <1/9) ) * ( (abs(y-2/3) < 1/9) + (abs(y) < 1/9) + (abs(y+2/3) < 1/9))) ) =0)&( (( ( (abs(x-2/3) <1/27)+(abs(x-2/3-2/9) <1/27) + (abs(x-2/3+2/9) <1/27) +(abs(x+2/3) <1/27) +(abs(x+2/3-2/9) <1/27) + (abs(x+2/3+2/9) <1/27) +(abs(x) <1/27) + (abs(x-2/9) <1/27) + (abs(x+2/9) <1/27)) * ((abs(y-2/3) < 1/27) + (abs(y-2/3-2/9) < 1/27) + (abs(y-2/3+2/9) < 1/27) +(abs(y) < 1/27) + (abs(y-2/9) < 1/27) + (abs(y+2/9) < 1/27) + (abs(y+2/3) <1/27) + (abs(y+2/3-2/9) <1/27) + (abs(y+2/3+2/9) <1/27) + (abs(z-2/3) < 1/27) + (abs(z-2/3-2/9) < 1/27) + (abs(z-2/3+2/9) < 1/27) + (abs(z) < 1/27) + (abs(z-2/9) < 1/27) + (abs(z+2/9) < 1/27) + (abs(z+2/3) < 1/27) + (abs(z+2/3-2/9) < 1/27) + (abs(z+2/3+2/9) < 1/27) )) +( ( (abs(z-2/3) <1/27) + (abs(z-2/3-2/9) <1/27) + (abs(z-2/3+2/9) <1/27) + (abs(z+2/3) <1/27) + (abs(z+2/3-2/9) <1/27) + (abs(z+2/3+2/9) <1/27) + (abs(z) <1/27) + (abs(z-2/9) <1/27) + (abs(z+2/9) <1/27) ) * ( (abs(y-2/3) < 1/27) + (abs(y-2/3-2/9) < 1/27) + (abs(y-2/3+2/9) < 1/27) + (abs(y) < 1/27) + (abs(y-2/9) < 1/27) + (abs(y+2/9) < 1/27) + (abs(y+2/3) < 1/27) + (abs(y+2/3-2/9) < 1/27) + (abs(y+2/3+2/9) < 1/27)))) =0), x^100 + y^100 + z^100 -1,1)
[x]: -1.001 , 1.001
[y]: -1.001 , 1.001
[z]: -1.001 , 1.001
;




Now, we still have only one level to reach the M4!
Cool
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jotero



Joined: 27 Jan 2007
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Location: Germany Hannover
PostPosted: Wed Sep 12, 2007 4:13 am    Post subject: Reply with quote
hello all Smile

i am speechless! very well made of you taha Embarassed Laughing

ciao
torolf
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nextstep
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PostPosted: Wed Sep 12, 2007 7:42 pm    Post subject: Menger Sponge level 4 Reply with quote
Hi all,
Thanks Torolf and here it is the famous Menger Sponge level 4 : Very Happy
Quote:
Name: MS4
F(): if(( abs(x) >1/3 | abs(y) > 1/3 ) &( abs(z) >1/3 | abs(y) > 1/3 ) &( abs(x) >1/3 | abs(z) > 1/3 )
&
( (( ( (abs(x-2/3) <1/9) + (abs(x+2/3) <1/9) + (abs(x) <1/9) ) * ((abs(y-2/3) < 1/9) +(abs(y) < 1/9) +(abs(y+2/3) <1/9) + (abs(z-2/3) < 1/9) +(abs(z) < 1/9) + (abs(z+2/3) < 1/9))) +( ( (abs(z-2/3) <1/9) + (abs(z+2/3) <1/9) + (abs(z) <1/9) ) * ( (abs(y-2/3) < 1/9) + (abs(y) < 1/9) + (abs(y+2/3) < 1/9))) ) =0)
&
( (( ( (abs(x-2/3) <1/27)+(abs(x-2/3-2/9) <1/27) + (abs(x-2/3+2/9) <1/27) +(abs(x+2/3) <1/27) +(abs(x+2/3-2/9) <1/27) + (abs(x+2/3+2/9) <1/27) +(abs(x) <1/27) + (abs(x-2/9) <1/27) + (abs(x+2/9) <1/27)) * ((abs(y-2/3) < 1/27) + (abs(y-2/3-2/9) < 1/27) + (abs(y-2/3+2/9) < 1/27) +(abs(y) < 1/27) + (abs(y-2/9) < 1/27) + (abs(y+2/9) < 1/27) + (abs(y+2/3) <1/27) + (abs(y+2/3-2/9) <1/27) + (abs(y+2/3+2/9) <1/27) + (abs(z-2/3) < 1/27) + (abs(z-2/3-2/9) < 1/27) + (abs(z-2/3+2/9) < 1/27) + (abs(z) < 1/27) + (abs(z-2/9) < 1/27) + (abs(z+2/9) < 1/27) + (abs(z+2/3) < 1/27) + (abs(z+2/3-2/9) < 1/27) + (abs(z+2/3+2/9) < 1/27) )) +( ( (abs(z-2/3) <1/27) + (abs(z-2/3-2/9) <1/27) + (abs(z-2/3+2/9) <1/27) + (abs(z+2/3) <1/27) + (abs(z+2/3-2/9) <1/27) + (abs(z+2/3+2/9) <1/27) + (abs(z) <1/27) + (abs(z-2/9) <1/27) + (abs(z+2/9) <1/27) ) * ( (abs(y-2/3) < 1/27) + (abs(y-2/3-2/9) < 1/27) + (abs(y-2/3+2/9) < 1/27) + (abs(y) < 1/27) + (abs(y-2/9) < 1/27) + (abs(y+2/9) < 1/27) + (abs(y+2/3) < 1/27) + (abs(y+2/3-2/9) < 1/27) + (abs(y+2/3+2/9) < 1/27)))) =0)
&
( ((
(
(abs(x-2/3) <1/81)+ (abs(x-2/3-2/27) <1/81)+ (abs(x-2/3+2/27) <1/81)+
(abs(x-2/3-2/9) <1/81) + (abs(x-2/3-2/9-2/27) <1/81) + (abs(x-2/3-2/9+2/27) <1/81) +
(abs(x-2/3+2/9) <1/81) +(abs(x-2/3+2/9-2/27) <1/81) +(abs(x-2/3+2/9+2/27) <1/81) +
(abs(x+2/3) <1/81) +(abs(x+2/3-2/27) <1/81) +(abs(x+2/3+2/27) <1/81) +
(abs(x+2/3-2/9) <1/81) +(abs(x+2/3-2/9-2/27) <1/81) +(abs(x+2/3-2/9+2/27) <1/81) +
(abs(x+2/3+2/9) <1/81) +(abs(x+2/3+2/9-2/27) <1/81) +(abs(x+2/3+2/9+2/27) <1/81) +
(abs(x) <1/81) + (abs(x-2/27) <1/81) +(abs(x+2/27) <1/81) +
(abs(x-2/9) <1/81) + (abs(x-2/9-2/27) <1/81) + (abs(x-2/9+2/27) <1/81) +
(abs(x+2/9) <1/81)+(abs(x+2/9-2/27) <1/81)+(abs(x+2/9+2/27) <1/81)
) *
(
(abs(y-2/3) < 1/81) + (abs(y-2/3-2/27) < 1/81) +(abs(y-2/3+2/27) < 1/81) +
(abs(y-2/3-2/9) < 1/81) + (abs(y-2/3-2/9-2/27) < 1/81) +(abs(y-2/3-2/9+2/27) < 1/81) +
(abs(y-2/3+2/9) < 1/81) +(abs(y-2/3+2/9-2/27) < 1/81) +(abs(y-2/3+2/9+2/27) < 1/81) +
(abs(y) < 1/81) + (abs(y-2/27) < 1/81) + (abs(y+2/27) < 1/81) +
(abs(y-2/9) < 1/81) + (abs(y-2/9-2/27) < 1/81) + (abs(y-2/9+2/27) < 1/81) +
(abs(y+2/9) < 1/81) + (abs(y+2/9-2/27) < 1/81) + (abs(y+2/9+2/27) < 1/81) +
(abs(y+2/3) <1/81) + (abs(y+2/3-2/27) <1/81) + (abs(y+2/3+2/27) <1/81) +
(abs(y+2/3-2/9) <1/81) + (abs(y+2/3-2/9-2/27) <1/81) + (abs(y+2/3-2/9+2/27) <1/81) +
(abs(y+2/3+2/9) <1/81) + (abs(y+2/3+2/9-2/27) <1/81) + (abs(y+2/3+2/9+2/27) <1/81) +
(abs(z-2/3) < 1/81) + (abs(z-2/3-2/27) < 1/81) + (abs(z-2/3+2/27) < 1/81) +
(abs(z-2/3-2/9) < 1/81) + (abs(z-2/3-2/9-2/27) < 1/81) + (abs(z-2/3-2/9+2/27) < 1/81) +
(abs(z-2/3+2/9) < 1/81) + (abs(z-2/3+2/9-2/27) < 1/81) + (abs(z-2/3+2/9+2/27) < 1/81) +
(abs(z) < 1/81) + (abs(z-2/27) < 1/81) + (abs(z+2/27) < 1/81) +
(abs(z-2/9) < 1/81) + (abs(z-2/9-2/27) < 1/81) + (abs(z-2/9+2/27) < 1/81) +
(abs(z+2/9) < 1/81) + (abs(z+2/9-2/27) < 1/81) + (abs(z+2/9+2/27) < 1/81) +
(abs(z+2/3) < 1/81) + (abs(z+2/3-2/27) < 1/81) + (abs(z+2/3+2/27) < 1/81) +
(abs(z+2/3-2/9) < 1/81) + (abs(z+2/3-2/9-2/27) < 1/81) + (abs(z+2/3-2/9+2/27) < 1/81) +
(abs(z+2/3+2/9) < 1/81) +(abs(z+2/3+2/9-2/27) < 1/81) +(abs(z+2/3+2/9+2/27) < 1/81)
)) +
(
(
(abs(z-2/3) <1/81) + (abs(z-2/3-2/27) <1/81) + (abs(z-2/3+2/27) <1/81) +
(abs(z-2/3-2/9) <1/81) + (abs(z-2/3-2/9-2/27) <1/81) + (abs(z-2/3-2/9+2/27) <1/81) +
(abs(z-2/3+2/9) <1/81) + (abs(z-2/3+2/9-2/27) <1/81) + (abs(z-2/3+2/9+2/27) <1/81) +
(abs(z+2/3) <1/81) + (abs(z+2/3-2/27) <1/81) + (abs(z+2/3+2/27) <1/81) +
(abs(z+2/3-2/9) <1/81) + (abs(z+2/3-2/9-2/27) <1/81) + (abs(z+2/3-2/9+2/27) <1/81) +
(abs(z+2/3+2/9) <1/81) + (abs(z+2/3+2/9-2/27) <1/81) + (abs(z+2/3+2/9+2/27) <1/81) +
(abs(z) <1/81) + (abs(z-2/27) <1/81) + (abs(z+2/27) <1/81) +
(abs(z-2/9) <1/81) + (abs(z-2/9-2/27) <1/81) + (abs(z-2/9+2/27) <1/81) +
(abs(z+2/9) <1/81) + (abs(z+2/9-2/27) <1/81) + (abs(z+2/9+2/27) <1/81)
) * (
(abs(y-2/3) < 1/81) + (abs(y-2/3-2/27) < 1/81) + (abs(y-2/3+2/27) < 1/81) +
(abs(y-2/3-2/9) < 1/81) + (abs(y-2/3-2/9-2/27) < 1/81) + (abs(y-2/3-2/9+2/27) < 1/81) +
(abs(y-2/3+2/9) < 1/81) + (abs(y-2/3+2/9-2/27) < 1/81) + (abs(y-2/3+2/9+2/27) < 1/81) +
(abs(y) < 1/81) + (abs(y-2/27) < 1/81) + (abs(y+2/27) < 1/81) +
(abs(y-2/9) < 1/81) + (abs(y-2/9-2/27) < 1/81) + (abs(y-2/9+2/27) < 1/81) +
(abs(y+2/9) < 1/81) + (abs(y+2/9-2/27) < 1/81) + (abs(y+2/9+2/27) < 1/81) +
(abs(y+2/3) < 1/81) + (abs(y+2/3-2/27) < 1/81) + (abs(y+2/3+2/27) < 1/81) +
(abs(y+2/3-2/9) < 1/81) + (abs(y+2/3-2/9-2/27) < 1/81) + (abs(y+2/3-2/9+2/27) < 1/81) +
(abs(y+2/3+2/9) < 1/81) + (abs(y+2/3+2/9-2/27) < 1/81) + (abs(y+2/3+2/9+2/27) < 1/81)
)
)) =0)

, x^1000 + y^1000 + z^1000 -1,1)
[x]: -1.001 , 1.001
[y]: -1.001 , 1.001
[z]: -1.001 , 1.001
;





Here you can see a detailed view ( 1/512) of the entire cube. This shows that this isosurface formulas is very accurate! I'll show later an algorithm for generating formulas for any level of the Menger Sponge. This maybe not the best description of the Menger Sponge but it has the advantage to be quite easy to write. Also, this opens new perspectives for new mathematical models!


Enjoy,
Taha
Very Happy
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Cheers,
Taha
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nextstep
Site Admin


Joined: 06 Jan 2007
Posts: 158
PostPosted: Thu Sep 13, 2007 5:47 pm    Post subject: Reply with quote
Hi all,
A new version of the Menger Sponge level 4. It's more than two times faster the previous one Razz
Quote:

Name: MengerSpongeL4_v2
F(): if(( abs(x) >1/3 | abs(y) > 1/3 ) &( abs(z) >1/3 | abs(y) > 1/3 ) &( abs(x) >1/3 | abs(z) > 1/3 ) ,
if(
( (( ( (abs(x-2/3) <1/9) + (abs(x+2/3) <1/9) + (abs(x) <1/9) ) * ((abs(y-2/3) < 1/9) +(abs(y) < 1/9) +(abs(y+2/3) <1/9) + (abs(z-2/3) < 1/9) +(abs(z) < 1/9) + (abs(z+2/3) < 1/9))) +( ( (abs(z-2/3) <1/9) + (abs(z+2/3) <1/9) + (abs(z) <1/9) ) * ( (abs(y-2/3) < 1/9) + (abs(y) < 1/9) + (abs(y+2/3) < 1/9))) ) =0),
if(
( (( ( (abs(x-2/3) <1/27)+(abs(x-2/3-2/9) <1/27) + (abs(x-2/3+2/9) <1/27) +(abs(x+2/3) <1/27) +(abs(x+2/3-2/9) <1/27) + (abs(x+2/3+2/9) <1/27) +(abs(x) <1/27) + (abs(x-2/9) <1/27) + (abs(x+2/9) <1/27)) * ((abs(y-2/3) < 1/27) + (abs(y-2/3-2/9) < 1/27) + (abs(y-2/3+2/9) < 1/27) +(abs(y) < 1/27) + (abs(y-2/9) < 1/27) + (abs(y+2/9) < 1/27) + (abs(y+2/3) <1/27) + (abs(y+2/3-2/9) <1/27) + (abs(y+2/3+2/9) <1/27) + (abs(z-2/3) < 1/27) + (abs(z-2/3-2/9) < 1/27) + (abs(z-2/3+2/9) < 1/27) + (abs(z) < 1/27) + (abs(z-2/9) < 1/27) + (abs(z+2/9) < 1/27) + (abs(z+2/3) < 1/27) + (abs(z+2/3-2/9) < 1/27) + (abs(z+2/3+2/9) < 1/27) )) +( ( (abs(z-2/3) <1/27) + (abs(z-2/3-2/9) <1/27) + (abs(z-2/3+2/9) <1/27) + (abs(z+2/3) <1/27) + (abs(z+2/3-2/9) <1/27) + (abs(z+2/3+2/9) <1/27) + (abs(z) <1/27) + (abs(z-2/9) <1/27) + (abs(z+2/9) <1/27) ) * ( (abs(y-2/3) < 1/27) + (abs(y-2/3-2/9) < 1/27) + (abs(y-2/3+2/9) < 1/27) + (abs(y) < 1/27) + (abs(y-2/9) < 1/27) + (abs(y+2/9) < 1/27) + (abs(y+2/3) < 1/27) + (abs(y+2/3-2/9) < 1/27) + (abs(y+2/3+2/9) < 1/27)))) =0),
if(
( ((
(
(abs(x-2/3) <1/81)+ (abs(x-2/3-2/27) <1/81)+ (abs(x-2/3+2/27) <1/81)+
(abs(x-2/3-2/9) <1/81) + (abs(x-2/3-2/9-2/27) <1/81) + (abs(x-2/3-2/9+2/27) <1/81) +
(abs(x-2/3+2/9) <1/81) +(abs(x-2/3+2/9-2/27) <1/81) +(abs(x-2/3+2/9+2/27) <1/81) +
(abs(x+2/3) <1/81) +(abs(x+2/3-2/27) <1/81) +(abs(x+2/3+2/27) <1/81) +
(abs(x+2/3-2/9) <1/81) +(abs(x+2/3-2/9-2/27) <1/81) +(abs(x+2/3-2/9+2/27) <1/81) +
(abs(x+2/3+2/9) <1/81) +(abs(x+2/3+2/9-2/27) <1/81) +(abs(x+2/3+2/9+2/27) <1/81) +
(abs(x) <1/81) + (abs(x-2/27) <1/81) +(abs(x+2/27) <1/81) +
(abs(x-2/9) <1/81) + (abs(x-2/9-2/27) <1/81) + (abs(x-2/9+2/27) <1/81) +
(abs(x+2/9) <1/81)+(abs(x+2/9-2/27) <1/81)+(abs(x+2/9+2/27) <1/81)
) *
(
(abs(y-2/3) < 1/81) + (abs(y-2/3-2/27) < 1/81) +(abs(y-2/3+2/27) < 1/81) +
(abs(y-2/3-2/9) < 1/81) + (abs(y-2/3-2/9-2/27) < 1/81) +(abs(y-2/3-2/9+2/27) < 1/81) +
(abs(y-2/3+2/9) < 1/81) +(abs(y-2/3+2/9-2/27) < 1/81) +(abs(y-2/3+2/9+2/27) < 1/81) +
(abs(y) < 1/81) + (abs(y-2/27) < 1/81) + (abs(y+2/27) < 1/81) +
(abs(y-2/9) < 1/81) + (abs(y-2/9-2/27) < 1/81) + (abs(y-2/9+2/27) < 1/81) +
(abs(y+2/9) < 1/81) + (abs(y+2/9-2/27) < 1/81) + (abs(y+2/9+2/27) < 1/81) +
(abs(y+2/3) <1/81) + (abs(y+2/3-2/27) <1/81) + (abs(y+2/3+2/27) <1/81) +
(abs(y+2/3-2/9) <1/81) + (abs(y+2/3-2/9-2/27) <1/81) + (abs(y+2/3-2/9+2/27) <1/81) +
(abs(y+2/3+2/9) <1/81) + (abs(y+2/3+2/9-2/27) <1/81) + (abs(y+2/3+2/9+2/27) <1/81) +
(abs(z-2/3) < 1/81) + (abs(z-2/3-2/27) < 1/81) + (abs(z-2/3+2/27) < 1/81) +
(abs(z-2/3-2/9) < 1/81) + (abs(z-2/3-2/9-2/27) < 1/81) + (abs(z-2/3-2/9+2/27) < 1/81) +
(abs(z-2/3+2/9) < 1/81) + (abs(z-2/3+2/9-2/27) < 1/81) + (abs(z-2/3+2/9+2/27) < 1/81) +
(abs(z) < 1/81) + (abs(z-2/27) < 1/81) + (abs(z+2/27) < 1/81) +
(abs(z-2/9) < 1/81) + (abs(z-2/9-2/27) < 1/81) + (abs(z-2/9+2/27) < 1/81) +
(abs(z+2/9) < 1/81) + (abs(z+2/9-2/27) < 1/81) + (abs(z+2/9+2/27) < 1/81) +
(abs(z+2/3) < 1/81) + (abs(z+2/3-2/27) < 1/81) + (abs(z+2/3+2/27) < 1/81) +
(abs(z+2/3-2/9) < 1/81) + (abs(z+2/3-2/9-2/27) < 1/81) + (abs(z+2/3-2/9+2/27) < 1/81) +
(abs(z+2/3+2/9) < 1/81) +(abs(z+2/3+2/9-2/27) < 1/81) +(abs(z+2/3+2/9+2/27) < 1/81)
)) +
(
(
(abs(z-2/3) <1/81) + (abs(z-2/3-2/27) <1/81) + (abs(z-2/3+2/27) <1/81) +
(abs(z-2/3-2/9) <1/81) + (abs(z-2/3-2/9-2/27) <1/81) + (abs(z-2/3-2/9+2/27) <1/81) +
(abs(z-2/3+2/9) <1/81) + (abs(z-2/3+2/9-2/27) <1/81) + (abs(z-2/3+2/9+2/27) <1/81) +
(abs(z+2/3) <1/81) + (abs(z+2/3-2/27) <1/81) + (abs(z+2/3+2/27) <1/81) +
(abs(z+2/3-2/9) <1/81) + (abs(z+2/3-2/9-2/27) <1/81) + (abs(z+2/3-2/9+2/27) <1/81) +
(abs(z+2/3+2/9) <1/81) + (abs(z+2/3+2/9-2/27) <1/81) + (abs(z+2/3+2/9+2/27) <1/81) +
(abs(z) <1/81) + (abs(z-2/27) <1/81) + (abs(z+2/27) <1/81) +
(abs(z-2/9) <1/81) + (abs(z-2/9-2/27) <1/81) + (abs(z-2/9+2/27) <1/81) +
(abs(z+2/9) <1/81) + (abs(z+2/9-2/27) <1/81) + (abs(z+2/9+2/27) <1/81)
) * (
(abs(y-2/3) < 1/81) + (abs(y-2/3-2/27) < 1/81) + (abs(y-2/3+2/27) < 1/81) +
(abs(y-2/3-2/9) < 1/81) + (abs(y-2/3-2/9-2/27) < 1/81) + (abs(y-2/3-2/9+2/27) < 1/81) +
(abs(y-2/3+2/9) < 1/81) + (abs(y-2/3+2/9-2/27) < 1/81) + (abs(y-2/3+2/9+2/27) < 1/81) +
(abs(y) < 1/81) + (abs(y-2/27) < 1/81) + (abs(y+2/27) < 1/81) +
(abs(y-2/9) < 1/81) + (abs(y-2/9-2/27) < 1/81) + (abs(y-2/9+2/27) < 1/81) +
(abs(y+2/9) < 1/81) + (abs(y+2/9-2/27) < 1/81) + (abs(y+2/9+2/27) < 1/81) +
(abs(y+2/3) < 1/81) + (abs(y+2/3-2/27) < 1/81) + (abs(y+2/3+2/27) < 1/81) +
(abs(y+2/3-2/9) < 1/81) + (abs(y+2/3-2/9-2/27) < 1/81) + (abs(y+2/3-2/9+2/27) < 1/81) +
(abs(y+2/3+2/9) < 1/81) + (abs(y+2/3+2/9-2/27) < 1/81) + (abs(y+2/3+2/9+2/27) < 1/81)
)
)) =0)
, x^1000 + y^1000 + z^1000 -1, 1), 1), 1), 1)
[x]: -1.001 , 1.001
[y]: -1.001 , 1.001
[z]: -1.001 , 1.001
;

_________________
Cheers,
Taha
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inode



Joined: 27 Jan 2007
Posts: 60
Location: Austria
PostPosted: Fri Sep 14, 2007 11:28 pm    Post subject: Reply with quote
MengerSponge + Pacman = MengerPacman (Razz)

Quote:
Name: MengerPacman
F(): if((abs(x)>t/2 | abs(y)>t/2)
&(abs(z)>t/2 | abs(y)>t/2)
&(abs(x)>t/2 | abs(z)>t/2)
& (((((abs(x-2/3)<t/6) + (abs(x+2/3)<t/6) + (abs(x)<t/6))
* ((abs(y-2/3)<t/6) + (abs(y)<t/6) + (abs(y+2/3)<t/6)
+ (abs(z-2/3)<t/6) + (abs(z)<t/6) + (abs(z+2/3)<t/6)))
+ (((abs(z-2/3)<t/6) + (abs(z+2/3)<t/6) + (abs(z)<t/6))
* ((abs(y-2/3)<t/6) + (abs(y)<t/6) + (abs(y+2/3)<t/6)))) =0)
, x^100 + y^100 + z^100 -1,1)
[x]: -1.001 , 1.001
[y]: -1.001 , 1.001
[z]: -1.001 , 1.001
;


set grid to 32x32x32 and start morphing Wink Gerd

ps. for me Menger Sponge L4 looks like a brain twister.
that can't be created by hand - or?

pps. formula definition would be clearer and easier to define using constants
(e.g. k1=1/3; k2=1/9; k3=2/3-2/27; etc...)
do you think calculation would be also faster with constant definitions?
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nextstep
Site Admin


Joined: 06 Jan 2007
Posts: 158
PostPosted: Sun Sep 16, 2007 6:08 am    Post subject: Reply with quote
Hi all,
The MengerPacman variation is very interesting. Thanks Smile
Quote:
ps. for me Menger Sponge L4 looks like a brain twister.
that can't be created by hand - or?

It's a brain twister however it can be made by hand and the proof is on the top Razz . This shows that there are plenty of discoveries to be made by using isosurfaces.
Quote:

pps. formula definition would be clearer and easier to define using constants
(e.g. k1=1/3; k2=1/9; k3=2/3-2/27; etc...)

Of course however this can hide some details to me in the process of optimization, so right now I prefer to let things as they are...
Quote:

do you think calculation would be also faster with constant definitions?

Yes and calculations will be much more faster with an optimization of the formulas itself (compare the two versions of the MS4 and I'm still working on the third versions Cool ).
_________________
Cheers,
Taha
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nextstep
Site Admin


Joined: 06 Jan 2007
Posts: 158
PostPosted: Fri Sep 21, 2007 5:58 am    Post subject: Reply with quote
Hi all,
Quote:

do you think calculation would be also faster with constant definitions?

K3DSurf 0.6.2 has an unique feature that make it extremely fast: I really think it's the fastest mathematical tool ever made (including the commercial ones like Mathematica) Cool . Why I'm saying that? because it includes a new mathematical parser technology for height speed calculations. As far as I know, there is no references to this on the specialized books or in the literature. So, lets play with it to see what can we get from this "secret" tool Razz .
First, this tool support all mathematical functions...except the "if" instruction Confused so we have to eliminate it from the MS4 formulas.
The first Menger Sponge level 4 isosurface can be changed to this (to eliminate the "if" instruction).
Quote:
Name: MS4
F(): -( x^100 + y^100 + z^100 -1)*
((abs(x) >1/3 | abs(y) > 1/3)*(abs(x) >1/3 | abs(z) > 1/3)*(abs(z) >1/3 | abs(y) > 1/3))

* ( (((abs(x-2/3) <1/9) + (abs(x+2/3) <1/9) + (abs(x) <1/9))*((abs(y-2/3) <1/9) + (abs(y+2/3) <1/9) + (abs(y) <1/9)+(abs(z-2/3) <1/9) + (abs(z+2/3) <1/9) + (abs(z) <1/9)) + ((abs(z-2/3) <1/9) + (abs(z+2/3) <1/9) + (abs(z) <1/9))*((abs(y-2/3) <1/9) + (abs(y+2/3) <1/9) + (abs(y) <1/9))) =0)


* ( (((abs(x-2/3) <1/27)+(abs(x-2/3-2/9) <1/27) + (abs(x-2/3+2/9) <1/27) +(abs(x+2/3) <1/27) +(abs(x+2/3-2/9) <1/27) + (abs(x+2/3+2/9) <1/27) +(abs(x) <1/27) + (abs(x-2/9) <1/27) + (abs(x+2/9) <1/27))*((abs(y-2/3) <1/27)+(abs(y-2/3-2/9) <1/27) + (abs(y-2/3+2/9) <1/27) +(abs(y+2/3) <1/27) +(abs(y+2/3-2/9) <1/27) + (abs(y+2/3+2/9) <1/27) +(abs(y) <1/27) + (abs(y-2/9) <1/27) + (abs(y+2/9) <1/27)+(abs(z-2/3) <1/27)+(abs(z-2/3-2/9) <1/27) + (abs(z-2/3+2/9) <1/27) +(abs(z+2/3) <1/27) +(abs(z+2/3-2/9) <1/27) + (abs(z+2/3+2/9) <1/27) +(abs(z) <1/27) + (abs(z-2/9) <1/27) + (abs(z+2/9) <1/27)) + ((abs(z-2/3) <1/27)+(abs(z-2/3-2/9) <1/27) + (abs(z-2/3+2/9) <1/27) +(abs(z+2/3) <1/27) +(abs(z+2/3-2/9) <1/27) + (abs(z+2/3+2/9) <1/27) +(abs(z) <1/27) + (abs(z-2/9) <1/27) + (abs(z+2/9) <1/27))*((abs(y-2/3) <1/27)+(abs(y-2/3-2/9) <1/27) + (abs(y-2/3+2/9) <1/27) +(abs(y+2/3) <1/27) +(abs(y+2/3-2/9) <1/27) + (abs(y+2/3+2/9) <1/27) +(abs(y) <1/27) + (abs(y-2/9) <1/27) + (abs(y+2/9) <1/27))) =0)


* ( ( ((abs(x-2/3) <1/81)+ (abs(x-2/3-2/27) <1/81)+ (abs(x-2/3+2/27) <1/81)+
(abs(x-2/3-2/9) <1/81) + (abs(x-2/3-2/9-2/27) <1/81) + (abs(x-2/3-2/9+2/27) <1/81) +
(abs(x-2/3+2/9) <1/81) +(abs(x-2/3+2/9-2/27) <1/81) +(abs(x-2/3+2/9+2/27) <1/81) +
(abs(x+2/3) <1/81) +(abs(x+2/3-2/27) <1/81) +(abs(x+2/3+2/27) <1/81) +
(abs(x+2/3-2/9) <1/81) +(abs(x+2/3-2/9-2/27) <1/81) +(abs(x+2/3-2/9+2/27) <1/81) +
(abs(x+2/3+2/9) <1/81) +(abs(x+2/3+2/9-2/27) <1/81) +(abs(x+2/3+2/9+2/27) <1/81) +
(abs(x) <1/81) + (abs(x-2/27) <1/81) +(abs(x+2/27) <1/81) +
(abs(x-2/9) <1/81) + (abs(x-2/9-2/27) <1/81) + (abs(x-2/9+2/27) <1/81) +
(abs(x+2/9) <1/81)+(abs(x+2/9-2/27) <1/81)+(abs(x+2/9+2/27) <1/81)
) *(((abs(y-2/3) <1/81)+ (abs(y-2/3-2/27) <1/81)+ (abs(y-2/3+2/27) <1/81)+
(abs(y-2/3-2/9) <1/81) + (abs(y-2/3-2/9-2/27) <1/81) + (abs(y-2/3-2/9+2/27) <1/81) +
(abs(y-2/3+2/9) <1/81) +(abs(y-2/3+2/9-2/27) <1/81) +(abs(y-2/3+2/9+2/27) <1/81) +
(abs(y+2/3) <1/81) +(abs(y+2/3-2/27) <1/81) +(abs(y+2/3+2/27) <1/81) +
(abs(y+2/3-2/9) <1/81) +(abs(y+2/3-2/9-2/27) <1/81) +(abs(y+2/3-2/9+2/27) <1/81) +
(abs(y+2/3+2/9) <1/81) +(abs(y+2/3+2/9-2/27) <1/81) +(abs(y+2/3+2/9+2/27) <1/81) +
(abs(y) <1/81) + (abs(y-2/27) <1/81) +(abs(y+2/27) <1/81) +
(abs(y-2/9) <1/81) + (abs(y-2/9-2/27) <1/81) + (abs(y-2/9+2/27) <1/81) +
(abs(y+2/9) <1/81)+(abs(y+2/9-2/27) <1/81)+(abs(y+2/9+2/27) <1/81)
)+((abs(z-2/3) <1/81)+ (abs(z-2/3-2/27) <1/81)+ (abs(z-2/3+2/27) <1/81)+
(abs(z-2/3-2/9) <1/81) + (abs(z-2/3-2/9-2/27) <1/81) + (abs(z-2/3-2/9+2/27) <1/81) +
(abs(z-2/3+2/9) <1/81) +(abs(z-2/3+2/9-2/27) <1/81) +(abs(z-2/3+2/9+2/27) <1/81) +
(abs(z+2/3) <1/81) +(abs(z+2/3-2/27) <1/81) +(abs(z+2/3+2/27) <1/81) +
(abs(z+2/3-2/9) <1/81) +(abs(z+2/3-2/9-2/27) <1/81) +(abs(z+2/3-2/9+2/27) <1/81) +
(abs(z+2/3+2/9) <1/81) +(abs(z+2/3+2/9-2/27) <1/81) +(abs(z+2/3+2/9+2/27) <1/81) +
(abs(z) <1/81) + (abs(z-2/27) <1/81) +(abs(z+2/27) <1/81) +
(abs(z-2/9) <1/81) + (abs(z-2/9-2/27) <1/81) + (abs(z-2/9+2/27) <1/81) +
(abs(z+2/9) <1/81)+(abs(z+2/9-2/27) <1/81)+(abs(z+2/9+2/27) <1/81))) + ((abs(z-2/3) <1/81)+ (abs(z-2/3-2/27) <1/81)+ (abs(z-2/3+2/27) <1/81)+
(abs(z-2/3-2/9) <1/81) + (abs(z-2/3-2/9-2/27) <1/81) + (abs(z-2/3-2/9+2/27) <1/81) +
(abs(z-2/3+2/9) <1/81) +(abs(z-2/3+2/9-2/27) <1/81) +(abs(z-2/3+2/9+2/27) <1/81) +
(abs(z+2/3) <1/81) +(abs(z+2/3-2/27) <1/81) +(abs(z+2/3+2/27) <1/81) +
(abs(z+2/3-2/9) <1/81) +(abs(z+2/3-2/9-2/27) <1/81) +(abs(z+2/3-2/9+2/27) <1/81) +
(abs(z+2/3+2/9) <1/81) +(abs(z+2/3+2/9-2/27) <1/81) +(abs(z+2/3+2/9+2/27) <1/81) +
(abs(z) <1/81) + (abs(z-2/27) <1/81) +(abs(z+2/27) <1/81) +
(abs(z-2/9) <1/81) + (abs(z-2/9-2/27) <1/81) + (abs(z-2/9+2/27) <1/81) +
(abs(z+2/9) <1/81)+(abs(z+2/9-2/27) <1/81)+(abs(z+2/9+2/27) <1/81))*((abs(y-2/3) <1/81)+ (abs(y-2/3-2/27) <1/81)+ (abs(y-2/3+2/27) <1/81)+
(abs(y-2/3-2/9) <1/81) + (abs(y-2/3-2/9-2/27) <1/81) + (abs(y-2/3-2/9+2/27) <1/81) +
(abs(y-2/3+2/9) <1/81) +(abs(y-2/3+2/9-2/27) <1/81) +(abs(y-2/3+2/9+2/27) <1/81) +
(abs(y+2/3) <1/81) +(abs(y+2/3-2/27) <1/81) +(abs(y+2/3+2/27) <1/81) +
(abs(y+2/3-2/9) <1/81) +(abs(y+2/3-2/9-2/27) <1/81) +(abs(y+2/3-2/9+2/27) <1/81) +
(abs(y+2/3+2/9) <1/81) +(abs(y+2/3+2/9-2/27) <1/81) +(abs(y+2/3+2/9+2/27) <1/81) +
(abs(y) <1/81) + (abs(y-2/27) <1/81) +(abs(y+2/27) <1/81) +
(abs(y-2/9) <1/81) + (abs(y-2/9-2/27) <1/81) + (abs(y-2/9+2/27) <1/81) +
(abs(y+2/9) <1/81)+(abs(y+2/9-2/27) <1/81)+(abs(y+2/9+2/27) <1/81))) =0)
[x]: -1.001 , 1.001
[y]: -1.001 , 1.001
[z]: -1.001 , 1.001
;

Make some tests with this formula with height grid (example: 100)
Now activate this button like in this image and recalculate. As you can see, calculations are from 2 to 3 times faster just by activating this button!

The best part however is that we can boost the speed to even greater record...
The MS4 formula on the top is equivalent to this :
Quote:
Name: MS4
F(): -( x^100 + y^100 + z^100 -1)*((Dxy * Dzy * Dxz) * ( (Cx*(Cy+Cz) + Cz*Cy) =0)*( (Bx*(By+Bz) + Bz*By) =0)*( (Ax*(Ay+Az) + Az*Ay) =0) )
[x]: -1.001 , 1.001
[y]: -1.001 , 1.001
[z]: -1.001 , 1.001
;

Now, go to the same page and fill the four fields Au, Bu, Cu and D{u,v} as shown in the image with theses values :

Quote:
A[u]: (abs(u-2/3) <1/81)+ (abs(u-2/3-2/27) <1/81)+ (abs(u-2/3+2/27) <1/81)+
(abs(u-2/3-2/9) <1/81) + (abs(u-2/3-2/9-2/27) <1/81) + (abs(u-2/3-2/9+2/27) <1/81) +
(abs(u-2/3+2/9) <1/81) +(abs(u-2/3+2/9-2/27) <1/81) +(abs(u-2/3+2/9+2/27) <1/81) +
(abs(u+2/3) <1/81) +(abs(u+2/3-2/27) <1/81) +(abs(u+2/3+2/27) <1/81) +
(abs(u+2/3-2/9) <1/81) +(abs(u+2/3-2/9-2/27) <1/81) +(abs(u+2/3-2/9+2/27) <1/81) +
(abs(u+2/3+2/9) <1/81) +(abs(u+2/3+2/9-2/27) <1/81) +(abs(u+2/3+2/9+2/27) <1/81) +
(abs(u) <1/81) + (abs(u-2/27) <1/81) +(abs(u+2/27) <1/81) +
(abs(u-2/9) <1/81) + (abs(u-2/9-2/27) <1/81) + (abs(u-2/9+2/27) <1/81) +
(abs(u+2/9) <1/81)+(abs(u+2/9-2/27) <1/81)+(abs(u+2/9+2/27) <1/81)


B[u]: (abs(u-2/3) <1/27)+(abs(u-2/3-2/9) <1/27) + (abs(u-2/3+2/9) <1/27) +(abs(u+2/3) <1/27) +(abs(u+2/3-2/9) <1/27) + (abs(u+2/3+2/9) <1/27) +(abs(u) <1/27) + (abs(u-2/9) <1/27) + (abs(u+2/9) <1/27)


C[u]: (abs(u-2/3) <1/9) + (abs(u+2/3) <1/9) + (abs(u) <1/9)


D[u,v]: abs(u) >1/3 | abs(v) > 1/3


Now, recalculate and enjoy the speed: You can be sure no one in the world had make the Menger Sponge Level 4 that fast (except me Razz)
In my case, it's more than 20 times faster .
Some precisions:
1) The grid value should be a multiple of 4 (ie, 4, 8, 12...)
2) There is other ways to make it even faster with this great K3DSurf tool ... Razz
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Cheers,
Taha
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