Marek14 wrote:2D:
Circle (II)
3D:
Sphere (III)
Torus ((II)I)
Multitoruses ([II]n I) based on multiple semicircles attached to a common line.
4D:
Glome (IIII)
Torisphere ((III)I)
Multitorispheres ([III]n I) based on multiple halfspheres attached to a common circle.
Spheritorus ((II)II)
Multispheritoruses ([II]n II) based on multiple semicircles attached to a common line. This may include both 2D and 3D arrangements of semicircles, creating a midcut with circular or polyhedral arrangement of spheres.
Ditorus (((II)I)I)
Dimultitoruses (([II]n I)I) based on multitoruses.
Multiditoruses ([(II)I]n I) based on multiple halftoruses attached to common pair of concentric circles.
Multidimultitoruses ([[II]m I]n I) based on multiple halves of multitoruses attached to common group of circularly arranged circles.
Tiger ((II)(II))
Multitigers ([II]n (II)) based on multiple halves of duocylinder margins (circle x semicircle) attached to a common pair of parallel circles. Mantis would belong here.
Hypertigers ([II]m [II]n) based on multiple quarters of duocylinder margins (semicircle x semicircle). Their attachment seems more vague to understand.
ICN5D wrote:Yes. Here's an animation of a=2 , b=2 , d=0 , t=0 , and animating c from 0 to 6.28. It's among the wilder things this equation makes. This is also the first gif I made using a screencapture program. The timing is a little tricky, but it's still much easier than manually one by one.
ICN5D wrote:So, this animation is when both directions of periodicity are mult by 2, and rotating the general rotation plane by 360 degrees, continuously. It basically morphs between a 3torus equivalent to a tiger equivalent. The output of the graph is pretty wild, and there's much more to it than what I made.
Teragon wrote:Ok, so you're not changing the point of view, but morphing the object between tiger and the 3torusconfiguration?
Theoretically it would be enough to stretch the whole object's angular dependence in the azimuth direction by a factor of 1.5, which is obviously not what the parameter "a" does. It's hard to figure out where the azimuthdependence is, with all those rotations in the equation.
mr_e_man wrote:(Notice, we couldn't have done this symmetrization so easily before I had expanded it. Or could we?)
ICN5D wrote:This plots a triangular array of 3 toruses standing on their rim. My next, more recent idea (experiment) was to define a mirrorimage (upsidedown) triangle array and call the 3rd variable "w" (when it's really z for the plotter) , and then compare it to this one above. Then, try to take all the terms containing w and insert them in to the equation above. It will produce a 4variable equation that should morph between the triangular arrays by rotation on plane zw. Or, so it seems. It's an experiment I've never tried before, and it's a total shot in the dark, but could totally work, too.
mr_e_man wrote:You have the right idea here. It's basically what I did with the symmetry that swaps a and b (the two major radii), and swaps z and w, and rotates in the x,yplane by 360°/(2n) (in this case 60°, thus replacing the triangle by its mirror image). But I didn't know how to enforce this symmetry and add the appropriate w terms without expanding the product.
Challenger007 wrote:I'm not an expert, but maybe it's worth adding some variables to correctly display the volumetric body and change the directions of the vectors in dynamics, so as not to increase the scale?
((x^2+y^2+z^2+w^2+4/3(a^2+ab+b^2)c^2)^2  (2bz)^2  (2aw)^2)^2 ( (x^2+y^2+z^2+w^2+4/3(a^2+ab+b^2)c^2)^2  (2bz)^2  (2aw)^2 + 8(a^2+ab+b^2)(x^2+y^2) )
+ 16 (x^2+y^2+z^2+w^2+4/3(a^2+ab+b^2)c^2)^2 (x^2+y^2) ( (a^2+ab+b^2)^2(x^2+y^2) + (4a^2+4ab+7b^2)(bz)^2 + (7a^2+4ab+4b^2)(aw)^2  (a^2+ab+b^2)(x^2+y^2+z^2+w^2+4/3(a^2+ab+b^2)c^2)^2 )
 64 (x^2+y^2) ( 3(b^6z^4+a^6w^4) + (x^2+y^2) ab(ab) ((a^2+3ab+2b^2)bz^2  (2a^2+3ab+b^2)aw^2) )
 16/(3 sqrt(3)) (x^2+y^2+z^2+w^2+4/3(a^2+ab+b^2)c^2) x(x^23y^2) ( 12(a^2+ab+b^2)((2a+b)(bz)^2(a+2b)(aw)^2) + (2a^3+3a^2b3ab^22b^3)((x^2+y^2+z^2+w^2+4/3(a^2+ab+b^2)c^2)^24(a^2+ab+b^2)(x^2+y^2)) )
+ 64 ( 1/27((2a^3+3a^2b3ab^22b^3)x(x^23y^2))^2  (ab(a+b)y(3x^2y^2))^2 )
( (x^2+y^2+z^2+4/3(a^2+ab+b^2)c^2  (2a+b)/sqrt(3) (2x))^2  b^2 ((2z)^2 + (2y)^2) )
* ( (x^2+y^2+z^2+4/3(a^2+ab+b^2)c^2  (2a+b)/sqrt(3) (x+y sqrt(3)))^2  b^2 ((2z)^2 + (yx sqrt(3))^2) )
* ( (x^2+y^2+z^2+4/3(a^2+ab+b^2)c^2  (2a+b)/sqrt(3) (xy sqrt(3)))^2  b^2 ((2z)^2 + (y+x sqrt(3))^2) )
+
( (x^2+y^2+w^2+4/3(a^2+ab+b^2)c^2  (a+2b)/sqrt(3) (2x))^2  a^2 ((2w)^2 + (2y)^2) )
* ( (x^2+y^2+w^2+4/3(a^2+ab+b^2)c^2  (a+2b)/sqrt(3) (x+y sqrt(3)))^2  a^2 ((2w)^2 + (y+x sqrt(3))^2) )
* ( (x^2+y^2+w^2+4/3(a^2+ab+b^2)c^2  (a+2b)/sqrt(3) (xy sqrt(3)))^2  a^2 ((2w)^2 + (yx sqrt(3))^2) )

( (x^2+y^2+4/3(a^2+ab+b^2)c^2  (2a+b)/sqrt(3) (2x))^2  (2by)^2 )
* ( (x^2+y^2+4/3(a^2+ab+b^2)c^2  (2a+b)/sqrt(3) (x+y sqrt(3)))^2  (b(yx sqrt(3)))^2 )
* ( (x^2+y^2+4/3(a^2+ab+b^2)c^2  (2a+b)/sqrt(3) (xy sqrt(3)))^2  (b(y+x sqrt(3)))^2 )
ICN5D wrote:Challenger007 wrote:I'm not an expert, but maybe it's worth adding some variables to correctly display the volumetric body and change the directions of the vectors in dynamics, so as not to increase the scale?
We don't need any more variables for this thing The shape curves around (is "embedded") in a 4d space, so we only need 4 variables: x, y, z, w . And since it (a multitiger) only has 3 radius sizes, we only need 3 parameters for them: a, b, c . And we don't have an issue with the size (scale) , really. We're trying to get a hexagon array of 6 circles (on plane xy) to join as two mirrorimage triangular arrays of toruses by rotation on plane zw (when we 'flip' the variables z and w back and forth by rotation). It's a hypothetical 4D donut shape that's cooler than anything I've rendered before. And we've been trying to define them with an equation for years.
ICN5D wrote:Sorry dude! Went to the beach and was away from computer.
ICN5D wrote:I really must know everything that you know about this geometric algebra, my dude. Serious voodoo magic going on there. And thanks for putting this info here. One day, when I'm better trained at this stuff, I'll be able to come back here and understand your process!
mr_e_man wrote:And here's the polynomial for a mantis, using the "expand product" approach (I've eliminated r² and d², and factored some more):
mr_e_man wrote:ICN5D wrote:Sorry dude! Went to the beach and was away from computer.
Well, it looked like you were here for some time after you saw the equation, and you might have shown us the first fruits from CalcPlot3D or POVRay. But no matter.
I guess you were hindered by the equation being in the wrong format: d² needs to be changed to 4/3(a²+ab+b²), and r² to (x²+y²+z²+w²), and (...)² to (...)^2, and [...] to (...), and √3 to sqrt(3). (I've made those changes now; see 'code' above. ) That's not to mention converting 4D to 3D.
ICN5D wrote:mr_e_man wrote:And here's the polynomial for a mantis, using the "expand product" approach (I've eliminated r² and d², and factored some more):
What program do you use to expand and simplify these equations? I'd like to try this out.
mr_e_man wrote:The product for the trigonal array of tori, n=3, is
[...]
It took me a week or two, but I was able to expand this (partially, keeping some terms like (r²+d²c²) intact) and simplify and refactor it.
mr_e_man wrote:mr_e_man wrote:(Notice, we couldn't have done this symmetrization so easily before I had expanded it. Or could we?)
Yes, we could! No need to take a month to expand and simplify the 4fold product for a spider; just leave the product as is.
mr_e_man wrote:ICN5D wrote:What program do you use to expand and simplify these equations? I'd like to try this out.
Nothing. Just lots of paper.
mr_e_man wrote:F(x,y,z,0) =
((x²+y²+z²+d²c²)²  4b²z²)⁴  8d² ((x²+y²+z²+d²c²)²  4b²z²)³ (x²+y²)
+ 4 ((x²+y²+z²+d²c²)²  4b²z²)² [ 5(A²+b²)²(x²+y²)²  (A⁴6A²b²+b⁴)(x⁴6x²y²+y⁴)  16A²b²z²(x²+y²) ]
+ 16 ((x²+y²+z²+d²c²)²  4b²z²) [ (A²+b²)(A⁴6A²b²+b⁴)(x²+y²)(x⁴6x²y²+y⁴)  (A²+b²)³(x²+y²)³ + 8(x²+y²)²(A²+3b²)A²b²z²  8(x⁴6x²y²+y⁴)(A²b²)A²b²z² ]
+ 512 (x²+y²)² A⁴b⁴z⁴  512 (x⁴6x²y²+y⁴) A⁴b⁴z⁴
 128 (x²+y²)³ (A⁴2A²b²+5b⁴)A²b²z² + 128 (x²+y²)(x⁴6x²y²+y⁴) (A⁴2A²b²3b⁴)A²b²z²
+ 4 ( (A⁴6A²b²+b⁴)(x²+y²)²  (A²+b²)²(x⁴6x²y²+y⁴) )².
Hmm... Would you like to try filling this in with some w terms, to make a spider?
ICN5D wrote:mantis:
X^3 3XY^2 2Y^3*cos(6θ) 12Z^2(X2Y) +4Z(4Z^23XY+3Y^2)*sin(3θ)
spider:
X^4 4X^2Y^2 +6Y^4 +2Y^4*cos(8θ) +16Z^2(2Z^2Y^2(XY)^2) 4((XY4Z^2)^22(Y^22Z^2)^2)*cos(4θ)
[...]
where,
X(r,z) => (r^2+z^2c^2)^2 +2a^2(2r^2+z^2+b^2c^2) 2b^2(z^2+c^2) +a^4+b^4
Y(r) => (a^2+b^2)r^2
Z(r,z) => ar(r^2+z^2+a^2+b^2c^2)
ICN5D wrote:A highly reduced polynomial that I found, for the 3D mantis solution (in cyl coords) is:
(r^4+2r^2z^2+62r^2+z^4+22z^2+361)^3 1200r^4(r^4+2r^2z^2+62r^2+z^4+22z^2+361) 16000r^6*cos(6θ) 192r^2(r^2+z^2+19)^2(r^4+2r^2z^2+22r^2+z^4+22z^2+361) +64r^3(r^2+z^2+19)(r^4+2r^2z^222r^2+z^4+278z^2+361)*sin(3θ) = 0
However, at the same time, I'm struggling to come up with the terms that contain w. This is where your process might come in Maybe a compatible part of your reasoning can add in the w terms to my equation above.
ICN5D wrote:If it's true, I recommend using wolfram alpha, and running chunks of terms through it and see what alternate expressions it comes up with. It can handle some fairly large equations.
I've made legit discoveries this way: the whole 'roots of unity/multitorus' thread and my factoring algorithm for the algebraic solutions of toratopes. That knowledge would not even exist, were it not for our electronic friends. I highly recommend it!
https://c3d.libretexts.org/CalcPlot3D/index.html?type=implicit;equation=((x^2+y^2+(z*cos(t)a*sin(t))^2+(z*sin(t)+a*cos(t))^2+145/3)^24(9(z*cos(t)a*sin(t))^2+16(z*sin(t)+a*cos(t))^2))^2*((x^2+y^2+(z*cos(t)a*sin(t))^2+(z*sin(t)+a*cos(t))^2+145/3)^2+296(x^2+y^2)4(9(z*cos(t)a*sin(t))^2+16(z*sin(t)+a*cos(t))^2))16(x^2+y^2)(x^2+y^2+(z*cos(t)a*sin(t))^2+(z*sin(t)+a*cos(t))^2+145/3)^2*(37(x^2+y^2+(z*cos(t)a*sin(t))^2+(z*sin(t)+a*cos(t))^2+145/3)^21369(x^2+y^2)7(225(z*cos(t)a*sin(t))^2+448(z*sin(t)+a*cos(t))^2))(16*sqrt(3))/9*(x^33xy^2)*(110(x^2+y^2+(z*cos(t)a*sin(t))^2+(z*sin(t)+a*cos(t))^2+145/3)^3148(x^2+y^2+(z*cos(t)a*sin(t))^2+(z*sin(t)+a*cos(t))^2+145/3)(110x^2+110y^2297(z*cos(t)a*sin(t))^2+480(z*sin(t)+a*cos(t))^2))64(x^2+y^2)(3(4096(z*sin(t)+a*cos(t))^4+729(z*cos(t)a*sin(t))^4)+168(15(z*cos(t)a*sin(t))^222(z*sin(t)+a*cos(t))^2)(x^2+y^2))+64(12100/27(x^33xy^2)^27056(3x^2yy^3)^2)592240896(z*cos(t)a*sin(t))^2(z*sin(t)+a*cos(t))^2;cubes=25;visible=true;fixdomain=false;xmin=8;xmax=8;ymin=8;ymax=8;zmin=8;zmax=8;alpha=255;view=0;format=normal;constcol=rgb(153,153,153)&type=slider;slider=a;value=0;steps=100;pmin=8;pmax=8;repeat=true;bounce=true;waittime=1;careful=false;noanimate=false;name=1&type=slider;slider=t;value=1.54461639;steps=60;pmin=0;pmax=pi/2;repeat=true;bounce=true;waittime=1;careful=false;noanimate=false;name=1&type=window;hsrmode=3;nomidpts=true;anaglyph=1;center=7.058302103282751,5.808293497004737,7.102400866711125,1;focus=0,0,0,1;up=0.45747881941680846,0.37013375975943846,0.8085258991963988,1;transparent=false;alpha=140;twoviews=false;unlinkviews=false;axisextension=0.7;xaxislabel=x;yaxislabel=y;zaxislabel=z;edgeson=false;faceson=true;showbox=false;showaxes=false;showticks=true;perspective=true;centerxpercent=0.5;centerypercent=0.5;rotationsteps=100;autospin=false;xygrid=false;yzgrid=false;xzgrid=false;gridsonbox=true;gridplanes=false;gridcolor=rgb(128,128,128);xmin=8;xmax=8;ymin=8;ymax=8;zmin=8;zmax=8;xscale=4;yscale=4;zscale=4;zcmin=16;zcmax=16;zoom=0.293333;xscalefactor=1;yscalefactor=1;zscalefactor=1
ICN5D wrote:So. I'm wondering what additional terms you would suggest at the end of this equation? Are there any we can use to refine the 45 degree oblique surfaces? Maybe something to smoothen out the 3 toruses as they begin touching at the poles of the cage structure  that is get rid of that little blob that appears right before they touch. I'm guessing that's what we can use them for. It all seems right so far. I wonder where Marek is, he'd get a kick out of this ......
ICN5D wrote:Alright, I checked out your equation some more, made a few more rotation animations. It's not so voodoo magic after all, just some very clever use of terms I never thought of! Now I can see how you use the alternating a,b with z,w terms to add in w. And I think this is the right equation.
ICN5D wrote:Which brings me to my next point: The next two pics below show these unique solutions, where the plane that we see slicing them is the zw plane. Both have a common solution, defined by only the zw terms: a square array of 4 circles.
Users browsing this forum: No registered users and 1 guest