Exploring a 3-Prong Multi-Tiger

Discussion of shapes with curves and holes in various dimensions.

Exploring a 3-Prong Multi-Tiger

Postby ICN5D » Wed Nov 07, 2018 3:12 am

All right, this ought to stir some things up around here :D I believe I've created a 3-prong structure based on a tiger. It's been about 3~4 years of blind experimentation, learning some stuff on wikipedia, getting more ideas, doing more poking about with equation stuff, and eventually coming up with something that resembles what we're looking for. It has interesting properties, and it's about as alien as anything we've seen.

I don't have a concise follow through with how I derived this equation, because my notes about this thing are scattered across several notepad files. Basically, it's a single set of points "d^6" defined over a product of 3 self-intersecting, non-orthogonal clifford toruses. It's made by the same technique that created a nice 3-prong multitorus, by converging 3 loop-like surfaces into place, to form a 3-prong cage, then fatten it up with a minor radius.

Now, by using 3 tigers, we should obviously expect to get six tori (in 3D), which overlap in place in 3 pairs (in triangular array). This is where I got the idea to tilt those overlapping tori, by changing the plane of rotation of the generating torus. The result was a triangle array of 3 tori that did some really funky stuff when we moved the 3D slice around.

Equation of a Tri-Tigroidal Cassini Surface a.k.a., 3-Prong Multi-Tiger , possible Mantis

• Adjustable Radius Parameters

(((sqrt(((sqrt(3)(x-a)-sqrt(y^2 + w^2)+c)/2)^2 + z^2) -b)^2 + (((x-a)+sqrt(3)(sqrt(y^2 + w^2)-c))/2)^2))*(((sqrt(((sqrt(3)((-x-sqrt(3)y)/2-a)-sqrt(((-sqrt(3)x+y)/2)^2 + w^2)+c)/2)^2 + z^2) -b)^2 + ((((-x-sqrt(3)y)/2-a)+sqrt(3)(sqrt(((-sqrt(3)x+y)/2)^2 + w^2)-c))/2)^2))*(((sqrt(((sqrt(3)((-x+sqrt(3)y)/2-a)-sqrt(((sqrt(3)x+y)/2)^2 + w^2)+c)/2)^2 + z^2) -b)^2 + ((((-x+sqrt(3)y)/2-a)+sqrt(3)(sqrt(((sqrt(3)x+y)/2)^2 + w^2)-c))/2)^2)) = d^6

• Hard-set Radius Values, for the most Mantis-like topology

a = 2.35 ; major radius
b = 2.5 ; secondary A
c = 4 ; secondary B
d = 2 ; minor radius

(((sqrt(((sqrt(3)(x-2.35)-sqrt(y^2 + w^2)+4)/2)^2 + z^2) -2.5)^2 + (((x-2.35)+sqrt(3)(sqrt(y^2 + w^2)-4))/2)^2))*(((sqrt(((sqrt(3)((-x-sqrt(3)y)/2-2.35)-sqrt(((-sqrt(3)x+y)/2)^2 + w^2)+4)/2)^2 + z^2) -2.5)^2 + ((((-x-sqrt(3)y)/2-2.35)+sqrt(3)(sqrt(((-sqrt(3)x+y)/2)^2 + w^2)-4))/2)^2))*(((sqrt(((sqrt(3)((-x+sqrt(3)y)/2-2.35)-sqrt(((sqrt(3)x+y)/2)^2 + w^2)+4)/2)^2 + z^2) -2.5)^2 + ((((-x+sqrt(3)y)/2-2.35)+sqrt(3)(sqrt(((sqrt(3)x+y)/2)^2 + w^2)-4))/2)^2)) = 2^6


Now for some cool stuff:


• Setting w=0 , translating the xyz slice up/down along the w-axis between -7.3 < w < 7.3

Image








• Setting w=0 , rotating on plane XW a full 360 deg

(((sqrt(((sqrt(3)((x*cos(b)+a*sin(b))-2.35)-sqrt(y^2 + (x*sin(b)-a*cos(b))^2)+4)/2)^2 + z^2) -2.5)^2 + ((((x*cos(b)+a*sin(b))-2.35)+sqrt(3)(sqrt(y^2 + (x*sin(b)-a*cos(b))^2)-4))/2)^2))*(((sqrt(((sqrt(3)((-(x*cos(b)+a*sin(b))-sqrt(3)y)/2-2.35)-sqrt(((-sqrt(3)(x*cos(b)+a*sin(b))+y)/2)^2 + (x*sin(b)-a*cos(b))^2)+4)/2)^2 + z^2) -2.5)^2 + ((((-(x*cos(b)+a*sin(b))-sqrt(3)y)/2-2.35)+sqrt(3)(sqrt(((-sqrt(3)(x*cos(b)+a*sin(b))+y)/2)^2 + (x*sin(b)-a*cos(b))^2)-4))/2)^2))*(((sqrt(((sqrt(3)((-(x*cos(b)+a*sin(b))+sqrt(3)y)/2-2.35)-sqrt(((sqrt(3)(x*cos(b)+a*sin(b))+y)/2)^2 + (x*sin(b)-a*cos(b))^2)+4)/2)^2 + z^2) -2.5)^2 + ((((-(x*cos(b)+a*sin(b))+sqrt(3)y)/2-2.35)+sqrt(3)(sqrt(((sqrt(3)(x*cos(b)+a*sin(b))+y)/2)^2 + (x*sin(b)-a*cos(b))^2)-4))/2)^2)) = 2^6

Image


Has the flip-flop triangular array of 3 tori, but only happens with 180 deg turns. This is what I theorized earlier, that it would not be a 90 deg rotation, but a full 180 deg flip. We see the same thing in the 2D slices of a 3-prong multi-torus.






• Setting w=0 , rotating on plane ZW a full 360 deg

(((sqrt(((sqrt(3)(x-2.35)-sqrt(y^2 + (z*sin(b)-a*cos(b))^2)+4)/2)^2 + (z*cos(b)+a*sin(b))^2) -2.5)^2 + (((x-2.35)+sqrt(3)(sqrt(y^2 + (z*sin(b)-a*cos(b))^2)-4))/2)^2))*(((sqrt(((sqrt(3)((-x-sqrt(3)y)/2-2.35)-sqrt(((-sqrt(3)x+y)/2)^2 + (z*sin(b)-a*cos(b))^2)+4)/2)^2 + (z*cos(b)+a*sin(b))^2) -2.5)^2 + ((((-x-sqrt(3)y)/2-2.35)+sqrt(3)(sqrt(((-sqrt(3)x+y)/2)^2 + (z*sin(b)-a*cos(b))^2)-4))/2)^2))*(((sqrt(((sqrt(3)((-x+sqrt(3)y)/2-2.35)-sqrt(((sqrt(3)x+y)/2)^2 + (z*sin(b)-a*cos(b))^2)+4)/2)^2 + (z*cos(b)+a*sin(b))^2) -2.5)^2 + ((((-x+sqrt(3)y)/2-2.35)+sqrt(3)(sqrt(((sqrt(3)x+y)/2)^2 + (z*sin(b)-a*cos(b))^2)-4))/2)^2)) = 2^6

Image

The result of z=0 is a very strange, cage-like object. Not at all what was expected, but makes sense considering the 3-prong skelet is a 3D object, and has a flatness in 4D space.





• Setting w=2.4 units from origin, rotating on plane ZW a full 360 deg

using same equation as above animation

Image

More wild stuff! Not sure what Mantis is supposed to have ..... but it's wild!



So, that's pretty good for now. I'll make some more later. There's still more slices to show that provide evidence that this is a 3-prong structure based on a tiger.
in search of combinatorial objects of finite extent
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Re: Exploring a 3-Prong Multi-Tiger

Postby Marek14 » Tue Nov 13, 2018 8:58 am

Yes, this looks pretty good! Is there an obvious way to generalize it for more prongs?
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Re: Exploring a 3-Prong Multi-Tiger

Postby ICN5D » Wed Nov 14, 2018 5:08 am

Really, you think so? But shouldn't the mantis have a 90 degree rotation morph between the 2 distinct triangle array of toruses (with a 6-bar cage at the 45 deg slice)? In this case, it's a 180 degree flip.

As for generalizing to n-prong multi-tigers, I guess it's as easy as converging more and more clifford toruses together, and making sure the plane of rotation is set to align them. Then, the radius parameters have to be at just the right ratios to overlap and fill in all the voids to form a smooth surface with no extra holes or dimples (highly exaggerated when the minor radius is too small).

Although, a problem I can see already is the rotating slice of a 4-prong multi-tiger, which won't show an alternating square array of 4 toruses. As in, you won't see the 2 distinct '+' and 'x' arrays (square and 45 degree tilted square arrays) from any rotation. It'll just be the same square array morphing to the shortened tube-like surface. I was able to get something like this though, when I made that tiger defined in spherical coordinates (which is actually very good at defining even-prong multi-tigers, despite the squished toruses for anything n<6).
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