The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.

Re: The Tiger Explained

Postby ICN5D » Fri May 09, 2014 1:07 am

Marek14 wrote:What if you use something like ((III)(II))(II), though -- there's no unique "first marker" to choose in the first part.



That's a very good question, Marek. I haven't explored lathing the tigroid prisms much, though I suspect they won't be too far off from what we see with cyltorinder ((II)I)(II). By that, I refer to how a cyltorinder can also be described as a duocylinder torus. So, let's take a looksy, here:

The simplest example is (tiger,circle)-prism : ((II)(II))(II)

Tiger is circle->duoring, and extruding into 5D makes ((II)(II))I, (tiger,line)-prism. This shape can be decomposed into cylinder->duoring, the extruded tiger. This prism has two flat endcaps of tigers, laced by an extruded version of circle->duoring. So, in between tiger || tiger , we get a [line-torus]-->duoring, the linear connection surface joining two parallel tigers. Spherating such a shape leads to (((II)(II))I) , torus->duoring, which conforms with what we already know, in how (II)I is cylinder and ((II)I) makes torus.

But, before spherating, let's lathe (tiger,line)-prism ((II)(II))I around a bisecting hyperplane, to make (tiger,circle)-prism. In the orientation of ((xy)(zw))v , a bisecting rotation around hyperplane xyzw will send the tiger endcaps tumbling end over end, along a circular path into 6D. This effectively creates a tiger->circle (((II)I)(II)) as one of the surtopes.

For the other surtope on ((II)(II))I , we have the [line-torus]->duoring. This linear attaching shape has a bisecting rotation, and becomes torus->duoring, the (((II)(II))I) surtope. This transformation follows the same rules as a bisecting rotation of a cylinder into duocylinder. The cylinder has two flat circle endcaps laced by a line-torus, the hollow tube. In the orientation (xy)z , rotating around bisecting hyperplane xy will turn the circles into a torus, and the line torus into another torus, ((II)I)+((II)I) the two ortho bound toruses.

So, resting on the surface of ((II)(II))(II) , the (tiger,circle)-prism, we have (((II)I)(II))+(((II)(II))I), a tigritorus ortho bound to a toratiger. Since our names are cris-crossing definitions, I'll just use (((II)I)(II))+(((II)(II))I). As for how to derive these surtopes in the new Surtope Algorithm, let's explore:

Starting off, we have ((II)(II))(II) again.

1) rewrite to form ((xI)(II))(II)

2) take the circle parameter and move in place of X, making (((II)I)(II)), the first surtope as a tigritorus

3) now use the form ((II)(II))(xI)

4) take the entire tiger parameter, and move in place of X, making (((II)(II))I), the second surtope as a toratiger

5) and, Voi-la ! We have both surtopes, correctly derived! :D Yay! (((II)I)(II))+(((II)(II))I)

---- Another important point to make is that lathing cylinder-->duoring will also make duocylinder-->duoring, by definition. This is pretty much what ((II)(II))(II) is, starting with (!!)(II) and inflate a duoring with a duocylinder. Minds blown, yet?

So, it would seem that ((II)(II))(II) has only two distinct solid shapes as parameters. Following the " replace first marker with X, then move other parameter in place of X " seems to conform fairly well, no matter how many parameters the tiger has. Though, I suppose BOTH circle parameters in the tiger can have the X wherever, and still produce the same result, given the commutative property of cartesian products.



Now, let's explore that ((III)(II))(II) , (cylspherintigroid,circle)-prism:

Cylspherintigroid decomposes into circle->(sphere x circle). A prism of this will produce ((III)(II))I, cylspherintigroid || cylspherintigroid, laced by a [line-torus]-->(sphere x circle). As stated before, the ((III)(II))I can also be decomposed into cylinder-->(sphere x circle).

In the orientation ((xyz)(wv))u , rotating around bisecting hyperplane XYZWV will, again, tumble the cylspherintigroid endcaps along a circular path, making a cylspherintigroid-->circle, (((II)II)(II)).

As for the linear connecting surface [line-torus]-->(sphere x circle), it predictably becomes torus-->(sphere x circle) , the (((III)(II))I) surtope. We all know how this works, it's the sock-rolling of the hollow tube, but a [hollow tube]-->(sphere x circle) prism. And, again, ((III)(II))(II) will lathe the inflating cylinder part into a duocylinder, making duocylinder-->(sphere x circle).

So far, it looks like ((III)(II))(II) will have a (((II)II)(II))+(((III)(II))I) as the two orthogonal bound surtopes.


Now, this brings me to my next point: A cylspherintigroid has TWO distinct types of parameters, and thus two types of toruses, (((II)II)(II)) type-1, and (((II)I)(III)) type-2. According to the orientation of ((xyz)(wv))u , it would seem that we have TWO distinct circular paths we can take during this bisecting rotation. One makes either cylspherintigroid torus type-1 or type-2. It depends on which parameter, sphere or circle, we tumble the cylspherintigroid endcaps along the rotating circular path. So, I guess ((III)(II))(II) has two states, depending on which lathing direction we use. This is new to me in regards to bisecting rotations. I just recently discovered this relationship with the topratope notation, the two types of cylspherinder toruses.


So, in the Cylspherintigroid Torus Type-1 case, let's apply it to the surtope algorithm:

1) Rewrite to form ((xII)(II))(II)

2) Move the circle parameter in place of X, making (((II)II)(II)) as the first surtope

3) Rewrite to form ((III)(II))(xI)

4) Move the cylspherintigroid parameter in place of X, making (((III)(II))I) as the second surtope

5) And, we end up with (((II)II)(II))+(((III)(II))I) a {cylspherintigroid-->circle} + {torus-->(sphere x circle)}, same as above derivations



In the case of Cylspherintigroid Torus Type-2:

1) Rewrite to form ((III)(xI))(II) , a new modification of the algorithm to address the recent findings

2) Move the circle parameter in place of X, making ((III)((II)I)) as the first surtope

3) Rewrite to form ((III)(II))(xI)

4) Move the cylspherintigroid parameter in place of X, making (((III)(II))I) as the second surtope

5) And, we end up with (((II)I)(III))+(((III)(II))I) a {cylspherintigroid-->circle} ortho bound to a {torus-->(sphere x circle)}, the OTHER surtopes of a ((III)(II))(II). This isn't all that startling after all :) . It follows the same principle of the type-1 and type-2 cylspherinder toruses, only they become the surtopes. Notice how both ((III)(II))(II) maintain the same (((III)(II))I) surtope, only differ in the (((II)I)(III)) and (((II)II)(II)) surtopes. This coming about from a [circle-->(sphere x circle)-->circle] being either a (((II)I)(III)) or a (((II)II)(II)).


In conclusion, an open toratope with different parameters in a tigroid factor will have as many distinct surtope pairs as the tigroid parameter has. Now, for a REAL challenge, we could try this method on something like a ((III)(II))(((II)I)(II)) with FOUR distinct factor parameters!!! In this case, we would get a combinatorial collection of different ortho bound surtopes. Not to mention an open toratope with three factors, as in a ((II)I)(III)(II), a (cylspherinder,circle)-prism --> circle. This would have only three surtopes, (((II)I)I)+(((III)I)I)+((II)II), all ortho bound to each other. Only in the case of an open toratope with a tigroid factor, and only in the case of that tigroid factor having different parameters, will we see alternate surtopes of the same shape come about.

And, yet again, we run into even more ambiguity when playing with high-D shapes. Only serving to further develop our cause of classification and computation, of course !! :D


-- Philip


PS: I can only hope that one of these posts will be notable enough for me to become a notable green person one day....
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Re: The Tiger Explained

Postby Marek14 » Fri May 09, 2014 4:52 am

Green person?
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Re: The Tiger Explained

Postby ICN5D » Fri May 09, 2014 5:21 am

Yes, the green-colored username, a Notable. It's no big deal, really. I'm not asking for it, since it doesn't happen that way. It'll probably happen one day, after enough awesome breakthroughs :)


But, other than that, how about the surtope algorithm? I think it's kind of neat. Still needs more clarification, especially with the three-factor opens. This was the first time I explored tigroid prisms. But, I was already well prepared for the lathing action, from my previous years of developing that notation I created. I might say I mastered the art of the bisecting rotation of rotopes, by deductive reasoning in the (n,circle)-prisms. But the toratopic notation taught me many more ways to rotate around a non-intersecting plane, hence the tiger.
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Re: The Tiger Explained

Postby Marek14 » Fri May 09, 2014 5:44 am

ICN5D wrote:Yes, the green-colored username, a Notable. It's no big deal, really. I'm not asking for it, since it doesn't happen that way. It'll probably happen one day, after enough awesome breakthroughs :)


But, other than that, how about the surtope algorithm? I think it's kind of neat. Still needs more clarification, especially with the three-factor opens. This was the first time I explored tigroid prisms. But, I was already well prepared for the lathing action, from my previous years of developing that notation I created. I might say I mastered the art of the bisecting rotation of rotopes, by deductive reasoning in the (n,circle)-prisms. But the toratopic notation taught me many more ways to rotate around a non-intersecting plane, hence the tiger.


Never actually knew there is a "notable" category of users... *shrugs*

As for the algorithm, it looks good, I just haven't been commenting much since I do not work with open toratopes that much (they are too drafty for me).

Here's another question I've been pondering, though: You certainly know that torus can be turned "inside out" and that if the original torus had horizontal circles drawn on it, the inverted one will have vertical circles. This seems to be simply based on torus as circle->circle, you swap the two circles. Similar logic would tell you that torisphere (circle->sphere) could be turned inside out into spheritorus (sphere->circle).
Now, would that work with tiger? Tiger as circle->duoring might not work, but maybe there's possible to turn a ditorus (circle->torus) inside out in two different ways, one leading to another ditorus (torus[major]->circle) and one leading to tiger (torus[minor]->circle)?

One more idea: since both ditorus and tiger are torus->circle, it seems there should be a way to transform ditorus into tiger or vice versa by simply rotating the torus.
The starting point could be the ditorus cut "two toruses". If you turned each of those toruses 90 degrees, you'd arrive to a cut of tiger. Could the ditorus and tiger equations be combined into a single one with parameter that would allow us to go from one to another by rotating toruses?
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Re: The Tiger Explained

Postby ICN5D » Fri May 09, 2014 7:02 am

Never actually knew there is a "notable" category of users... *shrugs*


Yep, apparently one has to be invited in by a notable.

As for the algorithm, it looks good, I just haven't been commenting much since I do not work with open toratopes that much (they are too drafty for me).


Well, being based solely off the closed, they shouldn't be too tough to wrap your mind around :) Now that you know what the edges look like ( closed ), and how they run together, the opens are nothing more than the solid, filled in forms, with curved rolling sides. I came to know those intimately first, including many others. And, I did end up achieving what I set out to do this year: learn the toratopic notation and cut algorithm. In fact, I surpassed what I expected to comprehend. Far surpassed. Just look at all of those awesome cross section renders. Four months ago, I didn't even know what ((I)I) meant. I "have a life", but discussing on this forum has greatly improved the quality of it, believe it or not. Now, I'm talking about sock rolling a cylspherintigroid prism. What?? That's crazy freaking stuff.


Here's another question I've been pondering, though: You certainly know that torus can be turned "inside out" and that if the original torus had horizontal circles drawn on it, the inverted one will have vertical circles. This seems to be simply based on torus as circle->circle, you swap the two circles. Similar logic would tell you that torisphere (circle->sphere) could be turned inside out into spheritorus (sphere->circle).
Now, would that work with tiger? Tiger as circle->duoring might not work, but maybe there's possible to turn a ditorus (circle->torus) inside out in two different ways, one leading to another ditorus (torus[major]->circle) and one leading to tiger (torus[minor]->circle)?



Hmmm. I've seen some other posts about this, it's one Keiji made about toratopic duals, and the addition of trios and quartets. I believe turning a ditorus inside out will only make 2 other ditoruses, since all three diameters are circles. Now, if this was a (((III)I)I), then there would be different toratopic trios, by interchanging the spherical diameter. I do not believe a ditorus can be turned inside out into a tiger. But, your special rotation reference can do so.

It seems that the duoring is some sort of un-deformable elementary structure, like an indestructible wireframe. It cannot be reduced any further, much like any margin of open toratopes. I hope that statement makes some caliber of sense. There is a very fundamental property with those margins, something strange about them. It seems to be that a duoring takes up a whole xyzw hyperplane, treated as one single diameter. A margin is but one, albeit a complex diameter, that gets inflated by some shape. That's what we see with (((II)(II))I), a torus along duoring. Or, even ((II)(II)), a circle along duoring. But, getting into much higher-D shapes, we have now seen that a duoring can be composed out of separate circle paths, as in tiger[maj1,2]-->duoring and tiger[maj1]-->duoring, (((II)I)((II)I) and (((II)(II))(II)) respectively. So long as they are orthogonal circular paths, a duoring has multiple constructions starting in 6D. Probably because of the three ortho 2-planes in 6D, it's a mater of combination in the two 2-planes out of three possible.


One more idea: since both ditorus and tiger are torus->circle, it seems there should be a way to transform ditorus into tiger or vice versa by simply rotating the torus.
The starting point could be the ditorus cut "two toruses". If you turned each of those toruses 90 degrees, you'd arrive to a cut of tiger. Could the ditorus and tiger equations be combined into a single one with parameter that would allow us to go from one to another by rotating toruses?



I can see that happening, visually. But, how to form an equation that does so, is beyond my knowledge. It is a very interesting idea, though. It seems to be a matter of rotating the final circle path 90 degrees. So, for both implicit equations:

(((II)I)I)
(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 - R3^2 = 0

((II)(II))
(sqrt(x^2 + y^2) - R1)^2 + (sqrt(z^2 + w^2) - R2)^2 - R3^2= 0


In both, we have three diameters, which is a good thing. I'm not sure where to start here, so I'll list the 3D cut equations:


(((I)I)I)
(sqrt((sqrt(x^2) - R1)^2 + y^2) - R2)^2 + z^2 - R3^2 = 0

((II)(I))
(sqrt(x^2 + y^2) - R1)^2 + (sqrt(z^2) - R2)^2 - R3^2= 0


I don't know how to generate a function that swaps parentheses. They seems to be fundamentally different, and probably for good reason. It might be possible with some sort of modified rotation equation, but how to plug it in, within the equation, is beyond my knowledge. I'd be interested to see if it's possible, for sure! An easier one to try would be turning a sphere into a torus, then extrapolate how the function works. Easy as pie, if possible.
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Re: The Tiger Explained

Postby Marek14 » Fri May 09, 2014 7:27 am

ICN5D wrote:
Never actually knew there is a "notable" category of users... *shrugs*


Yep, apparently one has to be invited in by a notable.

As for the algorithm, it looks good, I just haven't been commenting much since I do not work with open toratopes that much (they are too drafty for me).


Well, being based solely off the closed, they shouldn't be too tough to wrap your mind around :) Now that you know what the edges look like ( closed ), and how they run together, the opens are nothing more than the solid, filled in forms, with curved rolling sides. I came to know those intimately first, including many others. And, I did end up achieving what I set out to do this year: learn the toratopic notation and cut algorithm. In fact, I surpassed what I expected to comprehend. Far surpassed. Just look at all of those awesome cross section renders. Four months ago, I didn't even know what ((I)I) meant. I "have a life", but discussing on this forum has greatly improved the quality of it, believe it or not. Now, I'm talking about sock rolling a cylspherintigroid prism. What?? That's crazy freaking stuff.


Here's another question I've been pondering, though: You certainly know that torus can be turned "inside out" and that if the original torus had horizontal circles drawn on it, the inverted one will have vertical circles. This seems to be simply based on torus as circle->circle, you swap the two circles. Similar logic would tell you that torisphere (circle->sphere) could be turned inside out into spheritorus (sphere->circle).
Now, would that work with tiger? Tiger as circle->duoring might not work, but maybe there's possible to turn a ditorus (circle->torus) inside out in two different ways, one leading to another ditorus (torus[major]->circle) and one leading to tiger (torus[minor]->circle)?



Hmmm. I've seen some other posts about this, it's one Keiji made about toratopic duals, and the addition of trios and quartets. I believe turning a ditorus inside out will only make 2 other ditoruses, since all three diameters are circles. Now, if this was a (((III)I)I), then there would be different toratopic trios, by interchanging the spherical diameter. I do not believe a ditorus can be turned inside out into a tiger. But, your special rotation reference can do so.

It seems that the duoring is some sort of un-deformable elementary structure, like an indestructible wireframe. It cannot be reduced any further, much like any margin of open toratopes. I hope that statement makes some caliber of sense. There is a very fundamental property with those margins, something strange about them. It seems to be that a duoring takes up a whole xyzw hyperplane, treated as one single diameter. A margin is but one, albeit a complex diameter, that gets inflated by some shape. That's what we see with (((II)(II))I), a torus along duoring. Or, even ((II)(II)), a circle along duoring. But, getting into much higher-D shapes, we have now seen that a duoring can be composed out of separate circle paths, as in tiger[maj1,2]-->duoring and tiger[maj1]-->duoring, (((II)I)((II)I) and (((II)(II))(II)) respectively. So long as they are orthogonal circular paths, a duoring has multiple constructions starting in 6D. Probably because of the three ortho 2-planes in 6D, it's a mater of combination in the two 2-planes out of three possible.


One more idea: since both ditorus and tiger are torus->circle, it seems there should be a way to transform ditorus into tiger or vice versa by simply rotating the torus.
The starting point could be the ditorus cut "two toruses". If you turned each of those toruses 90 degrees, you'd arrive to a cut of tiger. Could the ditorus and tiger equations be combined into a single one with parameter that would allow us to go from one to another by rotating toruses?



I can see that happening, visually. But, how to form an equation that does so, is beyond my knowledge. It is a very interesting idea, though. It seems to be a matter of rotating the final circle path 90 degrees. So, for both implicit equations:

(((II)I)I)
(sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 - R3^2 = 0

((II)(II))
(sqrt(x^2 + y^2) - R1)^2 + (sqrt(z^2 + w^2) - R2)^2 - R3^2= 0


In both, we have three diameters, which is a good thing. I'm not sure where to start here, so I'll list the 3D cut equations:


(((I)I)I)
(sqrt((sqrt(x^2) - R1)^2 + y^2) - R2)^2 + z^2 - R3^2 = 0

((II)(I))
(sqrt(x^2 + y^2) - R1)^2 + (sqrt(z^2) - R2)^2 - R3^2= 0


I don't know how to generate a function that swaps parentheses. They seems to be fundamentally different, and probably for good reason. It might be possible with some sort of modified rotation equation, but how to plug it in, within the equation, is beyond my knowledge. I'd be interested to see if it's possible, for sure! An easier one to try would be turning a sphere into a torus, then extrapolate how the function works. Easy as pie, if possible.


Well, there's of course a simple way to find a transition between any two functions f = 0 and g = 0:
f*cos(t)^2 + g*sin(t)^2 = 0.
For t = 0, you'll get f = 0 and for t = pi/2, you'll get g = 0

But this is a bit of cop-out and doesn't work out too well.

Now, for our case:
The second equation should be probably a bit different. Instead of ((II)(I)), let's use ((I)(II):
(sqrt(x^2) - R1)^2 + (sqrt(y^2 + z^2) - R2)^2 - R3^2= 0

This way, we get the (sqrt(x^2) - R1)^2 in both equations. Let's mark this expression with A, and we have:

(sqrt(A + y^2) - R2)^2 + z^2 - R3^2 = 0
A + (sqrt(y^2 + z^2) - R2)^2 - R3^2= 0

I don't have time to develop this right now, but I'll have a look later...
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Re: The Tiger Explained

Postby ICN5D » Fri May 09, 2014 8:22 pm

Oh cool, that function looks relatively straightforward.

Im thinking of calling the multiple different surtopes of the same shape a " surtopic dual ", or trio, quartet, etc. So, a ((III)(II))(II) has the surtopic dual of (((III)I)(II))+(((III)(II))I) and (((II)II)(II))+(((III)(II))I).
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Re: The Tiger Explained

Postby Marek14 » Fri May 09, 2014 9:00 pm

ICN5D wrote:Oh cool, that function looks relatively straightforward.

Im thinking of calling the multiple different surtopes of the same shape a " surtopic dual ", or trio, quartet, etc. So, a ((III)(II))(II) has the surtopic dual of (((III)I)(II))+(((III)(II))I) and (((II)II)(II))+(((III)(II))I).


Maybe that should be "duet", not "dual"...

BTW, as for ditorus and tiger, maybe the solution could be this:

Tiger torus (((II)(II))I) has a cut that is minor pair of tigers (((II)(II))) and also a cut that is medium stack of two ditoruses (((II)(I))I). And there's a 4D rotation equation between these cuts: (((II)(Ix))x).

sqrt(sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b )^2) - R2)^2 + v^2 = R3^2

sqrt(sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2*cos(a)^2) - R1b )^2) - R2)^2 + w^2*sin(a)^2 = R3^2

Now, we can reduce these two cuts to a single tiger/ditorus by posing one of the diameters equal to zero.

For tiger:
sqrt(sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b )^2) - R2)^2 = 0^2

For ditorus:
sqrt(sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2) - 0)^2) - R2)^2 + v^2 = R3^2

So maybe if we did this:
sqrt(sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2*cos(a)^2) - R1b*cos(a)^2 )^2) - R2)^2 + w^2*sin(a)^1 = R1b^2*sin(a)^2

I.e., we rotate one dimension, but we also simultaneously shrink one diameter to zero and grow another one from zero.
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Re: The Tiger Explained

Postby ICN5D » Sat May 10, 2014 4:03 am

I plugged in the equation :

sqrt(sqrt((sqrt(x^2 + y^2) - 2.5)^2 + (sqrt(z^2 + b^2*cos(a)^2) - 2.5*cos(a)^2 )^2) - .075)^2 + b^2*sin(a)^1 = 2.5^2*sin(a)^2


and rotated A and B around, but only saw either tiger cut of a minor stack, or a torus, inflate to a spheroid, then back to minor stack. It's definitely not a minor stack to major stack morphing. Hmmmm? There were some other strange things in there, nothing remarkable, though. The equation seems to work in the fundamental way, but the end-result figure isn't a major stack from a minor stack, but something more sphere-like.



Duet is good, I'll use that. It better represents two individual states rather than inverted states, as dual suggests.
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Re: The Tiger Explained

Postby Marek14 » Sat May 10, 2014 5:24 am

Then we need to start with the equation of torus...

(sqrt(x^2 + y^2) - R)^2 + z^2 = r^2

What if we mix y and z here?

(sqrt(x^2 + (y*cos(a) + z*sin(a))^2) - R)^2 + (y*sin(a) - z*cos(a))^2 = r^2

This is the equation of a proper rotating torus, I checked.

So, now let's try to replace one of the coordinates with the expression for circle, (sqrt(x^2 + y^2) - R)^2...

(sqrt((sqrt(x^2 + y^2) - R1)^2 + (z*cos(a) + w*sin(a))^2) - R2)^2 + (z*sin(a) - w*cos(a))^2 = R3^2

I tried this, but it didn't work. What I got was, apparently, a rotating ditorus. So let's try to replace ANOTHER coordinate...

(sqrt(x^2 + ((sqrt(y^2 + z^2) - R1)^2*cos(a) + w*sin(a))^2) - R2)^2 + ((sqrt(y^2 + z^2) - R1)^2*sin(a) - w*cos(a))^2 = R3^2

The equation
(sqrt(x^2 + ((sqrt(y^2 + b^2) - 4)^2*cos(a) + z*sin(a))^2) - 2)^2 + ((sqrt(y^2 + b^2) - 4)^2*sin(a) - z*cos(a))^2 = 1^2

looks pretty fascinating; there is a morph between horizontal and vertical toruses, but it's not rotation -- you definitely have to see it. Other cuts are strange as well. This thing definitely IS a ditorus on a = 0 and tiger on a = pi/2 -- or something topologically like them, the ditorus looks a bit squashed, maybe it's parameters.

We're probably not completely there yet, but it looks promising.
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Re: The Tiger Explained

Postby wendy » Sat May 10, 2014 7:27 am

On relation to 'notables' etc, someome reccomends you and someone seconds it, and then someone else, usually higher up than me, implements it.
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Re: The Tiger Explained

Postby Marek14 » Sat May 10, 2014 8:02 am

Here's an interesting tidbit for today:

Code: Select all
2 (1/1):
2 - 1
circle (II): two points

3 (2/1):
3 - 1
21 - 1
torus ((II)I): four points

4 (5/2):
4 - 1
31 - 2
ditorus (((II)I)I): eight points
22 - 1
tiger ((II)(II)): 2x2 array of circles
211 - 1

5 (12/3):
5 - 1
41 - 5
tritorus ((((II)I)I)I): 16 points
tiger torus (((II)(II))I): 2x2 array of pairs of circles
32 - 2
torus tiger (((II)I)(II)): 4x2 array of circles
311 - 2
221 - 1
2111 - 1

6 (33/7):
6 - 1
51 - 12
tetratorus (((((II)I)I)I)I): 32 points
tiger ditorus ((((II)(II))I)I): 2x2 array of quartets of circles
torus tiger torus ((((II)I)(II))I): 4x2 array of pairs of circles
42 - 5
ditorus tiger ((((II)I)I)(II)): 8x2 array of circles
double tiger (((II)(II))(II)): 2x2 array of vertical stacks of two toruses
411 - 5
33 - 3
duotorus tiger (((II)I)((II)I)): 4x4 array of circles
321 - 2
3111 - 2
222 - 1
triger ((II)(II)(II)): 2x2x2 array of spheres
2211 - 1
21111 - 1

7 (90/13):
7 - 1
61 - 33
pentatorus ((((((II)I)I)I)I)I): 64 points
tiger tritorus (((((II)(II))I)I)I): 2x2 array of octets of circles
torus tiger ditorus (((((II)I)(II))I)I): 4x2 array of quartets of circles
ditorus tiger torus (((((II)I)I)(II))I): 8x2 array of pairs of circles
double tiger torus ((((II)(II))(II))I): 2x2 array of vertical stacks of two minor pairs of toruses
duotorus tiger torus ((((II)I)((II)I))I): 4x4 array of pairs of circles
triger torus (((II)(II)(II))I): 2x2x2 array of pairs of spheres
52 - 12
tritorus tiger (((((II)I)I)I)(II)): 16x2 array of circles
tiger torus tiger ((((II)(II))I)(II)): 2x2 array of vertical stacks of two major pairs of toruses
torus double tiger ((((II)I)(II))(II)): 4x2 array of vertical stacks of two toruses
511 - 12
43 - 10
ditorus/torus tiger ((((II)I)I)((II)I)): 8x4 array of circles
tiger/torus tiger (((II)(II))((II)I)): 2x2 array of vertical stacks of four toruses
421 - 5
4111 - 5
331 - 3
322 - 2
torus triger (((II)I)(II)(II)): 4x2x2 array of spheres
3211 - 2
31111 - 2
2221 - 1
22111 - 1
211111 - 1

8 (256/30):
8 - 1
71 - 90
hexatorus (((((((II)I)I)I)I)I)I): 128 points
tiger tetratorus ((((((II)(II))I)I)I)I): 2x2 array of 16-plets of circles
torus tiger tritorus ((((((II)I)(II))I)I)I): 4x2 array of octets of circles
ditorus tiger ditorus ((((((II)I)I)(II))I)I): 8x2 array of quartets of circles
double tiger ditorus (((((II)(II))(II))I)I): 2x2 array of vertical stacks of two minor quartets of toruses
duotorus tiger ditorus (((((II)I)((II)I))I)I): 4x4 array of quartets of circles
triger ditorus ((((II)(II)(II))I)I): 2x2x2 array of quartets of spheres
tritorus tiger torus ((((((II)I)I)I)(II))I): 16x2 array of pairs of circles
tiger torus tiger torus (((((II)(II))I)(II))I): 2x2 array of vertical stacks of two major/minor quartets of toruses
torus double tiger torus (((((II)I)(II))(II))I): 4x2 array of vertical stacks of two minor pairs of toruses
ditorus/torus tiger torus (((((II)I)I)((II)I))I): 8x4 array of pairs of circles
tiger/torus tiger torus ((((II)(II))((II)I))I): 2x2 array of vertical stacks of four minor pairs of toruses
torus triger torus ((((II)I)(II)(II))I): 4x2x2 array of pairs of spheres
62 - 33
tetratorus tiger ((((((II)I)I)I)I)(II)): 32x2 array of circles
tiger ditorus tiger (((((II)(II))I)I)(II)): 2x2 array of vertical stacks of two major quartets of toruses
torus tiger torus tiger (((((II)I)(II))I)(II)): 4x2 array of vertical stacks of two major pairs of toruses
ditorus double tiger (((((II)I)I)(II))(II)): 8x2 array of vertical stacks of two toruses
triple tiger ((((II)(II))(II))(II)): 2x2 array of 2x2 medium/minor stacks of ditoruses
duotorus double tiger ((((II)I)((II)I))(II)): 4x4 array of vertical stacks of two toruses
triger tiger (((II)(II)(II))(II)): 2x2x2 array of vertical stacks of two torispheres
611 - 33
53 - 24
tritorus/torus tiger (((((II)I)I)I)((II)I)): 16x4 array of circles
(tiger torus)/torus tiger ((((II)(II))I)((II)I)): 2x2 array of vertical stacks of four major pairs of toruses
(torus tiger)/torus tiger ((((II)I)(II))((II)I)): 4x2 array of vertical stacks of four toruses
521 - 12
5111 - 12
44 - 10
duoditorus tiger ((((II)I)I)(((II)I)I)): 8x8 array of circles
ditorus/tiger tiger ((((II)I)I)((II)(II))): 2x2 array of vertical stacks of eight toruses
duotiger tiger (((II)(II))((II)(II))): 2x2x2x2 A/A/B/B array of tigers
431 - 10
422 - 5
ditorus triger ((((II)I)I)(II)(II)): 8x2x2 array of spheres
tiger triger (((II)(II))(II)(II)): 2x2 array of 2x2 vertical stacks of spheritoruses
4211 - 5
41111 - 5
332 - 3
duotorus triger (((II)I)((II)I)(II)): 4x4x2 array of spheres
3311 - 3
3221 - 2
32111 - 2
311111 - 2
2222 - 1
tetriger ((II)(II)(II)(II)): 2x2x2x2 array of glomes
22211 - 1
221111 - 1
2111111 - 1


It's a semi-enumerated list of toratopes up to 8 dimensions including the basic toratopes of each dimension and their traces.
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Re: The Tiger Explained

Postby Marek14 » Sat May 10, 2014 8:29 am

And here's a look at 63 basic toratopes out of 723 in 9D.

Code: Select all
9 (723/63):
9 - 1
81 - 256
heptatorus ((((((((II)I)I)I)I)I)I)I): 256 points
tiger pentatorus (((((((II)(II))I)I)I)I)I): 2x2 array of 32-plets of circles
torus tiger tetratorus (((((((II)I)(II))I)I)I)I): 4x2 array of 16-plets of circles
ditorus tiger tritorus (((((((II)I)I)(II))I)I)I): 8x2 array of octets of circles
double tiger tritorus ((((((II)(II))(II))I)I)I): 2x2 array of vertical stacks of two minor octets of toruses
duotorus tiger tritorus ((((((II)I)((II)I))I)I)I): 4x4 array of octets of circles
triger tritorus (((((II)(II)(II))I)I)I): 2x2x2 array of octets of spheres
tritorus tiger ditorus (((((((II)I)I)I)(II))I)I): 16x2 array of quartets of circles
tiger torus tiger ditorus ((((((II)(II))I)(II))I)I): 2x2 array of vertical stacks of two major/minor/minot octets of toruses
torus double tiger ditorus ((((((II)I)(II))(II))I)I): 4x2 array of vertical stacks of two minor quartets of toruses
ditorus/torus tiger ditorus ((((((II)I)I)((II)I))I)I): 8x4 array of quartets of circles
tiger/torus tiger ditorus (((((II)(II))((II)I))I)I): 2x2 array of vertical stacks of four minor octets of toruses
torus triger ditorus (((((II)I)(II)(II))I)I): 4x2x2 array of quartets of spheres
tetratorus tiger torus (((((((II)I)I)I)I)(II))I): 32x2 array of pairs of circles
tiger ditorus tiger torus ((((((II)(II))I)I)(II))I): 2x2 array of vertical stacks of two major/major/minor octets of toruses
torus tiger torus tiger torus ((((((II)I)(II))I)(II))I): 4x2 array of vertical stacks of two major/minor quartets of toruses
ditorus double tiger torus ((((((II)I)I)(II))(II))I): 8x2 array of vertical stacks of two minor pairs of toruses
triple tiger torus (((((II)(II))(II))(II))I): 2x2 array of 2x2 medium/minor stacks of minor pairs of ditoruses
duotorus double tiger torus (((((II)I)((II)I))(II))I): 4x4 array of vertical stacks of two minor pairs of toruses
triger tiger torus ((((II)(II)(II))(II))I): 2x2x2 array of vertical stacks of two minor pairs of torispheres
tritorus/torus tiger torus ((((((II)I)I)I)((II)I))I): 16x4 array of pairs of circles
(tiger torus)/torus tiger torus (((((II)(II))I)((II)I))I): 2x2 array of vertical stacks of four major/minor quartets of toruses
(torus tiger)/torus tiger torus (((((II)I)(II))((II)I))I): 4x2 array of vertical stacks of four minor pairs of toruses
duoditorus tiger torus (((((II)I)I)(((II)I)I))I): 8x8 array of pairs of circles
ditorus/tiger tiger torus (((((II)I)I)((II)(II)))I): 2x2 array of vertical stacks of eight minor pairs of toruses
duotiger tiger torus ((((II)(II))((II)(II)))I): 2x2x2x2 A/A/B/B array of minor pairs of tigers
ditorus triger torus (((((II)I)I)(II)(II))I): 8x2x2 array of pairs of spheres
tiger triger torus ((((II)(II))(II)(II))I): 2x2 array of 2x2 vertical stacks of minor pairs of spheritoruses
duotorus triger torus ((((II)I)((II)I)(II))I): 4x4x2 array of pairs of spheres
tetriger torus (((II)(II)(II)(II))I): 2x2x2x2 array of pairs of glomes
72 - 90
pentatorus tiger (((((((II)I)I)I)I)I)(II)): 64x2 array of circles
tiger tritorus tiger ((((((II)(II))I)I)I)(II)): 2x2 array of vertical stacks of two major octets of toruses
torus tiger ditorus tiger ((((((II)I)(II))I)I)(II)): 4x2 array of vertical stacks of two major quartets of toruses
ditorus tiger torus tiger ((((((II)I)I)(II))I)(II)): 8x2 array of vertical stacks of two major pairs of toruses
double tiger torus tiger (((((II)(II))(II))I)(II)): 2x2 array of 2x2 medium/minor stacks of medium pairs of ditoruses
duotorus tiger torus tiger (((((II)I)((II)I))I)(II)): 4x4 array of vertical stacks of two major pairs of toruses
triger torus tiger ((((II)(II)(II))I)(II)): 2x2x2 array of vertical stacks of two major pairs of torispheres
tritorus double tiger ((((((II)I)I)I)(II))(II)): 16x2 array of vertical stacks of two toruses
tiger torus double tiger (((((II)(II))I)(II))(II)): 2x2 array of 2x2 medium/minor stacks of major pairs of ditoruses
torus triple tiger (((((II)I)(II))(II))(II)): 4x2 array of 2x2 medium/minor stacks of ditoruses
ditorus/torus double tiger (((((II)I)I)((II)I))(II)): 8x4 array of vertical pairs of two toruses
tiger/torus double tiger ((((II)(II))((II)I))(II)): 2x2 array of 4x2 medium/minor stacks of ditoruses
torus triger tiger ((((II)I)(II)(II))(II)): 4x2x2 array of vertical stacks of two torispheres
711 - 90
63 - 33
tetratorus/torus tiger ((((((II)I)I)I)I)((II)I)): 32x4 array of circles
(tiger ditorus)/torus tiger (((((II)(II))I)I)((II)I)): 2x2 array of vertical stacks of four major quartets of toruses
(torus tiger torus)/torus tiger (((((II)I)(II))I)((II)I)): 4x2 array of vertical stacks of four major pairs of toruses
(ditorus tiger)/torus tiger (((((II)I)I)(II))((II)I)): 8x2 array of vertical stacks of four toruses
(double tiger)/torus tiger ((((II)(II))(II))((II)I)): 2x2 array of 2x4 medium/minor stacks of ditoruses
(duotorus tiger)/torus tiger ((((II)I)((II)I))((II)I)): 4x4 array of vertical stacks of four toruses
triger/torus tiger (((II)(II)(II))((II)I)): 2x2x2 array of vertical stacks of four torispheres
621 - 33
6111 - 33
54 - 60
tritorus/ditorus tiger (((((II)I)I)I)(((II)I)I)): 16x8 array of circles
tritorus/tiger tiger (((((II)I)I)I)((II)(II))): 2x2 array of vertical stacks of 16 toruses
(tiger torus)/ditorus tiger ((((II)(II))I)(((II)I)I)): 2x2 array of vertical stacks of 8 major pairs of toruses
(tiger torus)/tiger tiger ((((II)(II))I)((II)(II))): 2x2x2x2 A/A/B/B array of major pairs of tigers
(torus tiger)/ditorus tiger ((((II)I)(II))(((II)I)I)): 4x2 array of vertical stacks of 8 toruses
(torus tiger)/tiger tiger ((((II)I)(II))((II)(II))): 4x2x2x2 A/A/B/B array of tigers
531 - 24
522 - 12
tritorus triger (((((II)I)I)I)(II)(II)): 16x2x2 array of spheres
tiger torus triger ((((II)(II))I)(II)(II)): 2x2 array of 2x2 vertical stacks of major pairs of spheritoruses
torus tiger triger ((((II)I)(II))(II)(II)): 4x2 array of 2x2 vertical stacks of spheritoruses
5211 - 12
51111 - 12
441 - 10
432 - 10
ditorus/torus triger ((((II)I)I)((II)I)(II)): 8x4x2 array of spheres
tiger/torus triger (((II)(II))((II)I)(II)): 2x2 array 4x2 vertical stacks of spheritoruses
4311 - 10
4221 - 5
42111 - 5
411111 - 5
333 - 4
triotorus triger (((II)I)((II)I)((II)I)): 4x4x4 array of spheres
3321 - 3
33111 - 3
3222 - 2
torus tetriger (((II)I)(II)(II)(II)): 4x2x2x2 array of glomes
32211 - 2
321111 - 2
3111111 - 2
22221 - 1
222111 - 1
2211111 - 1
21111111 - 1
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Re: The Tiger Explained

Postby ICN5D » Sun May 11, 2014 2:41 am

Awsome Marek! I've been wanting to see those +6D toratopes for a while. At least the interesting ones. And I noticed you neatly condensed the fat toratope combinations, to flesh out the cool ones. Though at this point, I could probably derive any of them! And know if it was valid.



I plugged in that rotation equation, and wow! That's the strangest rotation morphing I've seen. It really does go from a ditorus to a tiger, with translation. I played with the diameter values, trying to round out the toruses, but it still had that weird stretch to it. Couldn't make it perfectly round:

(sqrt(x^2 + ((sqrt(y^2 + b^2) - 2.5)^2*cos(a) + z*sin(a))^2) - 1.3)^2 + ((sqrt(y^2 + b^2) -2.5)^2*sin(a) - z*cos(a))^2 = 1^2




Here's something cool. The first animation is ((((II)I)(II))I) rotating one axis 90 degrees with the other two oblique. The second is ((((II)I)I)(II)), rotating from one empty cut to another: (((()I)I)(I)) to ((((I)I)I)())
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Re: The Tiger Explained

Postby Marek14 » Sun May 11, 2014 4:41 am

Looks like fun :)
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Re: The Tiger Explained

Postby ICN5D » Sun May 11, 2014 5:14 am

Here's something else:

It's an enumeration of the surtopes of all open toratopes 2D to 6D:

Code: Select all
Surtopes of the Open Toratopes
--------------------------------


2D:
II - [line || line]+[line || line]


3D:
III - [sqr || sqr] + [sqr || sqr] + [sqr || sqr] = [sqr || sqr]^3

(II)I - [circle || circle] + [ line-->circle ]


4D:
IIII - [cube || cube]^4

(II)II - [(II)I || (II)I]^2 + [square-->circle]

(II)(II) - ((II)I)+((II)I)

(III)I - [(III) || (III)] + [line-->sphere]

((II)I)I - cylinder-->circle , [((II)I) || ((II)I)] + [line-->torus]


5D:
IIIII - [tess || tess]^5

(II)III - [[(II)II || (II)II]^3 + [cube-->circle]

(II)(II)I - [(II)(II) || (II)(II)] + [((II)I)I+((II)I)I] 

(III)II - [(III)I || (III)I]^2 + [square-->sphere]

((II)I)II - cubinder-->circle , [((II)I)I || ((II)I)I]^2 + [square-->torus]

(III)(II) - ((III)I)+((II)II)

((II)I)(II) - duocylinder-->circle , (((II)I)I)+(((II)I)I)

(IIII)I - [(IIII) || (IIII)] + [line-->glome]

((II)II)I - spherinder-->circle , [((II)II) || ((II)II)] + [line-->spheritorus]

((II)(II))I - cylinder-->duoring , [((II)(II)) || ((II)(II))] + [line-->tiger]

((III)I)I - cylinder-->sphere , [((III)I) || ((III)I)] + [line-->torisphere]

(((II)I)I)I - cylinder-->torus , [(((II)I)I) || (((II)I)I)] + [line-->ditorus]


6D:
IIIIII - [penteract || penteract]^6

(II)IIII - [(II)III || (II)III]^4 + [tesseract-->circle]

(II)(II)II - [(II)(II)I || (II)(II)I]^2 + [((II)I)II+((II)I)II]

(II)(II)(II) - ((II)I)(II)+((II)I)(II)+((II)I)(II)

(III)III - [[(III)II || (III)II]^3 + [cube-->sphere]

((II)I)III - tesserinder-->circle , [((II)I)II || ((II)I)II]^3 + [cube-->torus]

(III)(II)I - [(III)(II) || (III)(II)] + [((III)I)I+((II)II)I] 

((II)I)(II)I - [((II)I)(II) || ((II)I)(II)] + [(((II)I)I)I+(((II)I)I)I] 

(III)(III) - ((III)II)+((III)II)

((II)I)(III) - (((III)I)I)+(((II)I)II) Type-1 , (((II)I)II)+(((II)II)I) Type-2

((II)I)((II)I) - ((((II)I)I)I)+((((II)I)I)I)

(IIII)II - [(IIII)I || (IIII)I]^2 + [square-->glome]

((II)II)II - cubspherinder-->circle , [((II)II)I || ((II)II)I]^2 + [square-->sphere-->circle]

((II)(II))II - cubinder-->duoring , [((II)(II))I || ((II)(II))I]^2 + [square-->tiger]

((III)I)II - cubinder-->sphere , [((III)I)I || ((III)I)I]^2 + [square-->circle-->sphere]

(((II)I)I)II - cubinder-->torus , [(((II)I)I)I || (((II)I)I)I]^2 + [square-->torus-->circle]

(IIII)(II) - ((II)III)+((IIII)I)

((II)II)(II) - (((II)I)II)+(((II)II)I)

((II)(II))(II) - (((II)I)(II))+(((II)(II))I)

((III)I)(II) - (((II)II)I)+(((III)I)I)

(((II)I)I)(II) - ((((II)I)I)I)+((((II)I)I)I)

(IIIII)I - [(IIIII) || (IIIII)] + [line-->pentasphere]

((II)III)I - glominder-->circle , [((II)III) || ((II)III)] + [line-->glome-->circle]

((II)(II)I)I - spherinder-->duoring , [((II)(II)I) || ((II)(II)I)] + [line-->sphere-->duoring]

((III)II)I - spherinder-->sphere , [((III)II) || ((III)II)] + [line-->sphere-->sphere]

(((II)I)II)I - spherinder-->torus , [(((II)I)II) || (((II)I)II)] + [line-->sphere-->torus]

((III)(II))I - cylinder-->(sphere x circle) , [((III)(II)) || ((III)(II))] + [line-->cylspherintigroid]

(((II)I)(II))I - cylinder-->duoring-->circle , [(((II)I)(II)) || (((II)I)(II))] + [line-->tiger torus]

((IIII)I)I - cylinder-->glome , [((IIII)I) || ((IIII)I)] + [line-->circle-->glome]

(((II)II)I)I - cylinder-->spheritorus , [(((II)II)I) || (((II)II)I)] + [line-->torus-->sphere-->circle]

(((II)(II))I)I - cylinder-->tiger , [(((II)(II))I) || (((II)(II))I)] + [line-->torus-->duoring]

(((III)I)I)I - cylinder-->torisphere , [(((III)I)I) || (((III)I)I)] + [line-->ditorus-->sphere]

((((II)I)I)I)I - cylinder-->ditorus , [((((II)I)I)I) || ((((II)I)I)I)] + [line-->torus-->torus]




I found it interesting that triocylinder, (circle^3)-prism , (II)(II)(II) , has three duocylinder toruses ((II)I)(II) on it's surface. I thought it was going to be three toruses, but that wouldn't make sense. It has to be a 5-D surtope, and turns out to be a duocylinder torus. This means that one three-factor shape I mentioned before, ((II)I)(III)(II) would actually have (((II)I)I)(III)+((II)II)((II)I)+(((III)I)I)(II) on it's surface, as the three rolling sides.
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Re: The Tiger Explained

Postby Marek14 » Sun May 11, 2014 11:23 am

Also interesting.
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Re: The Tiger Explained

Postby ICN5D » Sun May 11, 2014 6:37 pm

Oh, Toratope Notation, how can it be,
that you tell me what I will see,
when we cut down into 3D?
Being skeptical at first,
seen as technical and diverse.
Intricate with your parentheses,
it takes time before one sees,
the nature of your diameters,
sequestered into parameters.
Then came the moment of truth,
A graphing calculator from which we have proof,
A 3D cross section equation,
Comes out as expected, to my elation!
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Re: The Tiger Explained

Postby Marek14 » Sun May 11, 2014 7:22 pm

The notation, you have to see,
Is easy -- just like a, b, c.
Its holes and bulk, its higher rows
Show on the cave's wall as shadows
Like birds flying above the sea
They dive into our axes three
They roll and morph and slice and churn
While toratopic wheels do turn
Symbols' parenthetic bind
Will easily open your mind.
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Re: The Tiger Explained

Postby ICN5D » Wed May 14, 2014 6:22 am

I was thinking about how toratopes get their dimensions get expanded, and certain terminology around it. It seems like, if using a torus for example, the shape (((II)(II))(II)), double tiger, could be defined with something like torus^circle, and maybe what you call torus squared ((((II)I)((II)I))((II)I)), would be torus^torus. So, the tiger-torus^tiger-torus would be:

((((((II)I)(II))(((II)I)(II)))(((II)I)(II)))((((II)I)(II))(((II)I)(II))))

a 25 dimensional shape. Which would be the highest dimensional toratope I have written to date. Or, maybe a duotorus-tiger^(cylspherintigroid-torus type-1):

((((((II)I)(III))(((II)I)(III)))(((II)I)(III)))(((((II)I)(III))(((II)I)(III)))(((II)I)(III)))

a 36-dimensional toratope. I guess I could keep going, but the point's been made :)


You know, I kinda miss the cut algorithm homework...Hey, how about tossing me some huge toratopes, and I'll see if I can cut them down? If I build them up, I'll already know what the cut is, and it's no fun. I love analyzing the huge megatopes, what can I say?
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Re: The Tiger Explained

Postby Marek14 » Wed May 14, 2014 6:38 am

Well, I'm currently working on complete cut analysis of 7D cases down to 3D, but it's more of a hobby and I only do a bit every day, with results like:

Code: Select all
4. 4111-tritorus ((((IIII)I)I)I)
6D blobs:
Minor pair of 411-ditoruses ((((IIII)I)I)) -> 4111-tritorus
Medium pair of 411-ditoruses ((((IIII)I))I) -> 4111-tritorus
Major pair of 411-ditoruses ((((IIII))I)I) -> 4111-tritorus
3111-tritorus ((((III)I)I)I) -> 4111-tritorus
5D slabs:
Minor quartet of 41-toruses ((((IIII)I))) -> minor pair of 411-ditoruses, medium pair of 411-ditoruses
Major/minor quartet of 41-toruses ((((IIII))I)) -> minor pair of 411-ditoruses, major pair of 411-ditoruses
Minor pair of 311-ditoruses ((((III)I)I)) -> minor pair of 411-ditoruses, 3111-tritorus
Major quartet of 41-toruses ((((IIII)))I) -> medium pair of 411-ditoruses, major pair of 411-ditoruses
Medium pair of 311-ditoruses ((((III)I))I) -> medium pair of 411-ditoruses, 3111-tritorus
Major pair of 311-ditoruses ((((III))I)I) -> major pair of 411-ditoruses, 3111-tritorus
Tritorus ((((II)I)I)I) -> 3111-tritorus (x2)
4D slices:
Octet of glomes ((((IIII)))) -> minor quartet of 41-toruses, major/minor quartet of 41-toruses, major quartet of 41-toruses
Minor quartet of torispheres ((((III)I))) -> minor quartet of 41-toruses, minor pair of 311-ditoruses, medium pair of 311-ditoruses
Major/minor quartet of torispheres ((((III))I)) -> major/minor quartet of 41-toruses, minor pair of 311-ditoruses, major pair of 311-ditoruses
Minor pair of ditoruses ((((II)I)I)) -> minor pair of 311-ditoruses (x2), tritorus
Major quartet of torispheres ((((III)))I) -> major quartet of 41-toruses, medium pair of 311-ditoruses, major pair of 311-ditoruses
Medium pair of ditoruses ((((II)I))I) -> medium pair of 311-ditoruses (x2), tritorus
Major pair of ditoruses (((II))I)I) -> major pair of 311-ditoruses (x2), tritorus
Two ditoruses ((((I)I)I)I) -> tritorus (x3)
3D cuts:
Octet of spheres ((((III)))) -> octet of glomes, minor quartet of torispheres, major/minor quartet of torispheres, major quartet of torispheres
Minor quartet of toruses ((((II)I))) -> minor quartet of torispheres (x2), minor pair of ditoruses, medium pair of ditoruses
Major/minor quartet of toruses ((((II))I)) -> major/minor quartet of torispheres (x2), minor pair of ditoruses, major pair of ditoruses
Two minor pairs of toruses ((((I)I)I)) -> minor pair of ditoruses (x3), two ditoruses
Major quartet of toruses ((((II)))I) -> major quartet of torispheres (x2), medium pair of ditoruses, major pair of ditoruses
Two major pairs of toruses ((((I)I))I) -> medium pair of ditoruses (x3), two ditoruses
Four toruses ((((I))I)I) -> major pair of ditoruses (x3), two ditoruses
Empty cut (((()I)I)I) -> two ditoruses (x4)


So, let's have a look at something nasty.

(((((II)(II))((II)(II)(II))((II)(II)(II)(II))(((II)(II))(II)(II)))I)((II)I)I)((II)(II)))

It's a stream-of-consciousness toratope in 35 dimensions :)
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Re: The Tiger Explained

Postby ICN5D » Wed May 14, 2014 11:47 pm

All right,


(((((II)(II))((II)(II)(II))((II)(II)(II)(II))(((II)(II))(II)(II)))I)((II)I)I)((II)(II)))



• Cuts down to lowest non-empty trace of (((((I)(I))((I)(I)(I))((I)(I)(I)(I))(((I)(I))(I)(I))))((I)))((I)(I)))

Which is 262,144 (((II)(III)(IIII)((II)II))I)(II)) , a ((circle,sphere,glome,spheritorus)-toritetriger)-tiger


- A neat decomposition of this 16D trace-shape is a tiger-->glome-->{(circle x sphere^2 x glome)-margin torus}

Inflate ((!I)(II)) with a (IIII), then (((!!!!)I)(II)) with a ((II)(III)(IIII)((II)II)))




Arranged in a 2x2x2x2x2x2x2x2x2x2x2x2x2x1x4x1x2x2 sixteen axis array of 131,072 locations

• MAJ/MED/MIN{circle1/2, sphere, glome, torus[maj,min], spheritorus[maj,min]} stacks

• (circle,sphere,glome,spheritorus)-toritetriger[medium] pairs of 2




• In trace array notation, it would look like : ((((00)0,(000)0,(0000)0,((00)0,00)0,0)1,0)1,(00)0)0




• The construction process for this particular 35-D shape is straightforward. It would seem to have started life as a (((IIIII)II)I), a torispheripentasphere, then had many of its dimensions expanded

Starting with a (((IIIII)II)I) , a circle-->sphere-->pentasphere, the 521-ditorus:

1) 521-Ditorus[maj]-->circle-->duoring

2) 521-Ditorus[maj]-->sphere-->trioring

3) 521-Ditorus[maj]-->glome-->quattroring

4) 521-Ditorus[maj]-->spheritorus[maj,min]-->quattroring

5) 521-Ditorus[med]-->torus

6) 521-Ditorus[min]-->circle-->duoring



That one was tougher than I expected. I appreciate that, a good thinking puzzle :). I actually posted what I thought was the correct trace twice, then realized I was wrong. This is the third time, I think I got it......
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Re: The Tiger Explained

Postby Marek14 » Thu May 15, 2014 4:04 am

Well, there are some torus elements in the shape, so the trace should contain some pairs :)
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Re: The Tiger Explained

Postby ICN5D » Thu May 15, 2014 5:37 pm

Oh of course. I kept editing the post, so I probably deleted it. It would be the minor pair of (circle, sphere, glome, spheritorus) tetrigers.
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Re: The Tiger Explained

Postby ICN5D » Tue May 20, 2014 6:27 am

So, had a second look at a previously imaged shape: the Double Tiger (((II)(II))(II)). Found an amazing 3D structure that illustrates the coolest thing about 6D space. By this, I refer to the three orthogonal 2-planes of 6D.




This incredible structure is made by a duodecatangent cut. This single 3D slicing plane lies tangent to a whopping twelve points on the surface of a 6D Double Tiger. In the images, you can count the twelve narrow pinching points, joining the inflated masses. These are the twelve tangent points, in their rough location. Also take notice of how these 12 are arranged into the vertices of three squares. Those are reflecting the three orthogonal 2-planes!! And perfectly arranged in the proper manner, too. How nice to see that. Higher dimensions, now you're seeing them :)

Image

Image

Image

Image




My name wouldn't be Philip if I didn't try to bring you all something extra cool. This shape is an 8D toratope, notated as ((((II)I)(II))((II)I)). Simply put, this thing can made by starting with a tiger, running it along a circle to make a torus of a tiger : (((II)I)(II)), then inflating a duoring with this tiger torus making (((II)(II))((II)I)), then running this along another circle, making an 8D torus : ((((II)I)(II))((II)I)) . Or, more simply put as tiger[circle1]-->circle-->duoring-->circle. So, what we get is a 4x2x4 rectangular brick of 32 toruses:

Image



Now, let's isolate one of these columns of four. It's a whole tiger torus, (((II)I)(II)) !!

Image




And, each one is in place of the trace of a duoring torus. A duoring has the 2D cut of four points in a 2x2 array, the vertices of a square. So, a torus of this duoring would cut down as a side-by-side pairing of these 2x2 arrays, making a 4x2 array. This 4x2 of 8 points is the margin of a cyltorinder, a.k.a. duocylinder torus. This 8D toratope ((((II)I)(II))((II)I)) is made by inflating this duoring torus of 4x2, with an entire (((II)I)(II)), the tiger torus. So, that's why we get the 4x2 array of these vertical columns of 4. And keep in mind that each vert column of 4 has inflated a single point, as a 0D location along the torus of a duoring. What an awesome display of the structure of an eight dimensional shape!


I tried doing an actual tiger torus along tiger torus: (((((II)I)(II))I)((II)I)), but neither my computer, nor the rendering program could handle the function, when trying to separate the concentric pairing of toruses, in that 4x2x4 array. That's the shape I tried to do first, but then stepped down a dimension to just inflate the duoring torus. I really wish I could find a more sophisticated program, better suited for these larger, more complex functions. This one has worked out well, but it's holding me back from total awesomeness.
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Re: The Tiger Explained

Postby Marek14 » Tue May 20, 2014 6:52 am

Hm, 12 points in three orthogonal rectangles -- you might be able to inscribe an icosahedron there :D
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Re: The Tiger Explained

Postby ICN5D » Tue May 20, 2014 4:07 pm

How would that work? I haven't explored those types of shapes very much.
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Re: The Tiger Explained

Postby Marek14 » Tue May 20, 2014 4:46 pm

ICN5D wrote:How would that work? I haven't explored those types of shapes very much.


Well, icosahedron has 12 vertices which fall in 3 rectangles in 3 orthogonal planes.
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Re: The Tiger Explained

Postby ICN5D » Tue May 20, 2014 6:19 pm

Well now, that's interesting. There must be some relation in how that works. I have found many multitangent slicing planes with many toratopes. I wonder if they can be categorized in the same manner? A bitangent cut of a torus can inscribe a line where the two points are. I think we're on to something new here. Maybe, perhaps. I know there is a way to derive the angles of the slicing plane that produces the multitangent cuts. It's a little above my math skills now, but it's closely related to the Pythagorean Theorem of triangles.
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Re: The Tiger Explained

Postby ICN5D » Wed May 21, 2014 2:30 am

So, I did some investigating, and made a formula that gets close the rotation value for a bitangent cut of a torus. It would seem that ( pi * atan(R2/R1))/180 will give the correct value, but in experiment, it's slightly too small. But, in the neighborhood by a tenth, so it must be somewhat worthy. If R1 = 2, and R2 = 1, then the formula gives 0.51516. In practice, it really needs to be 0.52333 to make the two perfect circles. Any ideas, here?

So, for torus: (sqrt(x^2+(y*cos(a))^2) - 2)^2 + (y*sin(a))^2 - 1^2 = 0

Using ( pi * atan(R2/R1))/180 = a , where if R1 = 2 , R2 = 1 , then a = 0.51516 for bitangent cut

And, again, it could be a minor limitation of the program, like a round-off error or something.
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