Implicit equation for open toratopes are easy -- you just don't use all coordinates. For example, x^2 + y^2 = 1 is equation of circle in 2D, but equation of infinite cylinder in 3D and equation of circle x plane in 4D.

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Back to basics, good ole' Torisphere ((III)I) !

4D Torisphere : ((III)I)

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

• ((IIi)I) - torus

(sqrt(x^2 + y^2 + a^2) - 4)^2 + z^2 -0.75^2 = 0

Moving the torisphere up and down through 4-space, causing it to slide in and out of our thin 3D plane, with respect to 4D. This is cutting through its major diameter, having the shape of a sphere. This cut leaves its minor diameter alone, allowing us to see its circular shape

• ((III)i) - concentric spheres

(sqrt(x^2 + y^2 + z^2) - 4)^2 + a^2 -0.75^2 = 0

Moving the torisphere in and out at another angle, cutting it top to bottom in the other orthogonal 2-plane. This is cutting through its minor diameter, shaped like a circle. This cut leaves its major diameter alone, allowing us to see its spherical shape

• ((IIz)Z) - rotation

(sqrt(x^2 + y^2 + (z*cos(a))^2) - 4)^2 + (z*sin(a))^2 -0.75^2 = 0

Rotating 90 degrees from the torus to concentric sphere pair. At the moment of the tiny opening is the bitangent Villarceau section of two perfect spheres.

The bitangent Villarceau section, cut open. This is when the slicing plane runs tangent to two points on the surface of torisphere

4D Torisphere : ((III)I)

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

• ((IIi)I) - torus

(sqrt(x^2 + y^2 + a^2) - 4)^2 + z^2 -0.75^2 = 0

Moving the torisphere up and down through 4-space, causing it to slide in and out of our thin 3D plane, with respect to 4D. This is cutting through its major diameter, having the shape of a sphere. This cut leaves its minor diameter alone, allowing us to see its circular shape

• ((III)i) - concentric spheres

(sqrt(x^2 + y^2 + z^2) - 4)^2 + a^2 -0.75^2 = 0

Moving the torisphere in and out at another angle, cutting it top to bottom in the other orthogonal 2-plane. This is cutting through its minor diameter, shaped like a circle. This cut leaves its major diameter alone, allowing us to see its spherical shape

• ((IIz)Z) - rotation

(sqrt(x^2 + y^2 + (z*cos(a))^2) - 4)^2 + (z*sin(a))^2 -0.75^2 = 0

Rotating 90 degrees from the torus to concentric sphere pair. At the moment of the tiny opening is the bitangent Villarceau section of two perfect spheres.

The bitangent Villarceau section, cut open. This is when the slicing plane runs tangent to two points on the surface of torisphere

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Ah, now this is both interesting AND systematic

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Yep! Well, you've been telling me that I need to illustrate 4D first, then get into 5D and beyond. How is someone supposed to understand a tiger dance or rotation, if they've never seen the most basic example? I feel that with these tools at my disposal, the toratopes can be brought to life unlike anything seen before! So, onwards with 4D ....... ( hopefully today I can get the other 4 done, got plenty of time)

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

And next is spheritorus:

4D Spheritorus - ((II)II)

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

Spheritorus can be made by inflating the 1D edge of a circle with a 3D sphere, sphere --> circle

• ((Ii)II) - displaced spheres

(sqrt(x^2 + a^2) - 3)^2 + y^2 + z^2 - 1^2 = 0

Holding the main circle of spheritorus parallel to XW, and sliding through our 3-plane. This direction cuts through its major diameter, shaped like a circle. While leaving its spherical minor diameter alone, we see 2 spheres evolve like a circle cut down to 2 points

• ((II)Ii) - torus

(sqrt(x^2 + y^2) - 3)^2 + z^2 + a^2 - 1^2 = 0

Holding the main circle of a spheritorus parallel to XY, and sliding it through. This cuts through its minor diameter, and shows an evolution of a sphere cut to a circle. While leaving its major diameter alone, we see its full circular structure unchanged

• ((IY)yI) - rotation

(sqrt(x^2 + (y*sin(a))^2) - 3)^2 + z^2 + (y*cos(a))^2 - 1^2 = 0

Rotating in 4-space 90 degrees from hyperplane XY to XW. Our slicing plane passes along the bitangent villarceau section of two bananas, with very sharp tips!

Bitangent Villarceau Section:

Frozen in time during rotation, we see two points, as located on the surface of a spheritorus. One is on the +W top and one on the -W bottom of the spherical minor diameter, shown as equal locations along our 3-plane as two simultaneous narrowed down touching points

4D Spheritorus - ((II)II)

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

Spheritorus can be made by inflating the 1D edge of a circle with a 3D sphere, sphere --> circle

• ((Ii)II) - displaced spheres

(sqrt(x^2 + a^2) - 3)^2 + y^2 + z^2 - 1^2 = 0

Holding the main circle of spheritorus parallel to XW, and sliding through our 3-plane. This direction cuts through its major diameter, shaped like a circle. While leaving its spherical minor diameter alone, we see 2 spheres evolve like a circle cut down to 2 points

• ((II)Ii) - torus

(sqrt(x^2 + y^2) - 3)^2 + z^2 + a^2 - 1^2 = 0

Holding the main circle of a spheritorus parallel to XY, and sliding it through. This cuts through its minor diameter, and shows an evolution of a sphere cut to a circle. While leaving its major diameter alone, we see its full circular structure unchanged

• ((IY)yI) - rotation

(sqrt(x^2 + (y*sin(a))^2) - 3)^2 + z^2 + (y*cos(a))^2 - 1^2 = 0

Rotating in 4-space 90 degrees from hyperplane XY to XW. Our slicing plane passes along the bitangent villarceau section of two bananas, with very sharp tips!

Bitangent Villarceau Section:

Frozen in time during rotation, we see two points, as located on the surface of a spheritorus. One is on the +W top and one on the -W bottom of the spherical minor diameter, shown as equal locations along our 3-plane as two simultaneous narrowed down touching points

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

.... and for tiger .....

4D Tiger : ((II)(II))

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

A tiger can be made by inflating a duoring with a circle, commonly described as the inflated margin of a duocylinder. So, what the heck is a duoring? A duoring is a thin 2D sheet that curves into 3 and 4D. Its axial midsection in 3D is a vertical stack of two 1D curved edges. If cut at a 45 degree angle, we see two 1D rings intersecting at 90 degrees down the middle. The two distinct rims are the two major diameters, displaying their orientation very clearly. In fact, this structure I rendered is still a tiger, technically. I reduced the minor diameter to 1/40 th the size of its majors.

Multiple Angles of a Duoring

By inflating the duoring, we make a tiger, as shown in its 45 degree oblique angle cut. This is the tiger cage which contains the beast. In this cut, we can clearly see how the two torii are always attached. What's really neat is to compare this basic tiger cage to the tigric duotorus (((II)I)((II)I)) cage, made by 4 tigers in a cross combination of concentric major pairs.

Inflated Duoring makes Tiger Cage

• ((II)(Ii)) : 3D cut of two minor stacked torii

(sqrt(x^2 + y^2) - 2.5)^2 + (sqrt(z^2 + a^2) - 2.5)^2 -0.5^2 = 0

Here, we are holding one of the major diameters flat to XY, which causes its second one to run perpendicular. We will see two torii evolve in place of a circle cut to two points. Both axial cuts are the same, as both are circular. We cannot cut through the minor diameter, not until 5D Spheritiger ((II)(II)I), which only gives us two circular major diameters to cut through. I still find it fascinating how the torii remain flat and circular, and cassini deform only the minor diameter.

• ((IY)(Iy)) rotation

(sqrt(x^2 + (y*sin(a))^2) - 2)^2 + (sqrt(z^2 + (y*cos(a))^2) - 2)^2 -0.5^2 = 0

Rotating from hyperplane XY to XW, showing the symmetry of a tiger. In both axials, we can fit a 2D board or a 1D pole through its holes. The two separate torii are always joined by a solid wall on two sides, that curves into a higher dimension, into 4D.

Villarceau Sections of Tiger

The two instances of a bitangent cut of a tiger. Both 3-planes run tangent to the same two points, and show the structure of the duoring fairly well. The solid parts are polar opposite to the duoring path, and show an interesting alternation between the cuts. One could even go further to say that this cutting plane is tangent to the whole duoring itself, seeing how you could fit two orthogonal flat circles between the deforming torii

4D Tiger : ((II)(II))

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

A tiger can be made by inflating a duoring with a circle, commonly described as the inflated margin of a duocylinder. So, what the heck is a duoring? A duoring is a thin 2D sheet that curves into 3 and 4D. Its axial midsection in 3D is a vertical stack of two 1D curved edges. If cut at a 45 degree angle, we see two 1D rings intersecting at 90 degrees down the middle. The two distinct rims are the two major diameters, displaying their orientation very clearly. In fact, this structure I rendered is still a tiger, technically. I reduced the minor diameter to 1/40 th the size of its majors.

Multiple Angles of a Duoring

By inflating the duoring, we make a tiger, as shown in its 45 degree oblique angle cut. This is the tiger cage which contains the beast. In this cut, we can clearly see how the two torii are always attached. What's really neat is to compare this basic tiger cage to the tigric duotorus (((II)I)((II)I)) cage, made by 4 tigers in a cross combination of concentric major pairs.

Inflated Duoring makes Tiger Cage

• ((II)(Ii)) : 3D cut of two minor stacked torii

(sqrt(x^2 + y^2) - 2.5)^2 + (sqrt(z^2 + a^2) - 2.5)^2 -0.5^2 = 0

Here, we are holding one of the major diameters flat to XY, which causes its second one to run perpendicular. We will see two torii evolve in place of a circle cut to two points. Both axial cuts are the same, as both are circular. We cannot cut through the minor diameter, not until 5D Spheritiger ((II)(II)I), which only gives us two circular major diameters to cut through. I still find it fascinating how the torii remain flat and circular, and cassini deform only the minor diameter.

• ((IY)(Iy)) rotation

(sqrt(x^2 + (y*sin(a))^2) - 2)^2 + (sqrt(z^2 + (y*cos(a))^2) - 2)^2 -0.5^2 = 0

Rotating from hyperplane XY to XW, showing the symmetry of a tiger. In both axials, we can fit a 2D board or a 1D pole through its holes. The two separate torii are always joined by a solid wall on two sides, that curves into a higher dimension, into 4D.

Villarceau Sections of Tiger

The two instances of a bitangent cut of a tiger. Both 3-planes run tangent to the same two points, and show the structure of the duoring fairly well. The solid parts are polar opposite to the duoring path, and show an interesting alternation between the cuts. One could even go further to say that this cutting plane is tangent to the whole duoring itself, seeing how you could fit two orthogonal flat circles between the deforming torii

Last edited by ICN5D on Fri Jun 13, 2014 6:01 pm, edited 2 times in total.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

The Villarceau sections, then, are basically singular points where the topology of the shape changes?

In that case, maybe the cuts (not rotations) where topology changes could be interesting to showcase as well. In case of torisphere, spheritorus and tiger, this would be various rotations of lemniscate. For ditorus, fourth rotation of lemniscate is singular cut of the "major pair of toruses" type, and the "two toruses" cut should have three singular slices.

In that case, maybe the cuts (not rotations) where topology changes could be interesting to showcase as well. In case of torisphere, spheritorus and tiger, this would be various rotations of lemniscate. For ditorus, fourth rotation of lemniscate is singular cut of the "major pair of toruses" type, and the "two toruses" cut should have three singular slices.

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

The Villarceau sections, then, are basically singular points where the topology of the shape changes?

Yeah, I guess you could say that. The topology of the cut will change with rotation, and the bitangent section is the moment of the switch over. Basically, it's when the slicing plane touches two points on the surface, and happens to be a tangent line to those points. We've seen many instances of multitangent cuts, something I don't see anywhere else, other than this thread. Villarceau sections are on Mathworld, but only goes into bitangent cuts of a torus. It's also related to one of the Hopf fibration circles of the 3-sphere. So, that being the case, the higher dimensional multitangent sections could be higher order Hopf fibrations!

In that case, maybe the cuts (not rotations) where topology changes could be interesting to showcase as well. In case of torisphere, spheritorus and tiger, this would be various rotations of lemniscate. For ditorus, fourth rotation of lemniscate is singular cut of the "major pair of toruses" type, and the "two toruses" cut should have three singular slices.

Yes, in that case, I noticed that all 4D Villarceau sections and lemniscates can be made by a rotation of the bitangent section and lemniscate of a torus. This reflects the four rotations one can do to a torus to make the other 4 toroidals in 4D. Comparing them shows how the hyperplane transforms during a higher rotation, too, which is neat.

On another note, I tried to plug in the helix movement in the tiger function, but couldn't make it work properly. But, I did see some really bizarre, infinitely tesselated hyperbolic tiger-like things. I want to make an oblique translation GIF of the tiger, and others. How do I rotate a tiger 45 degrees, then slide it through our 3-plane?

Translation held parallel to XYZ

(sqrt(x^2 + y^2) - 2.5)^2 + (sqrt(z^2 + a^2) - 2.5)^2 -0.5^2 = 0

Rotation from XYZ to XYW

(sqrt(x^2 + (y*sin(a))^2) - 2)^2 + (sqrt(z^2 + (y*cos(a))^2) - 2)^2 -0.5^2 = 0

So, how to reorient the tiger in 4-space before translating through hyperplane XYZ? It seems like an additional sin/cos rotation point ought to work, but no luck. It needs an additional adjustable parameter " b ", but where would it go? I also tried out your helix function, but couldn't get it to work.

Last edited by ICN5D on Sat Jun 14, 2014 7:59 am, edited 2 times in total.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Well, the problem is the rotation equation is 3D. That's wrong, you're rotating a 4D object. So, you start like this:

(sqrt(x^2 + y^2) - 2)^2 + (sqrt(z^2 + w^2) - 2)^2 -0.5^2 = 0

Now you rotate 45 degrees in, say, YW plane:

(sqrt(x^2 + (y*sqrt(2)/2 + w*sqrt(2)/2)^2) - 2)^2 + (sqrt(z^2 + (y*sqrt(2)/2 - w*sqrt(2)/2)^2) - 2)^2 -0.5^2 = 0

And from this equation, you start doing cuts by replacing one coordinate with parameter:

(sqrt(a^2 + (x*sqrt(2)/2 + y*sqrt(2)/2)^2) - 2)^2 + (sqrt(z^2 + (x*sqrt(2)/2 - y*sqrt(2)/2)^2) - 2)^2 -0.5^2 = 0 -- this is the normal tiger cut, albeit the toruses will be in oblique orientation now.

(sqrt(x^2 + (a*sqrt(2)/2 + z*sqrt(2)/2)^2) - 2)^2 + (sqrt(y^2 + (a*sqrt(2)/2 - z*sqrt(2)/2)^2) - 2)^2 -0.5^2 = 0

(sqrt(x^2 + (y*sqrt(2)/2 + a*sqrt(2)/2)^2) - 2)^2 + (sqrt(z^2 + (y*sqrt(2)/2 - a*sqrt(2)/2)^2) - 2)^2 -0.5^2 = 0

This is what you've been looking for. These two cuts look the same, and they show very interesting evolution of "bent toruses".

Of course, the sqrt(2)/2 is just a convenient value for the generic equation:

(sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - 2)^2 + (sqrt(z^2 + (y*cos(b) - a*sin(b))^2) - 2)^2 -0.5^2 = 0

Try to explore it And, of course, this technique might lead to new interesting cuts of other figures as well.

(sqrt(x^2 + y^2) - 2)^2 + (sqrt(z^2 + w^2) - 2)^2 -0.5^2 = 0

Now you rotate 45 degrees in, say, YW plane:

(sqrt(x^2 + (y*sqrt(2)/2 + w*sqrt(2)/2)^2) - 2)^2 + (sqrt(z^2 + (y*sqrt(2)/2 - w*sqrt(2)/2)^2) - 2)^2 -0.5^2 = 0

And from this equation, you start doing cuts by replacing one coordinate with parameter:

(sqrt(a^2 + (x*sqrt(2)/2 + y*sqrt(2)/2)^2) - 2)^2 + (sqrt(z^2 + (x*sqrt(2)/2 - y*sqrt(2)/2)^2) - 2)^2 -0.5^2 = 0 -- this is the normal tiger cut, albeit the toruses will be in oblique orientation now.

(sqrt(x^2 + (a*sqrt(2)/2 + z*sqrt(2)/2)^2) - 2)^2 + (sqrt(y^2 + (a*sqrt(2)/2 - z*sqrt(2)/2)^2) - 2)^2 -0.5^2 = 0

(sqrt(x^2 + (y*sqrt(2)/2 + a*sqrt(2)/2)^2) - 2)^2 + (sqrt(z^2 + (y*sqrt(2)/2 - a*sqrt(2)/2)^2) - 2)^2 -0.5^2 = 0

This is what you've been looking for. These two cuts look the same, and they show very interesting evolution of "bent toruses".

Of course, the sqrt(2)/2 is just a convenient value for the generic equation:

(sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - 2)^2 + (sqrt(z^2 + (y*cos(b) - a*sin(b))^2) - 2)^2 -0.5^2 = 0

Try to explore it And, of course, this technique might lead to new interesting cuts of other figures as well.

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Oh, my gosh, this new function is gold. That's exactly what I've been looking for, for a while. There's things I've been wishing to see with oblique flythroughs, and I'm sure you all want to as well. I'm thinking an oblique translation of the tigric duotorus cage .....

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Here's another fun idea:

7D is the lowest dimension where you can have toratopes with 3D cuts that are empty in all 4 cardinal cut directions. An example can be the tiger tritorus (((((II)(II))I)I)I) and its cut ((((()())I)I)I).

Now, all nonempty cuts have to be outside the cardinal directions, which means that the diagonal cuts should be still nonempty and interesting

7D is the lowest dimension where you can have toratopes with 3D cuts that are empty in all 4 cardinal cut directions. An example can be the tiger tritorus (((((II)(II))I)I)I) and its cut ((((()())I)I)I).

Now, all nonempty cuts have to be outside the cardinal directions, which means that the diagonal cuts should be still nonempty and interesting

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Let's have a look at ditorus + torisphere lock, a combination I mentioned a while back.

We take a fairly thin ditorus:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + w^2 = 1

And a fairly thin torisphere:

(sqrt(x^2 + y^2 + w^2) - 9)^2 + z^2 = 3

Let's have a look at x-cut:

(sqrt((sqrt(a^2 + y^2) - 9)^2 + z^2) - 3)^2 + x^2 = 1

(sqrt(a^2 + y^2 + x^2) - 9)^2 + z^2 = 3

Here, you can see a beautiful image of torisphere and ditorus occupying the same region, but never intersecting.

The z-cut will be

(sqrt((sqrt(x^2 + y^2) - 9)^2 + a^2) - 3)^2 + z^2 = 1

(sqrt(x^2 + y^2 + z^2) - 9)^2 + a^2 = 3

And the w-cut will be

(sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + a^2 = 1

(sqrt(x^2 + y^2 + a^2) - 9)^2 + z^2 = 3

And of course, we can do various rotations:

XZ:

(sqrt((sqrt((x*cos(a))^2 + y^2) - 9)^2 + (x*sin(a))^2) - 3)^2 + z^2 = 1

(sqrt((x*cos(a))^2 + y^2 + z^2) - 9)^2 + (x*sin(a))^2 = 3

XW:

(sqrt((sqrt((x*cos(a))^2 + y^2) - 9)^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

(sqrt((x*cos(a))^2 + y^2 + (x*sin(a))^2) - 9)^2 + z^2 = 3 (can be simplified to (sqrt(x^2 + y^2) - 9)^2 + z^2 = 3

ZW:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (z*cos(a))^2) - 3)^2 + (z*sin(a))^2 = 1

(sqrt(x^2 + y^2 + (z*sin(a))^2) - 9)^2 + (z*cos(a))^2 = 3

The interplay between ditorus and torisphere in these pictures is amazing!

We take a fairly thin ditorus:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + w^2 = 1

And a fairly thin torisphere:

(sqrt(x^2 + y^2 + w^2) - 9)^2 + z^2 = 3

Let's have a look at x-cut:

(sqrt((sqrt(a^2 + y^2) - 9)^2 + z^2) - 3)^2 + x^2 = 1

(sqrt(a^2 + y^2 + x^2) - 9)^2 + z^2 = 3

Here, you can see a beautiful image of torisphere and ditorus occupying the same region, but never intersecting.

The z-cut will be

(sqrt((sqrt(x^2 + y^2) - 9)^2 + a^2) - 3)^2 + z^2 = 1

(sqrt(x^2 + y^2 + z^2) - 9)^2 + a^2 = 3

And the w-cut will be

(sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + a^2 = 1

(sqrt(x^2 + y^2 + a^2) - 9)^2 + z^2 = 3

And of course, we can do various rotations:

XZ:

(sqrt((sqrt((x*cos(a))^2 + y^2) - 9)^2 + (x*sin(a))^2) - 3)^2 + z^2 = 1

(sqrt((x*cos(a))^2 + y^2 + z^2) - 9)^2 + (x*sin(a))^2 = 3

XW:

(sqrt((sqrt((x*cos(a))^2 + y^2) - 9)^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

(sqrt((x*cos(a))^2 + y^2 + (x*sin(a))^2) - 9)^2 + z^2 = 3 (can be simplified to (sqrt(x^2 + y^2) - 9)^2 + z^2 = 3

ZW:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (z*cos(a))^2) - 3)^2 + (z*sin(a))^2 = 1

(sqrt(x^2 + y^2 + (z*sin(a))^2) - 9)^2 + (z*cos(a))^2 = 3

The interplay between ditorus and torisphere in these pictures is amazing!

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

All of those 4D toratope animations wouldn't be as informative without a little dimensional analogy,

3D Torus : ((II)I)

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

A torus can be made by inflating the 1D edge of a disk with a circle, shown as circle-->circle

Lucky for us, we can represent a torus in its 3D entirety. It makes you wonder how amazing a true +4D toratope is in its full glory.

• ((I)I) - displaced circles

(sqrt(x^2+a^2) - 2)^2 + y^2 - 0.5^2 = 0

Sliding a torus through a two dimensional plane, causing it to be cut along its major diameter. We see the classic cassini deformation, and the lemniscate of a torus. If we do a bisecting rotation of this cut animation, we would get the cassini deforming spheres cut of spheritorus ((I)II). If we do a non-intersecting rotation around an orthogonal circle, we get the cut evolution of a tiger ((II)(I)) ! If you inflate the circles as they are, you get a cut evolution of a ditorus (((I)I)I) .

• ((II)) - concentric circles

(sqrt(x^2+y^2) - 2)^2 + a^2 - 0.5^2 = 0

Sliding a torus through a 2D plane, cutting through its minor diameter. We see the first example of concentric toratopes appearing, dividing, then merging. If we were to do a bisecting rotation of this animation of plane XY, we would get the concentric spheres of a torisphere ((III)) . If you do a bisecting rotation of the minor diameter in plane XZ, you get the inflating torus cut of spheritorus ((II)Ii) .

• ((Iy)Y)

(sqrt(x^2+(y*cos(a))^2) - 2)^2 + (y*sin(a))^2 - 0.5^2 = 0

Rotating a torus 90 degrees from hyperplane XY to XZ, passing along the bitangent villarceau section of two perfect circles. We can do various rotations of this animation to make the higher toratope cuts. The Spheritorus, Ditorus, and the Torisphere all share this rotation animation.

Torus Lemniscate

All toratopes have at least one type of lemniscate. Many higher dimensional versions can be produced by various rotations of this one.

Villarceau Section of a Torus

All toratopes have at least one type of Villarceau section, except for the n-spheres. This section can be rotated in various ways to create the 4D sections showcased before. What's really neat to wrap your head around, is the positioning of the two points in this representation: one is on +Z top, one on the -Z bottom, of the circular minor diameter. Now, consider what the bitangent cut of the spheritorus looks like, and where the positions of those two points are: +W and -W of the spherical minor diameter ! Same case with all high-D villarceau sections: those narrowed down touching points are locations on the surface, that are in alternating positions in a higher dimension!

3D Torus : ((II)I)

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

A torus can be made by inflating the 1D edge of a disk with a circle, shown as circle-->circle

Lucky for us, we can represent a torus in its 3D entirety. It makes you wonder how amazing a true +4D toratope is in its full glory.

• ((I)I) - displaced circles

(sqrt(x^2+a^2) - 2)^2 + y^2 - 0.5^2 = 0

Sliding a torus through a two dimensional plane, causing it to be cut along its major diameter. We see the classic cassini deformation, and the lemniscate of a torus. If we do a bisecting rotation of this cut animation, we would get the cassini deforming spheres cut of spheritorus ((I)II). If we do a non-intersecting rotation around an orthogonal circle, we get the cut evolution of a tiger ((II)(I)) ! If you inflate the circles as they are, you get a cut evolution of a ditorus (((I)I)I) .

• ((II)) - concentric circles

(sqrt(x^2+y^2) - 2)^2 + a^2 - 0.5^2 = 0

Sliding a torus through a 2D plane, cutting through its minor diameter. We see the first example of concentric toratopes appearing, dividing, then merging. If we were to do a bisecting rotation of this animation of plane XY, we would get the concentric spheres of a torisphere ((III)) . If you do a bisecting rotation of the minor diameter in plane XZ, you get the inflating torus cut of spheritorus ((II)Ii) .

• ((Iy)Y)

(sqrt(x^2+(y*cos(a))^2) - 2)^2 + (y*sin(a))^2 - 0.5^2 = 0

Rotating a torus 90 degrees from hyperplane XY to XZ, passing along the bitangent villarceau section of two perfect circles. We can do various rotations of this animation to make the higher toratope cuts. The Spheritorus, Ditorus, and the Torisphere all share this rotation animation.

Torus Lemniscate

All toratopes have at least one type of lemniscate. Many higher dimensional versions can be produced by various rotations of this one.

Villarceau Section of a Torus

All toratopes have at least one type of Villarceau section, except for the n-spheres. This section can be rotated in various ways to create the 4D sections showcased before. What's really neat to wrap your head around, is the positioning of the two points in this representation: one is on +Z top, one on the -Z bottom, of the circular minor diameter. Now, consider what the bitangent cut of the spheritorus looks like, and where the positions of those two points are: +W and -W of the spherical minor diameter ! Same case with all high-D villarceau sections: those narrowed down touching points are locations on the surface, that are in alternating positions in a higher dimension!

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Now let's look at torisphere + spheritorus chain.

We'll take a standard thin torisphere

(sqrt(x^2 + y^2 + z^2) - 3)^2 + w^2 = 1

and a standard thin spheritorus

(sqrt(x^2 + w^2) - 3)^2 + y^2 + z^2 = 1

Of course, we need to move the spheritorus a bit.

(sqrt((x-3)^2 + w^2) - 3)^2 + y^2 + z^2 = 1

That's better. The final equations are:

(sqrt(x^2 + y^2 + z^2) - 3)^2 + w^2 = 1

(sqrt((x-3)^2 + w^2) - 3)^2 + y^2 + z^2 = 1

Now, for x-cut:

(sqrt(a^2 + y^2 + z^2) - 3)^2 + x^2 = 1

(sqrt((a-3)^2 + x^2) - 3)^2 + y^2 + z^2 = 1

y-cut (z-cut is the same):

(sqrt(x^2 + a^2 + z^2) - 3)^2 + y^2 = 1

(sqrt((x-3)^2 + y^2) - 3)^2 + a^2 + z^2 = 1

w-cut:

(sqrt(x^2 + y^2 + z^2) - 3)^2 + a^2 = 1

(sqrt((x-3)^2 + a^2) - 3)^2 + y^2 + z^2 = 1

Rotations are more complex here since this set of toratopes has no "true" center. If we use rotation algorithms from this set of equations, we'll get rotations centered on the torisphere: Note that YZ rotation is synmmetrical and leads to no changes:

XY (and XZ):

(sqrt((x*cos(a))^2 + (x*sin(a))^2 + z^2) - 3)^2 + y^2 = 1

(sqrt((x*cos(a)-3)^2 + y^2) - 3)^2 + (x*sin(a))^2 + z^2 = 1

XW:

(sqrt((x*cos(a))^2 + y^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

(sqrt((x*cos(a)-3)^2 + (x*sin(a))^2) - 3)^2 + y^2 + z^2 = 1

YW (and ZW):

(sqrt(x^2 + (y*cos(a))^2 + z^2) - 3)^2 + (y*sin(a))^2 = 1

(sqrt((x-3)^2 + (y*sin(a))^2) - 3)^2 + (y*cos(a))^2 + z^2 = 1

Now we'll center the rotation on spheritorus. This means the equations will change to:

(sqrt((x+3)^2 + y^2 + z^2) - 3)^2 + w^2 = 1

(sqrt(x^2 + w^2) - 3)^2 + y^2 + z^2 = 1

and rotations will be:

XY (and XZ):

(sqrt((x*cos(a)+3)^2 + (x*sin(a))^2 + z^2) - 3)^2 + y^2 = 1

(sqrt((x*cos(a))^2 + y^2) - 3)^2 + (x*sin(a))^2 + z^2 = 1

XW:

(sqrt((x*cos(a)+3)^2 + y^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

(sqrt((x*cos(a))^2 + (x*sin(a))^2) - 3)^2 + y^2 + z^2 = 1

YW (and ZW):

(sqrt((x+3)^2 + (y*cos(a))^2 + z^2) - 3)^2 + (y*sin(a))^2 = 1

(sqrt(x^2 + (y*sin(a))^2) - 3)^2 + (y*cos(a))^2 + z^2 = 1

And for our final trick, let's do rotations about the center midpoint:

(sqrt((x*cos(a)+1.5)^2 + (x*sin(a))^2 + z^2) - 3)^2 + y^2 = 1

(sqrt((x*cos(a)-1.5)^2 + y^2) - 3)^2 + (x*sin(a))^2 + z^2 = 1

XW:

(sqrt((x*cos(a)+1.5)^2 + y^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

(sqrt((x*cos(a)-1.5)^2 + (x*sin(a))^2) - 3)^2 + y^2 + z^2 = 1

YW (and ZW):

(sqrt((x+1.5)^2 + (y*cos(a))^2 + z^2) - 3)^2 + (y*sin(a))^2 = 1

(sqrt((x-1.5)^2 + (y*sin(a))^2) - 3)^2 + (y*cos(a))^2 + z^2 = 1

We'll take a standard thin torisphere

(sqrt(x^2 + y^2 + z^2) - 3)^2 + w^2 = 1

and a standard thin spheritorus

(sqrt(x^2 + w^2) - 3)^2 + y^2 + z^2 = 1

Of course, we need to move the spheritorus a bit.

(sqrt((x-3)^2 + w^2) - 3)^2 + y^2 + z^2 = 1

That's better. The final equations are:

(sqrt(x^2 + y^2 + z^2) - 3)^2 + w^2 = 1

(sqrt((x-3)^2 + w^2) - 3)^2 + y^2 + z^2 = 1

Now, for x-cut:

(sqrt(a^2 + y^2 + z^2) - 3)^2 + x^2 = 1

(sqrt((a-3)^2 + x^2) - 3)^2 + y^2 + z^2 = 1

y-cut (z-cut is the same):

(sqrt(x^2 + a^2 + z^2) - 3)^2 + y^2 = 1

(sqrt((x-3)^2 + y^2) - 3)^2 + a^2 + z^2 = 1

w-cut:

(sqrt(x^2 + y^2 + z^2) - 3)^2 + a^2 = 1

(sqrt((x-3)^2 + a^2) - 3)^2 + y^2 + z^2 = 1

Rotations are more complex here since this set of toratopes has no "true" center. If we use rotation algorithms from this set of equations, we'll get rotations centered on the torisphere: Note that YZ rotation is synmmetrical and leads to no changes:

XY (and XZ):

(sqrt((x*cos(a))^2 + (x*sin(a))^2 + z^2) - 3)^2 + y^2 = 1

(sqrt((x*cos(a)-3)^2 + y^2) - 3)^2 + (x*sin(a))^2 + z^2 = 1

XW:

(sqrt((x*cos(a))^2 + y^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

(sqrt((x*cos(a)-3)^2 + (x*sin(a))^2) - 3)^2 + y^2 + z^2 = 1

YW (and ZW):

(sqrt(x^2 + (y*cos(a))^2 + z^2) - 3)^2 + (y*sin(a))^2 = 1

(sqrt((x-3)^2 + (y*sin(a))^2) - 3)^2 + (y*cos(a))^2 + z^2 = 1

Now we'll center the rotation on spheritorus. This means the equations will change to:

(sqrt((x+3)^2 + y^2 + z^2) - 3)^2 + w^2 = 1

(sqrt(x^2 + w^2) - 3)^2 + y^2 + z^2 = 1

and rotations will be:

XY (and XZ):

(sqrt((x*cos(a)+3)^2 + (x*sin(a))^2 + z^2) - 3)^2 + y^2 = 1

(sqrt((x*cos(a))^2 + y^2) - 3)^2 + (x*sin(a))^2 + z^2 = 1

XW:

(sqrt((x*cos(a)+3)^2 + y^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

(sqrt((x*cos(a))^2 + (x*sin(a))^2) - 3)^2 + y^2 + z^2 = 1

YW (and ZW):

(sqrt((x+3)^2 + (y*cos(a))^2 + z^2) - 3)^2 + (y*sin(a))^2 = 1

(sqrt(x^2 + (y*sin(a))^2) - 3)^2 + (y*cos(a))^2 + z^2 = 1

And for our final trick, let's do rotations about the center midpoint:

(sqrt((x*cos(a)+1.5)^2 + (x*sin(a))^2 + z^2) - 3)^2 + y^2 = 1

(sqrt((x*cos(a)-1.5)^2 + y^2) - 3)^2 + (x*sin(a))^2 + z^2 = 1

XW:

(sqrt((x*cos(a)+1.5)^2 + y^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

(sqrt((x*cos(a)-1.5)^2 + (x*sin(a))^2) - 3)^2 + y^2 + z^2 = 1

YW (and ZW):

(sqrt((x+1.5)^2 + (y*cos(a))^2 + z^2) - 3)^2 + (y*sin(a))^2 = 1

(sqrt((x-1.5)^2 + (y*sin(a))^2) - 3)^2 + (y*cos(a))^2 + z^2 = 1

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Let's chain some tigers now.

I'm not sure if you can chain "square" tigers (where both major diameters are equal). We will use oblong tigers for now.

(sqrt(x^2 + y^2) - 6)^2 + (sqrt(z^2 + w^2) - 3)^2 = 1

(sqrt((x-6)^2 + z^2) - 6)^2 + (sqrt(y^2 + w^2) - 3)^2 = 1

Now, we can have cuts:

X:

(sqrt(a^2 + y^2) - 6)^2 + (sqrt(z^2 + x^2) - 3)^2 = 1

(sqrt((a-6)^2 + z^2) - 6)^2 + (sqrt(y^2 + x^2) - 3)^2 = 1

Y:

(sqrt(x^2 + a^2) - 6)^2 + (sqrt(z^2 + y^2) - 3)^2 = 1

(sqrt((x-6)^2 + z^2) - 6)^2 + (sqrt(a^2 + y^2) - 3)^2 = 1

Z:

(sqrt(x^2 + y^2) - 6)^2 + (sqrt(a^2 + z^2) - 3)^2 = 1

(sqrt((x-6)^2 + a^2) - 6)^2 + (sqrt(y^2 + z^2) - 3)^2 = 1

W:

(sqrt(x^2 + y^2) - 6)^2 + (sqrt(z^2 + a^2) - 3)^2 = 1

(sqrt((x-6)^2 + z^2) - 6)^2 + (sqrt(y^2 + a^2) - 3)^2 = 1

Rotations centered on one tiger (I canceled unit goniometric expressions):

XY:

(sqrt(x^2) - 6)^2 + (sqrt(z^2 + y^2) - 3)^2 = 1

(sqrt((x*cos(a)-6)^2 + z^2) - 6)^2 + (sqrt((x*sin(a))^2 + y^2) - 3)^2 = 1

XZ:

(sqrt((x*cos(a))^2 + y^2) - 6)^2 + (sqrt((x*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x*cos(a)-6)^2 + (x*sin(a))^2) - 6)^2 + (sqrt(y^2 + z^2) - 3)^2 = 1

XW:

(sqrt((x*cos(a))^2 + y^2) - 6)^2 + (sqrt(z^2 + (x*sin(a))^2) - 3)^2 = 1

(sqrt((x*cos(a)-6)^2 + z^2) - 6)^2 + (sqrt(y^2 + (x*sin(a))^2) - 3)^2 = 1

YZ:

(sqrt(x^2 + (y*cos(a))^2) - 6)^2 + (sqrt((y*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x-6)^2 + (y*sin(a))^2) - 6)^2 + (sqrt((y*cos(a))^2 + z^2) - 3)^2 = 1

YW:

(sqrt(x^2 + (y*cos(a))^2) - 6)^2 + (sqrt(z^2 + (y*sin(a))^2) - 3)^2 = 1

(sqrt((x-6)^2 + z^2) - 6)^2 + (sqrt(y^2) - 3)^2 = 1

ZW:

(sqrt(x^2 + y^2) - 6)^2 + (sqrt(z^2) - 3)^2 = 1

(sqrt((x-6)^2 + (z*cos(a))^2) - 6)^2 + (sqrt(y^2 + (z*sin(a))^2) - 3)^2 = 1

Definitely have a look at these

Rotations centered between tigers:

XY:

(sqrt((x*cos(a)+3)^2 + (x*sin(a))^2) - 6)^2 + (sqrt(z^2 + y^2) - 3)^2 = 1

(sqrt((x*cos(a)-3)^2 + z^2) - 6)^2 + (sqrt((x*sin(a))^2 + y^2) - 3)^2 = 1

XZ:

(sqrt((x*cos(a)+3)^2 + y^2) - 6)^2 + (sqrt((x*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x*cos(a)-3)^2 + (x*sin(a))^2) - 6)^2 + (sqrt(y^2 + z^2) - 3)^2 = 1

XW:

(sqrt((x*cos(a)+3)^2 + y^2) - 6)^2 + (sqrt(z^2 + (x*sin(a))^2) - 3)^2 = 1

(sqrt((x*cos(a)-3)^2 + z^2) - 6)^2 + (sqrt(y^2 + (x*sin(a))^2) - 3)^2 = 1

YZ:

(sqrt((x+3)^2 + (y*cos(a))^2) - 6)^2 + (sqrt((y*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x-3)^2 + (y*sin(a))^2) - 6)^2 + (sqrt((y*cos(a))^2 + z^2) - 3)^2 = 1

YW:

(sqrt((x+3)^2 + (y*cos(a))^2) - 6)^2 + (sqrt(z^2 + (y*sin(a))^2) - 3)^2 = 1

(sqrt((x-3)^2 + z^2) - 6)^2 + (sqrt(y^2) - 3)^2 = 1

ZW:

(sqrt((x+3)^2 + y^2) - 6)^2 + (sqrt(z^2) - 3)^2 = 1

(sqrt((x-3)^2 + (z*cos(a))^2) - 6)^2 + (sqrt(y^2 + (z*sin(a))^2) - 3)^2 = 1

I'm not sure if you can chain "square" tigers (where both major diameters are equal). We will use oblong tigers for now.

(sqrt(x^2 + y^2) - 6)^2 + (sqrt(z^2 + w^2) - 3)^2 = 1

(sqrt((x-6)^2 + z^2) - 6)^2 + (sqrt(y^2 + w^2) - 3)^2 = 1

Now, we can have cuts:

X:

(sqrt(a^2 + y^2) - 6)^2 + (sqrt(z^2 + x^2) - 3)^2 = 1

(sqrt((a-6)^2 + z^2) - 6)^2 + (sqrt(y^2 + x^2) - 3)^2 = 1

Y:

(sqrt(x^2 + a^2) - 6)^2 + (sqrt(z^2 + y^2) - 3)^2 = 1

(sqrt((x-6)^2 + z^2) - 6)^2 + (sqrt(a^2 + y^2) - 3)^2 = 1

Z:

(sqrt(x^2 + y^2) - 6)^2 + (sqrt(a^2 + z^2) - 3)^2 = 1

(sqrt((x-6)^2 + a^2) - 6)^2 + (sqrt(y^2 + z^2) - 3)^2 = 1

W:

(sqrt(x^2 + y^2) - 6)^2 + (sqrt(z^2 + a^2) - 3)^2 = 1

(sqrt((x-6)^2 + z^2) - 6)^2 + (sqrt(y^2 + a^2) - 3)^2 = 1

Rotations centered on one tiger (I canceled unit goniometric expressions):

XY:

(sqrt(x^2) - 6)^2 + (sqrt(z^2 + y^2) - 3)^2 = 1

(sqrt((x*cos(a)-6)^2 + z^2) - 6)^2 + (sqrt((x*sin(a))^2 + y^2) - 3)^2 = 1

XZ:

(sqrt((x*cos(a))^2 + y^2) - 6)^2 + (sqrt((x*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x*cos(a)-6)^2 + (x*sin(a))^2) - 6)^2 + (sqrt(y^2 + z^2) - 3)^2 = 1

XW:

(sqrt((x*cos(a))^2 + y^2) - 6)^2 + (sqrt(z^2 + (x*sin(a))^2) - 3)^2 = 1

(sqrt((x*cos(a)-6)^2 + z^2) - 6)^2 + (sqrt(y^2 + (x*sin(a))^2) - 3)^2 = 1

YZ:

(sqrt(x^2 + (y*cos(a))^2) - 6)^2 + (sqrt((y*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x-6)^2 + (y*sin(a))^2) - 6)^2 + (sqrt((y*cos(a))^2 + z^2) - 3)^2 = 1

YW:

(sqrt(x^2 + (y*cos(a))^2) - 6)^2 + (sqrt(z^2 + (y*sin(a))^2) - 3)^2 = 1

(sqrt((x-6)^2 + z^2) - 6)^2 + (sqrt(y^2) - 3)^2 = 1

ZW:

(sqrt(x^2 + y^2) - 6)^2 + (sqrt(z^2) - 3)^2 = 1

(sqrt((x-6)^2 + (z*cos(a))^2) - 6)^2 + (sqrt(y^2 + (z*sin(a))^2) - 3)^2 = 1

Definitely have a look at these

Rotations centered between tigers:

XY:

(sqrt((x*cos(a)+3)^2 + (x*sin(a))^2) - 6)^2 + (sqrt(z^2 + y^2) - 3)^2 = 1

(sqrt((x*cos(a)-3)^2 + z^2) - 6)^2 + (sqrt((x*sin(a))^2 + y^2) - 3)^2 = 1

XZ:

(sqrt((x*cos(a)+3)^2 + y^2) - 6)^2 + (sqrt((x*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x*cos(a)-3)^2 + (x*sin(a))^2) - 6)^2 + (sqrt(y^2 + z^2) - 3)^2 = 1

XW:

(sqrt((x*cos(a)+3)^2 + y^2) - 6)^2 + (sqrt(z^2 + (x*sin(a))^2) - 3)^2 = 1

(sqrt((x*cos(a)-3)^2 + z^2) - 6)^2 + (sqrt(y^2 + (x*sin(a))^2) - 3)^2 = 1

YZ:

(sqrt((x+3)^2 + (y*cos(a))^2) - 6)^2 + (sqrt((y*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x-3)^2 + (y*sin(a))^2) - 6)^2 + (sqrt((y*cos(a))^2 + z^2) - 3)^2 = 1

YW:

(sqrt((x+3)^2 + (y*cos(a))^2) - 6)^2 + (sqrt(z^2 + (y*sin(a))^2) - 3)^2 = 1

(sqrt((x-3)^2 + z^2) - 6)^2 + (sqrt(y^2) - 3)^2 = 1

ZW:

(sqrt((x+3)^2 + y^2) - 6)^2 + (sqrt(z^2) - 3)^2 = 1

(sqrt((x-3)^2 + (z*cos(a))^2) - 6)^2 + (sqrt(y^2 + (z*sin(a))^2) - 3)^2 = 1

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Here's another fun idea:

7D is the lowest dimension where you can have toratopes with 3D cuts that are empty in all 4 cardinal cut directions. An example can be the tiger tritorus (((((II)(II))I)I)I) and its cut ((((()())I)I)I).

Now, all nonempty cuts have to be outside the cardinal directions, which means that the diagonal cuts should be still nonempty and interesting

Well, that is interesting! I spent some time imagining a 5D intercept array, and how to explore it. After exploring the 4D lattice of glomes for of tetratiger ((II)(II)(II)(II)), I confirmed more of my beliefs in how 3D feels compared to 4D. So, learning from the experience and exploring the concept into a 5D array, I came to the conclusion that if two cardinal directions are empty, all single rotations and translations will be empty. We have to do a double rotation, or biaxial translation in order to make a scan along structures. Though, I haven't explored this idea with something like a (((((II)(II))I)I)I) yet, just a 5D array. But, I suppose it's worth a look. For this shape, even though the 3D plane cuts through both holes, there's a fresh supply of dimensions held up in smaller diameters, ready to be rotated back. We'll need two of those three, though! Unfortunately, the rendering program doesn't handle concentric major pairings in large arrays very well, the outer torii are always lumpy ribbons. I haven't attempted to render anything with an 8-plet of major pairs in anything larger than a vertical stack of 2, but we'll see. I did form the equation for ((((II)(II))I)(II)) lately, and resolved its diameters:

(sqrt(((sqrt(x^2 + y^2) - R1)^2 + (sqrt(z^2 + w^2) - R2)^2 - R3)^2 + v^2) - R4)^2 + (sqrt(u^2 + t^2) - R5)^2 - R6^2 = 0

• ((((I)(I)))(I)) , ((((Xa)(Yb))c)(Zd))

(sqrt(((sqrt(x^2 + a^2) - 4)^2 + (sqrt(y^2 + b^2) - 4)^2 - 4)^2 + c^2) - 2.5)^2 + (sqrt(z^2 + d^2) - 4)^2 - 0.8^2 = 0

• ((((Xa)(Yz))y)(Zx))

(sqrt(((sqrt((x*sin(d))^2 + a^2) - 4)^2 + (sqrt((y*sin(c))^2 + (z*cos(b))^2) - 4)^2 - 4)^2 + (y*cos(c))^2) - 2.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(d))^2) - 4)^2 - 1^2 = 0

• ((((Xa)(Yz))x)(Zy))

(sqrt(((sqrt((x*sin(c))^2 + a^2) - 4)^2 + (sqrt((y*sin(d))^2 + (z*cos(b))^2) - 4)^2 - 4)^2 + (x*cos(c))^2) - 2.5)^2 + (sqrt((z*sin(b))^2 + (y*cos(d))^2) - 4)^2 - 1^2 = 0

-- ETEs scan past horizontal square of 4 tigritoruses ((((I)(I))I)(II)) or vert sqr of 4 tritoruses ((((II)(I))I)(I))

I saw those diagonal scans in the function, cutting those two out of four 5d shapes! It was really cool, too. I rotated one way in a 3D hole, made a scan and saw cuts of two tigritoruses ((((I)(I))I)(II)), then rotated into another 3D hole and scanned past two tritoruses in ((((II)(I))I)(I)). I have never seen that level of complexity before, and it was kind of mindblowing! Both distinct diagonal scan evolutions came from the same, higher dimensional shape. I mean, I understand the where, why, and hows, but to see it in action, and know what I was seeing was amazing. It was the first time where I witnessed a transforming of the square arrangement of shapes by a rotation, then scanned past them after doing so, and highlighting different toratopes. But, it seriously pushes the limits of the rendering program.

As for the chaining of toratopes, I'd like to see it, but how do I represent two shapes in only one input field? I can't do multiple implicit equations, just multiple functions in Calc Plot 3D.

Plus, I still have some things left to post about ...... got super inspired and made some cool stuff.

Last edited by ICN5D on Sat Jun 14, 2014 9:20 am, edited 1 time in total.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

You can simply add implicit function twice. You'll have two input windows then.

Or, of course, you can use the trick of replacing (F(x) = 0 OR G(x) = 0) with F(x)G(x) = 0.

Or, of course, you can use the trick of replacing (F(x) = 0 OR G(x) = 0) with F(x)G(x) = 0.

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Ditorus checks:

(sqrt((sqrt(x^2 + y^2) - 4)^2 + z^2) - 2)^2 + w^2 = 1

XZ rotation:

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

X-cut (or Z-cut):

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

Y-cut:

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

W-cut:

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

XW rotation:

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

X-cut (or W-cut):

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

Y-cut:

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

Z-cut:

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

ZW rotation:

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

X-cut (or Y-cut):

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

Z-cut (or W-cut):

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

(sqrt((sqrt(x^2 + y^2) - 4)^2 + z^2) - 2)^2 + w^2 = 1

XZ rotation:

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

X-cut (or Z-cut):

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

Y-cut:

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

W-cut:

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

XW rotation:

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

X-cut (or W-cut):

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

Y-cut:

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

Z-cut:

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

ZW rotation:

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

X-cut (or Y-cut):

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

Z-cut (or W-cut):

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

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

And of course:

(sqrt((sqrt((x*cos(b) + z*sin(b)*cos(c) + w*sin(b)*sin(c))^2 + y^2) - 4)^2 + (x*sin(b) - z*cos(b)*cos(c) - w*cos(b)*sin(c))^2) - 2)^2 + (z*sin(c) - w*cos(c))^2 = 1

X-cut (also Z-cut and W-cut):

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

Y-cut:

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

Though it seems that no new cuts are actually produced by these obliques...

(sqrt((sqrt((x*cos(b) + z*sin(b)*cos(c) + w*sin(b)*sin(c))^2 + y^2) - 4)^2 + (x*sin(b) - z*cos(b)*cos(c) - w*cos(b)*sin(c))^2) - 2)^2 + (z*sin(c) - w*cos(c))^2 = 1

X-cut (also Z-cut and W-cut):

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

Y-cut:

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

Though it seems that no new cuts are actually produced by these obliques...

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Now, let's have a look at a simple chain of tiger and torisphere.

(sqrt(x^2 + y^2) - 3)^2 + (sqrt(z^2 + w^2) - 3)^2 = 1

(sqrt((x-8)^2 + z^2 + w^2) - 8)^2 + y^2 = 1

Ranges: x [-4,17], y [-4,4], z [-9,9], w [-9,9]

X-cut:

(sqrt(a^2 + y^2) - 3)^2 + (sqrt(z^2 + x^2) - 3)^2 = 1

(sqrt((a-8)^2 + z^2 + x^2) - 8)^2 + y^2 = 1

Y-cut:

(sqrt(x^2 + a^2) - 3)^2 + (sqrt(z^2 + y^2) - 3)^2 = 1

(sqrt((x-8)^2 + z^2 + y^2) - 8)^2 + a^2 = 1

Z-cut (and W-cut):

(sqrt(x^2 + y^2) - 3)^2 + (sqrt(a^2 + z^2) - 3)^2 = 1

(sqrt((x-8)^2 + a^2 + z^2) - 8)^2 + y^2 = 1

(sqrt(x^2 + y^2) - 3)^2 + (sqrt(z^2 + w^2) - 3)^2 = 1

(sqrt((x-8)^2 + z^2 + w^2) - 8)^2 + y^2 = 1

Ranges: x [-4,17], y [-4,4], z [-9,9], w [-9,9]

X-cut:

(sqrt(a^2 + y^2) - 3)^2 + (sqrt(z^2 + x^2) - 3)^2 = 1

(sqrt((a-8)^2 + z^2 + x^2) - 8)^2 + y^2 = 1

Y-cut:

(sqrt(x^2 + a^2) - 3)^2 + (sqrt(z^2 + y^2) - 3)^2 = 1

(sqrt((x-8)^2 + z^2 + y^2) - 8)^2 + a^2 = 1

Z-cut (and W-cut):

(sqrt(x^2 + y^2) - 3)^2 + (sqrt(a^2 + z^2) - 3)^2 = 1

(sqrt((x-8)^2 + a^2 + z^2) - 8)^2 + y^2 = 1

Last edited by Marek14 on Sat Jun 14, 2014 8:08 pm, edited 1 time in total.

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

XY rotation:

(sqrt((x*cos(a))^2 + (x*sin(a))^2) - 3)^2 + (sqrt(z^2 + y^2) - 3)^2 = 1

(sqrt((x*cos(a)-8)^2 + z^2 + y^2) - ^2 + (x*sin(a))^2 = 1

XZ(XW) rotation:

(sqrt((x*cos(a))^2 + y^2) - 3)^2 + (sqrt((x*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x*cos(a)-8)^2 + (x*sin(a))^2 + z^2) - ^2 + y^2 = 1

YZ(YW) rotation:

(sqrt(x^2 + (y*cos(a))^2) - 3)^2 + (sqrt((y*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x-8)^2 + (y*sin(a))^2 + z^2) - ^2 + (y*cos(a))^2 = 1

As always, the rotation animations are especially enchanting.

(sqrt((x*cos(a))^2 + (x*sin(a))^2) - 3)^2 + (sqrt(z^2 + y^2) - 3)^2 = 1

(sqrt((x*cos(a)-8)^2 + z^2 + y^2) - ^2 + (x*sin(a))^2 = 1

XZ(XW) rotation:

(sqrt((x*cos(a))^2 + y^2) - 3)^2 + (sqrt((x*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x*cos(a)-8)^2 + (x*sin(a))^2 + z^2) - ^2 + y^2 = 1

YZ(YW) rotation:

(sqrt(x^2 + (y*cos(a))^2) - 3)^2 + (sqrt((y*sin(a))^2 + z^2) - 3)^2 = 1

(sqrt((x-8)^2 + (y*sin(a))^2 + z^2) - ^2 + (y*cos(a))^2 = 1

As always, the rotation animations are especially enchanting.

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Heck yeah, awesome Marek! I've copy-pasted all those functions. I'll have to take a look at them some time. You should probably turn off smilies, or delete the space between the eight followed by a bracket It looks funny, though! Chillin' in the sun function, having a good time!

I wanted to get the other animations up from the new rot + trans function. It's awesome to have that much control over a 4D shape! I can pick it up and rotate it around at will, like it's in my hands! Now, I only need to use one function for a shape. Sadly, I run out of adjustable parameters after 5D, but no big deal. It's easy enough to alternate between a set value and a parameter. This definitely opens up a new way to explore high-D shapes.

I've been thinking about making 4D cut to 2D animations. They'd be really good at preparing for what you will see in 5D cut to 3D, by illustrating N-2 dimensional cut evolutions. A tiger rotation from four circles to an empty hole ((X)(Ix)) , is identical in every way to the the tigritorus rotation (((X)I)(Ix)). The four torii in the vertical square do the same morphing together, into 2 ear-shaped structures, that deflate and vanish while moving out from center. This is also equally identical to the N-3 rotation ((((X)I)I)(Ix)), which I animated here. Again, we see an identical morphing, showing the same stuff. But, in this case, it closer resembles an N-3 cut of 5D (((II)I)(II)), taking the form (((I))(I)), and rotating (((X))(Ix)).

In fact all N-1 cuts are identical, as well as all N-2, N-3, N-4, etc. It doesn't matter which dimension you start from, or which one you end up in, only the number of cuts. An N-D cut corresponds to a strict evolution sequence for all intercept arrays, across all toratopes. This holds true for rotation and translation.

I wanted to get the other animations up from the new rot + trans function. It's awesome to have that much control over a 4D shape! I can pick it up and rotate it around at will, like it's in my hands! Now, I only need to use one function for a shape. Sadly, I run out of adjustable parameters after 5D, but no big deal. It's easy enough to alternate between a set value and a parameter. This definitely opens up a new way to explore high-D shapes.

I've been thinking about making 4D cut to 2D animations. They'd be really good at preparing for what you will see in 5D cut to 3D, by illustrating N-2 dimensional cut evolutions. A tiger rotation from four circles to an empty hole ((X)(Ix)) , is identical in every way to the the tigritorus rotation (((X)I)(Ix)). The four torii in the vertical square do the same morphing together, into 2 ear-shaped structures, that deflate and vanish while moving out from center. This is also equally identical to the N-3 rotation ((((X)I)I)(Ix)), which I animated here. Again, we see an identical morphing, showing the same stuff. But, in this case, it closer resembles an N-3 cut of 5D (((II)I)(II)), taking the form (((I))(I)), and rotating (((X))(Ix)).

In fact all N-1 cuts are identical, as well as all N-2, N-3, N-4, etc. It doesn't matter which dimension you start from, or which one you end up in, only the number of cuts. An N-D cut corresponds to a strict evolution sequence for all intercept arrays, across all toratopes. This holds true for rotation and translation.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

And now we'll try an interlocking combination of tiger and ditorus. I'll use an oblique tiger to make it more fun.

(sqrt(x^2 + y^2) - 3)^2 + (sqrt(z^2 + w^2) - 9)^2 = 1

(sqrt((sqrt(z^2 + w^2) - 9)^2 + (y-3)^2) - 3)^2 + x^2 = 1

x [-4,4]; y [-4,7]; z [-13,13]; w[-13,13]

X-cut:

(sqrt(a^2 + y^2) - 3)^2 + (sqrt(z^2 + x^2) - 9)^2 = 1

(sqrt((sqrt(z^2 + x^2) - 9)^2 + (y-3)^2) - 3)^2 + a^2 = 1

Y-cut:

(sqrt(x^2 + a^2) - 3)^2 + (sqrt(z^2 + y^2) - 9)^2 = 1

(sqrt((sqrt(z^2 + y^2) - 9)^2 + (a-3)^2) - 3)^2 + x^2 = 1

Z-cut (+W-cut):

(sqrt(x^2 + y^2) - 3)^2 + (sqrt(a^2 + z^2) - 9)^2 = 1

(sqrt((sqrt(a^2 + z^2) - 9)^2 + (y-3)^2) - 3)^2 + x^2 = 1

x [-4,4]; y [-4,7]; z [-13,13]; w[-13,13]

XY rotation:

(sqrt((x*cos(a))^2 + (x*sin(a))^2) - 3)^2 + (sqrt(z^2 + y^2) - 9)^2 = 1

(sqrt((sqrt(z^2 + y^2) - 9)^2 + (x*sin(a)-3)^2) - 3)^2 + (x*cos(a))^2 = 1

XZ(XW) rotation:

(sqrt((x*cos(a))^2 + y^2) - 3)^2 + (sqrt((x*sin(a))^2 + z^2) - 9)^2 = 1

(sqrt((sqrt((x*sin(a))^2 + z^2) - 9)^2 + (y-3)^2) - 3)^2 + (x*cos(a))^2 = 1

YZ(YW) rotation:

(sqrt(x^2 + (y*cos(a))^2) - 3)^2 + (sqrt((y*sin(a))^2 + z^2) - 9)^2 = 1

(sqrt((sqrt((y*sin(a))^2 + z^2) - 9)^2 + (y*cos(a)-3)^2) - 3)^2 + x^2 = 1

The last one is absolutely hypnotizing.

(sqrt(x^2 + y^2) - 3)^2 + (sqrt(z^2 + w^2) - 9)^2 = 1

(sqrt((sqrt(z^2 + w^2) - 9)^2 + (y-3)^2) - 3)^2 + x^2 = 1

x [-4,4]; y [-4,7]; z [-13,13]; w[-13,13]

X-cut:

(sqrt(a^2 + y^2) - 3)^2 + (sqrt(z^2 + x^2) - 9)^2 = 1

(sqrt((sqrt(z^2 + x^2) - 9)^2 + (y-3)^2) - 3)^2 + a^2 = 1

Y-cut:

(sqrt(x^2 + a^2) - 3)^2 + (sqrt(z^2 + y^2) - 9)^2 = 1

(sqrt((sqrt(z^2 + y^2) - 9)^2 + (a-3)^2) - 3)^2 + x^2 = 1

Z-cut (+W-cut):

(sqrt(x^2 + y^2) - 3)^2 + (sqrt(a^2 + z^2) - 9)^2 = 1

(sqrt((sqrt(a^2 + z^2) - 9)^2 + (y-3)^2) - 3)^2 + x^2 = 1

x [-4,4]; y [-4,7]; z [-13,13]; w[-13,13]

XY rotation:

(sqrt((x*cos(a))^2 + (x*sin(a))^2) - 3)^2 + (sqrt(z^2 + y^2) - 9)^2 = 1

(sqrt((sqrt(z^2 + y^2) - 9)^2 + (x*sin(a)-3)^2) - 3)^2 + (x*cos(a))^2 = 1

XZ(XW) rotation:

(sqrt((x*cos(a))^2 + y^2) - 3)^2 + (sqrt((x*sin(a))^2 + z^2) - 9)^2 = 1

(sqrt((sqrt((x*sin(a))^2 + z^2) - 9)^2 + (y-3)^2) - 3)^2 + (x*cos(a))^2 = 1

YZ(YW) rotation:

(sqrt(x^2 + (y*cos(a))^2) - 3)^2 + (sqrt((y*sin(a))^2 + z^2) - 9)^2 = 1

(sqrt((sqrt((y*sin(a))^2 + z^2) - 9)^2 + (y*cos(a)-3)^2) - 3)^2 + x^2 = 1

The last one is absolutely hypnotizing.

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

I'm trying the ditorus/tiger morph.

We'll use these equations:

(sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + w^2) - 3)^2 = 1

(sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + w^2 = 1

With x = 0, they form identical couples of toruses, just rotated.

What does a ditorus and a tiger have in common?

Ditorus: (((II)I)I)

Tiger: ((II)(II))

Tiger torus: (((II)(II))I)

Tiger torus is the smallest toratope that includes both ditoruses and tigers in its cuts.

Tiger is tiger torus with minor diameter set to zero.

Ditorus is a tiger torus with second major diameter set to zero.

sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + w^2) - 3)^2) - 1 = sqrt(-v^2)

sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + w^2) - 0)^2) - 3 = sqrt(1-v^2)

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + (w*cos(a))^2) - 3*cos(a)^2)^2) - (cos(a)^2 + 3*sin(a)^2))^2 + (w*sin(a))^2 - sin(a)^2 = 0

(sqrt((sqrt(b^2 + y^2) - 9)^2 + (sqrt(z^2 + (x*cos(a))^2) - 3*cos(a)^2)^2) - (cos(a)^2 + 3*sin(a)^2))^2 + (x*sin(a))^2 - sin(a)^2 = 0

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(b^2 + (z*cos(a))^2) - 3*cos(a)^2)^2) - (cos(a)^2 + 3*sin(a)^2))^2 + (z*sin(a))^2 - sin(a)^2 = 0

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + (b*cos(a))^2) - 3*cos(a)^2)^2) - (cos(a)^2 + 3*sin(a)^2))^2 + (b*sin(a))^2 - sin(a)^2 = 0

Unfortunately, this doesn't work properly in Calc3D. The extreme corresponding to tiger doesn't work properly. I traced it to this:

sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + w^2) - 3)^2) - 1 = 0 - works

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + w^2) - 3)^2) - 1)^2 = 0 - doesn't work

Not sure why. Maybe it tries to draw two objects at the same spot?

We'll use these equations:

(sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + w^2) - 3)^2 = 1

(sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + w^2 = 1

With x = 0, they form identical couples of toruses, just rotated.

What does a ditorus and a tiger have in common?

Ditorus: (((II)I)I)

Tiger: ((II)(II))

Tiger torus: (((II)(II))I)

Tiger torus is the smallest toratope that includes both ditoruses and tigers in its cuts.

Tiger is tiger torus with minor diameter set to zero.

Ditorus is a tiger torus with second major diameter set to zero.

sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + w^2) - 3)^2) - 1 = sqrt(-v^2)

sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + w^2) - 0)^2) - 3 = sqrt(1-v^2)

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + (w*cos(a))^2) - 3*cos(a)^2)^2) - (cos(a)^2 + 3*sin(a)^2))^2 + (w*sin(a))^2 - sin(a)^2 = 0

(sqrt((sqrt(b^2 + y^2) - 9)^2 + (sqrt(z^2 + (x*cos(a))^2) - 3*cos(a)^2)^2) - (cos(a)^2 + 3*sin(a)^2))^2 + (x*sin(a))^2 - sin(a)^2 = 0

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(b^2 + (z*cos(a))^2) - 3*cos(a)^2)^2) - (cos(a)^2 + 3*sin(a)^2))^2 + (z*sin(a))^2 - sin(a)^2 = 0

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + (b*cos(a))^2) - 3*cos(a)^2)^2) - (cos(a)^2 + 3*sin(a)^2))^2 + (b*sin(a))^2 - sin(a)^2 = 0

Unfortunately, this doesn't work properly in Calc3D. The extreme corresponding to tiger doesn't work properly. I traced it to this:

sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + w^2) - 3)^2) - 1 = 0 - works

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (sqrt(z^2 + w^2) - 3)^2) - 1)^2 = 0 - doesn't work

Not sure why. Maybe it tries to draw two objects at the same spot?

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Here's some super cool oblique flythroughs of a tiger:

4D Tiger Oblique Fly-through

And, alas! The animation I've been waiting to see! Holding both major diameters at 45 degree angle to hyperplane XY, we slide it along 4-space through our 3D plane. And wow! This animation inspired me to make a 2D scan of a duoring, and display its amazing structure as well.

4D Tiger Tunnel to Tunnel Oblique Fly-through

This is slightly translating out, merging the two torii into a long tunnel. We then rotate 90 degrees to another long tunnel, in the other orthogonal direction. This confirms a suspicion I've had, about what it looks like to go through one of those holes. There really is structure there, like an impassable barrier! It's a wall that's there in one 2-plane, but not in the other! WOW! And, we can move around that wall, like traversing the skin of a torisphere. We have an extra direction to do so with a 4th spatial dimension. And, that's what this animation goes to prove. We are moving the infinitely thin slicing plane of XYZ, and tracing out the exterior of the central dual-chasm void. In fact, inside the tiger is nothing special. It's only the circular skin that has a magical ability. This is the first real illustration of those mechanics of a tiger, and what makes it so special.

4D Tiger Oblique Fly-through

And, alas! The animation I've been waiting to see! Holding both major diameters at 45 degree angle to hyperplane XY, we slide it along 4-space through our 3D plane. And wow! This animation inspired me to make a 2D scan of a duoring, and display its amazing structure as well.

4D Tiger Tunnel to Tunnel Oblique Fly-through

This is slightly translating out, merging the two torii into a long tunnel. We then rotate 90 degrees to another long tunnel, in the other orthogonal direction. This confirms a suspicion I've had, about what it looks like to go through one of those holes. There really is structure there, like an impassable barrier! It's a wall that's there in one 2-plane, but not in the other! WOW! And, we can move around that wall, like traversing the skin of a torisphere. We have an extra direction to do so with a 4th spatial dimension. And, that's what this animation goes to prove. We are moving the infinitely thin slicing plane of XYZ, and tracing out the exterior of the central dual-chasm void. In fact, inside the tiger is nothing special. It's only the circular skin that has a magical ability. This is the first real illustration of those mechanics of a tiger, and what makes it so special.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

How about the double ditorus lock?

This combination is made from two ditoruses which differ only in major diameters -- and in orientation. Even though two ditoruses are normally too "thin" to form a chain, this way should work.

(sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + w^2 = 1

(sqrt((sqrt(x^2 + y^2) - 6)^2 + w^2) - 3)^2 + z^2 = 1

X-cut:

(sqrt((sqrt(a^2 + y^2) - 9)^2 + z^2) - 3)^2 + x^2 = 1

(sqrt((sqrt(a^2 + y^2) - 6)^2 + x^2) - 3)^2 + z^2 = 1

Z-cut:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + a^2) - 3)^2 + z^2 = 1

(sqrt((sqrt(x^2 + y^2) - 6)^2 + z^2) - 3)^2 + a^2 = 1

W-cut:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + a^2 = 1

(sqrt((sqrt(x^2 + y^2) - 6)^2 + a^2) - 3)^2 + z^2 = 1

XZ rotation:

(sqrt((sqrt((x*cos(a))^2 + y^2) - 9)^2 + (x*sin(a))^2) - 3)^2 + z^2 = 1

(sqrt((sqrt((x*cos(a))^2 + y^2) - 6)^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

XW rotation:

(sqrt((sqrt((x*cos(a))^2 + y^2) - 9)^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

(sqrt((sqrt((x*cos(a))^2 + y^2) - 6)^2 + (x*sin(a))^2) - 3)^2 + z^2 = 1

ZW rotation:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (z*cos(a))^2) - 3)^2 + (z*sin(a))^2 = 1

(sqrt((sqrt(x^2 + y^2) - 6)^2 + (z*sin(a))^2) - 3)^2 + (z*cos(a))^2 = 1

The last one could be fun to see in translation:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (b*cos(a) + z*sin(a))^2) - 3)^2 + (b*sin(a) - z*cos(a))^2 = 1

(sqrt((sqrt(x^2 + y^2) - 6)^2 + (b*sin(a) - z*cos(a))^2) - 3)^2 + (b*cos(a) + z*sin(a))^2 = 1

Well, going to bed for now

This combination is made from two ditoruses which differ only in major diameters -- and in orientation. Even though two ditoruses are normally too "thin" to form a chain, this way should work.

(sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + w^2 = 1

(sqrt((sqrt(x^2 + y^2) - 6)^2 + w^2) - 3)^2 + z^2 = 1

X-cut:

(sqrt((sqrt(a^2 + y^2) - 9)^2 + z^2) - 3)^2 + x^2 = 1

(sqrt((sqrt(a^2 + y^2) - 6)^2 + x^2) - 3)^2 + z^2 = 1

Z-cut:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + a^2) - 3)^2 + z^2 = 1

(sqrt((sqrt(x^2 + y^2) - 6)^2 + z^2) - 3)^2 + a^2 = 1

W-cut:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + z^2) - 3)^2 + a^2 = 1

(sqrt((sqrt(x^2 + y^2) - 6)^2 + a^2) - 3)^2 + z^2 = 1

XZ rotation:

(sqrt((sqrt((x*cos(a))^2 + y^2) - 9)^2 + (x*sin(a))^2) - 3)^2 + z^2 = 1

(sqrt((sqrt((x*cos(a))^2 + y^2) - 6)^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

XW rotation:

(sqrt((sqrt((x*cos(a))^2 + y^2) - 9)^2 + z^2) - 3)^2 + (x*sin(a))^2 = 1

(sqrt((sqrt((x*cos(a))^2 + y^2) - 6)^2 + (x*sin(a))^2) - 3)^2 + z^2 = 1

ZW rotation:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (z*cos(a))^2) - 3)^2 + (z*sin(a))^2 = 1

(sqrt((sqrt(x^2 + y^2) - 6)^2 + (z*sin(a))^2) - 3)^2 + (z*cos(a))^2 = 1

The last one could be fun to see in translation:

(sqrt((sqrt(x^2 + y^2) - 9)^2 + (b*cos(a) + z*sin(a))^2) - 3)^2 + (b*sin(a) - z*cos(a))^2 = 1

(sqrt((sqrt(x^2 + y^2) - 6)^2 + (b*sin(a) - z*cos(a))^2) - 3)^2 + (b*cos(a) + z*sin(a))^2 = 1

Well, going to bed for now

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Wow, yeah that's a lifetime of exploratory functions, sitting right there! Man. I was up until 5 am last night ( this morning ) posting that stuff. I'm really enjoying my 2 week paid stay-cation between jobs, what can I say! But hey, sometimes I can't sleep because my mind won't shut off from thinking about 7D structures.

Especially that ((((II)(II))I)(II)), it's one of the most amazing shapes I've ever wrapped my head around, and explored its function in the plotter. It seems that 7D is the first instance where we can have four intercepts in a 5D hyperplane, and transform them by single rotation. This allows for more complex diagonal scans and empty translates, naturally. And very much worth documenting and showcasing.

Especially that ((((II)(II))I)(II)), it's one of the most amazing shapes I've ever wrapped my head around, and explored its function in the plotter. It seems that 7D is the first instance where we can have four intercepts in a 5D hyperplane, and transform them by single rotation. This allows for more complex diagonal scans and empty translates, naturally. And very much worth documenting and showcasing.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

I tried out the new translate + rotate function, and man I've been seeing some really amazing things I've never seen before. Mostly due to never having this much control over position in the shape. I've got some amazing things to show all .....

Oblique Structure Scan Past 2 Displaced (((II)I)(II))

This is from the shape (((II)I)((II)I)), where we have made the 5D cut of (((II)I)((I)I)), a vertical stack of two tigritoruses, like the 3D cut of a 4D tiger. We have positioned our 3D plane above these two, parallel to the empty axial cut (((II)I)(()I)) and tilted 45 degrees. We are scanning our 3D plane downwards, past the two in the column, and since we're holding a tilted plane steady moving downwards, we see the scan move from one side of the shapes to the other, in sequence.

Quadruple Tiger Cage Oblique Fly Through

This is from (((II)I)((II)I)) again. I've been wanting to see what this looks like, and it's absolutely amazing. This is a thin, mostly 2D laser beam, scanning top to bottom at a 45 degree angle through the oblique structure between (((II))((I))) and (((I))((II))), which I'm, inclined to call (((IO))((IO))), for representing an oblique structure in toratopic notation. The " O " stands for oblique, at an angle between axials of its 2 positions. Anyways, related to a duoring structure scan, this is displaying the curvature of a duoring inflated duoring, a di-duoring. This is the margin of a torus*torus prism, which becomes inflated by a circle to make a tiger duotorus. This margin is a 4D surface that curves into 5 and 6D. It intercepts a 2D hyperplane in a 4x4 array of 16 locations of a 0D point.

6D Tiger Duotorus Triaxial Rotation

A triple 90 degree rotation from an empty hole to an axial cut in (((II)I)((II)I)) . I think this is rotation (((yz)X)((Yx)Z)) , but there's a few combinations possible. It's a pretty wild transformation, for sure.

Oblique Structure Scan Past 2 Displaced (((II)I)(II))

This is from the shape (((II)I)((II)I)), where we have made the 5D cut of (((II)I)((I)I)), a vertical stack of two tigritoruses, like the 3D cut of a 4D tiger. We have positioned our 3D plane above these two, parallel to the empty axial cut (((II)I)(()I)) and tilted 45 degrees. We are scanning our 3D plane downwards, past the two in the column, and since we're holding a tilted plane steady moving downwards, we see the scan move from one side of the shapes to the other, in sequence.

Quadruple Tiger Cage Oblique Fly Through

This is from (((II)I)((II)I)) again. I've been wanting to see what this looks like, and it's absolutely amazing. This is a thin, mostly 2D laser beam, scanning top to bottom at a 45 degree angle through the oblique structure between (((II))((I))) and (((I))((II))), which I'm, inclined to call (((IO))((IO))), for representing an oblique structure in toratopic notation. The " O " stands for oblique, at an angle between axials of its 2 positions. Anyways, related to a duoring structure scan, this is displaying the curvature of a duoring inflated duoring, a di-duoring. This is the margin of a torus*torus prism, which becomes inflated by a circle to make a tiger duotorus. This margin is a 4D surface that curves into 5 and 6D. It intercepts a 2D hyperplane in a 4x4 array of 16 locations of a 0D point.

6D Tiger Duotorus Triaxial Rotation

A triple 90 degree rotation from an empty hole to an axial cut in (((II)I)((II)I)) . I think this is rotation (((yz)X)((Yx)Z)) , but there's a few combinations possible. It's a pretty wild transformation, for sure.

Last edited by ICN5D on Tue Jul 01, 2014 2:33 am, edited 1 time in total.

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

Well, the last one looks like it has some oblique elements, due to presence of "tiger cages" ...

- Marek14
- Pentonian
**Posts:**1148**Joined:**Sat Jul 16, 2005 6:40 pm

Yes it does It's really close to the oblique cut (((O)I)((O)I)) , between (((I)I)(()I)) and ((()I)((I)I)). The two tiger cages oblique. I've got an animation coming of that one, too! I rotated to 45 degrees between the square of four tigers (((I)I)((I)I)) , and translated out. Moving past the square at a 45 degree angle is super cool to see a 1x2x1 oblique evolution of the four tigers, cutting the square like a diamond. Again, that new function is absolutely amazing. One can actually pilot a 3D hyperplane around this way, without being tethered to the origin. Like steering an extra-dimensional spaceship, it's way cool going out to survey geometric structures in their home dimension Just wait, I'm cooking up some insane animation ideas .....

in search of combinatorial objects of finite extent

- ICN5D
- Pentonian
**Posts:**1077**Joined:**Mon Jul 28, 2008 4:25 am**Location:**Orlando, FL

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