From the two geometries of a Toroid and a Mobius ring: Torbus

[Ray Fischer]

I have made a geometry, see first attachment, I call the “Torbus”, from the two geometries of a Toroid and a Mobius ring, though the twisted ring cut from a torus is not the classical Mobius ring. I have not been able to derive the math (too many variables) that describes the movement of the twisted ring relative to the torus; though I have been able, from a model, to determine the movement of two unique points separated 1800 on the twisted ring. One point (red) travels 7200 through the torus and the other (green) travels around the torus 3600. I need help deriving the math. The animation on my website (myspace/torbuspuzzles) shows the twisted ring’s movement relative to the torus. In attachment 2 is a drawing of what the loci of the two points appear to be. The two points , red and green, are at opposite ends of what I call a “Torbus Baton” see attachment 3.
A thought:

 

[Ray Fischer]

I have made a geometry, see first attachment, I call the “Torbus”, from the two geometries of a Toroid and a Mobius ring, though the twisted ring cut from a torus is not the classical Mobius ring. I have not been able to derive the math (too many variables) that describes the movement of the twisted ring relative to the torus; though I have been able, from a model, to determine the movement of two unique points separated 180° on the twisted ring. One point (red) travels 720° through the torus and the other (green) travels around the torus 360°. I need help deriving the math. The animation on my website (myspace/torbuspuzzles) shows the twisted ring’s movement relative to the torus. In attachment 2 is a drawing of what the loci of the two points appear to be. The two points , red and green, are at opposite ends of what I call a “Torbus Baton” see attachment 3.
A thought:

I would like to add a picture to clarify the loci of the two point on the twisted ring as they move around and through the Torus. see attachment

 

[Ray Fischer]

I'm adding a math model of a Torbus need to know someone that can add a function to move the twisted ring around and through the Torus. The kerf will have a minimum width.
Select the Torbus animation.nbp file.Sorry it is a cdf file. If interested I can email it.

 

[Ray Fischer]

Another math expression is needed see attachment A.
Attachment B is a process to transform a plane to a sphere.

 

[Matterwave]

Are you making SxSxMo, or SxMo? Is this thing a 3-D volume or a 2-D surface?

 

[Ray Fischer]

Thanks for the question but I do not understand what is ment by SxSxMo, or SxMo.The Torbus I cut from a solid wooden Torus is 3-D. There are two linked pieces after cutting the Torus 1) twisted ring and 2) rest of the Torus. I then cut 2) the remaining Torus, to extract 1) the twisted ring which is the blue thing. It is like a thick Mobius ring. But it does not have the Torbus twisted ring geometry as noted when compared to the thinner pink Mobius ring. The bulges in the pink Mobius ring are due to stresses in the pliable material when twisted. A ribbon cannot be wrapped around the twisted ring as it can be wrapped around the Mobius ring, the “S” pattern had to be used for a laminate surface on the Torbus twisted ring.
The trasformed plane to a sphere is a 2D to 3D transform.

 

[Matterwave]

Well, is the representation of the thing you are making the 3-D structure, or the 2-D surface of that 3-D structure?

S is the circle, a torus is usually denoted SxS (as in 2 circles "multiplied" together), and Mo is the mobius strip. I'm wondering if you are making SxSxMo (torus "multiplied" by Mobius strip) or simply SxMo (circle multiplied by Mobius strip). The dimensionality of the 2 objects are different.

In other words, are you confined to a surface where you can only have 2 directions to go in, or can you move in all 3 directions?

 

[Ray Fischer]

Thanks again.
I do not know how to multiply geometries, perhaps that is why I cannot do the math. I will attach an animated Torbus to my post (if I figure out how to do that.). If I could get your email address I could send you the 3D math geometry of the “Torbus twisted ring” in a Torus. The one doing the math (in Mathematica) did not know how to move the “Torbus twisted ring” relative to the Torus in 3D
In making the sphere from a sheet of paper, movement on the paper, 2D, becomes a movement in 3D on the sphere. It would make a unique ciphering device.

 

[Ray Fischer]

Another thought- Laminates on a Mobius ring can move radially, 1D or circumferentially, also 1D,or in both direction at the same time on the laminated Mobius strip, 2D. When the laminated ring is separated, as in making the sphere a point can move in 3D

 

[Matterwave]

I'm probably not the best guy to help you in too much detail, my topology is pretty rudimentary, and I can't really visualize your torbus all that well since I can't even tell if it's SxMo or SxSxMo haha.

The cartesian product is not a very difficult concept, let's try the simplest example of RxR. R represents the real line (just think of a line), and then when you multiply it by R again, you get a plane.

RxR is a plane, and then RxRxR is a 3-volume space.

Let's start off in R, just imagine me being able to move on the real line, I can move back and forth only. If now I move into RxR, I can move back and forth and left and right. If I move into RxRxR I can move back and forth and left and right and up and down.

If I move in S, I can only move back and forth. The difference between R and S is that in S, eventually I'll get back to my starting point if I keep moving forward. If I move to SxS I can move back and forth or left and right. Eventually, if I move in any direction consistently I'll get back to my starting point. Interestingly though, SxS is not S^2 the sphere! It's the torus (in general). Perhaps this picture is easy to visualize: http://en.wikipedia.org/wiki/File:Torus_cycles.png

Notice though that you are ONLY allowed to move on the surface, you can't move inside of that torus.

I hope that made sense.

 

[Ray Fischer]

OK!
If you notice what is happening with the Torbus, a point in the torus is allowed to move outside of the torus on a restricted path, but the path it can move on depends on its position on the Torbus twisted ring in the tours. (Note the loci in post 7) The green dot is on the planar part of the Torbus twisted ring while the red dot is on the cylinder part of the Torbus ring. The red dot has to move through the torus twice to get the green dot back to its original starting point. One thing I would like to know is if a Torbus, of finite thickness, needs more than the space it occupies in the torus to move out of the torus, what I call kerf width. I’m assuming the points are on a Torbus ring of zero width when I say the red dot starts out form its origin travels 360o (it has returned to its starting position) then must travel another 360o to get the green dot (having traveled only 180o) back to its starting point. You can see this on myspace.com/ torbuspuzzles. I could not get it on this post. I don’t know what you would call this “dynamic geometric relative movement”.
Thanks for your interest not many, only you have an interest.

 

[quasar987]

Hello Ray,

It sounds like you could enjoy and benefit from the book "The shape of Space" by Jeff Weeks. (https://www.amazon.com/dp/0824707095/?tag=pfamazon01-20)

Among other things, it explains remarkably well the different topological (math) operations on geometries, but without any math.. i.e. just on the conceptual level.

 

[Ray Fischer]

Thanks for the suggetion.

 

[Ray Fischer]

Ref: #10
How would you describe (with your notations) a rectangle bounded by a circle moving around a ring; for each degree the rectangle moves around the ring the rectangle rotates .5 deg.in the circle (cw or ccw)? That is a "Torbus". If there is a minimum kerf, gap in the circle, between the rectangle and the circle (now in two parts but still forming a torus), The piece formed by the rectangle (a twisted ring aka "Torbus Ring")not a "Mobius" ring can be separated from the Torus yet remains linked to the torus. As shown in the first post. Another thought the "Torbus" ring is a folded "S" not a twisted ribbon, as is the Mobus ring.