A torus twist. A better method?

benstapptuba shared this question 1 month ago
Answered

I wanted to trace a twist around a torus going from the top of the torus (imagining the top as a clock) from one position to five positions over, example: 12 o'clock to 5 o'clock; 5 o'clock to 10 o'clock, 10 o'clock to 3 o'clock, etc. I plotted something manually, see the link below. Is there a more straight forward approach to this problem? Can someone do it with the 3d calculator for a more accurate representation? Can anyone help?https://www.geogebra.org/ca...

Comments (33)

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Hi,

do you want something like that ?

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This is perfect Patrick! Thank you! There is one thing I can't do and that is to continue to get the redline to extend until it gets back to 12 o clock. I don't see a slider for that but I'm not sure.

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Hi Patrick! I think I found the answer. I was looking for drag maybe? I looked at the very end of the equation where it said 2pi and replaced the the 2 with a variable and a slider.

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Hi,

a better solution: change maximum value of the slider ''e'' (end) to 24 instead 11.

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ah got it!

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I'm very inspired that you can so easily create these formulas! I did what you said which helped. I also had to take the 2 from the last 2pi in the parametric curve formula and make that a slider that goes up to 24 so that the parametric curve eventually gets back to 12 o'clock. One last request if you can help. Is it possible to rotate a pentagonal prism (standing like a house, pointed side up) around a dodecagon making a 60-cell torus? I think it is also called a duo prism? It would be 12 connected pentagonal prism placed around a dodecagon. I attached a picture but it is the wrong amount of sides. I would also need a line that connects the different points of the pentagons. Thank you again Patrick for your help. I'll send you the link to how I applied this stuff and give you many thanks if you want!

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Hi,

like this ?

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wow! what kind of degree in math do you need to be able to do this? Thank you Patrick!!! I might have another question for you but it is about bending this shape through other dimensions to connect sides that would normally be impossible in 3d...First I need to make sure I can ask the question clearly.

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Hi Patrick,

I don't know if this is possible? This is the final version of this model. Imagine that the line segment connecting to the different vertices around the pentagons is a tube. Now imagine that tube is secondary dodecagon/pentagon/torus/clock (like the primary but in tube form) - (image1). As it passes around the primary torus it is connecting a specific point of its tube clock to a specific point on the pentagon. The tube turns the following rotations for each point it touches as it goes around the initial torus: 30º, 30º, 30º, 30º, 90º(image2). After it gets back to the initial position, it folds into the initial torus connecting the positions of the (image3). The initial point connections: First point 5 o'clock, next point with 4 o'clock, next point with 3 o'clock, next point with 2 o'clock, next point with 1 o'clock, next point with 10 o'clock. The it does this 12 more times until it gets back to the starting point. Then it goes out of the tube and connects to line up with the primary torus. The connection is from the same direction so it would connect like a klein bottle.

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Patrick, about the torus o'clock version. How do I make the line segment rotate counter clockwise?

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Hi,

change minimum value of the slider ''e'' (end) to -12 instead 0.

I have a little trouble to understand the last request...(?)

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I put in -12 for e for the minimum but the line just vanishes. I'm not sure what I'm doing wrong.

I will work on the final image. I'm sill having trouble understanding myself. Apologies.

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Regarding the torus o'clock.

Can you plot a tube** that goes around the torus with the line segment? (maybe a different color?)

**It would not be a tube but a dodecagonal prism.

It should follow the red line segment you initially made.

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"Patrick, about the torus o'clock version. How do I make the line segment rotate counter clockwise?"

change minimum value of the slider ''e'' (end) to -12 instead 0.

It's work for this file : torusOclock.ggb

Of course, this method doesn't work with the file : pentagoneTorusOclock(2).ggb

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So, what do you want for the "red line" : curve or segment ?

In your last request, do you want surface or prism ?

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For the last request I would love dodecagon prism segments for the red line. this is for the pentagoneTorusOclock version.

My fault, what is the method for going counter clockwise for the pentagoneTorusOclock.

Thank you!!!

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Hi,

is this the right way?

I just joined points but I don't understand 30º, 30º, 30º, 30º, 90º(image2).

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I'm still amazed at your skills. Also thank you for your patience while I try to explain all this.

You have it correct but the smaller shape is a dodecagon not a pentagon. Again I'm sorry for not being clear. The 30 30 30 30 90 applies to that smaller dodecagon. On points of the dodecagon that would be starting at 12am, 30º > to 1pm, 30º > to 2pm, 30º > to 3pm, 30º > to4pm, 90º > to 7pm.

Also can you make the smaller dodecagon have a surface and can you make it also go counter clock-wise as well. I still can't make it do that by putting a negative for (e) for some reason.

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counter clock-wise for pentagoneTorusOclock(2).ggb :

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Ok.

Is it better (for a first time) ?

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yes this is the way!

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another idea

select tool1 in menu and click in last polygon

Files: foro.ggb
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thank you mathmagic. very cool.

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I think I follow. I'm just beginning to see something. If you join dodecahedrons the pentagon side naturally turns.

1. can you have two dodecahedron/ torus rings twisting around each other but joining into each other as well?

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Hi Patrick,

Friend! Please don't hate me. Here's a link with a small section on the scales so you know I'm actually going to do something with what you are helping me with and give you credit! https://www.jazzarium.pl/pr...

Okay! I think I've figured out a solution. Connecting dodecahedrons solves the rotation complications.

Can you do two twisting toroidal polyhedra (each one a link of 60 dodecahedrons) wrapped around each other?

Each position (e), would be adding another dodecahedron pair (one from each toroidal polyehedra) for a total of 60 positions? (can you also make it go counterclockwise too? Thank you :)

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I'm looking at these now! Thank you! I also included a picture of how they connect in a 4th dimension. I think this is just one concept of a hyperdodecahedron.

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Hi,

an other way to do ''last request'' :

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But I think it's not possible to finish with a ''klein bottle'' with this idea.

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I was wrong about that need for a klein bottle.

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Wow that is great Patrick!

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Hi benstapptuba,

One ''twisting toroidal polyhedra (each one a link of 60 dodecahedrons)'' : dodecaTor(1).ggb

''two twisting toroidal polyhedra (each one a link of 60 dodecahedrons) wrapped around each other '' : dodecaTor(2).ggb

''Connecting dodecahedrons solves the rotation complications '' (?) : dodecaTor(3).ggb

I think connecting dodecahedrons is more complicated than using the surface command.(GeoGebra has more difficulty calculating).

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I think the dodecaTor(3) is the one! but can you make a variable so it plots the pairs of dodecahedrons out from 0 - 60?

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my main concern is the latency on graphing several dodecahedron. I like how you represented it in the first two examples. I'm thinking that the second version is the best in terms of latency and representing the dodecahedron.

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Hello Patrick!

I think this might make the most sense for the model. I do not know what shape it can be. Maybe two mobius strips? I drew the model so that all 120 coordinates are clear and I drew where they should connect again. Do you have any ideas for how this would look on a 3d graph. Thank you for all of your help with this! The book is 2/3rds of the way done.

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