Cubinder's net

Discussion of shapes with curves and holes in various dimensions.

Cubinder's net

Postby icebreaker » Fri Sep 15, 2017 8:38 am

I'm glad to join the forum. My English knowledges are still not very good therefore I'm forced to write how I can do it.

I researched toratopes and found out that the net of Cubinder actually includes not a cube as it's imagined in the Wiki article (http://hi.gher.space/wiki/Cubinder) but a square prism whose length is equal 2пR where R is the radius of the cylinders. My pictures below show the analogy between Cylinder and Cubinder.

Sorry if I report about a mistake on the wiki in a wrong section. Then move it to the correct section.
Attachments
Cylinder_net.png
Cylinder_net.png (2.72 KiB) Viewed 1244 times
Cubinder_true_net.png
Cubinder_true_net.png (4.01 KiB) Viewed 1244 times
Cubinder_net.png
Cubinder_net.png (3.58 KiB) Viewed 1244 times
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Re: Cubinder's net

Postby Mercurial, the Spectre » Fri Sep 15, 2017 1:38 pm

The true net of a cubinder is four cylinders and a topological square prismatic 3-manifold with the four cylinders wrapped around the lateral faces of the 3-manifold.

To imagine this, consider a cubinder as the cartesian product of a circle and square. You would have four circle prisms (cylinders) and a square manifold, which is topologically a square prism that is actually curved in 4D. The square manifold can be unwrapped back to 3D to get a square prism (or a cube).

It's similar to the cylinder's 2-manifold that is topologically equivalent to the rectangle. To see, cut the cylinder in a direction perpendicular to the circles and unwrap it, the result is a rectangle or a square. The cylinder's net is basically two circles and a topological rectangular 2-manifold, and which the two circles are located opposite to each other.
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Re: Cubinder's net

Postby icebreaker » Fri Sep 15, 2017 3:00 pm

Mercurial, the Spectre wrote: The square manifold can be unwrapped back to 3D to get a square prism (or a cube).


Mercurial, the Spectre wrote: It's similar to the cylinder's 2-manifold that is topologically equivalent to the rectangle. To see, cut the cylinder in a direction perpendicular to the circles and unwrap it, the result is a rectangle or a square.


I don't understand how the net of Cylinder can include either rectangle or a square at the same time. It's obviously that the length of the rectangle must be 2пR, no more, no less. Otherwise the surface will not be long enough to stretch on the circles.

The same situation with Cubinder. Each line segment of the square prism, that is parallel to its 4 longest edges is converted to a circle in 4D. But the length of each circle = 2пR, therefore the length of the prism itself must be equal to 2пR. Or is it not true?

I suggest replacing the image in the article with this:
Attachments
Cubinder_true_net_blender.png
Cubinder_true_net_blender.png (41.53 KiB) Viewed 1232 times
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Re: Cubinder's net

Postby ICN5D » Sat Sep 16, 2017 1:23 am

Well, if you want to get down to exact proportions and all that, icebreaker is technically right. I don't think this was considered when the image was made, but moreso the topology of the unfolded net. In this case, rectangular prisms and cubes are the same.
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Re: Cubinder's net

Postby icebreaker » Sat Sep 16, 2017 4:17 am

Being honest and unbiased, we in fact understand that it is an mistake because all beginners who view this image believe that nothing can stretch and therefore they fall into a small misleading.

I made an alternative net for Cubinder and also for Duocylinder for comparison, that are more easy-to-understand. Are they suitable for the corresponding articles of Wiki?
Attachments
Cubinder_alternative_net_blender_.png
Cubinder_alternative_net_blender_.png (45.81 KiB) Viewed 1218 times
Duocylinder_net_blender_1.png
Duocylinder_net_blender_1.png (36.01 KiB) Viewed 1218 times
Duocylinder_net_blender_2.png
Duocylinder_net_blender_2.png (36.5 KiB) Viewed 1218 times
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Re: Cubinder's net

Postby icebreaker » Sat Sep 16, 2017 4:50 am

I forgot to make final touches. The net of the Cubinder is fixed.
Attachments
Cubinder_true_net_blender_fixed.png
Cubinder_true_net_blender_fixed.png (49.54 KiB) Viewed 1218 times
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Re: Cubinder's net

Postby icebreaker » Sat Sep 16, 2017 8:19 am

It's the final variant of Duocylinder's net (for Wiki).
Attachments
Duocylinder_net_blender_1_.png
Duocylinder_net_blender_1_.png (34.28 KiB) Viewed 1218 times
Duocylinder_net_blender_2_.png
Duocylinder_net_blender_2_.png (36.56 KiB) Viewed 1218 times
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Re: Cubinder's net

Postby icebreaker » Sat Sep 16, 2017 10:35 am

Well, Tesseract's net.
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Tesseract_net.png
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Re: Cubinder's net

Postby Mercurial, the Spectre » Sat Sep 16, 2017 1:25 pm

Great!
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Toratope's nets and cross-sections

Postby icebreaker » Sun Sep 17, 2017 3:26 pm

Here are a round cell-first, cylinder-first and face-first (is it a full list?) cross-sections of a Cubinder that are related to the previous net. It is fit to Wiki?
Attachments
Cubinder_c-s_round-first.gif
Cubinder_c-s_round-first.gif (19.81 KiB) Viewed 1201 times
Cubinder_c-s_cell-first.gif
Cubinder_c-s_cell-first.gif (10.54 KiB) Viewed 1201 times
Cubinder_c-s_face-first.gif
Cubinder_c-s_face-first.gif (22.14 KiB) Viewed 1201 times
Last edited by icebreaker on Mon Sep 18, 2017 4:16 am, edited 1 time in total.
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Re: Toratope's nets and cross-sections

Postby icebreaker » Sun Sep 17, 2017 4:39 pm

Cross-sections of Duocylinder:
Unfortunately I still can't make an face-first one (it looks like a tiger cross-sections when 45°). I will try to do it though.
Attachments
Duocylinder_c-s_red-first.gif
Duocylinder_c-s_red-first.gif (18.09 KiB) Viewed 1198 times
Duocylinder_c-s_blue-first.gif
Duocylinder_c-s_blue-first.gif (19.53 KiB) Viewed 1198 times
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Re: Toratope's nets and cross-sections

Postby icebreaker » Sun Sep 17, 2017 6:36 pm

I hope there're no mistakes here :D
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Duocylinder_c-s_face-first.gif
Duocylinder_c-s_face-first.gif (29.22 KiB) Viewed 1193 times
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Re: Toratope's nets and cross-sections

Postby icebreaker » Sun Sep 17, 2017 7:48 pm

Yes, there was a mistake, but now it's fixed.
As you see, both closed cylinders (tores) wrap themselves around each other like half of a tennis ball.
Attachments
Duocylinder_c-s_face-first_.gif
Duocylinder_c-s_face-first_.gif (32.72 KiB) Viewed 1191 times
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Re: Toratope's nets and cross-sections

Postby icebreaker » Mon Sep 18, 2017 4:14 am

I plan to make cross-sections for following toratopes (for Wiki articles):
    √ Tesseract
    Glome
    Spherinder
    √ Dicone
    √ Coninder
    Sphone
    Torinder
    Cyltrianglinder (+net)
You may suggest more.
Last edited by icebreaker on Sun Dec 24, 2017 9:20 am, edited 6 times in total.
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Re: Toratope's nets and cross-sections

Postby icebreaker » Mon Sep 18, 2017 6:34 am

Cell-first and face-first cross-sections of Tesseract. I will make an edge-first one but still haven't any ideas to visualize a corver-first one.
Attachments
Tesseract_c-s_cell-first.gif
Tesseract_c-s_cell-first.gif (9.66 KiB) Viewed 1185 times
Tesseract_c-s_face-first.gif
Tesseract_c-s_face-first.gif (19.67 KiB) Viewed 1185 times
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Re: Toratope's nets and cross-sections

Postby icebreaker » Mon Sep 18, 2017 11:08 am

Edge-first cross-section of a Hypercube. It seems that I already know how to make a corner-first one.
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Tesseract_c-s_edge-first.gif
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Re: Toratope's nets and cross-sections

Postby icebreaker » Mon Sep 18, 2017 12:54 pm

Uh, it was tediously. Corner-first cross-section of Tesseract.
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Tesseract_c-s_corner-first.gif
Tesseract_c-s_corner-first.gif (31.94 KiB) Viewed 1179 times
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Re: Cubinder's net

Postby wendy » Mon Sep 18, 2017 2:25 pm

This tesseract thing looks good.

Is there any way of colouring the faces. What i would look for is to start with a red tetrahedron, and end in a blue one. This shows the truncation sequence by larger and smaller dials gives the 'antitegum sequence'.
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Re: Toratope's nets and cross-sections

Postby icebreaker » Mon Sep 18, 2017 3:20 pm

Well, but what about another toratopes? Should I better make them (except of tesseract) with colored faces or only with colored edges?

It would look like this, only colors would be more pastel + edges as on cross-sections above.
Attachments
Spherelinder&Cubinder.png
Spherelinder&Cubinder.png (8.62 KiB) Viewed 1163 times
Spherelinder&Cubinder_cell-first.png
Spherelinder&Cubinder_cell-first.png (9.56 KiB) Viewed 1163 times
Cubinder_face-first.png
Cubinder_face-first.png (4.04 KiB) Viewed 1163 times
Duocylinder.png
Duocylinder.png (6 KiB) Viewed 1163 times
Last edited by icebreaker on Tue Sep 19, 2017 4:36 am, edited 2 times in total.
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Re: Toratope's nets and cross-sections

Postby icebreaker » Tue Sep 19, 2017 4:16 am

wendy wrote:What i would look for is to start with a red tetrahedron, and end in a blue one.

It means that this tesseract would be composed of 4 red-face and 4 blue-face cubes. But in the article http://hi.gher.space/wiki/Geochoron in chapter "Projection" there are only 1 red-edge cube and 1 blue-edge one + 8 connecting black edges. I made cross-sections (and a net) of a hypercube according to this because wanted to simplicity and transparency for understanding.

If I post on Wiki your variant of Hypercube cross-section, will it be accessibly? May be to do 1 red-face and 1 blue-face cross-sections and a net?

Your ideas are helpful for me.
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Re: Toratope's nets and cross-sections

Postby icebreaker » Sat Oct 21, 2017 2:40 am

Finally I return from a hospital where I found two more particular cross-section of a cubinder: round-face-first and edge-first.
As for tesseract, I will create it as 4 red-cell and 4 blue-cell one as a second variant of it. Another solids will be visualized as well as duocylinder and cubinder.
Attachments
Cubinder_c-s_edge-first.gif
Cubinder_c-s_edge-first.gif (33.95 KiB) Viewed 1060 times
Cubinder_c-s_round_face-first.gif
Cubinder_c-s_round_face-first.gif (41.38 KiB) Viewed 1059 times
Last edited by icebreaker on Sat Oct 21, 2017 6:13 am, edited 2 times in total.
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Re: Toratope's nets and cross-sections

Postby icebreaker » Sat Oct 21, 2017 5:00 am

I can't color the corner-first slices in red-blue. May be I will found a way to do it. While you may do it. Here is Blender file of it.
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Tesseract_cross-sections.blend.zip
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Re: Toratope's nets and cross-sections

Postby icebreaker » Mon Oct 30, 2017 1:57 am

Here is a 2D surface that separate one from another torus of a duocylinder. It looks like a tiger, but this has infinite small thickness. Is this a tiger too?
This body can roll as a duocylinder, only his footprint is a circle (the footprint of a duocylinder is a disk).
Attachments
Tiger_net.png
Tiger_net.png (25.28 KiB) Viewed 1025 times
Tiger_c-s_0degr1.gif
Tiger_c-s_0degr1.gif (35.46 KiB) Viewed 1025 times
Tiger_c-s_0degr2.gif
Tiger_c-s_0degr2.gif (30.19 KiB) Viewed 1025 times
Tiger_c-s_45dg.gif
Tiger_c-s_45dg.gif (46.75 KiB) Viewed 1025 times
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Re: Cubinder's net

Postby ICN5D » Mon Oct 30, 2017 4:09 pm

What you made here is almost a tiger! These are the 3D slices of a Clifford torus, or flat torus. If you embed a circle of smaller radius into this flat torus, you will get a tiger. Still, these are good visuals for people to see, as there aren't many that take 3D slices of the flat torus. Most people have done projections.
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Re: Toratope's nets and cross-sections

Postby icebreaker » Sat Nov 04, 2017 5:51 am

I found out that an another variant of the net of a Clifford torus is a simple square!
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Tiger_net2.png
Tiger_net2.png (19.13 KiB) Viewed 955 times
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Re: Toratope's nets and cross-sections

Postby icebreaker » Fri Nov 10, 2017 5:23 am

I need help for making rest slices of a coninder.
Attachments
Coninder-c-s_cylinder-first.gif
Coninder-c-s_cylinder-first.gif (22.12 KiB) Viewed 926 times
Coninder_c-s_cell-first.gif
Coninder_c-s_cell-first.gif (8.23 KiB) Viewed 926 times
Coninder_.png
Coninder_.png (3.81 KiB) Viewed 926 times
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Re: Cubinder's net

Postby ICN5D » Sat Nov 11, 2017 12:34 am

How are you defining the surfaces in blender? Are you using an implicit equation, then taking slices in 3d?

If you are (and this is a good way to do it), then you can use the full 4D equation for a coninder:

||sqrt(x^2 + y^2) + 2z| + sqrt(x^2 + y^2) - 2w| + ||sqrt(x^2 + y^2) + 2z| + sqrt(x^2 + y^2) + 2w| = a

You can set a=2 or whatever. The larger the number = larger shape.


Now, set the variable ' x ' to zero, or set it as an adjustable parameter. You can also use ' y ' instead, for the same results. When x = 0, you're slicing with plane yzw through the origin.

Setting x= ±1 or whatever will make the slices you're looking for. If a=2, then -2 < x < 2 is the range you want to take slices in, I believe.
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Re: Toratope's nets and cross-sections

Postby icebreaker » Sat Nov 11, 2017 3:24 am

I don't use any mathematic formula for making the slices, only my imagination.

To make the vertex-first slices of hypercube and face-first ones of duocylinder I used boolean operations (modificator).

Thank you very much for your explanation. Is it possible by using python that integrated to Blender?
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Re: Cubinder's net

Postby ICN5D » Sun Nov 12, 2017 1:06 am

I don't have any experience with either python or blender. I've been plotting implicit and parametric functions through a 3D graphing calculator, called calcplot3D.


I also started off with my imagination, too. After a long, continued interest, a lot of math experiments and playing around, along with input from others, I developed a few methods for getting equations for shapes, and using them to render slices and projections.

Don't get me wrong, you can still extrapolate a lot of info without the equation, and find accurate details about the shape. The equation ran through a computer is, however, much more revealing. It will show you things you didn't know, too!


If you want to use python/blender with an implicit equation, you'll likely have to implement a marching cubes algorithm.




-----------------



I just so happen to have rendered the coninder myself, recently. It was one of many experiments to see if my new parametric function algorithm is working. It seems to be doing a great job! Check out this projection of a rotating coninder:



Image



The parametric function I discovered for a unit coninder is :

r(x,y,z,w) = { (v-1)u*cos(t)√3 , (v-1)u*sin(t)√3 , 3v+1 , 2s√3 } | u,v,s ∈ [-1,1] ; t ∈ [0,π]

Now, you may notice this is a 4-manifold embedded in E^4 , so it's a solid 4D shape. I found another algorithm that decomposes this equation into 0,1,2, and 3-manifolds embedded in E^4 . These are the faces and edges that you see in the animation (only the 1D and 2D surfaces).
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Re: Cubinder's net

Postby ICN5D » Sun Nov 12, 2017 2:00 am

Here, check this out:

https://www.desmos.com/calculator/bjvsmpz8fg

It's a solid 3D cone sliced in 2D. Adjusting parameter 'a' will translate, 'b' will rotate.
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