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 195 times
Cubinder_true_net.png
Cubinder_true_net.png (4.01 KiB) Viewed 195 times
Cubinder_net.png
Cubinder_net.png (3.58 KiB) Viewed 195 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 183 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 169 times
Duocylinder_net_blender_1.png
Duocylinder_net_blender_1.png (36.01 KiB) Viewed 169 times
Duocylinder_net_blender_2.png
Duocylinder_net_blender_2.png (36.5 KiB) Viewed 169 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 169 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 169 times
Duocylinder_net_blender_2_.png
Duocylinder_net_blender_2_.png (36.56 KiB) Viewed 169 times
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Re: Cubinder's net

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

Well, Tesseract's net.
Attachments
Tesseract_net.png
Tesseract_net.png (42.83 KiB) Viewed 169 times
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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 152 times
Cubinder_c-s_cell-first.gif
Cubinder_c-s_cell-first.gif (10.54 KiB) Viewed 152 times
Cubinder_c-s_face-first.gif
Cubinder_c-s_face-first.gif (22.14 KiB) Viewed 152 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 149 times
Duocylinder_c-s_blue-first.gif
Duocylinder_c-s_blue-first.gif (19.53 KiB) Viewed 149 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 144 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 142 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 Mon Sep 18, 2017 1:01 pm, edited 1 time 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 136 times
Tesseract_c-s_face-first.gif
Tesseract_c-s_face-first.gif (19.67 KiB) Viewed 136 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
Tesseract_c-s_edge-first.gif (16 KiB) Viewed 131 times
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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 130 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 114 times
Spherelinder&Cubinder_cell-first.png
Spherelinder&Cubinder_cell-first.png (9.56 KiB) Viewed 114 times
Cubinder_face-first.png
Cubinder_face-first.png (4.04 KiB) Viewed 114 times
Duocylinder.png
Duocylinder.png (6 KiB) Viewed 114 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 11 times
Cubinder_c-s_round_face-first.gif
Cubinder_c-s_round_face-first.gif (41.38 KiB) Viewed 10 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
(107.88 KiB) Not downloaded yet
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