Square Rhombic Torus

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Square Rhombic Torus

Postby gmtrcply » Sat Oct 12, 2024 3:21 pm

This is a set of Square Rhombic Torus builds.

The base shape is a regular Octahedron. Such a Torus can be built from Octahedrons and Tetrahedrons.
You might think that the cross-section would be square, but the traverses run at 45° angle compared to the square outline of the Octahedrons.

Parts list, all regular ½" Spheres and 58mm Geomag Rods:
Order-0: 6 Spheres, 12 Rods (base element: Square Bipyramid of equal edge length, i.e. an Octahedron)
Order-1: 17 Spheres, 48 Rods
Order-2: 32 Spheres, 96 Rods
Order-3: 48 Spheres, 144 Rods

I have no mathematical explanation for the exceptional sphere count of the Order-1 builds. I would guess that the center point of these is not a vertex, but some kind of singular inner point.

It's really hard to get a good perspective on these, because of obstructed rods. The Order-3 is missing two orange rods, which I realized afterwards.
Order-0 emphasizes the build as Square Bipyramid (Base edges: Black, Lateral edges: Grey). The others have dark rods for the main edges of the trapezoid faces, and bright rods for the edges of the triangulation of the trapezoids.

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Re: Square Rhombic Torus

Postby Wagyx » Sun Oct 13, 2024 9:22 am

Here are the models. I'll also add them to the gallery.
https://asliceofcuriosity.fr/blog/posts/geometry19.html
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Square Rhombic Torus.zip
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Re: Square Rhombic Torus

Postby gmtrcply » Sun Oct 13, 2024 10:22 pm

Wagyx wrote:Here are the models. I'll also add them to the gallery.
https://asliceofcuriosity.fr/blog/posts/geometry19.html


Thanks again, the models in the gallery look great. I never have to worry about taking the photo from a bad perspective, again.
Just one annotation: The spheres count for the Triangular Order-1 is 13, the spheres count for the Square Order-1 is 17.

Today I have successfully built a Square Order-4 from long rods. But the available light was too bad for a photo, so I will give it a try tomoroow.
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Re: Square Rhombic Torus

Postby Peter Jepsen » Mon Oct 14, 2024 9:15 am

Maybe I'm repeating an exercise that someone else already did..

You can do the same exercise for the pentagonal equivalent.
GEOMAG Item 614.Model 07 Pentagonal Donut - PCS120B40S.05 Green.jpg
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Re: Square Rhombic Torus

Postby gmtrcply » Mon Oct 14, 2024 12:07 pm

Peter Jepsen wrote:You can do the same exercise for the pentagonal equivalent.


Yes, this is the third variant of this type of build.

The corner elements are based on bipyramids of respective polygons. The traverses have the same structure for all, just the ratio of the rhombic cross-section differs. Relative to the square build, the traverses are acuted for the triangular outline, and obtused for the pentagonal outline.

Oder-1 is irregular, but from Order-2 on, a higher order is built by inserting the same count of four Spheres and twelve rods for each traverse, i.e. +12/+36 for the triangle, +16/+48 for the square, and +20/+60 for the pentagonal outline.
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Re: Square Rhombic Torus

Postby gmtrcply » Mon Oct 14, 2024 4:54 pm

Here's the build of Order-4: 64 Spheres, 192 Rods.

Unfortunately, the contrast between the main edges in Fuchsia and the edges of the triangulation in Pink is not that big.

I started building with a blue sky outside, but then it became heavily cloudy. In some cases, I was unable to tell whether the rod would be fuchsia pr pink. To avoid erros, I checked the structure with my UV flashlight, and had to realize that some of the pink rods, ~20 out of ~200 have a different tone. I really need to check and sub-sort all of the rods using UV light.

The structure is surprisingly stable, although no supporting rods are used. Unlike the Triangular and the Pentagonal build, the traverses of the Square build have a width of one rod. This means that you could add a rod at each sphere pair between the inner and outer square of the middle layer (16 in total). In such cases, I use white rods.

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Re: Square Rhombic Torus

Postby Wagyx » Tue Oct 15, 2024 7:53 am

Hi Peter and Hans,
Here are the models and the website has been uploaded.
To answer your comment Hans, I should make a routine that removes duplicate points from the count or maybe correct the models.
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Square Rhombic Torus 2.zip
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Re: Square Rhombic Torus

Postby gmtrcply » Mon Oct 21, 2024 9:33 pm

Peter Jepsen wrote:GEOMAG Item 614.Model 07 Pentagonal Donut - PCS120B40S.05 Green.jpg


I just noticed: The filename is very revealing. So it's example 7 from the booklet of the Masterbox Compact Green. Very impressive photography.
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