In this tutorial, I'll show you two distinct techniques to modeling a toroidal ring of saddle surfaces, known as a Scherk-Collins sculpture, without any plugins, addons, or 3rd party software. This beautiful mathematical object could be useful as a pendant, earring, or any kind of modern art sculpture in a cg environment. The only limit is your own imagination.  This video is done in 3dsmax 2012, however the techniques and procedures therein are very universal, and could be translated easily to any capable polygonal modeling package, like Blender.

https://youtu.be/-pvJBP8KOhc

Please feel free to leave a comment, should you have any questions or input regarding the video, and please don't forget to check out my channel for more 3d  tutorials for 3dsmax, Blender, and more.

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That, Sir, is very classy. How on earth did you work those out. I'll have to try these when I can figure which way the normals are facing.

That, Sir, is very classy. How on earth did you work those out. I'll have to try these when I can figure which way the normals are facing.

Thanks, Mark.  Back in 2012, when I had first began using 3dsmax in school, I was using a interesting plugin called Mathsurf, which was a maxscipt that automatically produced mathematical shapes of all kinds in the Max environment.  This is an example of some objects I produced with it, back in 2012: http://luxxeon.deviantart.com/art/Toroidal-Hangout-321980449

After playing around with that plugin for a while, I began studying the surfaces it produced from the automated algorithms.  The output it produced often consisted of triangle topology, and I wanted to create the same shapes with logical quad surfaces, which are easier to edit and unwrap.

So, the first method illustrated in the video is a very commonly used technique I learned years ago from the Wolfram Mathematica community.  I was searching for a way to model a saddle surface without algorithms, and the people there were extremely helpful.

The second technique, however, is a variation I personally came up with, in a kind of "eureka!" moment, after experimenting with some older subdivision modifiers in 3dsmax.

PS:  It may not matter about the normals in your software, Mark.  The only reason it's an issue in 3dsmax, is because I'm using the "Editable Polygon" object type to create the shapes.  Editable Polygon, unlike the older "Editable Mesh" object types, will not allow you to model single-sided objects with invalid, or non-unified, normal faces.  It intentionally forces you to model proper objects, logically.  However, this is not actually the case in Blender, for example, where you have the freedom to weld points which reside on opposite face normals, without any problems.  As long as you thicken the surface into a solid object in the end, this shouldn't be problematic anyway.  If you were to leave it as a single-sided, non-manifold object, of course, the non-unified normals would become an issue.

I'm thinking there's a lot you could have done to improve both of those techniques.  Take the first one for example.  Why not just create one long plane with a cut segment through the middle, then twist the whole thing, instead of starting with two perpendicular planes, and welding the end points?  Seems like the end result of welding the points is the same as twisting one larger plane at the middle.  Also, it seems like the symmetry tool in Max has it's own built in pivot point. So instead of wasting time applying the symmetry, then moving the object's pivot every time, it seems you could have just added a few symmetry mods on top of each other, then moved their internal pivots to line things up.  Probably would have made the whole video about 10 minutes shorter, mate.

Nice tut Lux. Enjoyed it. Going to try that later. Looks like it will translate to Blender no problem. I got one for you though. There  is a thread in t he Blender forum I wish you would check out. http://www.renderosity.com/mod/forumpro/?thread_id=2889384

Because there is Picture of a Teapot in it. The basic shape would not be hard to model but there is raised detail along the bottom. I'm not sure how I would go about doing this other than a bump map. If you get a chance have a look. I'd like to know what you think.

Nice tut Lux. Enjoyed it. Going to try that later. Looks like it will translate to Blender no problem. I got one for you though. There  is a thread in t he Blender forum I wish you would check out. http://www.renderosity.com/mod/forumpro/?thread_id=2889384

Because there is Picture of a Teapot in it. The basic shape would not be hard to model but there is raised detail along the bottom. I'm not sure how I would go about doing this other than a bump map. If you get a chance have a look. I'd like to know what you think.

Thanks, Cybermonk. The teapot I saw there was indeed a tricky object.  That's definitely a case where I personally would create those specific details using either displacement or normal mapping, in a sculpting application.  However, it's not impossible to model it out either.  The only thing that would require some additional consideration there is the edge flow of the scalloped details.  You'd need to find a way to get the edge segments of the object to skew slightly towards a common centroid in those areas, while still maintaining the overall profile integrity of the object.  I suppose it could be done with some remeshing modifier tricks, but I'd have to experiment with some procedures to get the right ones.  The ornamental details themselves, however, could be easily brought out with some beveling, once you have the edges oriented properly.

In this case, I selected several edge loops halfway down the model, and quad chamfered them with two segments in 3dsmax, and raised up the interior segments to create the ornate scalloped effect after subdivision.

This obviously isn't a serious attempt to reproduce the effect, but it's the approach I'd probably take, once the topology was worked out.

I'm thinking there's a lot you could have done to improve both of those techniques.  Take the first one for example.  Why not just create one long plane with a cut segment through the middle, then twist the whole thing, instead of starting with two perpendicular planes, and welding the end points?  Seems like the end result of welding the points is the same as twisting one larger plane at the middle.  Also, it seems like the symmetry tool in Max has it's own built in pivot point. So instead of wasting time applying the symmetry, then moving the object's pivot every time, it seems you could have just added a few symmetry mods on top of each other, then moved their internal pivots to line things up.  Probably would have made the whole video about 10 minutes shorter, mate.

Sinner, I suppose after the first initial steps, it would be possible to use only the symmetry modifier's mirror gizmo to align the edges for automatic welding.  Yes, that's entirely possible, and probably faster.  However, for the purpose of a tutorial, I prefer to be very clear about how and why all the steps are done, and piling on symmetry modifiers in the way you suggest, while faster, could become confusing to someone trying to learn.  The chances of  an improper weld increase that way, and mistakes could be made which would result in a failed model.  So, I err on the side of caution here, and use the longer, but technically sound, technique instead.  If someone feels they could cut corners to speed up the process, at least they will know how the basic structure of the model is formed first, which would be most helpful in creating your own shortcuts or personalized procedure to achieving the object.

I knew you would have some good ideas Lux. The sculpt would work pretty good. In fact may be able to add some of that detail in with booleans.  I 'm thinking of using some curves with a bevel and taper applied. Convert them to mesh once you got them positioned and then boolean them in.  I know booleans make some crappy looking topology but I was thinking of blending and smothing in dyntopo sculpt mode. Then give it the old retopology. Thanks for the idea.

Seems like the end result of welding the points is the same as twisting one larger plane at the middle. 

Wrong!  You might think that, but I tried it, and it's not the same.  Maybe luxxeon could elaborate on the why better than I can, but I'm guessing it's because the twist modifier is twisting the object uniformly along a central axis (whichever you pick).  You can move the center point wherever you like, but then the twist won't be uniform anymore.  WELDING the two mid points together (ie. collapsing them) happens at the MIDPOINT of the two vertices, which is NOT necessarily the center point of the object itself in world space... in fact, it definitely isn't.  So doing the merge on the vertices like that results in a slightly different shape than simply applying a twist modifier.  Sure, you could go back in after applying a twist modifier and move points where you want them, but then that's going to be just as much, if not MORESO, than just welding them like lux did in the video. Correct me if I'm wrong, lux, but I tried what Sinner suggested, and it did NOT work.

I knew you would have some good ideas Lux. The sculpt would work pretty good. In fact may be able to add some of that detail in with booleans.  I 'm thinking of using some curves with a bevel and taper applied. Convert them to mesh once you got them positioned and then boolean them in.  I know booleans make some crappy looking topology but I was thinking of blending and smothing in dyntopo sculpt mode. Then give it the old retopology. Thanks for the idea.

That's a good idea.  Also, as long as this isn't an object which would see deformation in some way for animation, you could add those scalloped details as a kind of  floating geometry.  For instance, use retopo tools to paint some polygons along the surface of the modeled teapot as separate elements laying on the surface, then thicken it, and subdivide it smooth.   Since the details will be attached to the object as elements, no one will ever notice it's not part of the original base mesh. For example, I did this exact technique to create the scalloped-style feet on this piano stool model, as well as the ornamental work on the base.

http://www.renderosity.com/mod/freestuff/?item_id=73227

Again, as long as the model isn't required to animate in any way which would cause the surface deformation, or require the entire thing to be solid for something like 3D printing, then floating geo is often a great way to present details like this.  Retopology tools are also a great way to create those details directly on the surface.

Thanks Lux. That sounds like a good Idea. I have had fun following your tut. Even tried a double version.

Thanks Lux. That sounds like a good Idea. I have had fun following your tut. Even tried a double version.

I'm going to try doing this tut in Blender tonight as well.  I think all the techniques we discussed from the blender mobius tutorial will come in very handy here. This should also be a great model to test out in some Cycles renders too.

Here's my go at it this tutorial in Blender 2.74.  I've added some extra roundness with a cast modifier above the subdivision modifier, set to sphere, and adjusted to create just a little more roundness to the center.  Also, I've applied some laplacian smoothing earlier.  Very cool shape.  I see this as an awesome amulet or modern jewelry.

Wrong!  You might think that, but I tried it, and it's not the same.  Maybe luxxeon could elaborate on the why better than I can, but I'm guessing it's because the twist modifier is twisting the object uniformly along a central axis (whichever you pick).  You can move the center point wherever you like, but then the twist won't be uniform anymore.  WELDING the two mid points together (ie. collapsing them) happens at the MIDPOINT of the two vertices, which is NOT necessarily the center point of the object itself in world space... in fact, it definitely isn't.  So doing the merge on the vertices like that results in a slightly different shape than simply applying a twist modifier.  Sure, you could go back in after applying a twist modifier and move points where you want them, but then that's going to be just as much, if not MORESO, than just welding them like lux did in the video. Correct me if I'm wrong, lux, but I tried what Sinner suggested, and it did NOT work.

Ok, so I guess the twist doesn't work like I thought it did in 3dsmax.  Fair enough.  In Cinema, the twist deformer is very configurable, and centers can be offset.  I thought 3dsmax would have the same.  It looks like you can adjust the centers of the twist from the deformer's center.  I could have sworn I'd seen it done before.

Wowza!  Amazing work. 

Very, very cool renders, great job friend