Have Fun With Torus




Introduction: Have Fun With Torus

Hi, everyone. Here, we are going to explore a set of interesting tori. This Instructable is for people who love math and art.

As the fans of 3D Printing, many of us may be already used to plane modelling along with the 3D-printed polyhedral objects. So I was curious about how to apply 3D Printing into the fabrication of some charming spiral and hollow structures. Then I did a series of designs and tests about surface modelling, and finally successfully built the prototypes.

Expand your imagination. We will create some really fantastic topological shapes such as Mobius Strip, Interlocked Tori, and Trinity Knot by 3D Design and Printing. Followed by the tutorials and attached files, you will observe these toroidal objects more intuitively, and you can also take these tori as your accessories.

All the prototypes are one-time fabricated by Form 2 (Formlabs), the Desktop Stereolithography (SLA) 3D Printer, so you do not need to do any assembly after printing. You will obtain fresh skills about 3D Surface Modelling by software CAD Solidworks through the tutorial. I also hope to bring you inspirations for your future design and creation.

Step 1: Step 1: Materials, Tools and Operation Instruction


Clear Resin Cartridge (Form 2)

Alcohol Tank


1. Hardware

Desktop Stereolithography (SLA) 3D Printer, Form 2





2. Software


Operation Instruction

You are suggested to take a look at the operation instructions of Form 2 by the following link:


Step 2: Step 2: Design a Bagel Torus

Bagel is the most common torus. In Solidworks, A Bagel Torus can be formed by sweeping a profile through circular path. Here, I choose a circle as the profile. And if the diameter of the circle is equal to the diameter of the inner hole, you will get a very beautiful Bagel Torus shown in the photo. And after this, we are going to cut this Bagel into different shapes in the following steps.

Step 3: Step 3: Design a Mobius Strip

A Mobius Strip is ideally a surface with only one connected side. Imagine if there is an ant on the outside surface of a Mobius Strip, it can just hike through a certain direction along the strip to the same location on the inside surface without stepping over any boundary, and following the same direction of the path, the ant can travel back again to the same location on the outside surface.

The Mobius Strip can be formed by half-twisting a normal strip, namely twisting one end of an open strip by 180° and connecting the two ends of the strip. In Solidworks, a Mobius Strip can be formed by sweeping a profile through two semi-circular paths. In each path, the profile should also be twisted by 90°. So along the whole sweeping path, the profile is twisted by 180°. Here, I choose a flat ellipse as the profile so that we can obtain a solid Mobius Strip in Solidworks and it is feasible for 3D printing.

Step 4: Step 4: Construct a Mobius Space Inside a Solid Bagel

Imagine you have a bagel and you use a knife to slice the bagel through its reference circle. If you simply follow the above process, then you just cut the bagel into two separate pieces. But if you also carefully flip your knife by 180° until the complete cycle, you will still get a single bagel but with a loop tunnel inside. Assume your knife is thick enough, then what kind of materials or space does it cut off? It is actually a Mobius Strip as we mentioned.

The above geometry is really interesting. It is somehow like a model of the galaxy or universe. If you squeeze a ball with suitable size into the Mobius Space in the solid Bagel Torus, then in theory, the ball can roll freely through the whole inside loop without moving to the outside surface of the Bagel Torus.

In Solidworks, we choose “Swept Cut” to achieve this feature. Also, you can expand your imagination and make the geometry more charming. For example, you can redesign the Bagel Torus to be uneven in orbital diameter and then cut it by the same Mobius Space. Now you will obtain a shape like a sea shell. So try to create something better after the going through this step.

Step 5: Step 5: Make a Full-twisting Strip

If we twist one end of an open strip by 360° and connect the two ends of the strip together, now we obtain a full-twisting strip. In Solidworks, we use “Swept Base” to sweep a flat ellipse profile through a circular path by adding the twist value as 360°. This time, you will find that the developed geometry actually has two surfaces.

Step 6: Step 6: Make a Pair of Interlocked Tori

Again, let us imagine that we have a bagel and we use a knife to slice the bagel through its reference circle. This time, if we carefully flip our knife by 360° until the complete cycle, can you guess what kind of structure we are going to get? Now the bagel is indeed sliced into two pieces, but these two pieces are connected. So this is our Interlocked Tori.

Step 7: Step 7: Multi-twisting Torus

Inspired by the above ideas, we can just try to twist the strip by multiple turns. For example, if we twist the strip by two-and-a-half turns, then we will obtain a so-called Pentagon Torus. We can also remove a Pentagon Torus Space from a solid Bagel Torus to create another interesting pattern. Try to create more by yourself.

Step 8: Step 8: Trinity Knot

Trinity Knot is probably the most fascinating torus in this Instructable. Imagine that we twist a strip by three half-turns and connect the ends together. Then we cut the twisted strip through the central line by the complete cycle. Then you will obtain a connected strip with three cross-over points. If you stretch the developed shape properly and then you will make the Trinity Knot.

You can always better modify a Trinity Knot by your imagination. In this project, I just created a hollow tube to replace the strip feature. And the final product is like a cool roller coaster.

Form-2 prints the object by an upside-down way. The resin is filled in the tank at lower part of the printer where the laser comes out to solidify the resin to form the structures. The final product is attached to the upper board of the printer and you need to use a shovel to take it down. Support materials are attached to hold the whole structure stablely. Sometimes, these support materials make the final product more beautiful so you can just leave them there. For my Trinity Knot, I did not remove the support materials because they just like the holder of a roller coaster.

Step 9: Step 9: Enjoy Your Creation

Now you can create some cool tori by yourself. The modelling of torus is not really difficult but it requires your imaginations. You can easily fabricate an interesting torus by Solidworks and Form 2 within a few hours, so try and enjoy it!

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    3 Discussions


    Question 7 months ago on Step 4

    Your work is incredible, thank you for sharing it! Is it possible to create an interlocking bagel torus with a mobius strip and have it be 3D printed?


    3 years ago

    These are really neat! They'd be really awesome for a math class to show the principles in practice :)


    Reply 3 years ago

    Thank you!