Mathematical Art Project: Tom Hull's Five Intersecting Tetrahedra

[Kernel Sohcahtoa]

Introduction

 

Hello everyone.  I thought that some AF members may enjoy this particular arts and crafts project.  This model is called the Five Intersecting Tetrahedra by Tom Hull,  which is a modular origami model.  In modular origami, a specified number of modules are folded (the Fit module is relatively simple to fold), which contain hands and pockets that interlock with other modules (no glue is necessary) to form a solid or flat geometric shape.  Hence, if you enjoy building cool structures with your hands, then you might like this project.  Please note that you do not have to be familiar with origami to do this.  Also, if you are interested in doing this, I have provided a number of tips and tools (videos and diagrams) below in order to equip you for this project.  On the whole, this is a very rewarding project (most people stare at it with child-like curiosity and wonder), but it does require patience (this is the most important), spatial manipulation, and logic.  It is also an excellent gift: I have given Fits to my family members and to co-workers.
 

Tips 

 
Regular 8.5in by 11in printer paper will work fine.  I highly recommend that you use five different colors.  For each color, you will need two sheets (make sure you have backups).  Each sheet will be formed into a square (tools, section one) and then broken down into thirds (tools, section two).  After breaking down your original 10 pieces of paper, you should have six smaller pieces of paper for each of the five colors or 30 pieces of paper total.
 
Once you've broken down the paper into thirty smaller sheets, then watch the FIT video: single tetrahedral frame (tools, section three), which has better quality than the full length video.  Overall, the folder does an excellent job of clearly demonstrating how the module is folded (his creases are clear) and how six modules interlock to form a single tetrahedral frame (it may be useful to look at Hull's diagrams for folding the modules and interlocking them in order to complement your understanding of the video).  After you have completed this, I recommend that you view the full length assembly video in order to gauge what the assembly process will be like (fast forward to 5:07, which shows the assembly process after making the first tetrahedral frame and watch the video through to 16:37). Hence, once this step is completed, there are two main approaches that you can use to weave and interlock the remaining frames together (naturally, if you have a more novel and simpler approach, then please use that one).
 

A Possible Approach

The approach that I prefer is to use the diagrams, which show you how to weave the remaining tetrahedral frames together: they are woven together one frame at a time. For example, the second frame is woven into the first, then the third frame is woven into the first and second frames (etc.).   I found it useful to use each frame picture (Hull's diagrams) as a checkpoint and base image for how the Fit should look at a particular point.  For example, after weaving the second frame into the first, I would make sure that my point of view was identical to the picture and that everything matched up before moving on to the next image. Hence, all of the frames are in the exact same base position in each of his photos.  For example, the first frame stays in the same position when the second frame is woven into it.  Likewise, the first and second frames stay in the same positions when the third frame is woven into it and the same is true for the fourth and fifth frames.

 Once you have assembled the first and second frames, then the third frame will be a bit tricky, because the other two frames have a tendency to shift around. As a result, I placed books or whatever around it, which helped keep the frames stationary and made it easier for me to check the respective assembly picture and ensure that I was weaving the frames correctly (naturally, you will have to rotate the model in order to interlock the modules, but using books or whatever should minimize the shifting of the frames during these rotations).  Once you complete the third frame, then the fourth and fifth frames will be easier, as the frames will not shift around as much.  In addition, I found it useful (especially after weaving a few tetrahedra together) to place some modules in individually and then interlocking them. 

An Alternative Approach

An obvious approach is to simply assemble the model along with the user via the full length video and start and pause accordingly (this has the added advantage of seeing how the frames look at different vantage points). In addition, before moving on to the next frame, it may be helpful to pause the video at certain points to ensure that you have woven the frames correctly.  For example, at 5:55, the second frame is woven into the first frame and you can use this image as a checkpoint before moving on to the third.  9:19 is a checkpoint showing how the third frame looks in relation to the first and second frames. 12:20 through 12:21 (you do get decent vantage points within this second) and at 12:35 will be good checkpoints for how the fourth frame should look relative to the first, second, and third frames. Finally, 16:20-16:30 should be good checkpoints for ensuring that the fifth frame has been woven correctly in relation to the others.  However, you will need to do a lot of reorienting of your POV if you choose the video approach, because the object is continually rotated throughout the assembly process. As a result, I personally do not prefer this method, but if you possess superb rotational and spatial skills, then this approach may be simpler for you.

Conclusion

 
There are many logical, intelligent, and creative people on this forum.  These traits will come in handy when completing this project; however, patience and a genuine interest in building the Fit, are absolutely requisite for success.  In addition, please take your time when folding the modules (your model will look so much better at the end) and be sure to take advantage of the fact that paper is very flexible, so if you have to bend a module in order to get it to weave properly, then please do it.  Also, once you have finished  weaving and interlocking a particular frame, please be sure to check that your work is correct before moving on to the next frame (I often had to disassemble a frame or two and give it another try later on, because I failed to check my work accordingly).  More importantly, pay attention to your posture when doing this project and take breaks when necessary. Happy folding and building, and if you'd like, please post a picture of your completed model.  Thanks and live long and prosper, AF Members and anyone else. 
 

Tools

Section One: Making a rectangular sheet of paper into a square

Diagrams

Section Two: Folding a Square sheet of Paper into Thirds

Diagrams

*Please refer to method two, the origami method, which shows how to make a square into thirds.

Section Three: Folding the Modules, Interlocking the modules, and Assembling the Fit 

How to fold and Assemble a Single Tetrahedral Frame

Video: how to fold a module and assemble a single Tetrahedral frame

Full Length Video of the Assembly Process

Video: full length Fit Assembly

 P.S.  I was not able to post many videos for sections one and two, so please feel free to ask me to post other sources.  I'm happy to help, but again, patience is really the determining factor in building the Fit.

[A Handmaid]

Ooh, I will definitely try to do this!
I'll just try to make one out of just blank paper until I can get hold of some colored paper.

[Kernel Sohcahtoa]

Ooh, I will definitely try to do this!
I'll just try to make one out of just blank paper until I can get hold of some colored paper.

That is impressive.  After building your initial tetrahedral frame, I suppose you could assign an imaginary color to each new frame (or make helpful markers on your model), and along with following the tips mentioned in the first assembly strategy of the op (a possible approach), you may indeed have a workable approach (especially if you minimize the shifting of the initial frames).  In all honesty, If I only had blank paper available to me, then I'd break down the paper into the smaller pieces (5 groups, each group containing 6 pieces of paper, 30 pieces total) and then color them in accordingly.  I would then declare the Fit as a training/learning model.  But that's just me.  Good luck Handmaid

[vorlon13]