Back in 2017, I designed this tessellation based on the rhombile tiling, and I blogged about it here.  More recently, someone asked my permission to teach it in an origami convention.  I said, “Sure, but I only have crease patterns, no instructions.  Can you figure it out from there?”  This being an expert origamist, he figured it out alright, but he said it was very challenging.  I tried it myself, and I had to agree!

The challenge can be part of the fun, but I still wanted to make it a bit easier.  So I revisited the model to see what I could do.  I finally made some step-by-step instructions!  Very difficult to make instructions for tessellations, because every step involves multiple simultaneous folds.  I also added some steps to clean up the “edges” of the tessellation.  The result is what you see above.

2024 was a very artistically productive year for me.  I folded about 10 original designs, revisited my cube tessellation design, and made two group theory infographics. This is in addition to folding dozens of other people’s designs.  It helps a lot that I returned to origami meetups this year, which I had stopped attending during the pandemic.

This design is made by someone I know, a teenager that I see in the local origami space.  For him, this is a very simple design, something he can teach others in the space of an hour.  I often see him working on prototypes of much more complex designs, using massive yard-long paper.

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A few times before, I’ve posted origami failures, outtakes and disasters.  This time I’ll focus on some failed experiments–design ideas that didn’t quite work out.

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Many years ago, I designed the origami Arrow Illusion–an arrow that points in the opposite direction when viewed in a mirror.  It’s probably the biggest “hit” among my designs.  Recently, CFC (an Origami community) had a monthly challenge to create optical illusions, and it featured Arrow Illusion as an example.  So, I thought I’d design another.

I went through several ideas, but ultimately settled on the Reutersvärd Triangle.  It’s similar to the better-known Penrose triangle, but was actually created independently by Oscar Reutersvärd in 1934–before the Penrose triangle.

Like the Penrose triangle, the Reutersvärd Triangle is just an illustration–it’s impossible to actually make something shaped that way.  And yet, you can find many very convincing 3D-printed implementations of the Reutersvärd Triangle.  I felt this was a sign that I needed to origamize it.  (Note that the Arrow Illusion, too, was inspired by 3D printing!)

Actually designing the thing was fairly challenging, and I went through over half a dozen prototypes.  Even once I got the basic form down, I tried many ideas to optimize for simplicity and stability.  The end result is so elegant that its instructions fit on a single page.  Note: if you try to fold this, I suggest trying the “easy” version first, and make one with just 6 pieces of paper. It will look like this:

How does the illusion work?  The cubes are actually inverted.  Although it pretends to be a collection of cubes pointing outwards towards the camera, they’re actually internal corners pointing inwards.  It’s… not very convincing when you have the physical object right in front of you.  It’s most convincing when you take a photo in ambiguous lighting, and then sometimes it helps to turn the photo upside-down.  The lighting of the cubes doesn’t match the shadows behind the model, which may have caused some of you to see through the illusion already.  But in case you don’t see it, here’s a photo from the side:

I’m saddened to hear that Meenakshi Mukerji recently died.  I almost had an opportunity to meet her at a convention, but there was a pandemic and it never happened.  I’d give her a lot of credit for getting me into modular origami, and many of my earliest models were from her books.  I love her work.  This is a simple model that I have folded many times; it can be found in Ornamental Origami.