I might be less active this month, so I thought I’d make up for it with an origami model you can dig into. This model is called “Six Intersecting Pentagonal Prisms” because it’s literally made from six pentagonal prisms. I’ve got some photos below the fold showing the step by step addition of each pentagonal prism.

The idea of weaving six pentagonal prisms together isn’t totally novel. There’s an older model by Daniel Kwan (diagrams, video) which is mathematically equivalent, but with a few cosmetic differences. There are also two more distinct ways to weave six pentagonal prisms (here and here).

Why six pentagonal prisms? Why not five or seven? Or why not six triangular prisms or hexagonal prisms?

I won’t answer that question in full detail, but it has to do with symmetry. If we want to create a model with a high degree of symmetry, the primary options are the five platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. This model is based on the dodecahedron. Each pentagonal prism has two pentagonal faces, which appear on opposite sides of the model. So 6 pentagonal prisms leads to 12 pentagonal faces, like a dodecahedron.

We could apply the same logic to the other platonic solids (except for the tetrahedron, since the triangular faces aren’t opposite to each other). A cube has 6 square faces, so there’s a model with 3 intersecting square prisms. An octahedron has 8 triangular faces, so there’s a model with 4 intersecting triangular prisms. An icosahedron has 20 triangular faces, so there’s a model with 10 triangular prisms. Feel free to look each of these up, I guarantee that someone has made them.

This method won’t produce an exhaustive list of the possibilities, though. For a more exhaustive list, I’d refer to Robert Lang’s list of polypolyhedra.

One thing I like about this model is that it’s basically a realization of a hemi-dodecahedron. A hemi-dodecahedron is one of those exotic mathematical concepts you read about on Wikipedia, which must surely have no useful application. It’s an “abstract polyhedron”, which is a generalization of polyedra unconstrained by such things as “euclidean geometry”. The hemi-dodecahedron consists of six pentagonal faces (each of which can be identified with a hexagonal prism), and all the faces are adjacent to each other. It’s just like how each of the hexagonal prisms is adjacent to the other five.

]]>There are really two color theories, the scientific theory of color perception, and the aesthetic theory of choosing color palettes.

The former is quite interesting, containing some surprising facts: yellow is the brightest color, many shades of green can’t be produced by modern displays, white is defined arbitrarily.

The latter is a hodgepodge of various historical ideas and a collection of overgeneralized advice. When I’ve read about aesthetic color theory online my impression is that much of it is either already taught to children or else it is not very good. Here is my attempt to identify some non-bullshit principles of color theory.

1. Recognize prior meanings and associations. Making a color palette is an exercise in referencing prior meanings, and also generating new meanings for other people to reference (or possibly you want a low-key color scheme that is very difficult to reference). Contrary to certain advice on the internet, single colors do not have very strong meanings–they’re simply used in too many different contexts. The strongest meanings in my experience come from holidays, flags, and natural objects. All these meanings are of course culturally and contextually dependent.

2. Choose contrast level carefully. High-contrast colors emphasize boundaries, while low-contrast colors may make them difficult to see. For some reason, many people think that high-contrast high-saturation colors are ugly, although I think it depends a lot on context. I’d put that in the category of “prior associations”–people have some prejudices about what brightness levels are appropriate where.

3. Know at least a little about color perception. Most importantly, be aware that about 5% of people are red-green color-blind.

4. Some possibilities that people sometimes forget about: Black and white are colors too. Not all the colors need to be fully saturated. You can completely change the color palette simply by changing how much of each color there is.

5. For large numbers of colors (at least 3 or 4), choose color combinations with low information density. Two common techniques are analogous colors (which are nearby on the color wheel) or triads/tetrads (which are evenly spaced along the color wheel). The particular technique isn’t important, but you want the color scheme to be something people can process easily. For contrasting colors, it helps if most of the colors are “canonical” colors, the kind that can be described in one word. For instance, “lime green, dusty blue, mustard yellow, maraschino cherry red” is just too complex and weird. “Green, blue, yellow, red”, on the other hand, is the color scheme of Google. (Although to be fair, even if Google’s colors are weird we’ve had a long time to get used to them.)

]]>I dabble in many kinds of origami, especially in the context of interacting with origami groups. I created this model, because someone at the origami group was really excited about making Pokemon, and asked if anyone could help him with this one. It was difficult because we didn’t know the folding steps, we only had a crease pattern.

Unfortunately, the website that hosted the crease pattern is down, so I can’t even share the crease pattern. But here’s a crease pattern for another Pokemon (Latios) that’s along similar lines.

As you can imagine, folding this without further instructions is a bit difficult. But we put our heads together and managed to make something passable. There were three of us working on it, and each of us brought experience with one of the three basic steps. The first step is to pre-crease along all those lines indicated in the diagram above (the step I was good at). The second step is to collapse the model into a base, which will have appendages of all the right lengths, but won’t look like a Pokemon yet. The third step is to mold the model, shaping it and adding frills until it looks right.

Of course, since it was just my first attempt, and we were muddling through it, I wouldn’t say that it’s a very *good* Skarmory. We certainly glossed over a few of the components–I didn’t bother trying to make clawed feet. For an example of a good Skarmory, look here and scroll down.

And for the record, I’m not actually familiar with any of these Pokemon. Back in my day we only had 151 Pokemon, and Skarmory certainly wasn’t among them. When we set to work on Skarmory, I briefly looked up an image so I could choose the right color of origami paper. Reddish brown seemed like the correct choice.

]]>Today’s model is one of my earliest original designs. This is a truncated octahedron, which is the shape you get when you take an octahedron, and chop off the 6 tips.

I was interested in designing a model with this particular shape, because it has some special significance in condensed matter physics. There’s a certain kind of crystal structure, called the “body-centered cubic structure”, which looks like this:

We can define the **Wigner-Seitz cell** as the volume of space that is closer to a particular atom, than it is to any other atom in the crystal. The Wigner-Seitz cell for this crystal, is shaped like a truncated octahedron.

For technical reasons that I’m not going to bother explaining, my primary association with a truncated octahedron is not a Wigner-Seitz cell, but instead a Brillouin Zone. Curious readers are encouraged to become physics majors and study condensed matter physics.

As far as the origami design goes, I’ve lost the design for this one! I made it back in the day when I didn’t think to document the folding steps. I could probably figure it out if I tried to reproduce it though. I believe it’s made from 56 pieces: 1 per hexagonal face, 2 per square face, and 1 per edge.

]]>Ah, so here’s a really old original design that I made in 2013. The story goes that I have a copy of Meenakshi Mukerji’s *Ornamental Origami*, which has a chapter on planar models. These are models where the folded form consists of multiple intersecting planes. One of my favorite models of all time is Tung Ken Lam’s WXYZ Triangles, which consists of four intersecting triangles. Later origamists would take this idea even further. What if you had 5 intersecting planes, or 6 intersecting planes, or more? So I made a bunch of planar models with different numbers of planes.

So as part of that whole thing, I wanted to make at least one model with 5 intersecting planes. There is a design, by Francis Ow called VWXYZ Squares that does just that. But either I couldn’t find diagrams, or I wasn’t satisfied with it, or else I just wanted to try designing my own. So, that’s what we have here.

Mind you, this is not especially great origami design. I would describe it the model as “delicate”, but it hasn’t fallen apart yet, so that’s something. The fact that the points of the squares pull apart (thus the “double-pointed” in the name) was actually a surprise to me. The hole at the center of the model was deliberate though. The fact that such a simple design works so well is a testament to the power of planar models.

For the geometry geeks: can you figure out what polyhedron this is based on? It’s not one of the named polyhedra, and it’s not very symmetrical.

]]>Ah, here’s a lovely origami model that I have not yet shared. Not much to say about this one. It’s a 30-unit model with icosahedral symmetry. I used 3 distinct colors, and the petals are curled with a toothpick.

]]>This is going to be one of those origami posts where I talk way too much about math. But before I get to the math, I will explain how you can make one of these things entirely with ordinary arts and craft tools.

“Ordinary tools” is the relevant bit here, since my understanding is that experts in curved-crease origami don’t use ordinary tools, they use things like vinyl cutters. When I first tried making these, I could not find any instructions for how to make these models using ordinary tools (I later found an article by Ekaterina Lukasheva), so when I finally figured out a method, I wanted to share it.

Before we draw the creases directly on the paper, we need to make a template. The template ensures that each of the four curves are identical to each other.

There are several ways you can make a template. My method is to take a thick piece of paper (card stock), draw the desired curve onto it, and then cut it out. Ekaterina Lukasheva also suggested using a French Curve. Personally I don’t like the idea of a French Curve because I want to know what kind of mathematical curve I’m drawing, but I’m sure the French Curve suffices.

Here I’m going to explain how to draw a **logarithmic spiral** on a piece of paper. Why a logarithmic spiral? Because I know a logarithmic spiral will work, whereas I’m not sure about other kinds of spirals. You can, of course, just go and look up a logarithmic spiral and then print it onto your paper. Here’s a spiral I generated from an online calculator, using a = 2.71828. Feel free to use this one.

But you can also draw a logarithmic spiral using a ruler and pencil. To do this, I first cut out a wedge in the blue shape shown below. The length of the wedge should be at least 75% of the length of your origami paper. The “pitch angle” can be any angle, but I strongly recommend trying an angle of about 45 degrees first. I draw the wedge using a protractor, but you can just approximate if you don’t have one. (Larger angles work too, but sometimes smaller angles will not fold.)

Next, we take a separate piece of paper, and draw a big dot on it to indicate the center of the spiral. We place the wedge onto the paper such that one of the wedge’s sides intersects with the spiral center. Then we draw a small line segment, which is a piece of our spiral. We continue drawing the spiral by adjusting the wedge just a bit (keeping the spiral center intersecting with the wedge’s side), and drawing more connecting line segments. Technically a logarithmic spiral will never reach its own center, but you can go ahead and connect the spiral to the center when you get close enough.

Finally, we cut the spiral out. I use an Exacto knife, but you can use scissors too. Make sure that the piece you cut out has the spiral center clearly marked.

Next, we need to trace the spiral onto the origami paper. You should pick origami paper that’s light in color on at least one side, so you can make visible pencil marks on it. Use your pencil and ruler to draw the horizontal and vertical bisectors of the square. Then you place your spiral template onto the origami paper such that the spiral center is in the center of the square.

The tricky part is to draw several spirals that are equally spaced from one another. What I do is I mark the template where the bisector lines meet the edges of the template. Then, when I rotate the template 90 degrees, the bisector lines meet the template at the same points.

Of course, this is assuming that you’re drawing only 4 spirals. You can draw any number of spirals, as long as there’s an even number, and at least 4 of them. But the geometry might be a bit trickier if you do more than 4.

Once we trace the spirals using pencil, we take away the template, and trace over the spirals again, now with a **scoring tool**. I used to use a ball point pen, but now I use a stylus that was originally intended for ceramics. You score the paper by tracing the spirals while pressing hard on the paper. Make sure that you have something behind the paper that can handle the pressure! I wouldn’t want people making marks all over their library books, or poking holes through their paper.

If you’ve scored the paper correctly, you should still be able to see the score marks when you flip over the paper.

Once you’ve finished scoring, you’re nearly done. With some coaxing, the paper should fold along the score marks.

To fold the paper, it’s important to keep in mind that the creases should alternate between mountain and valley. Furthermore, the paper between the creases is also curved, alternating between concave up and concave down. One way to make the fold is by pinching the crease, just a little bit at a time. Another way to make the fold is by applying pressure so that the paper on one side of the crease is concave up, while the paper on the other side of the crease is concave down.

It’s quite difficult to make the creases near the center of the paper. Just start from the outside, work your way towards the center, working over each of the creases multiple times.

That’s the end of my instructions. Here are some more spirals (I made a bunch of them).

Curved-crease origami is mathematically very complicated. There are a couple papers by Erik Demaine that describe some of the rules, but I feel like this is still only scratching the surface. I’ll pull out a few highlights from those papers.

If we consider uncreased paper (i.e. the behavior of the paper between creases), the paper can always be divided into two types of regions. There are **planar** regions, and **curved **regions (Demaine calls them parabolic regions). Mathematically, the curved regions are a kind of ruled surface. A ruled surface is a surface that can be decomposed into a set of straight lines (called rule segments).

Hypothetically, if you traced out all the rule segments, and unfolded the paper, then you would have a bunch of straight lines, each drawn from one crease to another crease. The rule segments never cross over one another.

Now, part of the reason I was interested in these spiral creases, is that in some ways, these are just about the most mathematically basic example of curved creases. I chose logarithmic spirals because they even have scaling symmetry; if you shrink down the paper and rotate it a little bit, you get the same pattern as the one you started with. The scaling symmetry greatly simplifies things, and I hoped that the problem would be solvable. (Also, I tried a freely-drawn spiral first, but that didn’t work.)

But I still could not solve this problem. Basic question: Is the spiral mathematically possible, or does it require stretching the paper just a tiny bit? I don’t know! Basic question: What angle do the rule segments make with the creases? I don’t know!

One thing I discovered from experimentation, is that sometimes the spiral doesn’t fold, especially when the pitch angle is too small. Here’s one that uses a small pitch angle of about 12 degrees.

As hard as I tried, I could not get the inner part of the spirals to fold. Is that because it’s mathematically impossible? Or do I just need to try harder? Does folding the spiral suddenly become impossible at a particular pitch angle, or does it just slowly become more and more difficult to fold the spiral? If it suddenly becomes impossible, what’s the critical pitch angle? I tried to answer these questions, but I could not.

So how’s that for an interesting math problem? I hope to find a solution some day.

]]>Meenakshi Mukerji featured some of my origami on her guest gallery page, including this model. Then I realized I hadn’t posted this one on my blog yet, so here it is.

Regular readers have probably noticed that I feature an awful lot of models designed by Meenakshi Mukerji. That’s because I have four of her books! The Star with Spirals is featured on the cover of *Origami Inspirations*, which I would say is the best of the four. Although, if you’re reluctant to commit, you might consider one of the *Exquisite Origami* books, which are cheaper because they’re in grayscale.

I always like to comment on the geometry/coloring of these things. This is a 30-unit model, with each unit corresponding to the edge of an icosahedron. There’s also a dual arrangement I could have used, witch each unit corresponding to the edge of a dodecahedron instead. I use 5 colors, in a standard symmetrical pattern. (Using the terms that I defined last year, the pattern symmetry group is the same as the shape symmetry group.) The color scheme is supposed to be 4 analogous colors (pink, white, yellow, brown) and one contrasting color (blue).

]]>I was looking through my photos, and I realized that there are several large models that I never got around to sharing. This is one of them. The Three-Form consists of 24 little cubes, assembled into a larger mathematical design.

This one is inspired by General Relativity, Einstein’s theory of gravity. At the time I was reading Sean Carroll’s textbook on the subject, and I was lamenting how difficult it was to visualize the mathematical concepts therein.

As some of you may know, General Relativity makes use of tensors, which are generalizations of vectors. If we have a simple vector in 4 dimensions (3 dimensions of space, and one dimension of time), we could represent the vector with its four components: v_{x}, v_{y}, v_{z}, and v_{t}. On the other hand, a “rank-two” tensor might have 16 components: T_{xx}, T_{xy}, T_{xz}, T_{xt}, T_{yx}, T_{yy}, T_{yz}, T_{yt}, T_{zx}, T_{zy}, T_{zz}, T_{zt}, T_{tx}, T_{ty}, T_{tz}, T_{tt}. My model represents a kind of rank-three tensor, which has 64 components.

More specifically, I was trying to represent a “fully antisymmetric” tensor. If you swap any two of the letters, you get a negative sign. For instance, T_{xyz}= -T_{xzy}. By swapping around letters, we can prove that most of the components of the tensor are zero (e.g. T_{xxy}= 0). What we have left are 24 components that are *not* zero, and each of these components is represented with a little cube.

Even though there are 24 non-zero components, most of those components are identical to each other (perhaps with a negative sign attached). There are only four independent components, each of which is represented with a different color scheme and design.

Two explain it with less math, basically I have a 4x4x4 cube, where I’ve removed all the diagonals. As a result, if you look at the model from any of its three main axes (x, y, or z), you see gaps along its diagonal.

All in all, there are 126 separate pieces of paper.

]]>Over a year ago, I designed an origami arrow that points in the opposite direction when looked at in a mirror. I wanted to revisit the design, and make it easier to fold, because this is basically the most popular origami thing I have ever done. Below, I show detailed diagrams, and a sneak peek behind the curtain.

This model was based on a paper by Kokichi Sugihara, explaining a particular optical illusion. You can watch an amazing video showing a cylinder that has a square reflection. My model is optimized for viewing at a 45 degree angle, and it works best at a distance (because of field of view effects).

You are free to print out these instructions. But if you plan on distributing copies, please ask permission. E-mail me at skepticsplay at gmail dot com, so I can give you a proper pdf.

What does the illusion look like from a different angle? The illusion breaks down fairly quickly, as you can see here:

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