Zing-Man Origami - Polyhedra and Tessellations
Tessellations are very popular in origami these days, although personally I havenít done much with them. Here is a tessellation that features all pentagons. It doesnít use pentagonal symmetry however, the underlying grid is square. Itís an ancient pattern used in Moorish and Mideastern art and architecture.
My first origami tessellations was a grid of pentagons. Now Iíve made a tessellation of triangles and squares to go with it. And I mean that in a very specific way. This mesh is the dual of the other. If you draw a dot in the center of each cell of the pentagon grid and connect them all, youíd get this pattern. Five cells converge at each vertex. In the other pattern all the cells were the same shape, but either three or four cells converged at a vertex. The two of them make a nice set.
I created this mandela-like pattern at 2011 convention. It's a spiral matrix of parallelograms that features fivefold symmetry. At first I thought it was a Penrose tessellation, but the joining rules do not conform to the Penrose tiling. I have since folded several true Penrose quasicrystal tilings, but have yet to fold one out of good paper. They are more challenging because several different angles come together at the vertices.
This was a precursor to the Pseudo-Penrose Star. It features a fourfold symmetry that is easer to fold and has fewer cells (24 vs. 40). In fourfold geometry this is an allowable quasicrystal tiling.
An elaboration on the Cairo Tessellation, this pattern features a grid of alternating pentagons and hexagons.
This fascinating shape is something like a sunken icosahedron, and can also be seen as twelve intersecting pentagons with a raised star on each face. I tried several iterations of the layout to get the details right. The basic idea is fairly straightforward. I use fivefold polar symmetry, and the whole pattern embedded in a single pentagon that takes up pretty much the entire square sheet. I divided it into a grid of parallelograms using simple ratios. Each parallelogram then gets subdivided into the triangles that form the faces of the shape. The pentagon's height is slightly less than its width, which results in a then strip of unused paper at the bottom edge of a square sheet. I decided to try folding the strip around all five sides (except where it gets truncated at the corners), and that turned out be just the trick.
Iíve seen this kind of thing done as modulars and thought it was doable as a single sheet. Single-sheet stellated polyhedra are pretty advanced but the color change brings it to a whole ënuther level of complexity. It turns out to be a very rewarding shape to fold, and the design is replete with all kinds of interesting symmetries. The first challenge was to work out how to achieve the arrangement of alternating colors. Once Iíd worked that out the resulting (flat) shape would serve as the base for the 3-d phase. I needed two corners to come to the center like a blintz, but offset. Working out the amount of offset for the grid to be the right size was the key problem. It turns out the key angle is 67.5 degrees, which is 3/4 of 90 degrees and easily derived. It also turns out the angle has a slope of 3/2, which is also easily derived from a square grid. From this I was able to work out the arrangement of the squares in the inner rotated grid and the outer triangular grid areas, which correspond to the blintzed flaps. The 3/2 slope made was convenient because the grid is has an integer relation of the unit the whole. Each square of the grid has a length of 2/13 the edge of the paper, as you can see in the crease pattern.
This fascinating shape can be seen either as a stellated octahedron or two mutually intersecting tetrahedra. This model is a compound, made of four sheets of square paper, each containing one corner of each of two tetrahedrons. Different methods of creating the module render the final shape in similar or contrasting colors. I have been working a version from a single sheet with the color change.
2005 - 2007
One of my all-time favorite shapes has always been the Stellated Dodecahedron. I've designed versions of this shape over time. The first is from a single square. It is very challenging to fold because there are many flaps of extra paper to deal with by the time you get to the end, and the model wants to spring apart. Still, I like the CP because the layout maximizes the root pentagon and underlying square. Vertices of the finished form touch 3 edges of the paper. Next up is the one folded from a 2:1 rectangle. This one is remarkably efficient in it's use of paper, to the point where I had to set it into a larger area to have paper to do the joining. It's also kind of cool because it has a sort of zigzag layout. Last is a version made from 2 squares, each of which comprise half the finished model. This is much easier to fold because you can reach inside each half as you're making it, and the leftover flaps of paper become tabs that fit into the opposite half, nicely solving the problem of what to do with the leftover bits. At the end, the two halves lock together tightly and securely. The resulting model is quite attractive, because it's much easier not to crush it as you're putting it together.
Also called Three Intersecting Octahedra, or the TriOcathedron, this polyhedron sits atop one of the towers in M.C. Escher's Waterfall. Folded from a single square sheet.
Crease pattern for Stellated Rhomibic Dodecahedron
This fourteen-sided Archimedean solid has six regular hexagons and eight squares as faces. I designed this on a trip to Washington D. C. Folded from a single square sheet.
Crease pattern for Truncated Octahedron
One of only a handful of unique hexahedra, this is a very beautiful shape, whose construction is replete with the golden ratio. Way cool.
Crease pattern for Pentagonal Pyramid
Original method for folding the twenty-sided platonic solid from a single rectangular sheet of a 2:1 ratio. My method features good utilization of the paper and optimization for size, a easy and straightforward prefold sequence, and a very secure lock. In general, folding a polyhedron from a 2:1 rectangle is easier than a square and better utilizes the paper. This is because you have more edge relative to area, and you can use something like a Mercator projection for the layout, or a two-hemisphere clamshell arrangement. On the other hand, the square is more elegant, so after this design I stopped using a rectangle unless absolutely necessary.
Crease pattern for Icosahedron
I discovered an interesting shape: an octahedron made of three regular pentagons and five equilateral triangles. I've never seen this shape before and don't know if it has a name, so I'm calling it the Tri-Pent Octahedron. It has the interesting property that if you replace each pentagon with five equilateral triangles you get an icosahedron. Several other shapes have this property, including the pentagonal antiprism and a pentagonal pyramid. There's another shape with 15 triangles and one pentagon, and yet another still with two pentagons and 10 triangles, but it's not an antiprism because the pentagons are not in parallel planes. Here are CP's for the complete set of shapes "Icosahedra with One or More Slices Cut Out of Them, Who's Faces are Composed of Pentagons and Triangles."
Crease pattern for Tri-Pent Octahedron
Crease pattern for Tri-Pent Dodecahedron I
Crease pattern for Tri-Pent Dodecahedron II
Crease pattern for Tri-Pent Heptadecahedron
Crease pattern for Tri-Pent Hexahedron
In an effort to complete a Periodic Table of Polyhedra in origami, I have begun to concentrate on producing all of the lower polyhedra that I have not seen created elsewhere. There are two unique pentahedra (polyhedra having five sides). One is a pyramid with a square base, and the other is the triangular prism. The one shown here is a "regular" triangular prism, having all edges of unit lengths and faces of regular polygons, and only a single type of vertex. I not sure why is it not considered an Archimedean solid; I guess because it bears no particular relationship to any Pythagorean solid. Folded from a single square sheet.
Another pentahedron, the faces of this solid include a square, two equilateral triangles, and two trapezoids whose long edge is twice as long as the other three edges. It has the interesting property of being able to form a tetrahedron when joined to another of the same shape by the square face. Folded from a single square sheet..
This intriguing hexahedron has as faces two equilateral triangles, two rhombi, and two trapezoids, and it has the interesting property that it can be used as a module to tile space. I do not know if it has a formal name among mathematicians, so I'm calling is a Nugget. Folded from a single square sheet.