## Zing Origami - More Polyhedra |

Stellated Dodecahedra
*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.

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.

Icosahedral Slices (Tri-Pent Polyhedra)
*2001*

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, Whose Faces are Composed of Pentagons and Triangles."

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, Whose Faces are Composed of Pentagons and Triangles."

Crease pattern for: Tri-Pent Octahedron .
Tri-Pent Dodecahedron I .
Tri-Pent Dodecahedron II .
Tri-Pent Heptadecahedron .
Tri-Pent Hexahedron

Stellated Rhombic Dodecahedron
*2004*

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.

Also called Three Intersecting Octahedra, or the TriOcathedron, this polyhedron sits atop one of the towers in M.C. Escher's

Truncated Tetrahedron
*2003*

This eight-sided Archimedean solid has four regular hexagons and four equilateral triangles as faces. My model is folded from a single square sheet. Very neat.

This eight-sided Archimedean solid has four regular hexagons and four equilateral triangles as faces. My model is folded from a single square sheet. Very neat.

Pentagonal Pyramid
*2003*

One of only a handful of unique hexahedra, this is a very beautiful shape, whose construction is replete with the golden ratio. Way cool.

One of only a handful of unique hexahedra, this is a very beautiful shape, whose construction is replete with the golden ratio. Way cool.

Half Tetrahedron
*2002*

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.

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.

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.

Triangular Prism
*2002*

In an effort to complete a*Periodic Table of Polyhedra* in origami, I began to by producing the lower polyhedra. There are two unique pentahedra. 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.

In an effort to complete a

Dual Tetrahedron
*2001*

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.

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.

Icosahedron
*2001*

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.

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.

Cuboctahedron
*2001*

This fourteen-sided Archimedean solid is one of my favorite shapes, and closely related to the Truncated Octahedron, but easier to fold. One of my first successful polyhedra designs.

This fourteen-sided Archimedean solid is one of my favorite shapes, and closely related to the Truncated Octahedron, but easier to fold. One of my first successful polyhedra designs.