Continuing to explore the potential for icosahedral single-sheet polyhedra, this model manipulates the proportions of Semi-Sunken Icosahedron and uses a grid of alternating hexagons and triangles to achieve a shape like a soccer ball.

After the success of the Stellated Icosahedron Starball, I explored a number of other single-sheet polyhedral models using the icosahedron as a base. The Semi-Sunken Icosahedron seems to arrive at the essence of the series, as it can be folded from a basic 12x12 triangular grid on a hexagonal sheet. The finished object is fascinating to look at, revealing an octahedral geometry at the level of individual faces. In addition, the model is easier to fold and lock is very strong.

The original inspiration for this model was taking a fresh look at one of the forms I was trying to create in my Starball explorations. The key idea was twelve star shapes embedded on a single-sheet polyhedron the resembles a ball or sphere. Early attempts were from a dodecahedron base, using pentagonal symmetry on the folded sheet, but this proved to be very complicated and the stars did not have enough depth to really pop. I was on the plane to CFC3 in Bogota, Colombia, when I reimagined the subject using an icosahedron as a base. Mathematically the model is a stellated icosahedron, but the sunken inner flaps are allowed to open up a little to reveal the star-shaped areas. In addition to capturing the look I was going for, it is much easer to fold than its predecessor, as it develops from a triangular grid on a hexagonal sheet of paper.

I have another single-sheet dodecahedron layout folded from a pentagon that has the center of a face at the center of the paper. That arrangement very symmetrical but has the drawback the model closes with five flaps that pinwheel together in the middle of the opposite face. In a quest to have a reasonably symmetrical model with twelve clean faces, I explored the possibility of having a vertex be at the center of the paper. This model is folded from a decagon rather than a pentagon, since the layout has a tenfold rather than fivefold symmetry.

The shape is two intersecting cubes, and the color change makes each one of the cubes a different color. This means that the color changes need to alternate in a checker or zigzag pattern. The model begins with a 7-by-9 rectangle. It could be folded from a square, but then the first step is to fold two edges under.

The shape is based on the regular dodecahedron, a platonic solid composed of twelve regular pentagons. The stellated version replaces each face with a five-sided pyramid, resulting in a star-like shape composed of sixty triangles. My Stellated Dodecahedron is the result of a long and arduous quest. I made several attempts at it a few years back, but it was beyond my skill at the time, and I only barely managed to make one out of a giant sheet of foil, and it didn't hold together too well. For the new attempt I began with a pentagon sheet of paper rather than a square. When I got to the end I had a nice array of identical flaps that made perfect tabs, and I was able to use a twist-lock to finish the model.

The shape is a complement to the Stellated Dodecahedron. Both are composed of sixty triangles and form star shapes out of sets of coplanar faces. With the Great Dodecahedron, the coplanar faces form a pentagon with the star rising out of the middle in the negative space. This model is a development of my previous version, folded one from a square. The new is folded from a pentagon, and has a refined CP so the corners provide nice flaps and the model goes together well and holds its shape quite strongly.

The shape is a shape based on the regular dodecahedron, a platonic solid composed of twelve regular pentagons. The dimpled version replaces each face with a smaller, inverted pentagon, and each vertex with a sunken pyramid. This shape is equivalent to a semi-sunken variation of the Archimedean solid the Icosidodecahedron. It is also interesting in the the convex edges describe six great circles.

A 3-D evolution my Penfractal Tessellation. Each face of the dodecahedron is subdivided into six smaller pentagons. The model also makes a reasonable representation of the dodecahedral analog of the tesseract, albeit in 3-d hyperbolic space rather than true 4-d space (trust me on this; I looked it up). This model was the beginning of a whole series folded from pentagonal sheets of paper.

The shape is two intersecting cubes. I first saw it in an M. C. Escher print many years ago. The model is folded from a single sheet of paper, a rectangle with a 2:1 ratio. Still, this is only a preliminary study. My goal is to fold it from a square and with a color change so that the two cubes are in contrasting colors. This will require six color change regions. Fortunately I have a method for this, similar to the way I affect the color change in my Stellated Octahedron.

Single-sheet stellated polyhedra are pretty advanced but the color change brings it to a whole 'nuther level of complexity. The first challenge was to work out how to achieve the arrangement of alternating colors. 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 angle is 67.5 degrees, which is 3/4 of 90 degrees and easily derived. It also 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.

An approximation of a sphere. I developed 32- and 72-facet versions. It's a nice layout but difficult to fold well. Fun fact: "Sphere" is Thelonious Monk's middle name.

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.