The Balloon Icosahedron

The icosahedron is another platonic solid that has all of its vertices with odd degree. What makes it a little trickier than the cube is that it has twelve degree 5 vertices and 30 edges (instead of 8 vertices of degree 3 and 12 edges of the cube). To ease in complexity of remembering the balloon design, symmetry in the use of balloons is preferred. Essential this means that each balloon could rotated into the position of an other balloon. Symmetry dictates the number of balloons should divide the number of edges so 3 balloons is a good choice. This would have ten segments per balloon.

With three balloons, we have six ends that can take care of six of the twelve degree 5 vertices. The other six vertices with odd degree can be turned into degree 6 vertices by joining them to an interior point. The video below demonstrates the construction but perhaps the psuedo-gauss code is more explicit.





From the graph of the balloon, you can see that symmetry of the balloons since you can rotate the graph 120 degrees and the shading of the lines (which represent the colour of the balloon) all become swapped. A further symmetry is that it can be inverted from start to end. This describes a D3 symmetry group.

Here is the psuedo-gauss code:

1^z 3 9 10 5_a 6 9_c 8_d 2 6 11 12_z

2^z 1 5_a 11 7_b 8_d 0 5 4_e 3 8 12 10_z

3^z 2 7_b 12 9_c 4_e 0 7 6 1 4 10 11_z

The Balloon Cube

The cube balloon, like the tetrahedron balloon, has the problem of having odd degree vertices. By having two balloons, four of the problem vertices can be dealt with. The other four will be dealt with by going through the interior point once for each balloon turning those vertices into degree 4 vertices. To symmetrically space these degree 4 vertices, I used the vertices so that none share an edge (the points form a tetrahedron).



3 4 1 5_a 0_b 2 6 7 8
6 5 8 4_a 0_b 7 3 2 1

This design thus use two balloons with eight segments each, which allows for large segments for a large cube. Notice that the two internal segments would theoretically be 0.8660 (root 3 over 2) the length of the larger sections but I think in practice it's more like 0.7 because of the bulging properties of balloons. With only 4 internal balloons this design is very sensitive to the ratio of the external and internal balloons. This allows for some interesting patterns that are topologically equivalent but that don't look like a cube.

An alternative design that would ensure a cube look would have all 8 vertices connected to the centre point:

0^z 3 4 1 5_a 0_b 2 6 7 8 0_z
0^z 6 5 8 4_a 0_b 7 3 2 1 0_z

The Balloon Octahedron

The octahedron was the first balloon I ever made. It is by far the easiest one to make since all the vertices have degree 4. There are many one balloon paths that construct the octahedron but the one I describe here is nice because it does not require a balloon to be pulled through a triangle.

To describe this I have the following youTube videos.



Here's the original with a 350 balloon.



The psuedo-gauss code for this is

1 2 3 4 5 2 6 4 1 5 6 3 1

The Balloon Tetrahedron

The first balloon design is this 1-balloon tetrahedron design. It's not the easiest design but it's the easiest shape. The tetrahedron is composed of four triangles but I like to focus on the edges and vertices (I like to think in terms of graph theory). There are six edges and four vertices. The problem with the making the tetrahedron is that the vertices have degree three--this means that every vertex has a problem (see Eulerian path). A balloon comes into a vertex and leaves the vertex which takes care of two of the three edges but the remaining one has to be dealt with. So you can either have the balloon start or end at that vertex (but each balloon only has two ends) or you could cheat. There a number of ways to cheat but the one used here is to join each balloon with an interior point. With this design, each of the vertices now has degree four.



As well as a video of the construction, I'll include a diagram and a sequences of that describes the design similar to Gauss code. This is perhaps more to introduce the method of description rather than to be useful in this case. So here I'll have a planar graph with the vertices labeled but the interior point(0) omitted (which would make it non-planar).



0^z, 1, 2, 3, 1, 4, 2, 0, 4, 3, 0_z

The z superscript indicates that the 0 is a delayed connection until the z subscript.

Balloon polyhedra

Since I haven't been making many polygon patterns I've decided to put my balloon designs up on my blog. I'll try to space them out a bit. These will hopefully end up in a paper soon. Here are the platonic balloons that I have designed. Aside from the tetrahedron, these are a little tricky to design. But for now enjoy the picture. Eventually I'll have a photographer take pictures of these. I'll post the designs soon.

S tiles

What else was I going to do with triangles, squares, and heptagons. This pattern was based on taking a clock-wise loop of ten heptagons (it's like a loop of six pentagons--not very circular) and unraveling it with a clock-wise loop of triangles and a clock-wise loop of squares. That is what makes the octagonal shape that surrounds the "S". So then it's just a matter of a little decoration and tiling. I'm surprised no one has really blasted me on my choice of colours. Bring the flame war on!

X Bug

I'm calling this one X Bug since it is a checker board of X's and Bugs. The pattern was developed by unraveling a loop of six heptagons. Six sets of a square and two triangles were interlaced with the heptagons to create the outline for the bug. The bugs were connected by overlapping the two triangles at the corners. After the checker board of bugs were in place the holes that were left were decorated with a few squares and triangles to make the holes look like X's. This pattern is similar in construction to this pattern and many others. Perhaps the asymmetry (or rotation symmetry) makes it seem more like this pattern.

It's patterns like this one that make me wonder if I need to create another interface for PolygonR&D that incorporates TileLands interface of easily making polygon paths....

Here's close up of one bug. It's the hole on the inside that I consider the bug.