On a very long train ride from Buffalo to Albuquerque, I ended up chatting with a rapper on his way back to LA. We both had a lot a hours to fill so I showed him some of my polygon programs. Nearly every time I play with TileLand I end up creating something new (at least to me). This time was no different as you can see from the picture on the left.
This pattern with pentagons and triangles started out as a big 30-sided polygon (constructed by alternating pentagons and triangles) and slowly turned into a triangular grid of green triangles that has been augmented with zig-zag spacers that resemble purple and red dresses.
I'm not yet finished with playing with the fruits of this play. I'm going to attempt to make something different with the E shaped white space. The interesting thing about this E is that it can be completely surrounded by polygons by inserting another pentagon where you have the two kissing red triangles. Check back in a couple of days and I should have the new pattern completed.
Here is a blurred version to use as a background. One of these days I'll have to eventually address the idea of colours--but that day is not here yet. :)
I was about to progam a PolygonR&D program to compute with Fibonacci codings but I got distracted with an old pattern. I get so sucked in with all things pentagon. I wonder if there is some occult connection... pentagram etc. Kidding.
OK, here's my procrastination pattern. It's related to leaves. I thought I had a similar pattern posted but I couldn't find it (I probably should have a better index of images). Each square in leaves is replaced with a pair of triangles. The colouring adds a little bit of complexity as well but nothing too crazy.
With a small modification I'll post the pattern that I thought I had previously posted.... If you are the one person following this blog and happen across this pattern, like me know.
Here is the design of the rhombic dodecahedron that I promised for a long time. Since it's tricky to follow I have a diagram below that may help you create it. The structure probably looks best when you use one to two inch bubbles rather than the longer ones I used in the video.
Here is the psuedo-gauss code for this design and the pattern. Probably I need to include a sequence of diagrams to really make this dodecahedron. I find these useful too. The diagrams perhaps are more useful in the design phase rather than the construction phase....
The * is there to indicate a bit of a tricky spot. For these you must pull the balloon through a square. It's not that tough on the balloon but it is important to know that it must come through the square an not just wrap around--the geometry changes when the balloon is complete and it would come apart otherwise.
Also note that 0 is mentioned but it is not in the diagram. The reason for this is that it would clutter up the diagram (it would turn the graph into a non-planar graph). The 0 point represents the centre of the dodecahedron.
It's been a while since I made a zigzag pattern (a post for that matter). This octagon grid interlaced with a zigzag of triangles and heptagons leaves very small holes or gaps. You can see the alternation of the colours and polygons by following a loop around one of the holes. Notice that there are two types of holes: one that goes up into the right and another that goes up and to the left. These are mirror images of each other.
The slight skew produced by the heptagon-triangle spacers makes the pattern shift slightly up. Perhaps I should rotate the image a little so that pattern can be used as wallpaper....
Hmmmm probably I should make PolygonR&D have a save as background options. So many ideas so little time.
I originally made this pattern with TileLand but later switched to PolygonR&D. I've included the program below.
zig { sides 3 paint black sides 7 paint olive sides 8 2 right ifon none { paint gray create zag 4 right create zag } { } }
zag { sides 3 paint olive left sides 7 paint black left sides 8 2 right ifon none { paint green create zig 4 right create zig } { } }
I'll have to make my next pattern with brighter colours. Perhaps I'll return to some winding and unwinding patterns...
The dodecahedron is a tricky structure to make out of a small number of balloons. There are twenty degree 3 vertices and thirty edges. For symmetry, we need the number of balloons to divide the number of edges but as well figure out how to deal with the odd degree vertices. I have designed a 2-balloon dodecahedron but it was not very aesthetically pleasing. The design I currently prefer is a 3-balloon design. The trick was to instead of using all the ends of the balloons to each take care of an odd degree vertex leaving fourteen (which is not divisible by three), was to have two vertices where three ends come together leaving eighteen vertices of degree 3 (which is divisible by three).
In this design, I use nine doubled edges. The doubled edges stabilize the shape as well as locking the balloons local geometry. To make the weaving of the balloons easier, the doubling of the edges are delayed until the other balloons are in place. This delay reduces the tension on the twisting.
Below is a video of the construction of a dodecahedron.
Here is the psuedo-gauss code for this design.
1a 2 5 2d 6 12 6e 7b 14 15e 16 17 16 20 1a 3 7b 3 8 15 8e 9c 17 18e 19 11 19 20 1a 4 9c 10 18e 10 5d 11 12e 13 14 13 20
* the superscript means a delayed connection. For example in the first balloon, the first line, the sequence 2 5 2d 6 12 6e 7b can be created all at once as a string of 7 segments like a string of sausages. Even though the 2 is repeated, it is not joined until the third balloon reaches the d (see the 5d). Then the two segment 2-5 and 5-2d can be twisted together to form a double edge.
The graph has a three-fold rotational symmetry (C3). To keep the symmetry of the graph, node 20 has three copies but really it is a single node at infinity.
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.
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 5a 0b 2 6 7 8 6 5 8 4a 0b 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:
