What are blurred images good for?
Are they just there to cover up blemishes of reality or the lack of skills of the photographer?
Or is having more information always better? Shouldn’t at least something be in focus, always?
Or better, everything, with absolute clarity, so that nothing is hidden and no question remains?
Sometimes, I think, it is necessary to reconsider everything.
Whenever you show a mathematician two examples, s(he) wants to know them all. So, after the introductory examples of Butterfly and Fractal it’s time to make something more complicated. Jiangmei and I started by classifying all possible vertex types that can occur when you build polyhedra using only translations of four of the six types of faces of the rhombic dodecahedron (and make sure they attach to each other as they do it there). We found 14 different ones, and a particularly intriguing one is what we called the X:
The central vertex has valency 8, and we were wondering whether we could use it to build a triply periodic bifoldable polyhedron. It is easy to combine two such Xs to a Double X:
One can then put a second such Double X (with the order of the Xs switched) in front. Note that these are still polyhedra. Below are two deformation states of these quadruple Xs. We see that they are quite different.
So far, the construction can be periodically continued up/down and forward/backward. It is also possible to extend to the left/right, and there are in fact two such possibilities, allowing for infinite variations, because one has this choice for every left/right extension. They are indicated by the arrays below.
If you don’t have the time to build your own model, here again is a movie showing the unfolding/folding of a rotating Dos Equis.
The canyonesque nature of the Fern Canyon State Park makes one forget that this place is next to the coast, which empasizes the contrast between the complexity of the ferns and the simplicity of the beachscape.
This is a place to enjoy the lack of presence, which has dissolved into a beautiful grainy gray.
Time passes gently,
and allows for modest reflection.
The second bifoldable object Jiangmei showed me was this:
You can find a movie showing how this folds together in two ways here. To understand how and why this works, let’s first look at a simple saddle:
This is a polyhedron with a non-planar 8-gon as boundary. Its faces are precisely the four types of faces that are allowed in our polyhedra: All others have to be parallel to these four. The four edges that meet at the center of this saddle constitute the star I talked about the last time. Again, all edges that can occur must be parallel to one of these four. One can fold the saddle by moving the upwards pointing star edges further up (or down), and the downwards pointing edges further down (or up), thereby keeping the faces congruent. This works locally everywhere and therefore allows a global folding of anything built that way.
For instance, the hollow rhombic dodecahedron above can be bi-folded. Now note that this piece is also a polyhedron with boundary. In fact, its boundary is exactly the same octagon as the boundary of the saddle.
Observe also that at the center of this piece we have a vertex in saddle form. This suggests to subdivide all rhombi into four smaller rhombi, remove the saddle an the middle vertex of the doubled hollow dodecahedron, and replace it by a copy of the standard hollow dodecahedron. This gives you Jiangmei’s fractal. Repeating this is now easy. Below is the generation 2 fractal (animation):
And, just for fun, the generation 10 fractal:
You can see it being bifolded here. So far, the two completely folded states of our polyhedra looked very much the same. We will see next week that this doesn’t need to be the case.
After the brief excursion to Northern California, we are back to Indiana. This state also has a Fern dedicated nature preserve, the Fern Cliff Nature Preserve, located a little west of Greencastle. Things are a little different, though.
We do have a lot spooky cliff and very little path, mostly dead ends.
But this means one can explore, and the patient visitor will find ferns that mingle with liverworts.
I took these pictures 9 years ago, but I don’t expect that much has changed.
A little bit off the (beaten) path are some views of the sandstone cliffs I really liked:
I will pay this place another visit as soon as the eternal heat wave ends.
We all know that cardboard cubes are rigid, which is why we get our packages in boxes. We also all know that if we remove two opposite faces from a cube, we can fold it together. This started to interest me when I noticed that the polyhedral approximation of the Schwarz P surface is surprisingly flexible. This summer, I showed this to our local Origami and Paper Folding expert, Jiangmei Wu from our School of Art and Design, and she became interested. A few days later she came with a paper model that looked like this:
She called it a simple variation of the polyhedral P-surface. Hmm. This is a triply periodic polyhedral surfaces tiled with rhombi. To understand it, we build it out of smaller units (which we called butterflies):
The really cool thing about it is that it can be folded together in two different ways, like so:
You can find an animation showing the continuous deformation here. We stared at this for a (long) while, until we realized that this has to do with rhombic dodecahedra. The structures up above are composed of the rhomboids from last week that tile a rhombic dodecahedron. The latter has, as the name hints, 12 faces, which occur in opposite pairs. Like the cube, it is rigid per se, but becomes foldable if we remove two pairs of parallel faces, leaving us with four faces to use, which are distinguished by color up above.
Above you can see the four hollow parallelepipeds (which we called hollowpeds). The almost trivial but nevertheless mind bending realization is that everything you build out of these hollowpeds becomes a structure foldable in two different ways. Next week I’ll show Jiangmei’s second model, a foldable fractal… If you can’t wait, check out this.
After the modest ferns from last week, let’s indulge. One of the places to be is the Fern Canyon in the Prairie Creek Redwoods State Park, which lies, alas, in Northern California.
Most people who make it into the coastal redwoods that far north don’t bother to take the windy detour to that state park and trail head. Here is what they miss.
The vertical walls of the short canyon are packed with ferns.
Their gentle motion is impossible to capture in a photograph.
At some places, they appear to float before a darker background.
After seeing this, you will keep dreaming of a house with walls like this.
Last week we learned how Rototiler moves can unpack a cube. As a warmup, below are the moves for a 2x2x2 cube projected parallel along a cube diagonal onto hexagons:
We start at the left, remove the frontmost cube, and keep going. The solution is far from being unique, but not too complicated. Today, we do the same with a hypercube. The projection of a 1x1x1x1 into 3-space along a main diagonal is a rhombic dodecahedron, tiled by four rhomboids. These rhombic dodecahedra have 8 obtuse, 3-valent vertices at the corners of a cube, and 6 acute, 4-valent vertices at the corners of an octahedron.
There are two ways to tile the rhombic dodecahedron with these rhomboids, and changing one to the other corresponds to a rototiler move in space. Let’s do this with the 2x2x2x2 hypercube, whose projection is a rhombic dodecahedron tiled by 32 rhomboids.
At first it seems as if there is no swappable rhombic dodecahedron available, but if we remove three rhomboids and look inside (which is the direction of the fourth dimension, after all), we can see it. After swapping it, we also remove the frontmost rhomboid of the swapped dodecahedron.
We then see that the four removed rhomboids together make up another swappable dodecahedron. We replace it by its swap. The same can be done at three other places.
The next thing to do is to swap 6 more dodecahedra. One of them is the one which shows yellow and purple rhomboids in the right figure above, sitting between the red and blue “vertex”. All these dodecahedra correspond to the edges of the tetrahedron whose vertices are the already swapped four peripheral dodecahedra. Doing these six swaps leads to a tiling very much like the one above to the right, where now the other four obtuse vertices mark swappable dodecahedra. Swapping these and finally the hidden central dodecahedron completely unpacks the hypercube. It took us 1+4+6+4+6+1 = 32 swaps, as expected.
Next week we’ll see what this is good for…