It is also this time of the year to chase away the dark hours by making presents. As in previous years,
we will make a stellation out of paper without glue. This year, we are going to stellate the Icosidodecahedron, one of the fancier Archimedean solids.
The stellation is quite simple, it is also a compound of the dodecahedron and icosahedron. The simpler compounds of a Platonic solid with its dual are also doable, see the post from two years ago.
To make it, we will need 20 triangles and 12 pentagons, so printing and cutting two of the templates below will do. I suggest to print four templates in two different colors and to make two models.
Then we start by sliding five triangles into one pentagon like so:
Then we add five pentagons between two adjacent triangles.
Next another five triangles:
Now we have finished one half of the model. This already would make a nice dome for the backyard.
You can make another half and try to attach them, but I think it is easier to just keep going.
This next step is a little tricky, because to prevent the polygons from falling out, it is best to add a ring of alternating pentagons and triangles. When done, it looks like this:
The last two steps (add five more triangles and one more pentagon) are then pretty clear, but still tricky because you have to insert the new polygons in four or five slits essentially at once.
There are many good places to contemplate the clashes between old and new in Berlin, and one of them is the area along the Spree near the U-Bahn station Schlesisches Tor. This is where the world ended for people living in West-Berlin while the city was divided. Now one can walk across the bridges and admire the construction circus on both sides.
Herbert George Wells might have thought that his phantasies have come true. When they are done with all this, will it looks like this?
And will we get more playful little sculptures like the Molecule Man by Jonathan Borofsky?
There is some obvious resistance. It feels like the perfection of a finished building is stifling the creativity.
Who wouldn’t want to defend the octagonal brick building below?
Do we really want to lose all this?
My taste is more for blending old and new and let them coexist.
I like architecture, or, to be precise, certain states of buildings. Ruins are fascinating, but even more so construction sites. Both are usually off limits (as are the corresponding states of human affairs, death and conception, unless you are involved one way or the other). So I am often forced to trespass a little.
In this case, as you can see, the door was open, and I just couldn’t resist.
Views like the one above make it instantly clear that we are not on a generic construction site. Somebody with taste has been designing this, and whoever is doing the construction work, is doing an excellent job by creating crystal clear previews of what’s to come.
Wondrous tools are on display too, just for me. I can only guess their purpose by looking at the ornamented concrete slabs. Everything is purposeful, even the occasional leftover tile.
What fascinates be most at places like these is the tension between the clarity of the present and the vagueness of an undefined future.
Berlin has changed a lot since the wall came down in 1989. Most notably the constricted architecture from before finds its counterpoint in buildings that show a liberated sense of what can be done with space.
One of my favorites is the Libeskind addition to the Jewish Museum from 2001.
You can only enter it underground and are confronted immediately with long and slanted corridors.
I felt the natural way to photograph this is by slanting the camera as well. There is a lot of narrow vertical space,
admitting just enough light so that we don’t feel claustrophobic.
Then there are the Voids, most of them inaccessible, but present through views and gaps in our perception.
We lose the distinction of being inside or outside, but we learn that is us who create the space around us.
Today we look at tilings that utilize just the four other squares. The first step in classifying these is again a simplification, making the split corner squares uniformly green. This leaves us with two tiles:
Ignoring the pink triangles for the moemnt, we recognize the problem we solved last time: The green squares need to occur in shifted rows or columns, like in the example below. Here we have four rows of green squares. Rows 1 and 2 are shifted, as are rows 2 and 3, but rows 3 and 4 are alined.
To add the pink triangles, note that two pink triangles fit together to a pink diamond, and each grid cell needs to have one of those, but we can only use those edges that are not already adorned with a green square. This leaved us with the following possibilities: If two consecutive rows of squares are aligned, we have place two diamonds in the square space between four squares, and we can do this horizontally or vertically. This can be done independently of neighboring squares, as shown between the two bottom rows below.
If the rows are shifted, we also have two possibilities to place the diamonds, but each choice affects the entire row, again as show above in the top rows.
Finally, we need to undo the merging of the orange and blue triangles into green squares, and we can do so by splitting each square either way and independently.
Below is an example how teh corresponding polyhedral surfaces will look like. The horizontal squares correspond to the green squares of the tiling. They are the floors and ceilings of rooms that have two opposing walls and two openings. I start seeing applications to randomly generated levels of video games here…
Squaring the circle is easy, you just need to know what you want to do. My personal favorite method is to use elliptic functions defined on rectangular tori to map rectangles to disks, as shown below for a square. These maps don’t preserve area (which is what the Greeks had wanted), but they preserve angles.
I had some leftover architecture images from Columbus and wanted to see how they look when made circular. Here, for instance, is the AT&T building
and this is a circular version:
There are three degrees of freedom one can play with (the dimension of the automorphism group of the hyperbolic plane), which means that one can squeeze parts of the image towards the boundary cirle. Here are two other versions of the same image.
Another favorite of mine is the atrium of the Cummins office building with its wonderfully intricate play with straight lines and black and white.
Now we only have to find architects and builders who create buildings that have these curves in reality.
The little town Holbrook in Arizona offers convenient accommodation after visiting the Petrified Forest National Park. This is not a wealthy town, but the downtown area has its own nostalgic charm. You wonder what life was like here a hundred years ago.
Then you come across this street sign. Choosing a name is a delicate thing. Apparently, in the good old times a saloon shooting ended in such a way that the establishment was renamed the Bucket of Blood Saloon. In the long run, this didn’t help much, and after the building fell apart, the name survived as the street name, to this day.
Other local attractions allude to that bit of the town’s history in appropriate color.
The moral? Appearances change, names stay. But it seems the town hasn’t quite figured out whether that name is a curse or an opportunity.