Nut Domains

The square nuts from our Meccano set. Pushed together randomly, they form nice orientation domains - little areas where each square lines up with its neighbors.  Gaps open up between these domains, since they are not  lined up with each other.

The pattern reminds me of a city map, where rectangular buildings on a block are lined up, and then at a bigger street, the angle changes.

Favorite Games: Nethack

Nethack A classic. You are an @-shaped hero, and your goal is to explore a dungeon filled with monsters (mostly letters) and treasures (gold $ and gems *) and other objects. I haven't played this one seriously for a while - it is much nicer to play on a full keyboard with a number pad, and I seem to usually sit at a laptop. This is a very addictive game, once you get past the initial steep learning curve. For this post we wanted a couple of screenshots, and once we started playing we got hooked again. We even learned to play without the number pad.

While exploring the dungeon one finds weapons (, armor [, and spell books +, and gets tougher by gaining experience and by learning new skills. Nethack is fun because it is so unpredictable. The ascii graphics is very simple, but the world of Nethack is complex. Monsters can interact with each other and with the world around them. You never know what will happen next, and the text description of the events in the dungeon forms an unpredictable story - sometimes a completely absurd fairy tale.

I grew up playing Nethack on DOS, and liked the way the map is drawn there with full lines for the walls. It is tricky to get this graphic mode to work in a unix terminal, for some reasons connected with different character sets. Anyway, there are several solutions. I installed the program konwert, and ran Nethack in this way on one of the text mode consoles (control+alt+f1):

nethack | konwert cp437-utf8

Then simply enabling IBMgraphics in the in-game options make the graphics look as it was meant to.



Taking screen shots of a text mode terminal is something I have never needed to do before, but it is possible with the snapscreenshot program. I ran it in this way, in a text-mode console, with Nethack running in console 1:

sudo snapscreenshot -f8 -c1 -x1 > screenshot1.tga

In an X-term window snapscreenshot does not work.

The Favorite Game series

• OpenTTD a.k.a. the Train Game
• Nethack
• Widelands

Circle Stickers - Lots of Dots

One great thing about Germany is the abundance of small, inexpensive round paper stickers - supposed to be used for archiving, I guess, but in my experience, they're also great for super-fast artwork creation, and for arranging nicely on a table.





On a side note, this company has a very impressive sticker repertoire. I haven't yet seen a shop that would sell all of these, not even in Germany.

Not a sixty degrees angle

The previous kind of salmiak wasn't, and this one isn't - maybe it just shouldn't be, a sixty degrees angle on the salmiak rhombus. Too bad! But at least it leaves a nice star-shaped space where the corners don't fill up in the above pattern.

Botany

Yesterday, I finished working on an idea for I had for learning to use Blender, an open-source program for making three-dimensional models. The result is called Botany, it's also on my art page.

Basically, I wanted to make something that utilizes and shows off the flat surfaces used in 3D modelling. Usually, one tries to hide away the sharp edges and use tons of triangles and effects to make it look organic. The other main idea was to make an image that's supposed to be two-dimensional, and not a snapshot of a 3D scene (which it is, too, nevertheless).

The idea came from designing boxes for the platform game. The thought is to draw simple sectors on a square, and shade them in a way that's compatible with a three-dimensional interpretation.
I thought up as many of these sectored squares as I could, and constructed and arranged them in Blender. The main part of the work was placing the nodes and connecting the right ones, to form the geometry I wanted.

As a bonus, I took some landscape snapshots as well. I used Blender's depth-of-field feature, which I probably don't know enough about, but as you can see, the result isn't that great on edges with a high contrast.

Decagon Girih Solutions

In an earlier post, I wrote about two solutions for the pattern on the ten-sided Girih tile.

In the middle is the standard one, also in paper in Science. The one on the right, with straight lines, I saw in a post on Robodino about laser cutting Girih tiles. The one on the left I might have seen somewhere, or made up myself... Anyway, these are three different solutions with tenfold rotational symmetry.

Wikipedia says: "Most tiles have a unique pattern of girih inside the tile which are continuous and follow the symmetry of the tile. However, the decagon has two possible girih patterns one of which has only fivefold rather than tenfold rotational symmetry." - but it doesn't say which ones they mean.



After some playing around, I realized that there are all kinds of ways to connect the patterns while still (I think) following the rules. The ones above have only a twofold rotational symmetry. These are probably not the only ones, but with their low symmetry, they are not the most interesting...

Instead, I'm really happy about these! They all have fivefold rotational symmetry. The upper left one is the one we cut in acrylic, and the rest are new. The lower left one might not quite conform, since it has another type of crossing in the middle, but who cares? It's pretty!

All in all, these are eleven possible girih patterns for the decagon. Could Wikipedia be wrong on this?

Bricks in the Wall

Memory game favourites

Beaded bowl surfaces

Hexagons side by side forms a flat surface, with zero curvature. Reading Make Magazine's Math Monday, I learned that a pentagon among the hexagons makes the surface  curvature positive, like the surface of a sphere. A heptagon does the opposite - it creates a saddle surface, which has a negative curvature.

Here, I've built the same bowl-shaped trial surface from white glass beads and from Magnetic spheres. The surface is formed from hexagons, with a pentagon in each 'corner', to make the surface curve.

In an earlier post, I used only pentagons, which shapes the surface into a sphere.

Girih tile math

All angles that appear in the girih tiles are multiples of 36 degrees, an angle that appears in a regular decagon. The girih pieces are strongly related to Penrose tiles: each girih tile can be decomposed into dart and kite Penrose tiles. The Penrose tiles are famous for creating aperiodic tilings, patterns that do not repeat themselves. Aperiodic patterns are possible to create with the girih tiles as well - for example, girih may be laid in a pattern of fivefold rotational symmetry. Fivefold symmetry is impossible in periodic tilings.

Each girih tile can be constructed of smaller girih tiles. The Penrose tiles have the same property. If this subdivision is repeated, it may lead to an aperiodic tiling - depending on the rules for replacing large tiles with smaller ones. More on this in this article by Raymond Tennant (pdf).

From the paper in Science.

We first heard about these tiles in a paper in Science (here without subscription)

We wanted to give our pieces some jigsaw-puzzle-like tabs to keep the tiles aligned when building, but the pentagon creates a parity problem. Instead we drew a zigzag shape on each side. Now all the sides are identical - no parity problems - and the sides align nicely.

The knot pattern on each piece is two straight lines in from the middle of each edge, at 54 degree angles. Where these lines meet inside the tile, they are joined. For the other tiles, the rules are unambiguous, but for the ten-sided pieces, Wikipedia mentions that there should be two solutions (but all pictures I've seen show only one) Well, after some thinking, we found another solution (probably 'the' other solution), so we're happy to show our ten-sided pieces with their different knot patterns.

EDIT: Some more decagon solutions.



The girih drawings for laser cutting and more pictures of the tiles and of the laser cutting process.

File for laser cutting girih tiles

The laser cutter reads vector graphics; a red line means 'cut' and a black surface means 'engrave'. I made an svg file with the girih tiles placed side by side. You can download the file, visit your local Fab Lab, and make your own girih tiles! There is some room for improvement in the file - each side is cut twice, which is a waste of time and possibly burns the acrylic more than necessary. This file works fine, but it would be even better if one would remove those double lines.


View Fab Labs on Earth in a larger map

Contents of the file:



At the sides of each piece, there is a 'teeth' pattern, which I put there to make the pieces align better. Another 'innovation' is the double black line that forms the outline of a rope tied in an infinite knot, with a crossing at the sides of each girih piece. It turned out that it is possible to design the tiles so that the rope regularly passes above, then below, then above... for any pattern that one builds with them.

More pictures of the tiles and of the laser cutting process.

Laser cutting Girih tiles

The laser cutter at Fab Lab Groningen.


The machine can both engrave and cut. It is almost magical to see one's design gradually appear as a physical object. Here the laser is cutting our girih tiles from a 3 mm acrylic sheet. The machine does the engraving first, one sees the knot pattern formed by the pieces appear. This is how girih patterns typically look when they are used for decoration, you see the knot pattern but not the borders between the pieces. Then the pieces are cut. The cut lines are quite different from the lines drawn on the tiles. Probably this is part of the reason for the complexity and beauty of girih patterns.


I find the Fab Lab concept fantastic, giving anyone the chance to use this kind of professional fabrication machines. They had 3D printers and a CNC mill as well. Not to mention the nice people at the Fab Lab, guiding me through the process of using the laser cutter!

Someone else also made a set of  laser cut girih tiles, at the Fab Lab in Lille. Some more pictures of our tiles, and the svg file for the laser cutter.

Platform game, level 4

After many months, we finally have a new version of the platform game! As always, it comes with a new level.

Download the game here: for Windows, Linux(32) and Linux(64). See these instructions for installing it on different platforms.

Acrylic Girih Tiles

We stumbled across the local Fablab on our holiday in Groningen, the Netherlands. We wanted to try their laser cutter, so I designed some girih tiles in Inkscape. These things have so many wonderful features, so more posts to are sure to follow - the laser cutting process, and the svg file for the laser cutting machine.



A nice collection of girih cut out of paper.

The New Grey

Since I had almost no grey Lego bricks in my old collection, I ordered some new ones from Pick-A-Brick. I realized that the color was different - the new grey bricks, on the left, make the old ones on the right look even older and dirtier than they are. According to Brickipedia, this new color, bley, has some blue in it and replaced the old one in 2004. A pity, since the old grey bricks are among the most stylish objects I know.

UPDATE. Fascinated by this shape, I made a painting of it.

Lego Storage

Organizing Lego bricks has always been something of a problem, if you're inclined to categorize. We took a clear plastic lidded box, a SmartStore Classic 31, and cut clear plastic sheets into the proper shape, to divide the box into sections. The sheets are polycarbonate Lexan sheets, 1.5 mms thick, just thin enough to be cuttable with household scissors. Five sheets were used in one direction, giving one compartment for each of the six traditional Lego colors. Compartment width was chosen to match the relative frequency of each color.



Where the sheets cross each other, we cut out a thin rectangle from both sheets, from the top to the center in one sheet, and from the bottom to the center in the other, to form a cross halving joint. It was helpful to drill a 2 mm hole at the inner edge of the rectangle, and then make two parallel cuts from the side to the hole. We fastened the plastic sheets with hot glue to the box. One long sheet was placed perpendicular to the six shorter ones, to stabilize and to separate thin and thick bricks.



Here is the box with the divisions in place and filled with our supply of bricks.

Dot grid beads

Attempting to make Fimo clay beads with a simple repeating pattern. They turned out to serve as illustration of how reduction distorts a pattern. I thought I was using a lot of dark brown buffer clay around the gray dot, but in the first step (4 dots) we already see the dots beginning to square up. When reducing the 4-dot cane, and forming the 16-dot cane, the corners have escaped further into the corners... Possible remedies: more buffer, and perhaps making the canes on a larger scale in the first place.


The ends of the dot grid cane contained patterns which are perhaps more interesting than the actual dot grid. Made some round beads with slices on the surface.

Noncanonical

New Legos

From the wonderful Pick a Brick machine. Adds to the old stash.

Magnetic Dodecahedron

Magnetic spheres forming a dodecahedron. Made from twelve five-sphere rings. The rings turn into five-sided polygons when placed side by side in the dodecahedron - this is seen with a flux detector (right).

 A larger one, made from ten-sphere rings.

The largest one I could make with my set, with fifteen-sphere rings. The dodecahedron was still surprisingly stable, but softer, like an over-ripe orange.