Airport luggage cones

Found these beauties at the airport. My theory is that they have something to do with luggage transport.

Wintry Moiré

I saw an awesome Moiré pattern, formed from two perforated sheets of metal at slightly different distances from the camera. This gives a slight size difference, and therefore a slight frequency mismatch for the two sheets. It's nice how the original hexagonal pattern of the holes is repeated, hugely, in the Moiré pattern (much nicer than the boring examples on Wikipedia)!

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.

Five-sided Christmas Star Pastries

This is my version of a julstjärna, a traditional scandinavian Christmas pastry, which always has four arms. I wanted to expand the concept a bit.



Here is how to make it! The traditional four-sided julstjärna is made from a square piece of dough, so our five-sided ones are made from pentagons. A pentagon is of course a much less optimal shape for cutting up the dough sheet, but this is compensated for by the sublime beauty of the finished product!

Cut from each corner towards the middle, add a blob of plum jam. Fold one corner from each flap over the middle of the pastry. Adding some water to the surface as glue, squeeze until the folded corners stick together.

Girih Flower

The fivefold rotational symmetry of Girih tiles is just the thing for creating nice flowery shapes. Happy birthday Fredrik!

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.

WiFi screen

We live on the edge of wireless reception. Even with an external antenna in the window, the signal was not quite strong enough and the connection was frequently dropped. Our friend Halza had the idea to use a strainer as a reflector. This trick works great for us too, and improves the reception considerably.
The WLAN antenna inside the strainer is one of these.

The WLAN signals are in the 2.4 GHz band, which means that one wave length is 12.5 cm. Already a reflecting plane, placed 1/4 wavelength (3.1 cm) behind the antenna should provide constructive interference and some signal gain. A parabolic reflector with the antenna in the focal point provides more. The strainer not exactly a parabola, but has the right general looks.

Lots of variants on this idea are available.

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?

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.

Red flower cane beads

A flower cane made from the magenta and orange Fimo polymer clay from this set. The center is made of scraps from that checkered cane. First, I made some simple flat round beads by cutting slices from the reduced cane.


I then reduced the cane further, and put thin slices on a white clay sphere, made from half a Fimo strip. Twelve 1 cm slices fit covered the sphere nicely, in a dodecahedral formation.


Quarter-strip spheres were covered by six slices (cubic formation) and half-strip cylinders carried two rows of four slices. Leftover flower canes I stacked and rolled until they were tiny, wrapped them in white and cut into small flat beads.

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.

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.

Striped Blue Beads

An experiment with making 'length-wise' striped beads of Fimo polymer clay. The transparent greenish stripe is 1 part transparent white, blue, and green, correspondingly.

Another set of beads - dark sparkly green-blue with some pink mixed in.

Bonus - wave patterns forming in the sink when sanding these blue beads.

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.

Salmiak

Salmiak-shaped candies of, well, salmiak. They make nice mosaic blocks as well.  Unfortunately the angles are slightly off: I thought three fat or six thin corners would fit together, that would have given more pattern options.

Bubbles II

A raft of bubbles observed one day while doing the dishes. For some reason, many bubbles of equal size appeared. Here they have arranged themselves in a crystal. A few impurities (bubbles with a different size) cause crystal defects.

Bubbles have actually been used to study and visualize the physics of crystals. Here some videos, and an early article about the technique by Bragg.

Street Things Belgium, part three

Part of the Atomium.