Mathemagical Pixie Castles

Four different Pixie castles and an accompanying wall can be designed using the Customizer:

—PixieEye Castle
The PixieEye has a hexagonal pattern on top, inspired by a pattern often found i nature. As you are reading this, you are actually seeing through just such a hexagonal pattern, hence the name of the castle. On the rear side of our cornea, we have a monolayer of cells in exactly this pattern. The pattern is of course also popular among bees. The reason may be that it is the 2D structure with the least amount of wall relative to the size of the cells, and it is also the structure with the least amount of stress on the wall junctions.
[http://www.nature.com/news/how-honeycombs-can-build-themselves-1.13398]

—PixieIce Castle
The pattern on the PixieIce castle resemble a snowflake and is constructed with a fractal known as the Koch curve. The triangle in the middle is the initial structure, and surrounding that is the first iteration, and the outermost pattern is the second iteration. In every iteration, each line of the previous structure, is replaced with a new line that has a triangular kink in the middle. When the number of iterations increase, a pattern is formed, that is repeated within the pattern itself. This type of self similar pattern, or fractals, are often found in nature.
[https://en.wikipedia.org/wiki/Koch_snowflake]

—PixieShell Castle
The Nautilus mollusc is famous for its impressive shell. The shell can be perfectly described with a logarithmic spiral, and such a spiral is found on top of this castle. As the spiral turns, the distance from the center to the spiral increases exponentially. This pattern is found in the smallest biological systems, in weather systems like hurricanes and all the way up to the arms of galaxies.
[http://www.daviddarling.info/encyclopedia/L/logarithmic_spiral.html]

—PixieFlower Castle
Many have wondered at the pattern in the sunflower and spotting spirals form in many directions and angles. The pattern is know as the Fibonacci pattern, and is a pattern that very efficiently pack the small flowers (florets) that make up the large familiar sunflower. The pattern is formed by placing the small florets along a spiral, every time 137.5 degrees has passed. If this value is changed even by a fraction of a degree, a totally different pattern is formed. This value can be changed in the Customizer, and different patterns can be made.
[http://algorithmicbotany.org/papers/abop/abop-ch4.pdf]

—Castle shape:
The castles can be printed with walls at different angles, so that castles of different polygonal shapes can be made. Whenever the kids knock out the sand of the mold, they learn or are reminded of the angle required to build the current polygonal shape, as it printed in the bottom of the sand molds. There are also options for a castle with two, three or four walls for a line, tee or cross construction.

—Lanterns
All castles and the wall can be printed as hollow lanterns when using transparent filament. Just choose Lantern in Customizer. If they are designed larger than 38mm an LED tea light can fit inside. They can of course also be printed in solid colors, and used in board games, landscape models or desktop ornaments. When printed as lanterns, the walls are vertical whereas when printed as molds, the walls are slightly sloping, to easier get the sand out.

—Printing
A lot of effort was put into making the molds easy to print without any need for support or special angles. The sand molds can simply be printed upside down, and the lanterns right way up. All test and prototype prints were made by first uploading the design to Thingiverse, and then configuring the print in Customizer before downloading them again for printing, to make sure all settings worked as intended. A minimum of 15% infill gives the best results. All prints in the pictures were made with 0.2mm layers and an 0.5mm nozzle.

—Castle building in sand
Making the molds easy to print was only half of the job, as the real purpose was to make cool sand castles. Here I had great help from my son, who made lots of castles, and found the weak parts of the design. We found the best way to get nice sharp sand castles, is to repeatedly throw small amounts of sand into the bottom of the mold, until it is full. Not until the mold is full should gentle pressure be applied to the sand. Scrape off any excess sand, and firmly knock down the castle mold. We also found out that you do not need a beach to play with these sand molds. We just bought a bag of fine sand at our local hardware store, and put a piece of wood on our garden table.

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