It is always snowing over Riemann-Lebesgue Hill. This snow is likely a bit different from what you see falling in your neighborhood. That snow is formed from the condensation of water vapor into droplets that then freeze into a crystalline seed for more water vapor to freeze onto. Over and over over until, voila! The snowflakes in the Mathemalchemy exhibit are formed using mathematics and lasers.
Snakes on Plane (The Geometric Kind)
First a design is generated by a computer program composed in MATLAB, Mathematica, SAGE or we can have a friendly snake assist us – Python. A nice introduction to generating Koch Snowflakes can be found at Trinket – # Your basic Koch triangle.
This is a nice introduction to Python and iteration with a little help from a turtle. The exhibit has a turtle named Tess on a journey to infinity so it is only fitting that a turtle might advise us to see how mathematics can make this beautiful shape.
It All Starts With A Triangle
Our snowflake designs are generated by Ingrid Daubechies. The examples I am going to work with start with an equilateral triangle as described by Helge von Koch in his 1904 paper “Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire” or in English “On a continuous curve without tangents constructible from elementary geometry“. Notice that French was the international language of the sciences at the time.
I am not certain if he used a straightedge and compass or a ruler but he was able to draw and trisect some very nice lines.
Ingrid supplied me with five iterations of Koch snowflakes which I imported into a free graphics application called Inkscape. It can do just about anything programs you might pay hundreds of dollars for. Lots of helpful videos on YouTube also.
Next I opened the 4th iteration in another instance of Inkscape and “copy pasted” it into the 5 th iteration drawing. After a little resizing I placed it inside the outline of the 5th iteration. Next applied the same process to the 3rd, 2nd and 1st iterations.
In this picture you can see the innermost snowflake is simply an equilateral triangle with smaller equilateral triangles of edge length ⅓ of the original place at the center of those edges.
The same process Koch painstakingly performed in pen and ink.
Snowflakes and Lasers
Ok time for lasers! Laser cutting that is. I use Ponoko because I have found that I can navigate their process well and their prices are reasonable.
Let’s upload a design. Ponoko requires the design to be a scalar vector file. A better name for this might be “Scalable Vector File” because that is the thing it does so well, it preserves the shape of the design throughout enlarging and shrinking.
Here is our snowflake post uploading. Ponoko goes through a few steps like analyzing and finishing up so the first time I did this it took about two minutes. I was concerned I had overtaxed its computing service or broke the internets… nope it just takes some time.
Important steps now are to define which color line does what. I created the outline in blue and the interior in black so that as you can see below blue is for cutting and black is engraving.
After confirming the design it is time to pick the material. We decided on 3 mm or ~⅛” clear plexiglass. Just a few more clicks and this design is ready for cutting. Confirm the material. Ponoko is verbose when it comes to the number of steps. You can also go backwards in the process. Their customer support is quite good also.
Now it’s pretty much the same as any other online shopping site. One gotcha though. Engraved lines do not show up real well in the final stages of their process. As you can see in the pic below. Their customer service assured me they were in fact still there.
OK the order is placed the capacitor banks are charging up in anticipation of big laser action. So now we wait. I could have opted for next day but the surcharge is hefty. I chose five days. Ponoko sends tracking information whether they ship UPS or USPS. Knowing the where and when of package delivery is always nice.
Good Things Come In Large Packages
Sure enough here it is!
The acrylic is covered with paper on both sides with some very tenacious adhesive to protect it from scratching.
Pop out the snowflakes and then it is bath time.
Snowflakes’ Bath Time!
I let them soak for about three hours and paper slid right off. On other orders they have used a brown paper which apparently has more glue. Takes overnight in soapy water for that stuff.
Voila! Part 2
And here we are, a lovely snowflake at the seashore. Beautiful!
However, at one time mathematical functions like this were considered “Monsters!”
A very good read on this topic can be found at MacTutor History of Mathematics Archive at University of St Andrews, Scotland – A History of Fractal Geometry