The Wooden Torus

Sept, 2011. My wife's birthday was approaching, and I decided to create a piece of desk art for her. Since she runs a math tutoring center, I chose something mathematical. For a while I've been interested in creating wooden shapes on a laser cutter that can be assembled using slots. After some doodling of ideas I decided to make a wooden torus which finally came out like this (click art images for larger versions):

This page describes how I designed and made it and presents the DXF and Corel draw files.

Design

Making a 3D torus at some point requires being able to create the geometry, so I started with the equation for a torus in X,Y,Z coordinates: (x^2+y^2+z^2+R^2-r^2)^2=4R^2(x^2+y^2) where r is the radius of a cross section of the tube and R is the radius that tue tube makes. Since I wanted cross sections, I chose to walk along fixed x values for each slice. Then for a fixed xi, solve for y to get the top and bottom curve equations. I wrote a custom C#/WPF program to do the work, complete with sliders for various attributes. The number of slices is how many to fit in each direction, and the thickness is for the thickness of the gaps, which corresponds to the thickness of the material I want to use. Here is a screenshot and a tiny piece of the C# code.

 <img alt="Small pieces" height="630" src="Screenshot.png" width="708" class="style1" /><p>
  &nbsp;</p>
 <img alt="Code" height="802" src="Code.png" width="874" class="style1" /><p>
  &nbsp;</p>
 <p>&nbsp;</p>
  <p>I wanted 32 slices for my wife's 32nd birthday, and I wanted to use 
  1/4&quot; wood, since that is about as thick as the laser cutter cuts. I wanted 
  the inner hole to be the size of a tube in case I want to nest some (plus 
  I find that ratio pleasing). I also wanted the final item to be roughly 
  square gridded, meaning the gaps and the solid parts should be pretty 
  close in size. This leads to an outer dimension of about (16*2+1) *1/4 = 
  8.25&quot;, a pretty nice size.</p>
  <p>So the code takes the values from the sliders, walks the corresponding 
  torus in constant x slices, and for each slice uses the equation for z in 
  terms of y and walks y in 1/100&quot; increments, computing a list of z values. 
  The values are stitched together along the top, and the bottom has notches 
  inserted where the corresponding slices should go. The result is cleaned 
  and written out to a DXF file. All this is done in custom code since I 
  don't know how to make such things in any solid package (although someone 
  probably does). <a href="MakeTorusDXF.zip">Here is the zipfile</a> of the 
  VS2010 project files.</p>
  <p>This above values led to the various radii. Tweaking was needed to ensure the outer 
  lips were thick enough to hold the pieces together. Here is the resulting<a href="TorusFinal.dxf"> 
  DXF</a> and <a href="TorusFinal.cdr">Corel Draw</a> file. The DXF is 
  pretty spaced out so you'll want to edit it. The Corel Draw file is for a 
  12&quot;x24&quot; piece of 1/4&quot; wood. Each file creates enough for 1/2 of the final 
  torus, so you'll need to cut two. </p>
  
 </div>

Assembly

Here are some assembly shots.

 

 

 

 

 

<p>To keep the pieces snug, I used wood glue. The gluing of pieces has to be 
done very carefully and in a precise order to ensure all pieces are snug. 
The rough idea is to assemble the shape without glue so the pieces are all 
held in a good position. Then glue the first 4 above since they rest on the 
lowest surface. Once they are dry, glue the top layer of pieces to only 
those 4 cross pieces. Once that is dry, turn over, and glue everything. </p>
  <p>The order is to make sure you don't glue any pieces that are not fully 
  snug and held in place by gravity.</p>

  </div>
  
  <div class="boxed"><h3>Final</h3>

  <p>Here is the final result, and my wife's birthday present.</p>
  <p><a href="Finished1.JPG">
  <img alt="Finished torus" height="480" src="Finished1_Small.jpg" width="640" class="style1" /></a></p>
  <p>&nbsp;</p>
  <p>I made two matching torii, one for each of us. People at work saw my 
  prototype and liked it so I decided I'd like one for my desk also.</p>
  <p><a href="Finished2.JPG">
  <img alt="Finished 2" height="480" src="Finished2_Small.jpg" width="640" class="style1" /></a></p>
	  <p>Here are the two final ones and the initial prototype (the wood of 
	  which is missing parts, is cut funny, etc.).</p>
  <p><a href="Finished3.JPG">
  <img alt="Finished 3" height="480" src="Finished3_Small.jpg" width="640" class="style1" /></a></p>
  <p>&nbsp;</p>
  <p>That is all. I hope you enjoyed this, and happy hacking!&nbsp;&nbsp;</p>


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