Sum of Squares

There is a lovely formula for the sum $latex S_n$ of the first $latex n$ square integers.  Namely

$latex S_n = 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n(n+1)(2n + 1)}{6}.$

Sum of Squares

The sum of the first 6 cubes represented as a solid.

 

Imagine the sum $latex S_n $ as the volume of the pyramid of 1 x 1 x 1 cubes with one cube on the top layer, 4 on the next, 9 on the next and so on up to $latex n^2$ cubes on the bottom layer as seen in the figure above. From the above expression, we see that the sum is one sixth the volume of a box with dimensions  $latex n$, $latex n + 1$, and $latex 2n + 1$. So it is at least conceivable that six of these pyramids could be packed into a rectangular volume of that size.  And as the pictures indicate below, there is such an arrangement.

ClockTHREEjr circuit validated

ClockTHREEjr is on its way to reality.  This week we’ve validated the matrix archetecture, finalized the schematic, designed the enclosure, and are nearly complete with the layout!

FlorinC is the project lead on this one.  Anool is doing a bang up job on the hardware design, I’ve been heading up the enclosure design.  Go Go WyoLum!



The First Cut… goes deeper in.

 

After a few tries Shyama and I got the router actually cutting a real part, of our own design!  Ok, granted it was a simple part, but its a start.  The part, a  ClockTHREE stand consists of two end pieces connected by a couple of dowels.  It was cut from 1/4 inch plywood.  I think it would be really nice out of 1″ hardwood and skip the dowels.  Have to see if Angus Hines has any suggestions.

Here is what we learned.  In MeshCam, you can define part supports that hold the part in place for the entire cut.  Because, as we discovered, the router does not work just as soon as it becomes free from the stock material!  It turns out this option is only available from STL generated files.  Our first runs  from bitmaps were doomed from the start.

We also were seeing some strange cuts until we realized that a “ballmill” bit had been selected rather than an “endmill”.  Fixing that also decreased the total cut time.

Here is the tool chain:

  • Python to generate OpenScad code
  • OpenScad generates STL
  • MeshCam generates GCode (ngc)
  • EMC2 executes GCode on Router

It works, but may not be the most efficient.

We have a long way to go, but we are making progress!

MORE pics

ClockTHREE Jr w/ FlorinC!

FlorinC has come up with ClockTHREE Jr! This is a smaller PCB with 112 individually addressable mono LEDs.  Anool and I have agreed to team up with him.  We still need to decide the final architecture.  I think it would be super easy to base it off of ClockTHREE with only one rtcBOB.  Florin has been working with another design.

Should be great fun.

UPDATE:

Florin decided to to go with the ClockTHREE design.  Anool pointed out that rowBOB will actually support 16 rows of LEDs and we only need 8.  He also pointed out the perfect replacement part.  Should save money and be easier to build.  Also, this will be our first board with the new OSHW logo!