Determining the proper angles to cut pyramid faces

I have built a few projects that involved the construction of a hollow pyramid by assembling triangular faces. As the construction progressed, I quickly realized that the proper miter and bevel angles of each pyramid face are not trivial. ie They are not 45*, 22.5*, 60*, or any other common construction number. The reason is that the faces of the pyramid meet each other in a compound angle, thus making it difficult or impossible to cut the pieces without first calculating the miter and bevel using geometry.

I made this diagram in Google SketchUp with the dim_angle.rb plugin (http://www.crai.archi.fr/RubyLibraryDepot/Ruby/em_geo_page.htm) . The angles are as follows:

VA = Vertex angle
SA = Slope angle
HA = Hip angle
DA = Dihedral angle (measured in a plane that is orthogonal to the pyramid's sloping edge)
MGA = Miter gauge angle

When building pyramids, the VA and SA are inputs to the following equations. HA is an intermediate step, and the MGA and DA are outputs. These two output angles indicate how the miter and bevel adjustments must be set on the saw to cut the pyramid faces.

For a pyramid with a square base, the VA is 45*. The SA is chosen by the designer to give the pyramid a desired aspect ratio. High SA values, would make the pyramid very tall relative to the base. For this example, let's choose 60*. The first calculation is for MGA:

VA= 45
SA = 60
MGA = arctan(tan(VA)/cos(SA))
MGA = arctan(tan(45)/cos(60)) = 63.43 (not a "common" angle at all!)

Now, the HA is not really useful for setting the saw, but makes the final equation easier to manage.
HA = arctan(sin(SA)*tan(MGA)*cos(VA))
HA = arctan(sin(60)*tan(63.43)*cos(45)) = 50.77

DA = 2(90-(arctan(cot(VA)*tan(HA)*cos(HA)))
DA = 2(90-(arctan(cot(45)*tan(50.77)*cos(50.77) = 104.48

Depending how your saw measures bevel cuts, the proper bevel angle setting is given by just this part of the equation:

arctan(cot(VA)*tan(HA)*cos(HA))
arctan(cot(45)*tan(50.77)*cos(50.77) = 37.76

So, set your saw's miter gauge to the MGA (63.43) and the bevel to 37.76, and cut away. I'm sure you math guys that really loved the "fun" trig identities will have a more compact form for these equations, so feel free to post them.

Attaching copper wires to flat plastic ribbon cables

I am going to leave this post published, but I do not recommend this method anymore. The connection is just too fragile. I'll post a better method later.

Many modern electronic device use flat plastic ribbon cables to connect one circuit board to another. These cables are constructed by applying conductive polymer traces to a plastic backing. They cannot be soldered, and are usually connected to the circuit board with a special connector that uses spring pressure to create an electrical contact. If the connector cannot be removed from the board or bought from a supplier, the following technique can be used to attached wires to the ribbon.

First, tape the ribbon cable securely to a work surface (paper-covered tabletop). Then, strip the ends of some small-gauge solid copper wires (I like to use wire-wrap wires). Tape each wire in place, so that the wire is securely positioned over a trace on the ribbon
Mix up a batch of silver, conductive epoxy. This stuff is pretty expensive, but I'm always surprised how little I use for each project, so the supply of it lasts a long time. Load the well-mixed epoxy into a syringe, and use a luer-lock blunt needle tip. The diameter will depend on the pitch of your ribbon cable, but I would say .020" is a good start. The luer-lock tip is important since the syringe needs to be squeezed quite hard to get the viscous epoxy out. Just imagine that needle tip suddenly coming loose, and the syringe squirting epoxy all over!
Most conductive epoxies have a pot life of about 5 or 10 minutes (depending on ambient temperature). All epoxy must be applied before before the pot life ends, or it will not stick well even though it seems like it might be okay when it is applied. Once the epoxy is applied, I usually use an oven to accelerate the curing. I did not in this case because I was afraid of melting the plastic ribbon cable. Instead, I waited overnight to make sure the epoxy had set.


Finally, I use a hot glue gun to completely cover the wiring and epoxy joints. This provides a surprising amount of strength, and allows the part to be hanlded carefully without any other mechanical support.

Metal inlay technique using solder

Here is a technique that I developed to make a metal inlay with solder. I used soft lead solder for this project because it's easy to melt, and I am currently out of silver solder. I imagine the results will be slightly better with silver, since it is harder and will have a better shine.

I attached a piece of .030" thick brass sheet to a thick acrylic spoilboard with double-stick tape (Permacel). Using a 3/64" stub end mill, I CNC cut lettering. I started this cut with WD-40; however, the chip buildup was so large, I switched to water-based flood coolant to get the chips out of there. Cutting specs are: 4000 RPM, about 2.5 inches per minute, .020 depth of cut, 2-flute.


Some edges turned out very clean. For unknown reasons, other edges had a big lousy burr where the cutter pushed metal out of its path (instead of cutting it).
I used a MAPP gas torch to heat small sections of the brass and applied solder after fluxing the piece with acid paste.
I sanded the piece starting with 60 grit (just until I hit the brass), then worked up through each grit until I was using 600 with water. Yes, I use water with the orbital sander, it actually works pretty well! I reattached the piece to the spoilboard with more tape. This makes sanding much easier.
I made a custom polishing tip from aluminum, and a foam-padded piece of felt that was intended to be placed on the ends of table legs. First I used rubbing compound with the custom tip, then white rouge with a loose cotton wheel. I learned that using brown rouge on a spiral-sewn wheel was too aggressive on the soft solder. It preferentially chewed away the solder, leaving the piece uneven and not attractive. I developed the custom tip to distribute the polishing force as much as possible.



The biggest problem is the voids in the solder. I think these are bubbles caused by boiling flux or impurities while the solder is molten. They might also just be low spots where I did not apply enough solder. Oh well, I think it turned out pretty nice. I have also done this technique with acrylic and epoxy instead of brass and solder.

MIlling tiny parts from thin brass sheet

I recently completed a project that required me to replace a steel part with a brass one for a small game-controller joystick. In the picture below, there are two brass replacement parts mounted on a small circuit board with the original steel part on the right. The part itself started life as a flat piece of metal that was bent on four axes to produce a 5-sided cube (the bottom face of the cube is open and fits into a white plastic base).




Here are is the original steel part and its brass replacement, which I machined from sheet stock. The stock is about .025" thick, which is just slightly thicker than the original steel. I began this project by unfolding the original part, and scanning it with a flatbed image scanner. I cleaned up the image in the GIMP, and converted it into a vectorized image format. Then, I created a toolpath for my CNC mill. This sounds relatively easy, but it took many hours to accomplish the whole series of tasks. I started experimenting with speeds/feeds/lube/depth of cut, etc, etc and I eventually found a winning combination:

Use stub-length micro endmills. I used a .020" dia end mill, which often broke when I breathed on it too hard. Using stub-length end mills helps prevent breakage.

The depth-of -cut should be about half the tool's diameter, so .010", in this case. I used three passes to cut through the .025" thick stock.


The chipload should be the tool diameter divided by 130. So, .020/130 = .00015. My mill is limited to 4500 RPM, so this means, I can only feed the tool at about 1.4 inches per minute (2 flute cutter). In reality, I chose to use an even slower feed rate of about .5 to .75 inches per minute.

Galling the brass was a big problem, and I found that WD-40 with occasional air blasts to clear the chips worked very well. Probably, a mist coolant system would be the best, but I don't have one.

Workholding was accomplished by using double-stick tape to adhere the brass stock to a thick acrylic spoil board.


I also tried making a fixture to hold my Dremel tool to the mill spindle. This would allow me to spin the tiny end mills at 30,000RPM (proper speed). It was a good idea, but the runout in the Dremel tool (.005") caused the .020" end mills to break very often. It just didn't work, so I would advise against using a Dremel for this kind of milling.