The Pythagorean Goblet

You probably heard about the Pythagorean Cup, a famous practical joke supposedly conceived by Greek philosopher Pythagoras. It uses a siphon effect to empty the cup completely once it gets filled high enough. Let me show you how to make one out of wood!

If you want to make it, as much as I appreciate taking action right away, read through the whole thing first to make sure you can plan this in a way that fits your needs and design desires.

Step 1: How It Works

The "mechanics" of this are pretty simple. The only tricky thing about that project is to create the "piping", but it is really not that complicated.

What happens is that as the liquid rises in the cup (or goblet), it also rises in a small tube connected to it at the bottom. This tube is at some point bent downward, and as soon as the liquid level reaches that bend it will start running out through the downspout section of the tube, emptying the whole cup.

The idea behind it is that you should not be too greedy, otherwise you end up with nothing. While I enjoy the sentiment, the knob in the center of the cup/goblet is a dead giveaway, and after all, you can easily refill the goblet later to half full as often as you want. In theory, it would be easy enough to create a cup that works without the centerpiece by putting the piping in its walls. But that sounds more like something you would have to 3D-print than turn from wood.

If you want to know more about the mechanism and get a good demonstration, check out this video by Practical Engineering about the Bell Siphon.

Let's move on to the actual build!

Step 2: Tools and Materials

For this project, I will assume that you know your way around a lathe. You can find plenty of turning tutorials and practice projects both here as well as on YouTube for that.

Here is what you will need in terms of materials:

• a large piece of wood for the goblet (I used willow). This should obviously be large enough for the entire goblet unless you want to add a block of wood for the stem. You should read ahead and decide what you want to make first. It also needs to fit into your lathe.
• a smaller piece of wood for the "centerpiece" which will house most of the piping (I used maple).
• a smaller piece of wood to fix the stem (optional, I used walnut). See above and a few steps down.
• wood glue - for obvious reasons.
• surface finish. I used beeswax because I almost always do, but you can use anything that you enjoy applying and that is, preferably, food-safe.

And these are the tools that you will need:

• a lathe. To turn the wood into shape (literally). Seriously, this is mandatory for any woodturning project.
• something to hold the wood on the lathe. I used a scroll chuck, but a screw chuck or a face place will work as well. I also used spur and live center to hold one piece.
• a Jacobs chuck for your tailstock. This can be replaced using a drill press, a handheld drill or by actually turning if you have the skills.
• turning tools. I use a mix of traditional and cutting insert tools, the later mainly for the hollowing part of the goblet.
• a drill press for drilling perfectly aimed holed. You can make due with a handheld drill, but a drill press would be the easier choice.
• a long brad point bit to drill all the way through the goblet. I have a rather long one for that purpose, and it is about 1/4" (or 6mm) in diameter.
• a brad point bit of about 1/4" (or 6mm) size. You can use the long one mentioned above, but it is a lot more manageable to use a shorter one for shorter holes.
• forstner bits, the sizes depending on both what you have and whether you want to do the stem-fix.
• a rotary tool with a mill bit. I use a long thin one.

Before we start, there is something I need to clarify about that walnut plug so you can better plan your project.

Step 3: Side-note - How This Idea Started Out

This step is not important to the project in itself, but I wanted to add it to clarify a few things. First, this woodturning started out as me trying to turn a goblet with two captive rings. That refers to a turning technique where you create a ring around a stem of sorts from one piece of wood, and in such a way that it cannot come off the finished piece.

That did not work out (for lack of skill that I do not want to get into at this point. Well, I guess I just did), and I left the half-finished goblet on the shelf for a couple of months. Which is also the reason why I do not have any footage of the basic shape being turned.

When I finally picked it back up with the intention of finishing it, I discovered that there was a defect in the wood smack at the center of the stem. So I decided to forego the captive rings and replace the stem with another piece of wood. And since this goblet would then be a tad boring, I went through my list of ideas and found a note regarding the Pythagorean Cup.

If you are only interested in the creation of the piping for the siphon effect, you can skip over the steps called "Fixing the Stem". But if you have a piece that needs fixing in a similar way, or simply like how that fix looks, keep reading!

Step 4: The Basic Goblet Shape

Like I said, I did not record anything as I turned the basic shape of the goblet. You are pretty much free to pick any shape you want here. The only mandatory part of this project is the cup section, i.e. something to pour beverages into. You can completely omit the stem if you want to.

As for the shape, other than enough room in the center to accommodate the centerpiece and a decent amount of liquid, you can pick any design you like. No matter whether you pick straight lines, angles or curves - you will have chosen wisely (and yes, that is a movie quote).

Here is the problem I ran into - a defect right in the center of the stem. You can read about how I fixed this in the following three steps, or skip ahead if your wood is more sound than mine. Of course, you can also use this as an artistic element (or two or three of them) without the need to fix anything.

And something else, if you are going with a screw chuck on this, make sure to make the pre-drill hole for that as deep as you can. You will thank me later - in Step 8, actually.

Step 5: Fixing the Stem 1 - Preparation

For this "interlude", you will need a block of wood to replace the defect (or satisfy your artistic needs), a Forstner bit of about half the diameter you want for your stem, and a small drill bit long enough to drill through the replacement block. Calipers come in handy to get the size right.

First, I turned the stem down to the diameter of the Forstner bit. I used a square cutting insert turning tool to get a crisp edge, but a parting tool will work just as well. I got lucky. The defect went all the way through at this point. A few millimeters further and the piece would have broken. So work your way down slowly and keep checking so you can stop and pick a larger Forstner bit if possible.

This part will serve as tenons for the replacement piece, and you need to cut the goblet in two for that to work. I did that on the lathe, but you can just as easily use any kind of saw if you do not want to risk the piece breaking prematurely.

Step 6: Fixing the Stem 2 - the Replacement

To replace the stem portion of my goblet I chose a piece of figured walnut. To plug it in between the goblet parts you need to drill into it with the Forstner bit from both sides to create mortises to the goblet's tenons. And since we want the piece to remain as true as possible, you need to keep the holes aligned.

To that end. I used a smaller drill bit to drill all the way through the block of wood. Then I could use these holes to center my Forstner bit. It does not matter if you go a bit too deep with that, just as long as you go deeper than the tenons left on the goblet.

Step 7: Fixing the Stem 3 - Glue-Up

Gluing up the pieces should be straightforward if the fit between hole and tenon is good. In any case, use enough glue so that it squeezes out when you put the pieces together and add some pressure - either using a clamp or your lathe, which should be no problem if you are planning for this step.

What I mean is that if you plan for this, you will do it when the goblet is still in a shape where you can use your tailstock to clamp it together. When I did the fix on mine, I had already hollowed it out further than my tailstock could reach, so I used an additional clamping block that may or may not have been completely in line with the rest of the goblet.

Let it sit for a while to make sure the glue dries completely. Also, if you do have defects in there you might consider adding some additional glue to fill those up as well. It will not give you the same effect as epoxy resin when it comes to filling something up, but it will give you some additional strength nonetheless.

Step 8: Drilling the Through Hole

A warm welcome back to all those readers who skipped over the stem fix. You really missed something there, but let's not look back now. Instead, we now need a through hole through the whole goblet. Since the lathe is spinning already this is easy to do with a long enough drill bit. And this is also why I told you to make the pilot hole for your screw chuck (if that is what you are using) as deep as you can make it because you do not want to drill into your screw chuck.

Here is how I did it. I used a long drill bit (I estimate it to be maybe 30 cm/12" long) which I held in my hands. Once I found the center point of the lathe (which was not as simple as I would have expected), I could easily hold the drill and push it all the way through the wood until it appeared at the bottom (which is an advantage of scroll chucks - they have an opening there.

If you cannot do it that way for whatever reason, you can use a normal-length drill bit and a handheld drill, or a drill press and go from both sides. If you realize during planning that you do not have a bit long enough even for that, you can use the stem-fix method to cut the goblet in two and drill the through-hole before you glue the fix in. Just make sure to keep the hole open so glue squeezeout does not block it shut. A skewer should do the trick.

Step 9: Adding a Forstner-Step

The next step is best done on the lathe with a chuck in the tailstock. A drill press works too as long as you can get the alignment close enough to right. Pick a Forstner bit of a diameter that is 4-5 times the diameter of the brad point bit you are using. The reason for that is that you need to drill two holes into the center knob, and need to leave wood between them and some wall thickness. If you fall between two Forstner bits in your collection, pick the larger one.

The step should be in the vicinity of 1/4"/5 mm deep to facilitate a good glue-up.Again, you should choose deeper over shallower just to be on the safe side.

And if you are skilled enough, you can actually turn this step into the goblet. It is certainly possible, but I prefer having the flat surface of a Forstner hole as opposed to trying to get something flat and in the process removing more and more material until the goblet is gone.

Step 10: Turning the Centerpiece

For the actual knob that goes in that step, I used a piece of maple. It does not look as stunning as either willow or walnut, but it does the job just fine. Coming at it from a design standpoint I would argue that the goblet does not need any more visual distraction at this point.

At this point, I simply turned a cylinder that had the diameter of the Forstner bit used for the step at the tail end. I would recommend going for a snug fit. Then I placed it into the goblet and used a pencil to mark where the top of the cup part is.

Using a parting tool, I then cut the center knob in two a bit below that line. I did not go all the way through but left a small knob standing that I then broke off. This will come in handy in the next step.

Step 11: Off-center Drilling

The center knob is "where it's at" as far as the piping for this goblet is concerned. Two holes need to be drilled along its length, both of them off center - which is basically the only way to fit them both in there. That is also why I left that knob standing. It helps in spacing the holes by aiming next to the knob on both sides.

One of the holes needs to be stopped so it doesn't get away. Actually, that means it should not go all the way through. That is the one I start with, and I set my drill press to that depth. You can do the same with a hand drill (if you believe you can aim straight enough) and a piece of tape as a mark on your bit.

The other hole does need to go through, but instead of resetting the drill press depth (which would work, too), drill to that stop, then add a spacer underneath and finish the hole.

Step 12: Going Sideways

The stopped hole needs an opening at the bottom. I placed the knob on its side and aimed to where I expected the non-through hole would end. Then I worked my way into it until I found it. But be careful. Especially with a hand drill, your hole might not be quite where you thought it would be, and drilling into something round is never easy, to begin with.

While I drilled this side hole without help, you could make your life easier using a clamp to keep it from rolling.

Now, the two holes at the top need to be connected. The easiest way to do that is to create a channel between them. I used the same drill to hog out the majority of the material, which also takes care of the knob.

But the bit left a rather torn and rough surface, which would not work well with being wet. So I smoothed it out using a mill bit. I have one that is long enough to also round over the corners of the bent, but the mechanism should work even if the shape is not completely optimized as far as flow dynamics go.

Step 13: Centerpiece Glueup and Shapedown

To finish this part of the goblet, the top needs to be closed off. What better way to do that than by gluing the piece I parted off on the lathe earlier? It will also give me continued grain flow, even though with a wood as subtle as this, nobody will actually notice.

Apply the glue to the face with the channel routed out, and use it sparingly to avoid clogging the piping with squeezeout. Once the glue has dried I shape the knob, I just need to make sure not to turn into the holes. For that, I recommend leaving a bit of material standing above the glue line.

At this point, I sand this part and apply beeswax as a finish.

Step 14: Finishing the Rest

The same - the need for sanding and finishing - also applies to the goblet itself. In the video, you can actually see a quick comparison between the walnut sanded, burnished with a hand full of shavings, and then finished with beeswax.

Make sure to finish the inside of the goblet thoroughly because that is where the wood will see most "stress" as far as the surface is concerned.

Step 15: Final Glue-up

The knob needs to be relieved of its top part if you did not outright part it off, and the tip probably requires some sanding and a bit of finish. Then, I use the same mill bit as before to elongate the off-center hole at the bottom to connect with the center hole in the stem.

And now it is time for the final glue-up. Put glue into the step in the goblet but make sure to leave the hole open. Then pace the knob into it and apply pressure. If the fit is not too snug you might want to use clamps to keep it in place as the glue does its thing.

Step 16: Pythagorean Goblet in Action

Check out the video if you want to see the siphoning effect in action.

I hope you enjoyed this project! If you make one yourself make sure to let me know and use the "I made it!" feature just below. If you have any questions feel free to ask, and as always, remember to Be Inspired!