Introduction: Euler Vase

About: SLO MakerSpace offers tools, training, and consultation services that enable community members to experiment with and master a wide variety of Maker skills. We are dedicated to the following values: Commun…

This instruction set is intended to help you get started woodworking at SLO MakerSpace, in San Luis Obispo, California.

The project is heavily inspired by "Geometric Faced Vase" by FaustWoodWorks.

Check out his Instructable at:

The title "Euler Vase" refers to the math behind the Euler Characteristic, which describes the relationship between the vertices, edges, and faces. We will discuss this more in step 5 (optional).

Your kit should include one block of wood, and a plastic tube. Don't be distressed if the tubes look ugly. Only one of them will be used, and it will be out of sight in the hole.

Step 1: Cut Your Wood Block to Size

Your block starts about 3 1/2" squared, and approximately 10" to 12" in length.

You need to cut this down to a manageable size so we can shape it on the band saw. First, attach a shop vacuum to the tube opening under the bandsaw table, and turn the shop vacuum on. Next, grab a push stick and a push block.

Align the fence at the correct distance. I chose to remove about 3/4" from each side. The less you remove, the bigger your vase will be. Larger blocks are more difficult to manage on the belt sander, and take much longer to shape. Remember that you can have a vase that is narrow in one axis, and wide in another.

Place the blade guard about a quarter inch above your block so it can fit through the saw.

After the vacuum, fence, blade guard, and push blocks are in place, push the block through. SLOW DOWN NEAR THE END OF THE CUT! Slow way, waaaaaay down, until the block is almost stopped by the time you finish the cut. If you are pushing too hard at the end of a cut, you risk jerking your hand toward the blade. We do not want this to happen. If you push lightly at the end of the cut, there is no problem. Fortunately, the push stick gives you an extra 12 inches of stopping distance.

After you make the first cut, turn off the bandsaw and wait for the blade to completely stop before removing your piece. Turn the piece sideways, grab your sticks again, and make the second and final cut.

Once you are finished, turn off the bandsaw. Disconnect the vacuum hose and use it to suck up any extra dust on and around the bandsaw.

Now that you've cut the width. You're ready to cut it to length.

Step 2: Cut the Piece to Length

I chose 9 inches. You can measure it to be precise, or use your intuition. Use the Dewalt ChopSaw to make the cut. Remember, keep your hand out of the no-hand zone. Clamp the lumber from the side as shown with the orange clamp, or clamp it from above, or you can use both. The ChopSaw should be constrained so that it does not slide to the end of the track, and both axes should be set to zero degrees.

Hold onto your stump. You may want to use it later to test wood stains and finishes.

Step 3: Make the Hole

Use the forstner bit in the drill press to make the hole. Forstner bits are in the black metal cabinet near the drill press. Make sure the piece is CLAMPED to the drill press. Use the vertical brace with a clamp or the vice and the bolts that go thru the table. Adjusting the bolts is easiest with a flathead screwdriver while you hold the black handle up. Note: If you just hold your block by hand, you will gouge the edges of the hole every time you plunge the forstner bit back in. Go down slowly and pull out often to let the chips clear out. If you want to make a deeper hole, you can switch to a spade bit to descend even farther. Please sweep or vacuum around the drill press when finished.

Step 4: Shape the Faces

This is the must fun step, but perhaps the most challenging. Sand the faces down to create the artistic shape. Use the belt sander or the sanding disk -- BUT FIRST -- set-up a system to remove dust. At one side of the sander, you will see two round holes a few inches wide. Attach this to either a shop vac or a dust collector and turn it on. Now you're ready to turn on the sander.

Please do NOT wear gloves. We don't want gloves to get caught in any moving parts. I recommend safety goggles (as opposed to safety glasses) because they are better at preventing dust from blowing into your eyes.

Grab a disposable dust mask too if you prefer.

As you sand, use moderate pressure and move the piece back and forth across the belt. If you push too hard, the piece will quickly go out of control! Do not sand when you are in a hurry. Try not to wiggle or turn the piece as you sand, as this will produce curved faces. This vase has 31 flat faces. If you need a better view of each face, you can get on your knees to put your head eye-level with the belt (this is where the goggles and dust mask help).

Avoid placing any two faces at angles that are too similar. We want the edges between faces to be visible when you turn it in the light. If you have two faces that “blend together” as you turn it, consider sanding the faces together by pressing down at the edge between until they disappear into one larger face. You can also change the angle of one or both of the faces, producing a sharper edge. If any corners appear too prominent or too sharp, you can create a new face by sanding down that corner or edge. If you see a face is too small and draws the eye, you can do two things: You can eliminate that face by sanding down the surrounding faces, or you can press that small face into the sander to enlarge it until it looks like it fits in.

You can also use a surform as an alternative to the belt sander. (see image)

This is the most tedious step, and you may find quickly that the picture you had in your head doesn't match the shape in your hands, but don't worry. It takes a while to get used to this method of shaping. If you start getting frustrated, take a break. Take your block home with you and look at it in your spare time. Think about how you want it to look. When you come back the next day, you'll know where you need to sand next.

Remember: This step isn't engineering - it's art. Have fun.

Step 5: The Math of Euler Characteristic (optional)

The title "Euler Vase" refers to the mathematical relationship between Vertices, Edges, and Faces. Regardless of how you choose to shape the planes of the vase, the Euler characteristic, X, will always be the same.

X = V - E + F = 2 where V is the number of vertices, E is the number of edges, and F is the number of faces. For this vase, X = 2, the same Euler Characteristic. You can count the edges and faces you just sanded to verify this for yourself. Use pieces of tape to mark which faces/edges/vertices you count as you go, record the number, then remove the tape and start again. You can pretend the hole isn't there and count the top as one face, or you can subdivide the cylindrical portion into faces and edges and vertices and count them. However, if you make your vase into a "doughnut" by drilling all the way through, then the Euler characteristic becomes zero! Zero is the Euler characteristic for all toruses, such as doughnuts and coffee mugs. For more information on topology, check out "A History of Mathematics" by Carl B Boyer. Call Number 510.9 in Adult Non-Fiction at the San Luis Obispo County Library.

Step 6: Modify the Object (optional)

This step offers suggested corrections to undesired results in previous steps:

Wobbly Piece: If you sand a face on the bottom too far and the piece becomes wobbly, you can clamp it in the Dewalt chop saw and cut off some from the bottom. I chose to do this and enlarge my hole on the top from ⅞” to 1” with a new forstner bit.

Holes or Cracks: My particular piece of scrap wood had a tiny hole hidden in it and a tiny crack. I patched them both with wood putty. This is stored in the yellow fireproof cabinet in the metal shop. To apply the putty, wear gloves. Push the wood putty into the hole or crack. Then use a putty knife to flatten it and scrape away the excess.

Step 7: Cut the Tube

This vase itself is NOT waterproof. If you pour straight in, water will soak into the grain and run out the bottom. Therefore, you will use one of the plastic tubes we've provided. Find the one that fits your hole best and cut it to length. Your kit should include one for a 7/8" hole, 1" hole, and 1+1/8" hole. If your hole is a different size, let us know and we will find the right diameter from our inventory. If you give back whatever extra tubes you have, that helps us make kits for the next person.

Measure the depth of the hole in the wood. If the tape measure is too wide to fit, grab a screwdriver and use your thumb to compare against the tape measure.

In the labeled metal drawers behind the table saw select a 3D printed tube of the correct diameter. The tube you choose should be just wide enough to slide down the hole easily. Use a fine-blade hacksaw with a mitre box to cut it to length. You may choose to subtract about an inch if you want some of the wood inside the hole to be visible. Fine blades are for metals or hard materials. This tube is soft. However, in our case we need a fine blade because we are cutting through a very thin wall tube. While using a hand-saw, where every stroke is controlled, you can place your fingers close to the blade when you grip your plastic. A tight clamp will crush the tube, use care and control when handling and cutting the tubes.

Step 8: Smooth the Faces

The belts we put on the sander have a very coarse grain and the shaped sides aren't smooth. Grab some high-grit sand paper from the drawer under the reciprocating belt sander. A higher number means a smoother finish. I used 150 grit.

Wear gloves to protect the surface from grease on your hands.

Don’t sand recklessly or you will remove the crisp edges between the faces. To preserve the edges, hold the paper on the table with one hand, press your piece into the table with the other until it rests flat and secure on one of its faces. Then drag/push away while pressing into the table. After several passes, switch to the next face.

Step 9: Apply a Finish

I used a polyurethane that can be found in the yellow cabinet in the metal shop. Please prepare your workstation and do not stain our tables. I used a piece of particleboard scrap from the alley.

Put on plastic gloves.

Use a can-key or a flat-head screwdriver to remove the lid. Use a rag to apply the finish to the piece. Roll the piece around to make sure it spreads evenly in the hole. Leave the piece drying in an upright position out of the way so others can use the shop.

Replace the lid and use a rubber mallet (not a hammer) to seal the top. Rinse the rag in the closet in the hallway, then discard rag in the red bin under the metal shop sign. Dispose of your gloves.

Before you leave, write a note on a piece of paper or a piece of scrap with your name, your contact info (phone and email), and time when you will be back to collect it. Place the note next to your vase so nobody moves it or throws it away.

Step 10: Closing Notes

Come back later when the finish is dry. Add a second coat if desired and allow it to try again.

Fill your plastic tube with water and carefully drop into the hole. Try to not spill too much on the wood itself.

Now lets get some flowers! Here are some local florists we recommend:

Wilder Floral

1349 Chorro St


Sprigs Floral Design

788 Pismo St, Ground Fl


Adornments Flowers & Finery

680 Morro St

Unit D


Open Air Flowers

1055 Osos St



Woodworking Contest 2017

Runner Up in the
Woodworking Contest 2017