We’re all familiar with a doughnut/donut (ring torus) and it’s plain that all you need to define it are the diameters of two circles, one for the cross section and the other to give the overall size. But who knew that hidden within is another circle, the Villarceau circle, named after a 19th Century French mathematician?

A pair of Villarceau circles are revealed when a torus is sliced by a plane which passes through the centre point of the torus and touches its surface in two places. We can use this hidden geometry to create a 3D light shade from 2D sheet material. Read on to find out how.

This shade is best made using a laser cutter, especially if you're making it from plywood. The laser produces really accurate cuts which means that the individual pieces slot together well without any sloppiness. But it should be possible to make a shade using a bandsaw or jigsaw, and if you use cardboard or perhaps polypropylene sheet you could cut it by hand with scissors or a craft knife. Whatever sheet material you choose, don't pick one that's totally rigid or you won't be able to assemble the pieces - you'll need to bend them a little to get them to interlock with each other.

If you're using a sheet material that's 1.5mm thick and you want a shade of the same dimensions as mine - 375mm (15") diameter and 175mm (7") tall with a 125mm (5") diameter hole in the middle - then you can just download the files attached to Step 4 and either use the DXF vector files for laser cutting or the PDFs to cut the shapes by hand. Otherwise, follow Steps 1-3 to design a torus shade to suit your needs.

I used a laser cutter with a 600mm x 300mm bed and I was able to cut 4 of the 10 crescents I needed (see Step 3) at a time. In other words, a piece of plywood measuring 600mm x 900mm was ample, but that will depend on the size of your laser's bed as well as the size of light shade you want to make.

Supplies

Thin plywood (I used 1.5mm 3-ply from James Latham), or other sheet material with a very low power bulb

A laser cutter

3D CAD software - if you're designing your own shade

An empty aluminium drinks can

Approx 50cm/20” of steel wire, about 2mm diameter (AWG 11-12)

2-part epoxy glue

A drill (preferably a pillar drill) and a HSS bit the same size as your wire

A hole cutter 29mm diameter or a little smaller

Plaster of Paris or filler

A protractor

Flat and round needle files

2 pairs of pliers, one with wire-cutters

Wood finish and metal paint - optional

Step 1: Deciding on the Dimensions

Work out how wide your shade needs to be overall, ie the outer diameter of the torus. I went for one that’s 375mm in diameter. If in doubt, find something of a similar size (or just cut a circle from a sheet of cardboard) and get someone to hold it up in front of the light while you view it from the other side of the room. Because this shade isn’t solid, a large one isn’t particularly heavy so you can go big and dramatic if you wish. However, you may want to come back and adjust this dimension later, after working out how much plywood you’d need to make it and whether it will be cut wastefully because of the size of the laser cutter’s bed.

Then choose the diameter of the hole in the centre of the doughnut. It needs to be large enough to get your hand in to change a bulb easily, and sufficiently bigger than a bulb to allow plenty of clearance around it. I went with 125mm.

These two dimensions determine the diameter of the circular cross-section of the torus – it’s half the difference between them, ie (375-125)/2 = 125mm. This would mean that the shade is only 125mm high, which isn’t really tall enough to hide the pendant lampholder and the bulb when the shade is viewed from the side. I decided I wanted my shade to be 175mm high. Again, hold something up against the bulb to judge what height is suitable, and think about your ceiling height if this is for a pendant fitting rather than a table lamp or standard lamp – you don’t want it to hang down too low, and it will look best if the bulb is in the centre rather than too far down or up.

So how to get extra height without affecting the other dimensions? Easy, we just need to squash the doughnut so that the cross section becomes an oval that is taller than it is wide. This will have the effect of making the Villarceau circles into ellipses too.

Step 2: Checking the Slot Width for Laser Cutting

The plywood I used is nominally 1.5mm thick but it measures nearer 1.6mm. In theory, therefore, you'd need to design a slot 1.6mm wide in one piece of ply in order for a second piece of ply to pass through it perpendicularly. But that doesn't allow for the laser cut's finite width - the kerf - nor for the fact that it might be quite hard to push two slotted pieces together if the slots are exactly the same size as the thickness.

Before making the test slots on the laser cutter as described below, do a series of cuts with different speed and power combinations to decide what's the best setting. You want to achieve a reasonable speed of cut without excessive blackening/charring of the edges.

To check what size slots to cut, I first cut two interlocking test pieces of plywood with a series of slots from 1.50 to 1.65mm wide. Pushing them together (perpedicularly), I found that the pair of 1.55mm slots meshed best - not too difficult to slide into place but no sloppiness to the joint once they were pushed home. You might want to do the same exercise if you're designing your own shade from scratch, or even if you're intending to use the PDFs or DXF files attached to Step 4, to avoid wasting material by creating a shade that doesn't fit together properly. Apply wood finish before slotting your test pieces together, assuming you intend to finish the actual shade pieces before assembly. (I decided late in the day to Danish oil my light shade and only did so post assembly, which was much more fiddly, hence my recommendation in this Instructable to apply any treatment while the pieces are still separate - see Step 5.)

Of course, none of the joints in this light shade are perpendicular, not even the one at the centre of each crescent unless the torus cross-section is circular rather than an oval. I worked on the basis that, if 1.55mm wide slots were right for a perpendicular join, then joints at shallower angles would also work as long as the shade was designed for making from 1.55mm thick material. That is what I did (see next step) and it was fine.

Step 3: Modelling the Light Shade

Having decided on the size and shape of the shade and the material thickness to use for the design, we can start modelling it. I used Fusion 360 which is free for students, educators and hobbyists.

Because each of the ten pieces of this shade intersect with 5 others, at 3 different angles, the slot widths need to vary according to that angle – the shallower the angle, the wider the slot needs to be to allow the intersecting piece to pass through it. And none of the three angles is 90°, which means that all of the widths have to be greater than the thickness of the sheet material from which the shade is constructed. All of this explains why the PDF and DXFs attached to the next step are only suitable for use with a sheet material of thickness 1.5-1.6mm. Unless you want to make a light shade of the same dimensions as mine from material of the same thickness as mine, you will need to design your own as explained below.

The stages I went through in Fusion 360 were as follows – if you use a different CAD system you’ll be following essentially the same method:

Create a sketch (Sketch 1) in the XY plane and draw a circle of the overall diameter of the shade (375mm in my case), centred at the origin.
Create a second sketch (Sketch 2) in the XZ plane and project the Sketch 1 circle into it. Then draw an ellipse of the correct width and height (125mm and 175mm, respectively, in my case) touching (but inside) one end of the projected line. Draw a line that goes through the origin and is a tangent to the ellipse.
Create a torus by revolving the ellipse around the Z axis.
Construct a plane through 2 edges: the tangent line and the Y axis.
Use the new plane as a tool to split the torus, and remove one of the two resulting bodies.
In the Surface workspace, unstitch the remaining body then remove its curved surface(s) to leave just two flat crescent faces that touch each other. Remove one of them as well.
Thicken this face symmetrically to create a new body of the thickness arrived at in Step 2 and return to the Solid workspace.
Do a circular pattern of the new body around the Z axis to create 5 solid bodies in total. Then mirror the one that is bisected by the YZ plane in that plane to create a copy. Pattern that copy as before to create 5 bodies leaning in the opposite direction to the first 5. At this point you will get a pretty good idea of how the finished shade will look, especially if you change the appearance of all 10 solid bodies to look like the material you intend to use. (See rendered image above of a flattened version of the torus.)
Each crescent is intersected by all 5 of the ones that lean in the opposite direction. Select a crescent and split it using the 5 intersecting ones as the splitting tools. Remove all the bodies apart from the new crescent-shaped one with 5 slots in it that have been created by this splitting action.
Select the new split crescent and use Look At to view it straight on. Save this view as a Named View.
Create a new sketch (Sketch 3) in the plane of the top surface of the split crescent and project its outline (by using Project/Include, Intersect then selecting the body) into it, to get the outline of just the top surface that is in the sketch plane.
Do exactly the same on the underside of the split crescent, ie create a new sketch (Sketch 4) in that plane and project into it the outline of the lower surface that's in the same plane.
The outline in both of the new sketches should be a crescent with slots in it where other crescents will pass through it. Note that the slots get wider towards the tips of the crescent, because the other crescents passing through it near the tips do so at a shallower angle. We now need to combine the two sketches to maximise the slot widths. Working in Sketch 4, project into it those straight Sketch 3 lines - just 1 for each of the 5 slots - that are outside the pair of existing Sketch 4 lines marking the sides of each slot. Zoom into each end of each slot in turn and then project in also the Sketch 3 curves that are needed to give a complete crescent outline with no gaps in either the concave or convex side. After doing that I was left with a small gap at the concave end of the central slot, which I joined with a straight line.
Edit Sketch 4 to get rid of the central line of each slot and also any duplicate curved lines - take care to delete the right ones or you'll create gaps in the outline and will no longer have a full, crescent-shaped profile. Then extend the straight lines that create the slots, where necessary, to reach the curved outline (you'll first need to break the projection link for lines that are to be extended), and also delete any slanting lines at the ends of these short lines. Cut each slot in half by joining the centre points of the pair of lines.
Copy Sketch 4 (select all, right-click, copy, paste into a new Sketch 5 created in the same plane), then edit Sketch 5 to make it into a crescent with slots only reaching half way in from the concave side.
Copy Sketch 4 again to create a new Sketch 6, also in the same plane, and edit it to make the other type of crescent, one with slots that go half way in from the convex side.

In stage 8 above, there’s no reason why you shouldn’t choose to make a number of copies other than 5 of each type.

Step 4: Cutting the Crescents

Save each of the two crescent sketches as DXFs (or SVGs if that's what your laser cutter needs and the CAD system you're using will do it) so that they can be laser cut, or create a drawing and save it as a PDF if you intend to cut them by hand. The DXFs I used and the equivalent PDFs are attached to this step. 4 crescents will fit within a 600mm x 300mm rectangle (which is the size of the laser cutter bed I used) and each PDF crescent will just fit on an A4 sheet of paper - the tip-to-tip measurement should be 241.1mm.

Thin plywood tends to be more flexible in one direction than the other. Orientate the crescents with the bendier direction from tip to tip. They will nest together quite closely, but remember to allow for the kerf of the laser.

Using the laser settings you arrived at in Step 2, cut 5 of each type of crescent.

Step 5: Wood Treatment and Assembling the Crescents

Wipe round all the edges of the laser-cut crescents with a barely damp cloth to remove the sooty residue, or lightly sand the edges to smooth them if you've cut them from plywood by hand.

Now is the time to apply a wood finish if you want. I gave my plywood two coats of Danish oil.

Once everything is dry, assemble the shade using the photos and CAD renderings as guidance. Each crescent that leans to the right interlocks with 5 left-leaning ones and vice versa. A slot of a particular length interlocks with a slot of the same length in another crescent. Don't push the interlocking joints fully home until you have all the crescents in place, you'll need to keep some looseness in the structure in order to get the last few crescents in. Once they are all in, work round the shade pushing every pair of interlocking slots fully together. There shouldn't be any need for glue.

Step 6: Making the Centre of the Support "spider"

I used the base of a beer can as the centre of the "spider" from which the shade hangs. The standard size for the hole in a UK shade ring is 29mm diameter. If lampholders in your country are a different size then you may need to find something bigger than a standard drinks can.

Rinse out the empty can then cut off the base along its edge - the aluminium is so thin that an ordinary pair of scissors will do it. File the cut edge smooth so you don't cut yourself. Mark the centre point with a pen - I found the centre using a paper circle, folded along 2 diagonals, and a pin, as in the 2nd photo. Then use a protractor to make 5 evenly spaced marks around the ridge (the angle from one mark to the next is 72°) through which you'll drill holes for the wires, as shown in the 3rd photo.

With the base of the can upside down (ie the central dish facing up) and supported by a piece of scrap wood under the centre, drill a pilot hole through the centre point and then use a hole cutter to make the full size hole. File it out until it fits easily over the lampholder, and then a bit more if you intend to paint it.

Turn the base the right way up and fill the void with plaster of Paris or filler, level with the top edges, taking care to push the material well into the ridge so that its walls won't distort when you drill through them.

While the plaster/filler is setting, test what size drill bit to use for the spider's legs by drilling holes in a spare piece of aluminium from the remainder of the can. You want the wire to be a fairly tight fit in the hole. If in doubt go for a smaller bit as the holes can always be enlarged a little with a needle file. Then, when the plaster/filler is good and firm but not rock hard, clamp the can base vertically with one of the marked points on the ridge facing upwards. Drill down through both walls of the ridge. Rotate the can base to the next marked point and repeat until each of the 5 pairs of holes has been drilled.

Break the plaster/filler out of the aluminium ring and wash it clean. Use a needle file to remove any sharp edges from the holes and enlarge them if needed until a piece of wire will just go through.

Step 7: Finishing the Spider

For a ceiling light

Place the spider's centre over the light side and measure roughly how long each wire leg needs to be to reach from (almost) the hole in the middle of the spider to about an inch (25mm) beyond the nearer joint at the top of the shade. Add on a little extra to be on the safe side - I went for 90mm - then cut 5 wires to that length.

File one end of each wire to remove any sharp burrs, then straighten them if necessary. (You can do that by hammering them lightly, one at a time, on a flat surface while rotating them.) Then bend the tip down slightly - see 1st photo - with 2 pairs of pliers so that the wire gets some support from the dish in the can base after it has passed through the pair of holes in the ridge.

With the ridge uppermost, fit the legs through their holes. Draw a star shape with 5 equally spaced legs on paper or card and use it to check that the wire legs stick out at the right angle and that they lie flat on the table - see 2nd photo. Adjust the bend at the tips if necessary. Then mix up some 2-part epoxy glue and put a blob of it around each wire where it passes through the holes, both on the inside and the outside of the ridge.

When the glue has completely set, hold the spider centrally over the light shade. Mark the point towards the free end of each leg where it's level with the tip-to-tip joint of two crescents. This is where the wire needs to bend upwards so that the shade won't fall off the spider if it gets knocked. For the shade to hang level, all of the legs need to be the same length to this point, as measured from the point where they emerge from the ridge in the spider's centre. Adjust the marks to get these lengths the same and double-check that the bends will then be in the right place by holding the spider centrally over the shade again. Err on the side of making the straight length of wire a little too long rather than bending the tip upwards too early.

Before making the bends, cut off excess wire about 18mm (3/4") beyond where the bend will be. Then, when you've bent the tips up - use 2 pairs of pliers again - you should be able to manipulate the spider into place quite easy to check the bend angles are suitable. Tweak them until the wire tips follow the same angle at the interlocking joint to provide the maximum contact area and support. Once you're happy, fille the cut ends smooth and then give the whole spider a wash in soapy water to degrease it prior to painting.

For a standard lamp or table lamp

The spider will need to support the shade lower down than for a pendant lampholder or the bulb will stick out of the top of the shade. The spider’s legs will have to tuck under the 5 joints between two crescents nearest their lower tips (see last photo of this step and the final, rendered image) instead of the identical joints nearest their uppermost tips. The only problem with that arrangement is that the low level joints slope downwards rather than upwards, meaning that the tips of the legs will also need to slope downwards and the shade won’t be nearly as secure as when it’s hung from a pendant fitting.

If you do decide to make a shade that’s supported from below, simply make the spider as described above for a ceiling light then flip it over to use it. It needs to go over the lampholder so that the bulb protrudes from the side of the can base opposite the ridge, otherwise it can be hard to get your fingers in to do up the lampshade ring.

Painting

Painting isn't essential as long as you've used non-rusting wire, but it does help to make the spider look more professional and less like something cobbled together from a drinks can and wire coathangers. Using white paint also makes it less obtrusive when viewed from below against a white ceiling. Galvanised wire has a zinc coating which means that, like the aluminium can, most paints won't stick to it well. You'll need to use a specialist primer and top coat - I primed with Hammerite Special Metals Primer and put 2 coats of white Hammerite smooth metal paint on top.

Step 8: Assembly and Hanging

All the remains to be done is to slip the spider into position so that the upturned tips of its legs support the 5 uppermost joints of the shade. (Or the downturned tips support the 5 lowest joints for a standard or table lamp.) Then fit the spider over the lampholder and do up the shade ring to hold it securely.

Introduction: Torus Light Shade - Laser Cut