Polyhedron Light Shade

I’ve always been fascinated by polyhedra. As a child I remember building paper models of Platonic solids with their identical, regular, polygonal faces. (For the benefit of younger readers, there was no internet in those days, we had to make our own amusement.) But sometimes a little irregularity adds to the charm, which is where Catalan solids come in. Named after the 19th Century Belgian mathematician Eugène Catalan, these polyhedra have identical, but non-regular, polygonal faces.

Last year I started using the CAD/CAM package Fusion 360 and modelled several Catalan solids as a learning exercise, including two of the hexecontahedrons (60-sided solids). They reminded me of something, particularly the pentagonal hexecontahedron. It took me a while to realise that its geometry is – I think – the basis of a light shade I’ve admired for a while. The shade in question, called Coral, is by the New Zealand based (but British born) designer David Trubridge. Inspired by his work, I decided to make a laser-cut plywood shade based on another 60-sided Catalan solid, the deltoidal hexecontahedron.

If you want to understand how I designed this light shade, or make your own using differently shaped plywood pieces, or even one based on a different polyhedron, then you’ll need to read through all the steps. But if you just want to make one of the same type, then jump to Step 3 where you can download the DXF vector file or PDF for the unit piece (I’ll call this an “element”) that’s based on a deltoid, a kite-shaped polygon. Then beg or hire time on a laser cutter to make the job of producing 60 plywood elements quick and easy, or cut them by hand if you have the patience. You could perhaps make them from polypropylene sheet or thin card instead of plywood to make cutting out by hand easier, and join them using brads or paper fasteners, but then you’d need to use a very low wattage LED bulb.

We start by creating a virtual, 3D model of a deltoidal hexecontahedron, with its 60 identical kite-shaped faces and 62 vertices. Then we create an interesting shape based on the geometry of a single face. When 60 of these elements are joined together at the same vertices as the underlying polygon, a lattice-work lamp shade is produced that will filter the light and create interesting patterns in the room.

You'll need:

• a sheet of “aviation” ply, no more than about 1.5mm thick. An area of about 0.9m² is needed for a 400mm diameter polyhedron (actual size required will depend on the dimensions of the laser cutter bed).
• 62-66 white/natural nylon M3 nuts and bolts, at least 12mm long and preferably with a hex or pan head – don’t get recessed heads.
• access to a laser cutter – without one, you’ll need a coping saw or a bandsaw, a 3mm drill bit and a lot of patience!
• a sharp knife
• wire-cutting pliers
• flexible paint suitable for wood and a mini paint roller (optional)
• a nylon strain relief cable gland sized to suit your pendant fitting flex (also optional). I bought a PG7 gland for my 6mm diameter flex, similar to this.

Aviation ply is thin but surprisingly strong – it’s so called because biplanes were made from it in the early days of aviation and it’s still used by model plane enthusiasts. I used a 3-ply Finnish birch from James Latham that’s nominally 1.5mm thick but actually measures more like 1.65mm. They supply even thinner birch ply, down to well below 1mm. Choose a high grade ply that's free from defects such as knots that could crack when it's bent.

A deltoidal hexecontahedron has 60 identical faces, each face being a deltoid (kite) with the ratio of short to long sides of 1:(7 + √5)/6 (approx. 1:1.539). The angle between the two short sides is 118.267°, the angle between the two long sides is 67.783° and the remaining two angles (between a short and a long side) are 86.974°. The dihedral angle (the angle between any two faces) of the polyhedron is 154.12°.

I modelled my polyhedron in Fusion 360 (which is free for hobbyists, students and educators) as below – it won’t be very different in other CAD packages. Refer to the screenshots for guidance, but a basic familiarity with Fusion 360 is assumed.

1. In the Design workspace, create a sketch in the X-Y plane and draw a 4-sided kite shape, adding constraints to make both short sides the same length. Do the same with the pair of long sides. Position the kite such that the long diagonal down the centre is on the Y axis, and draw in that diagonal as a construction line.

2. Now make use of this Rechner online calculator to work out what length the pair of long sides should be. Insert the required radius of the finished light shade as the midsphere radius – 200 in my case - hit Calculate, make a note of the long edge size and then dimension the sketch accordingly. (The midsphere is the sphere to which every edge of the polyhedron is tangent. The finished radius won’t be exactly the midsphere radius, but close enough.) Dimension a short edge too, again using the figure from the calculator.

3. Make the angle between the two short sides 118.267°. (If the kite shape had become a concave deltoid, that should sort it out.) Check that the other angles are then as they should be (see above).

4. Now we need to place the point on the diagonal that will lie on the surface of the insphere, which is the sphere that just fits within the polyhedron, touching every face. We know from the online calculator what the radius of that sphere is, 194.922 in my case, and fortunately the midsphere and the insphere are concentric for this polyhedron. That means that there is a right angled triangle whose hypotenuse is the midsphere radius and another side is the insphere radius. The third side is the line we’re about to draw, from the point where the insphere touches the face to the point where the midsphere touches a side. Its length can, of course, be calculated using Pythagoras’s theorem. Place a random point (not at the centre) on the diagonal. Let’s call it point A. Draw a construction line from point A that is perpendicular to a long side of the kite, and dimension it to be √(midsphere radius squared - insphere radius squared). Finish the sketch.

5. Create a new sketch in the Y-Z plane and Project point A and the two corners at the ends of the diagonal into it. Draw a vertical construction line upwards from point A. Dimension the length of this line to the insphere radius from the online calculator. (The point at the top of this line is the centre of the insphere.) Draw construction lines from the insphere centre to the corners at each end of the central diagonal. Finish the sketch.

6. Create a body by Lofting the kite profile, choosing the centre of the insphere as the second profile and Centerline as the guide type.

7. Change from the Solid to the Surface tab and Unstitch the new body before deleting its 4 triangular faces to leave just the kite-shaped one. Then return to the Solid tab.

8. Create a Circular Pattern of the kite face, selecting the line from its narrow corner to the insphere centre point as the axis. Make the pattern of 5 kites including the original one, through the full 360°. They should fit together perfectly to make a sort of an umbrella shape.

9. The full hexecontahedron is made of 12 such umbrellas. Make an extra 2 by patterning the first one around the insphere radius from the widest corner of the original kite shape – the opposite end of its centre line from the first pattern axis. This time the quantity is 3, not 5.

10. We need to pattern twice more to get all 60 faces, but to do that we must draw in two more axes for the patterning. Create a new sketch in the Y-Z plane and Project the insphere centre point into it. Then turn off the visibility of the first 2 sketches, to avoid confusion. Draw in the 2 required axes as construction lines, as shown in the screenshot – one ends at the junction between 2 umbrellas, the other at the wide corner of a kite on the opposite side. The second line is out of the plane of the sketch and must therefore be created as a 3D sketch by first drawing a line from the centre of the insphere to a random point in the sketch plane and then performing a point-to-point move to relocate that point to where it needs to be. Finish the sketch.

11. Create a 4th umbrella by patterning one of those next to the first new axis around it, choosing quantity 3 but supressing one of the two copies to avoid duplication. See screenshot whch shows the umbrella that is to be patterned and the axis in blue. Now we have 4 umbrellas and 20 faces.

12. The final pattern is of all 20 existing faces, around the second of the newly-created axes, with quantity 3 and no suppression. The end result is a deltoidal hexecontahedron shell.

Now for the fun part, designing an element based on the kite. A flat kite-shaped piece of plywood can’t be bent in two directions to make a spherical shade. Instead, a somewhat spindly shape needs to be created, with legs that will flex. The holes for assembly will be centred on the four corners of the kite, which means the legs need to extend a short way beyond the corners.

1. Turn off the visibility of everything, sketches and bodies. Create a new sketch in the X-Y plane and Project the 4 corners of the kite into it. We’re going to design the element in this sketch. Start by drawing in the sides of the kite and its diagonals as construction lines.
2. A good place to start is by Offsetting lines either side of both diagonals and then rounding the ends with the Fillet tool. You could make your shape curvy like mine, or straight-sided, but sharp internal corners are best avoided because the plywood will likely crack there when it’s bent into its 3D form. The first screenshot shows a simple cross-shaped element which has then had unwanted lines removed, internal corners radiused (with the Fillet tool) and holes added for assembly in the second screenshot. (In reality, this cross shape might be unsuitable in most materials – it could be too spindly and liable to break when forced into a 3D curve.) Experiment with different designs, the element doesn’t even need to be symmetrical.
3. To get an idea of what an element you’ve drawn will look like as a shade, delete any screw holes or other holes over the corner points before finishing the new sketch - you’ll lose one of the pattern axes if you don’t. Then move the sketch back in the timeline to immediately after the first sketch that contains the kite. Roll the History Marker back to that point, just after the new sketch, and create a new body by Extruding the element’s profile upwards (ie away from the centre of the polyhedron) by 0.05mm. Modify its appearance to a pale coloured wood to mimic plywood, then roll the History Marker fully forwards and edit the first circular pattern to select the new body as the pattern type instead of the original kite profile. Repeat for the remaining three circular patterns. The third screenshot shows the result for the simple cross shape. Remember, the actual light shade will be less pointy, more spherical, because the tips of adjoining elements are held flat against each other where they’re fastened together.
4. You can also go into the Render workspace in Fusion 360 to model the final design and get an even better impression of how it will look. The 5th picture in this step is a rendered image of the design I made using the curvy shape with a central hole in the 4th picture, and looks very like the actual light shade (6th picture). The 7th image is a rendering of an asymmetrical, curvy design painted white on the inside surfaces.

When you have a shape you’re happy with, add the 4 assembly holes centred on the four corners of the kite.

Finally, create a vector file (SVG or DXF) of the shape if you intend to laser cut the elements, or print it as a 1:1 scale PDF if you're going to cut it by hand. In Fusion 360 you right click on the sketch in the browser and select Save As DXF. (This method of creating a DXF file still works for those with a Personal Use licence even after the recent changes.) For a PDF you'll need to select the appropriate body, then change to the Drawing workspace to create a drawing from it before using File, Print to print that drawing.

I know from my Fusion 360 model, the online deltoidal hexecontahedron calculator and from measuring the finished article, that a kite-based element of the dimensions shown in the PDF attached to this step will produce a sphere of approximately 38cm diameter. (NB. This is a little smaller than the 40cm diameter polyhedron modelled in Step 1.) This kite has long sides that are 107.886 mm long and short sides of 70.086mm (measuring from the centre of the hole in each leg of the element). To scale up or down, all you need to do is multiply by the relevant factor. For example, say you want a 50cm diameter shade instead, then the dimensions of the kite need to be:

long side = 107.886mm x 50/40 = 134.858mm

short side = 70.086mm x 50/40 = 87.608mm

(Edit: The denominator in the above equations should of course be 38, not 40.)

If you’d rather not do the maths, you can produce a sphere of whatever size you want using the online calculator to calculate the kite’s dimensions – just enter half the desired diameter as the “midsphere radius” and that will be close enough. (The assembled shade will curve itself into a near-spherical shape rather than staying polyhedral with flat faces, so the size will be approximate.)

Keep in mind the weight of the finished lamp shade, especially when resizing or re-designing the element. My 38cm diameter plywood shade weighs about 350g (12oz), including the weight of the nylon nuts and bolts. The strength of the electrical flex isn’t likely to be a problem for larger shades as even a slim 0.5mm² flex can carry 2kg, but the ceiling rose, and potentially the lampholder too, will need to have built-in strain relief, and the ceiling rose should be screwed to a joist or some other firm fixing, not just plasterboard.

Also, consider the possibility of the shade getting too hot if you reduce its size. Keep the risk to a minimum by using a low wattage LED bulb and, if necessary, positioning the bulb in the centre of the shade rather than near the top – see Step 8. Another issue with a small shade is that the individual elements need to curve more tightly at the ends of their legs where they are bolted together, and consequently there will be a size below which even thin plywood is insufficiently flexible.

Lastly, you may need to re-size the assembly holes if you aren’t able to source suitable 3mm (M3 – that’s about 1/8”) nylon bolts or other fasteners. Ideally, buy your bolts before you cut the pieces and check the hole size you’re planning to use is suitable by laser-cutting (or drilling, if that’s what you will do) a test hole first. You want a little clearance, but not so much that the joints are sloppy. The holes in the attached DXF file are 3.0mm diameter, but they come out slightly larger because of the laser cutter’s kerf (width of the cut). This size worked fine with my M3 bolts but may not be right for all nominally M3 fasteners, nor for all laser power and speed settings.

To make a light shade like mine, based on a curvy element with a central hole, import the attached DXF file into Fusion 360 (use the Upload command in the Data Panel) or other suitable software and tweak it as needed. Or for a hand-cut version, adjust the printing scale of the PDF to give the size you want.

You should now have a vector file of the element you are going to cut, either one that you have designed yourself or the DXF downloaded from the previous step, resized if necessary. Alternatively, print out the PDF from the previous step and use it as a template to cut the elements by hand. Perform a test cut, with the shape orientated on a piece of plywood such that the direction of maximum flex is parallel to the short axis of the kite, ie the line between the two side legs. This will minimise the chance of cracking.

Assuming your sheet of plywood has been rolled up and therefore has a tendency to curve in a particular way, lay it on the laser cutter bed such that the inside of the curve (which will be the inside of the shade) is the side that will suffer more from scorch marks during cutting. You may need to tape the plywood down if it wants to curve. Alternatively, I've found that using a steam iron on a wool setting is an effective way of flattening aviation ply – press it for a minute or so on both sides with plenty of steam then lay it with the concave face downwards and place a weight in the middle to keep it flat while it dries and cools.

After cutting a single element, check that it can be bent gently in two directions into part of a spherical surface of approximately the right radius - shaping it over a bent knee works well. To be on the safe side, you may wish to cut five first and join them into an “umbrella” – see Step 7 - to test whether the plywood you are using will work. If it's too stiff, or it breaks, you’ll either need to find an alternative material or apply some steam to make it more flexible.

To minimise wastage, plan the layout of multiple elements on rectangles of plywood that will fit onto the bed of the laser cutter. The long central axis of each one should run perpendicular to the direction of greatest bendiness. I found that with a 600mm x 300mm (2ft x 1ft) bed I could fit the greatest number (13) onto each sheet of that size if I orientated them with the long central axis parallel to the short sides of the rectangle, so I cut up my large sheet of plywood accordingly. It cut using a Stanley knife run against a straight edge, although it did need several passes of the knife to cut right through.

To get the most efficient layout, you can position individual elements in whatever CAD system you use, or use free nesting software such as Deepnest or Nesting Center, or simply draw around the test element you cut out onto a sheet of paper the same size as the laser bed. Remember to allow for twice the kerf thickness between any two elements.

The bigger the bed of the laser cutter, the less wastage there will be. I cut 13 elements from each of five 550mm x 300mm pieces of plywood, giving a total area of 0.825m². This gave me 5 extra in case of breakage, although none actually broke. I started with a 1,530mm x 1,530mm sheet, plenty for two 38cm diameter lamp shades.

I suggest cutting a maximum of 55 elements until you have decided how the shade will be attached to the pendant light fitting – see Step 8. If you want it to hang so that it's symmetrical, and/or you want the bulb to be in the centre rather than towards the top, then the final 5 elements will need to be modified to attach to a spider.

Set up the laser cutter’s software to cut out all the bolt holes and then the central voids before moving on to cut around the outlines. Make sure you lay the plywood on the bed the right side up, so that any burn marks will be on the inside of the shade (and so less obvious, even if you’re not going to paint the inside). Cut the elements, retaining the “negatives” (ie what’s left of each bed-sized sheet of ply) because they may be useful when you come to paint or stain the elements.

With a barely damp cloth, wipe around the edge (including the edge of the central void) of each element to remove the sooty residue. If there are any smudges on the flat surfaces, they can be removed with a pencil eraser, or by light sanding, but that is best done after painting because sanding will also be needed to remove any paint that’s in the wrong place.

There’s no real need to paint the inside of the shade, but doing so covers up any scorch marks from the laser and - assuming you use white or another pale colour – helps to bounce the light around. It would look pretty cool to paint the inside a bright or dramatic colour that goes with the décor of the room.

Alternatively, you could stain the outside surface of each element a darker colour.

Depending on how thick the paint you are using is, the easiest method of applying colour might be to place the elements back in the “negative” sheet from which they were cut, then use a mini roller over the whole sheet. Having them sitting within their outlines means they move around less and consequently there’s less chance of getting paint or woodstain on the underside. However, if the paint is quite thick, the elements will stick to the roller and lift out of position. That happened with me and I had more success picking up each one in turn and painting it using a foam “brush”.

I used a flexible paint from Wickes that’s meant for use on ceilings and plasterboard walls that could move a little. It’s called (no prizes for guessing this) Flexible Paint and only comes in brilliant white, but it could possibly be tinted using artists’ acrylic paint – you’ll need to experiment if you want to go down that route. Otherwise, the safest option would be to use a stain or several coats of watered-down emulsion that will soak into the surface of the wood rather than lying on the surface, because a layer of ordinary paint will most likely crack or flake off when you come to assemble the shade and force the flat plywood elements into a curve. But be cautious about over-wetting the plywood in case you cause the layers to separate or raise the grain.

After painting, you’ll probably need to clean paint from the edges here and there using a knife or fine sandpaper. If this results in the dark, scorched colour disappearing in places then it can be touched up with a permanent marker pen or artists’ acrylic paint mixed to match the colour.

A deltoidal hexecontahedron has 12 vertices at which 5 faces join, 30 at which 4 join and 20 at which 3 join. The 5-face vertices are at the narrow corner of the kite where the two long sides meet, the 3-face ones at the wide corner where the two short sides meet and the 4-face ones the other, medium-angle corners. Fortunately, you don’t really need to think about this when assembling it, just proceed as follows.

1. Cut the bolts to length. Place a single bolt through 4 pieces of scrap ply or elements – don’t worry about which hole in each to use, you’re just working out how thick 5 layers are. Put on a washer to give a little clearance if you have one, or if not a sheet of thick paper, then a nut (preferably a steel one). Do up the nut just short of finger tight so that the 4 bits of plywood can be fanned out and rotated individually around the bolt. Then place a 5th one on the bolt, on top of the nut. Cut off the surplus piece of bolt flush with the surface of the plywood using sharp side-cutters or end-cutters. Take off the top element and undo the nut, which should clear up any minor damage to the thread. Cut another 11 bolts to the same length (or just 10 more if you're using a 5-legged spider), again with nuts on them. Then do the same with 30 bolts of the right length to go through 4 layers of ply and 20 for 3 layers. For a spider, you’ll also need 5 bolts to go through 2 layers. Keep each group in a separate pile.
2. Join the end of the long leg (equivalent to the narrow corner) of each of 5 elements together with a bolt of the right length, doing up the nut finger tight. The nut should be on the inside, the bolt head on the outside, unless you want to make a feature of the nuts. Place a 4-layer bolt through the hole in one of the side legs of an element (equivalent to the medium corner) and then through the adjacent side leg of the element next to it. Put on a nut but leave it very loose, you’re going to be undoing it again soon. Do the same with another three pairs of side legs. When you come to the final pair, there will be a gap between the bolt holes. You need to gently coax this umbrella-like assembly of 5 elements into a concave bowl shape, to bring the bolt holes together. A second pair of hands will be useful, and a former of some sort – I used my knee. If you're afraid of breaking the plywood, try steaming the assembly over a kettle for a couple of minutes to make it more flexible.
3. Repeat for another 10 groups of 5 elements. Now there are eleven umbrellas.
4. Working with 6 umbrellas, 5 of them need to surround a central one. Start by bolting 3 umbrellas together using the hole in one middle leg (the hole at the wide corner of the kite) from each of them and a 3-layer bolt, doing up the nut finger tight. Then connect the side legs together on each side of this new junction, using the bolts that already hold the central umbrella together. Work around the remaining sides of the central umbrella, joining 3 more to it, and to their neighbours, in the same way. As you add the fourth element to each junction that has a 4-layer bolt, do up that nut finger tight. When you get to the last junction, you should find that this assembly of 30 elements is approximately hemispherical.
5. Work around the edge of the hemisphere adding the remaining 5 umbrellas. As before, the middle legs join only to other middle legs in groups of three while the side legs of each shape join to the side legs of the adjacent shape and to the junction of two side legs in the already-completed hemisphere.

With 11 umbrellas (55 elements) connected together, you should still be able to get your hand inside holding a light bulb. Check all the nuts are sufficiently tight. Then insert the lampholder from the outside through the central void in one of the elements. (NB. This will only work with a standard UK 29mm diameter lampholder. See next step if yours is a different size.) Fit the lamp shade retainer ring from inside the sphere to hold it in place on the lampholder, then insert the bulb.

If you’re happy with the appearance of the light shade suspended in this way, then go ahead and cut another 5 elements and use them to close the sphere. It should be just possible to hold a nut in place with the index finger of one hand, inserted through a gap in the sphere, while doing up the screw on the outside with the other hand. But if you’d rather that there was a vertical axis of rotational symmetry, or your lampholder is a different size, or you want the bulb to be in the centre of the sphere, then go to the next step.

In the previous step I explained how to suspend the shade by treating the central void of one of the elements as if it were the ring on a lamp shade that fits over the lampholder. You may not want to do that, either because the lampholders in your country are not 29mm diameter, or because you’d prefer the bulb to be on an axis of rotational symmetry, or you’d rather it was at the centre of the spherical shade instead of near the top. There are solutions to those issues.

For a smaller lampholder, it may be possible to use a laser-cut plywood ring of the correct diameter under the central void of one of the elements, to trap it in place when the lamp shade retaining ring is tightened. Otherwise, simply alter the central void of one element to make it a circle of the appropriate size. That method should also work for larger lampholders, as long as they aren’t too big.

Another solution is to take the flex from the ceiling rose through the sphere, so that the lampholder and bulb are in the middle of it. To do that you'll need to enlarge the hole in either the long leg of each of 5 elements, or the central short leg of each of 3, to allow the flex to pass through the centre of a 5- or 3-element group instead of using a bolt. The five or three legs still need to be held together to keep the spherical shape, and they need to grip the flex to prevent the shade sliding down it. A suitable cable gland (clamping, strain relief type) could be used to hold the leg ends together and keep them at the correct height above the pendant lampholder, but it will need to have a threaded section long enough to cope with all the plywood layers.

The solution I went for, which I think is the neatest and also makes changing the bulb relatively easy, is to shorten the long leg of 5 elements to make them stumpier, and then create a 5-legged spider that these stumpy legs will connect to instead of connecting to each other. The spider has a central hole through which the pendant flex passes. In my first version I fitted a cable gland onto the flex below the spider to support the weight of the shade so that the lampholder can be positioned to place the bulb in its centre. Later, I drilled out the hole to take the threaded body of the gland itself, which looks better. A 13mm hole is suitable for a PG7 gland.

The DXF and PDF files attached to this step are for the stumpy element and the spider with a 13mm central hole. The spider can be removed by undoing the bolts connecting it to 4 of the 5 stumpy elements, giving access to the lampholder, and the resulting gap in the sphere is big enough for a 55mm diameter bulb to squeeze through.

It may be possible to do something similar at a three-element junction instead, but I doubt it with my design of element because the leg that leads to the widest corner of the kite is already short.