Interlocking Icosahedron Light Shade

A couple of years ago I produced an ’ible called Polyhedron Light Shade for a near-spherical shade made out of plywood and based on a deltoidal hexecontahedron, a 60-faced Catalan solid. It got a fair bit of attention and was even featured as a Top Project in HackSpace magazine. But making and assembling 60 pieces of ply is a bit of a faff, especially if you don’t have a CNC machine to do it with. Which started me wondering how to create a similar shade, still nearly spherical, but with fewer faces.

This interlocking icosahedron light shade is the result. It's based on a regular icosahedron, a Platonic solid with 20 identical faces, all equilateral triangles. It's still much easier and quicker to make if you have access to a laser cutter or possibly a CNC router. But if you don’t and are prepared to cut polypropylene or cardboard pieces individually with scissors, at least you’ll only have 20 to cut to produce quite a dramatic shade that casts interesting shadows. That means it's economical to make too, the material for mine cost just over £7.

For those interested in the design process or who perhaps want to create their own variation, carry on reading. But if you just want to make a shade that’s approximately 300mm diameter from 0.75mm (0.03”) polypropylene sheet, then jump to Step 7 where you'll find the DXF files or PDFs needed to make one.

All you need is:

• a sheet of 0.7-0.8mm (0.03") thick polypropylene or single walled cardboard, 1100mm x 650mm is plenty
• access to a laser cutter or ...
• ... a printer, cardboard, pencil and scissors

The previous hexecontahedron design needed so many faces because they are joined at their corners, and the only sensible way to join flat pieces of material at their corners – ie single points – is to lap one over the other. If you try to do that with fewer, larger pieces of relatively stiff material like plywood then each one has to bend quite a lot and they will likely break. And if you do it with a bendier material like polypropylene then it may well flop and sag instead of holding a spherical shape.

The solution is to join the individual elements that make up a shade at their sides, not their corners. When the zone of contact is a line rather than a point, it stiffens the structure and makes it better able to support its own weight. Joining the polyhedron along its edges means it’s possible to interlock adjacent elements rather than overlapping them, and that gives quite a different appearance to the overall design. It also means that the faces can remain flatter, so it’s not necessary to have a hole in the middle of each one to allow it to bend without snapping – although I think it looks better if the centre of each face isn’t solid.

For me, 20 faces is a good compromise which avoids having to cut and assemble dozens of elements but still produces an interesting, near-spherical shape. Having settled on a regular icosahedron, I again used Rechneronline's geometry calculator as a shortcut method of working out how long the triangles’ sides should be to give me a shade of the size I wanted – although I did need to adjust things later to fit the elements on the laser cutter’s bed in a non-wasteful way.

I used Fusion 360 to design this shade. The instructions that follow assume you are using it too, but you should be able to follow the same process in any CAD package. NB The design shown in the Fusion 360 screenshots that follow is not exactly the same as the shade I made which is shown in the photographs and rendered images, but the DXF files and PDFs attached in Step 7 are for the version I made.

The first job is to produce an icosahedron in your CAD package, to enable you to see what the assembled shade will look like. There are several online Fusion 360 tutorials for this that you can find with a Google search, such as Kristian Laholm's, but in the end I did it by creating a triangular pyramid and patterning it multiple times, much as in my Polyhedron Light Shade 'ible. This method produces lines (the three shorter edges of the pyramid) from the centre point of the icosahedron to every vertex, and these lines are useful as axes of rotation when you come to pattern the individual elements that make up the shade in Step 4.

This is the process:

1. Create a sketch in the XY plane and draw a horizontal line of length 158mm to form the base of a triangle. (Referring to the Rechner calculator, this is the edge length necessary to give a circumsphere radius of approx 150mm, ie a diameter of about 300mm / 12".) Now draw the other two sides of the triangle and constrain them (with the equal constraint) to make them the same length as the first side. Draw in construction lines from each corner to the midpoint of the opposite side. Place a point where these lines intersect at the centroid of the triangle, and centre the triangle at (0,0) by making the centroid point coincident with the origin. Finish the sketch. (Screenshot 1)
2. Create a new sketch in the YZ plane and draw a vertical construction line upwards from the origin. Make it at least 170mm long, it needs to be long enough to go through the centre of the icosahedron. Project the centre line of the triangle in Sketch 1 - the centre line that's along the Y axis - into Sketch 2 and draw a line from the apex of the triangle of length d1 * sqrt(10 + ( 2 * sqrt(5) )) / 4 where d1 is the length of the side of the triangle, ie 158mm. You'll see that this line is the same length as the circumsphere radius given in the Rechner calculator, just over 150mm. Place the centre of the circumsphere by making the free end of this line coincident with the vertical construction line. (Screenshot 2) I'll call this point the Centre Point of the icosahedron - and the shade - from now on. Finish the sketch.
3. Create a new pyramidal body by Lofting using the triangle as Profile 1 and the Centre Point as Profile 2. (Screenshot 3)
4. Create 4 more pyramidal bodies by using Circular Pattern on the first one with quantity 5 and selecting the circumsphere radius line you drew in Sketch 2 as the axis. (Screenshot 4)
5. Working from the Centre Point side of this group of 5 bodies, create another Circular Pattern by selecting one of the bodies and one of its long (58mm) edges - not the one that isn't touching another body - as the axis. Quantity 5 again, but supress the instance that's not needed. You should have 8 pyramidal bodies. (Screenshot 5)
6. Continue selecting individual bodies and patterning with quantity 5 around the appropriate edge of the body, supressing the unwanted copies, until you have 20 bodies forming a complete icosahedron. (Screenshot 6) It helps to have an image of an icosahedron to look at while you're doing this, such as the one in the Rechner calculator. It took me 7 circular patterns in all, but I wouldn't be too surprised to find it can be done with fewer. It's not the end of the world if you end up with duplicate bodies.

That done, we can move on to the fun part, designing the triangular face.

Create a new sketch in the XY plane and project everything from Sketch 1 into it. Working in one third of the triangle only, design away.

I started by drawing a construction circle centred on the triangle’s centroid and passing through its corners. I offset it by 15mm to give the outer boundary of my element – the offset is necessary to keep the boundary away from the corners of the triangular faces of the polyhedron, things could get messy if the interlocking edges of the elements extended to those vertices. Then offset again to give the width of the curvy “arm” of the element. (Screenshot 1) With 0.8mm thick polypropylene and a shade of this size, you probably don’t want to go any less than about 12-15mm. (In my initial design the arms are only 11.8mm wide, which makes them a little too spindly and floppy for my liking. I’ve gone up a bit in the version shown in the screenshots. There are other small differences too.)

Mirror the inner and outer curve of the arm in the side of the triangle then cut a gap in the outer piece. This is necessary to allow the arm of one element to be inserted through the arm of the adjacent element during assembly, and it can be done by drawing a radial line through the arm where you want to cut it, trimming the two curves at that point, filleting the cut end to give a rounded shape and then mirroring that side of the outer arm to make the other side of it the same. (Screenshot 2)

Now you need to put in connectors to join the inner section of the arm in one third of the triangle to the one next to it. I drew a circle centred on the corner of the triangle and offset it by the same amount as before to give the connector width. If you first draw a small construction circle in the centre of the triangle you can make the other circle touch it, which will (after patterning in the next stage) give a pleasing curvy hexagonal hole in the middle with equal length sides.

Trim away the unwanted lines and mirror the connector. (Screenshot 3) Then pattern around the centre point to create the full shape (Screenshot 4). I’d suggest finishing the sketch and proceeding to the next step now, to get an impression of how the finished shade will look, before creating the slots for assembly. You may want to come back to this sketch and revise it if you’re not happy with how it looks, and the slots are best left until the rest of the element has been finalised - see Step 6.

Extrude the element you drew in the previous step outwards, away from the Centre Point, to give a solid body of whatever thickness of material you intend to use, and give it an appropriate appearance (Screenshot 1). Turn on the visibility of the 20 pyramids that make up the solid icosahedron and make sure that their hidden edges are visible by adjusting the display settings if necessary, because you're going to need to use them as rotational axes.

Create a Circular Pattern using the new body, with quantity 5 as before, choosing the line from one corner of the triangle to the Centre Point (ie an edge of one of the pyramids created in Step 3) as the axis (Screenshot 2). Keep making circular patterns in this way, always choosing quantity 5 and an axis from the corner of the relevant triangle to the Centre Point, supressing the unwanted copies where necessary, until you have 20 elements covering all 20 triangular faces. This took me 7 goes, as before.

Turn off the visibility of the pyramids to leave just the 20 elements (Screenshot 3). Now you should be able to see quite well what a shade composed of these elements would look like. You can get an even better impression if you render it with a fake bulb as in the next step, or even cut a few shapes in cardboard and assemble them as in the photo above.

Fusion 360's rendering capability works really well for light shades if you put a light source inside the shade and provide some surfaces for the shadows to fall on.

Create a 60mm diameter sphere in the centre of the modelled shade to represent a light bulb, then give it the appearance of a 1,500 lumen emitter. This can be found under Miscellaneous, Emissive in the Appearance menu.

Create two more bodies in the X and Y planes, fairly close to the light shade, to represent walls, and a 4th to represent a ceiling over it. Apply whatever pale colour you wish to them using Appearance.

Now go into the Render workspace and adjust the view to show the shade (and bulb) with the walls and ceiling in the background. In Setup, Scene Settings choose a very low level of background brightness. Then perform the render and let it go to at least the Excellent level. You’ll see the shadows the shade casts, as in the images above.

Having seen a rendered image of what the light will look like, go back to your design and make any changes you feel are necessary to improve the aesthetics and the practicalities, eg does it look good from all angles? is the bulb well enough obscured to prevent glare? Also check how well multiple elements will fit into your sheet of polypropylene, and onto the laser cutter's bed if you intend to use one. It may be that by reducing the size by a small amount you'll be able to reduce wastage considerably. On the other hand, you may decide to scale things up to make the most of whatever material you have.

Once the scale is fixed, you can add assembly slots to the design. The elements need to slot together reasonably tightly or the shade will fall apart every time it's touched. But on the other hand, they shouldn't be so tight that it's a struggle to interlock the elements and they distort rather than lying where they should, on the faces of the underlying icosahedron. So it's important to make the slots the right width.

Polypropylene melts readily and as a result the kerf (cut width) when it's cut with a laser is relatively large. The elements attached to Step 7 are designed with slots just 0.1mm wide, despite the fact that the thickness of the sheet I used is about 0.75mm, and they are a good fit. I cut a pair of identical test strips with slot widths varying from zero (ie just a single cut line) to 0.7mm in 0.1mm increments to work out which was best - see photo. Ideally, you should do the same and resize the slots if necessary, because the kerf will depend on the laser power and speed used and probably the particular brand of polypropylene sheet and even its colour too.

Go back into Sketch 3 (or whatever sketch contains the latest, tweaked and re-sized version of your design) and draw a construction line across the pointy bit of each arm in one third of the design. Offset these lines on either side by 0.05mm, or half of whatever measurement your slots test has shown is right. Then draw a line joining the mid points on either side (Screenshot 1) and trim away the unwanted lines to make slots. Replace the straight line at the base of each slot with a tangent arc (Screenshot 2) to avoid sharp internal corners that could possibly tear. One slot in each arm must open outwards and the other inwards (Screenshot 3). Finally, pattern the slots using a Circular Pattern onto the other two arms (Screenshot 4).

Think about how your shade will fit over a pendant light fitting or the lampholder of a table or standard lamp. One of the 20 elements needs to have a suitably sized circular hole in the centre instead of the curvy-sided hexagon. I created a copy of my final design in a new sketch and replaced the curvy hexagon with a 28mm diameter hole to suit standard UK fittings.

With access to a laser cutter, producing the elements is a quick process. I was able to fit 8 elements at a time on the 600mm x 300mm bed which meant I cut a group of 8 twice and then did a final cut of the 17th-19th ordinary elements and the 20th one with a circular hole for the light fitting. Using a 60W carbon dioxide laser (CO₂ lasers are very suitable for cutting polyprop) I found a power level of 20% (ie 12W) and a speed of 20mm/s worked well.

It is apparently possible to cut polypropylene sheet with a CNC router if an appropriate bit is used, but I haven't tried it. You'd probably need to leave connecting tabs to be cut off manually afterwards, to prevent the elements shifting around as the cut nears completion.

The old-school method would be to print the PDFs attached to this step on cardboard (or stick a paper print onto cardboard), cut them out and use them as templates, drawing around them onto the polypropylene sheet. A soft pencil marks well, especially if the sheet has an orange-peel finish, but is easily rubbed off afterwards. Then cut all 19+1 elements out using scissors and/or a craft knife. Alternatively, just print all 19+1 copies onto cardboard and make a cardboard shade.

Attached are two DXF files and two PDFs, one each for the ordinary element with a curvy hexagonal centre (PP icos element, cut 19 of these) and the element with a 28mm diameter hole to go over a standard UK lampholder (PP icos element ring, cut 1). The PDFs are sized to print on A4 paper. Print at actual size and check the dimensions before using them to create elements.

It's a good idea to give polypropylene elements a wash if they've been laser cut, to remove the slightly sticky dust that the process seems to generate. Just swish them about in warm water with a little washing up liquid (aka dish soap) added, rinse them and then let them dry.

Examine the elements to see if one side of the material is different from the other and, if so, which you’d rather have on the outside of the shade. The white polyprop I used is smooth and quite shiny on one side but has a very fine orange-peel surface on the other. I opted to have the shiny side on the inside, to reflect as much light as possible. If the difference is subtle, don’t worry about having to ensure that you assemble all the elements the right way around because once you’ve put the first two together you fix the orientation - the slots will only interlock when they are all the same.

(NB. The photos of the part-assembled shade attached to this step show it from the inside for clarity, but you will be assembling from the outside with the ends of the "arms" sticking outwards.)

Start with the element with a round hole in the centre, which will go over the light fitting. Attach an element to each of its 3 arms by interlocking the arms together. One pointy end of each element's arm will need to be inside the one belonging to the adjoining element and the other outside, in order to get the slots to fit together. The arms protrude on what will be the outside of the shade, just check you have the right side facing outwards before going any further.

You will now have the round-hole element surrounded by 3 others (2nd photo). Interlock two more elements to each other, then put them between one pair of the 3 surrounding elements, interlocking the arms at each side. Note as you do so that a gap is created (which is at a vertex of the icosahedron) surrounded by 5 “arrows” formed by the pointy ends of the arms. Look out for this pattern appearing as you progress – there should always be 5 arrows pointing towards these gaps (3rd photo).

Do the same with the other two pairs of the 3 surrounding elements. The result is that 10 elements have been used and half of the shade has been constructed (4th photo). Put this half to one side for now.

Assemble the remaining 10 elements to build the other half in the same way, ie surround one element with 3 others then put a pair of elements between each pair of those three (5th photo).

You now have two halves, each with 6 unconnected arms sticking out. Connect the halves at these 6 places to complete the assembly (6th photo).

FOR SHADES MADE OF ANY FLAMMABLE SUBSTANCE, INCLUDING POLYPROPYLENE, USE ONLY A LOW WATTAGE LED BULB AND ENSURE THERE IS ADEQUATE CLEARANCE BETWEEN THE SHADE AND THE BULB TO AVOID ANY RISK OF FIRE. THE FIRST TIME YOU USE IT, FEEL THE INSIDE OF THE SHADE THAT'S NEAREST THE BULB WHEN IT HAS BEEN ON FOR 30 MINUTES TO CHECK IT HASN'T GOT NOTICEABLY WARM.

Remove the bulb from your light fitting and unscrew the shade ring. Take the new shade and slide one of the joints apart in the ring of elements surrounding the one that will go over the lampholder, so you can slip your hand inside. Fit the shade in the usual way, holding it in place with the shade ring, then replace the bulb. Finally, reconnect the two elements to complete the shade, stand back and admire it.

I think these spherical shades look best in a pendant fitting when the bulb is in the centre, particularly for larger shades. You can achieve that by using a cable gland - see Step 8 of Polyhedron Light Shade.