And now for a little diversion from the waistcoat.

Last weekend, a friend asked me if I could make a Dr Who fez for his cuddly orang-utan. (Yes, I know. I have some really interesting friends!) I said that would be no problem, but I would need to know what it looked like, as I’m not a Whovian myself. He obligingly linked me to some screencaps, and told me that the hat needed to be about 6 cm in diameter at the base to fit the orang-utan. From the pictures, that suggested it needed to be about 5 cm in diameter at the top and 5 cm high along the slope (not vertically).

Now this requires a little bit of maths, but for the sake of the maths-phobic I won’t go into the details here. Suffice it to say that what you need is a cone with the top cut off (the fancy mathematical word for this is a *frustum* of a cone), and with some fairly basic algebra and geometry – you don’t even need any trigonometry – it’s possible to work out how to construct it. The best thing, though, is that the same basic drawing works for a fez of any size, up to and including a large adult. In this post I’m going to show you how.

The first thing you need is a very large sheet of paper, or, ideally, thin card. I’m using thin card here because it just so happened that I had bought a large polythene artist’s bag for the purpose of carrying my embroidery around, and it had a large piece of card inside it that was perfect for what I needed. A piece of paper 110 x 80 cm will be large enough for a fez to fit any adult human being I’ve ever seen; for a child, you can get away with a smaller piece. If you need to stick two pieces together to get the right size, start working from one of the sides opposite the join – the aim is, if possible, to avoid any joins on the finished pattern piece.

Take your piece of paper and measure halfway along one of the shorter edges, and then draw a point just inside the edge from there. If it helps, draw a short line perpendicular to the edge through that point, though depending on what sort of protractor you’re using, you may not need to. Now draw a line on either side of the point angled at 18 degrees from the vertical, so that the total angle between the lines is 36 degrees. Measure this as carefully as you can, because any error in the angle will alter the size of the hat and may cause it to fit badly. You should end up with something that looks like this:

The next thing to do is to find a hat that fits you well and measure the internal diameter. If it’s not quite circular, pull it gently into as circular a shape as possible before you measure.

This is just an example, since I’m making the fez for the orang-utan here and I already know that the diameter I need is 6 cm. However, if I had wanted to make one for myself, the diameter of my hat is 18 cm, so I will also give the figures that I would use to make a fez that fitted me. If you don’t have a hat, or you do but it’s stretchy and can’t be measured, then measure around your head, get your calculator, and divide by pi. If you’re not quite sure about that, you can do this instead: tie a piece of string round your head so it fits snugly but not too tightly, lift the loop off your head, arrange it into the best circle you can make, and then measure the distance across it.

Once you’ve got your diameter, then all you do is multiply it by 5. It doesn’t matter whether you work in centimetres, inches, or for that matter Gallifreyan… er… what do they have on Gallifrey again?, provided you’re consistent. For the orang-utan’s fez, that gives me 30 cm, and if I were making one for myself it would be 90 cm. Using the point where the lines meet as the centre, draw an arc of a circle at that distance. You can do this either by measuring a series of points between the two lines at the same distance from the point where they meet and then connecting those points up into an arc, or by tying a piece of sewing thread to a pencil at one end and a pin at the other, adjusting till you have the right length, putting the pin through the point where the lines meet, and using this arrangement like a very large pair of compasses. This gives you the line which will form the base of the side piece.

Now you want to draw the arc for the top of the pattern piece. Do this in exactly the same way as before, but this time the distance from the point where the lines meet is 5/6 of the distance you used for the first arc. For the orang-utan, that is 5/6 of 30 cm, which is 25 cm. For my hat, it would be 5/6 of 90 cm, which is 75 cm. Here’s what it looks like for the orang-utan, and I’ve airbrushed the photograph so you can clearly see the area between the two arcs, which we are going to be using as the side piece.

Trace this pattern on a piece of greaseproof paper. Also draw two circles with diameter equal to the length of one of the straight sides; for the orang-utan that is 5 cm, and for me it would be 15 cm. Draw seam allowances round the pieces. You’ll see that the seam allowance I have used for one of the circles is smaller than the one I have used for the other two pieces. That is because felt doesn’t fray, so you don’t need so much seam allowance on it as you do for the lining. The pieces with the wider seam allowances are for the lining, so you may be wondering why I haven’t bothered to trace a piece for the felt side of the hat. The reason is simple: I’m going to use the original cardboard pattern for that in a few moments.

Just like this, in fact. Cut out the pattern piece that you constructed; if you used paper rather than card, stick it onto thin card now. It will make it easier to draw round it, and you will need it to be fairly stiff in order to firm up the fez.

Remember to draw a seam allowance round it, as you see in the photo. Cut out this piece and the circle with the smaller seam allowance from felt.

Fold the side piece so that the short edges match and the marked side faces outwards, and then stitch the side seam. (Sorry about the blurred picture, and one or two others that follow.)

Now open it out, press the seam open, and pin the circle into place at the top with the marked side facing out. Stitch the top of the fez into position and clip notches along the seam to stop it from bunching up when you turn it the right way out. When you have finished, you should have something like this:

Turn it the right way out, and, behold, you have the shell of your fez!

Take the two pieces with the wider seam allowances, and pin them to the lining fabric. The centre line of the side piece should run along the grain of the fabric. Cut them out.

Make up the lining exactly as you did for the shell of the fez, but don’t turn it the right way out when it’s finished. Instead, take the original cardboard side piece and tape it together at the edges (you can see I’ve gone a little bit wild with the parcel tape here), and push the lining up through the middle so that the seam meets the top of the cardboard, like this:

Turn the edge of the lining up over the bottom of the cardboard – you will need to do this carefully to avoid pulling down the lining inside the card – and tape it in place. If you’re not sure you can do this easily, you can always put a few small stitches in to hold the top seam to the upper edge of the cardboard before you start.

Now push the whole arrangement inside the felt shell, matching the seams, and stitch together at the edge. There are various ways of doing this, and the one you choose will depend on the size of your hat as well as your sewing skill. I’m just oversewing here – nothing fancy. Make sure you pull the felt firmly over the cardboard and lining so that there is a little bit of overlap at the bottom, so that you can keep your stitching on the inside rather than right on the edge.

Congratulations – you have now very nearly finished! All you need to do now for complete authenticity is to cut a short strip of felt and sew one end of it to the centre of the top of the hat so that it sticks up. If you find this difficult, cut two identical short strips, sew them together down the centre line with just a little bit left over at one end, open out these short ends like an angled bracket, and sew each of them to the top of the hat close to the join between the pieces.

Here is my glamorous assistant, Wilfred the penguin, who kindly agreed to model the finished fez before it went off to its new owner.

All he needs now is a TARDIS, and he’s hot to trot. 😀

Posted by The Mathematics of Doctor Who’s Fez | Harmonic Mama on 03/11/2011 at 12:18 pm

[…] one great site with a fantastic, mathematical approach to making the Doctor’s Fez over at Elinor’s Crafts. Elinor saved the day! Her instructions are great and more importantly the creation of the for a […]

Posted by gift card on 11/09/2013 at 9:56 pm

My spouse and I stumbled over here from a different website and thought I might as

well check things out. I like what I see so now i

am following you. Look forward to looking into your

web page yet again.

Posted by baroquemongoose on 11/09/2013 at 10:01 pm

Many thanks! 🙂