As promised, in this post, we'll discuss the mathematics behind sewing with pleats!
Before we begin, I will be heavily referencing the types of pleats discussed in my last post, Sewing Pleats 101. If you've not yet read it, I highly suggest taking a few minutes to do so and then returning. The math included in this post applies to ALL folded pleats and there is just a bit of math at the very end which applies to cartridge pleats only.
I will be referencing the formula sheet which I have developed, available for download below. In this post, we'll be going in the same order as presented on the formula sheet.
All caught up and ready to go? Alright, let's do it!
Variable Definitions and Notes
While we will follow the formula sheet directly, there will be some visual aids included in this post which are NOT included in the downloadable formula sheet (making it printer-friendly).
Let's discuss variables.
If you've ever taken an algebra class that included word problems, you will be familiar with the concept of defining your variables first. If you've not, a variable is any alphabetical representation of a number which can change (vary) depending on the situation. While x is most commonly used, any letter can function as a variable, including letters outside of the Latin alphabet (Greek letters being common examples).
When we use a variable in these formulas, it lets us write the formula in a way where we can use it over and over again, even though the numbers will change from scenario to scenario. This will make more sense as we go through a few examples below.
We will use the following variable definitions:
w = width of one pleat*
n = number of pleats
L = length of fabric used (total length BEFORE pleating)*
F = finished fabric length (total length AFTER pleating)*
s = amount of space between pleats
m = number of spaces*
*Any unit of measurement can be used (e.g. inches, centimeters, etc.) as long as the SAME unit of measure is used for all variables. If you will measure in inches, ALL measurements must be in inches and the same is true for any other unit of length.
We will encounter a handful of other variables as we go into more specific cases, below, but these are our essentials.
Before we move onto the formulas, let's talk about some important notes on measuring the variables.
(Measuring F and L) F, the finished length of the fabric after pleating, is measured from the beginning of the first pleat to the end of the last pleat. This is essential for about 1/3 of the formulas to work since they rely on (m = n - 1).
This also means that any "extra" fabric at either end of the pleats needs to be subtracted from L for the math to work out. In the photo example below, the first pleat begins 1/2 inch from the edge of the fabric while the last pleat ends 1 1/4 inch from the end of the fabric. This means a total of 1 3/4 inches need to be ignored to properly measure L.
I recognize this seems a bit confusing - you will typically only measure ONE of these numbers as you will either know L, how much fabric you're starting with, or F, the amount of space the pleated fabric needs to fit in.
(The Fudge Factor) If you take the example above and use it in the formulas below, you'll find the math is just a little bit off. This is what I refer to as the "fudge factor" and it is NOT included in the formulas. The fudge factor exists because these formulas account for horizontal distance only and do not include the amount of fabric used up in each "turn" of the pleats. To include this would require fabric density and careful measurements of fabric.
When sewing, you can avoid the need to account for this entirely by calculating and marking the pleat width before actually making the pleats. This will result in pleats which are ever so slightly smaller than the calculated width, but will take up exactly the right amount of fabric.
If you want to be more mathematical, this is technically referred to as a margin of error.
(The disappearing m) You may wonder why m, the number of spaces between the pleats, appears on the variable list but not in any of the formulas. This is because I have used a simple substitution to eliminate it. The number of spaces between the pleats will always be the number of pleats minus one (m = n - 1). Should you wish to derive the formulas yourself, this will be necessary to put L in terms of F.
(Cases) Each of the cases discussed below uses a UNIQUE set of formulas. While they may appear (and are) similar, they are not identical. This makes it crucial to identify which set of formulas will apply in each case. To help with this, I have included a worked example with a picture for each case.
No Space between Pleats and No Stacking/Overlapping
This is the simplest case and stems from the fact that each folded pleat uses 3 times the amount of fabric it covers. This is easy to see when looking from the side of the pleats as shown above. Knowing this fact, we can derive our essential pleats formulas:
L = 3nw
L = 3F
F = nw
Let's use the example shown in the photo. Let's say that in this case, I was solving for the appropriate number of pleats. I already knew my total fabric length L = 12 inches and decided on a pleat width of w = 1 inch.
By rearranging the first formula, I can solve for n:
n = L / 3w = 12 / (3*1) = 12 / 3 = 4 pleats
After making my four pleats, I can verify the other formulas by measuring that F = 4 inches (which I could also have solved for using the second formula).
With Space between Pleats and No Stacking/Overlapping
This case is an extension of the case above so the base formulas are very similar. Essentially, the only change is that we need to account for the amount of fabric between each pleat by adding sm AKA the amount of space between the pleats times the number of spaces between each pleat. Remember also that m = n - 1 which allows for a substitution. This provides the following base formulas for this case:
F = nw + s(n - 1)
L = 3nw + s(n -1)
L = [F(3w + s) +2ws] / (w + s)
*If you're not already referencing the formula sheet (download at the top of the post), I highly recommend it as the formatting is much easier to read.
Let's use the photo above as an example. This time, let's say I know that the finished pleats need to take up F = 4.25 inches and have decided that my pleats will be w = 0.75 inches wide and I want n = 4 pleats. I need to find out how much space to leave between my pleats, s, and how much fabric I will need in total, L.
Since I'm solving for two variables, I will need to use two equations. Let's start with s. On the formula sheet, I have rearranged the first equation to put s in terms of F, n, and w. This means I can solve for s even though I don't know L:
s = (F - nw) / (n - 1) = [4.25 - (4*0.75)] / (4 - 1) = (4.25 - 3) / 3 = 1.25 / 3 = 0.416
--> round up to 0.5 inches
Now we can find L using either the second or third equations. I've picked the second as it's a bit simpler:
L = 3nw + s(n -1) = (3*4*0.75) + 0.5*(4-1) = 9 + 0.5(3) = 9 + 1.5 = 10.5 inches
No space between Pleats and Complete Stacking (e.g. double, triple pleats)
Now that we're past the basics, we can start to get a little tricky. We will now need a new variable: k = stacking factor (e.g. 2 for a double pleat, 3 for a triple pleat, etc.)
The impulse here is to assume that a double box would multiply the amount of fabric used by a factor of 2, but that would be incorrect. Look closely at the photo above - one triple box pleat is on top while one double box pleat is on the bottom. Count the layers of fabric used in each of the pleats. There are five layers for the double pleat and 7 layers for the triple pleat. Recall that for a single pleat, we use 3 layers.
This gives us the pattern of 3, 5, 7, etc. Each time we increase the stacking factor, k, by 1, we add two layers of fabric. What this means formulaically is that instead of multiplying by 3 to get the amount of fabric used in a single pleat, we will multiply by (2k + 1).
Importantly, the length of the finished fabric, F, will not be affected by this change at all. F continues to rely only on the width and number of pleats in this case.
So we have the following formulas:
F = nw
L = nw(2k + 1)
L = F(2k + 1)
Take a look at the example shown below:
In this case, I used k = 2 for double box pleats, starting with L = 13 inches of fabric. I wanted each box pleat to be w = 1 inch wide, so the question is how many pleats, n, can I create?
To solve, I'll rearrange the second formula:
n = L / [w(2k + 1)] = 13 / [1(2*2 + 1)] = 13 / 5 = 2.6
--> round to n = 2.5 pleats to account for the fudge factor
In making the pleats, as you can see in the photo, this works out correctly to 2.5 double box pleats.
Just for fun, you can try this same math using k = 2 and w = 0.5 inches which would be the math if we counted each part of the box pleat as a separate knife pleat. The results work in this case too, with n = 5.2 pleats (round to 5).
With Space between Pleats and Complete Stacking (e.g. double, triple pleats)
Now we can get extra tricky by adding spaces between our stacked box pleats. If you compare the set of formulas for "With space between pleats and no stacking" to the formulas for "No space between pleats and complete stacking," this next set of formulas is the natural progression.
Note that these formulas will only work for cases where n is great than or equal to 3 pleats. A fun little quirk I discovered when working up some sample swatches.
After adding the spaces between the pleats, sm, and substituting m = n - 1, we get the following formulas:
F = nw + s(n -1)
L = nw(2k +1) + s(n -1)
L = ([2kw(F + s)] / (w + s) ) + F
Let's look at an example. To make the sample here, I had to join two pieces of scrap together which is why there is a slight color change in the middle.
For this sample, I used inverted double (k = 2) box pleats with a width of w = 1 inch. I started with a fabric length of L = 22.25 inches (yes, I know it's a weird number, my join was messy) and wanted to make n = 4 even box pleats. The question then is how much space, s, to leave between each pleat.
To find the answer, I use a solved version of the second formula:
s = [L - nw(2k + 1)] / (n - 1) = [22.25 - (4*1)(2*2 + 1)] / (4 - 1) = [22.25-20] / 3 = 0.75 inches
I can then solve for F using the first formula:
F = nw + s(n -1) = (4*1) + 0.75(4 - 1) = 4 + 2.25 = 6.25 inches
You can see from the photo above that this calculation of F is spot on!
No space between Pleats and Partial Overlapping
Mathematically, this was the most challenging set of formulas for me to find (it took me 3 tries!). This last case for folded pleats is another odd one which requires its own variable: a = width of the pleat still showing after the overlap.
In defining the variable, I found this was the easiest to measure while working with the fabric. For example, 1 inch wide knife pleats where each pleat overlaps by 3/4 inches, there would be 1/4 inch showing so a = 0.25 inches. It's also needed when making the pleats, as you'll see in a minute.
Let's start by making an adjustment for F, the finished length of the fabric. Looking at the diagram above, the finished length of the pleated section is a function of the new variable, a, and the number of pleats, n. However, notice that the topmost pleat (here shown on the left), provides its entire width, w, to the finished fabric length. This gives us the formula:
F = a(n - 1) + w
Now for the total fabric length, L. This one requires another visual explanation as it's not abundantly obvious. When making a new pleat, you must begin the first fold at a distance a inches away from the end of the previous pleat. Once folded up, the new pleat is then laid over its neighbor. This creates the spacing between the pleats. This is no different from our previous spacing variable, m, and you'll see the substitution of m = n - 1 below.
So to construct the formula for the total fabric length, L, notice that each pleat takes up the spacer length, a, plus twice the pleat width, w. In the photo above, you can actually see this by following the fabric from the end of the previous pleat around and over to my index finger which marks the end of the pleat being folded. However, we also need to address the first pleat which did not have a spacer and instead takes up the regular amount of fabric, 3w. This gives us the following formula:
L = 3w + (a + 2w)(n - 1)
When combined with the first formula, we get our last base formula for this case which does not rely on the number of pleats:
L = F + 2w + [(2w(F - w)/a]
Let's apply these new formulas by looking at the example below. Notice that this pleating pattern creates a dense fabric which can be difficult to stitch.
Here, I've decided I want w = 1 inch wide pleats that overlap most of their neighbor pleats so that only a = 0.25 inches remain showing. I know the finished length must be F = 4 inches as prescribed by my pattern. How much fabric do I need to pleat this section?
To solve, we'll use the third formula since we don't need to know the number of pleats, n:
L = F + 2w + [(2w(F - w)/a] = 4 + 2(1) + [(2*1)(4 - 1) / 0.25] = 6 + [(2*3)/0.25] = 6 + 24
= 30 inches
Full disclosure on the margin of error (AKA the fudge factor) here - I made the 4 inch sample above using only 26 inches of fabric. The margin here, however, is not as extreme as it would seem as 26 inches of fabric should have given me a 3.5 inch sample. The difference is accounted for as I am not a robot and this fabric is pretty thick. That said, were I pleating a skirt and had an extra 4 inches of fabric after forming the pleats, I would simply stitch the extra into the seam and trim it after sewing.
Cartridge Pleats Only
We come to our final case and it is, perhaps, the simplest of them all!
Some important notes as we delve into the only basted pleat in this grouping:
Here, w = length of the basted stitch = pleat width
To use these formulas, you must first make a gauge swatch and measure the number of pleats per inch
The math here is approximate due to rates of compression of the fabric. That is, the margin of error is slightly higher here than with our folded pleats. Be ready to adjust the fit slightly after pleating.
We will also need a new variable: g = gauge, number of pleats per inch (or per cm).
The first formula here is easily determined using our units of measure. For this example, I'm using inches:
gauge = (number of pleats) / inch
--> g = n / F
For our second formula, we want to determine the total length of fabric, L, needed to make the pleats. This is similar to our folded pleats, except instead of taking up 3 times the width of fabric, each pleat only uses twice its pleat width. You can see this by looking at the photo above - each pleat is one up and down. Each up and down is also the pleat width. This gives us the following:
L = 2nw
Combining with the first formula to eliminate the number of pleats gives us:
L = 2Fwg
Using the above photo's swatch as an example, the gauge here measured g = 5 pleats/inch when made with w = 1 inch basting pleats. Let's say that I am making a cartridge pleated skirt with this fabric and my skirt needs to measure F = 30 inches after pleating. I can solve for the total length of fabric I will need using the last formula:
L = 2Fwg = 2*30*1*5 = 300 inches = 8 1/3 yds
Remember, however, there is a margin of error here. Were I buying fabric for this fictional skirt, I would round up to 9 yard to purchase my fabric.
That's all for this week, friends! Next time, we'll be taking a turn back over to knitting with a new pattern release for the knit version of my Soft Waves crochet blanket.
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Until then, happy crafting!