Up till now I’ve been making crosses by cutting fabric into three inch squares with my rotary cutter and then using scissors to cut a one inch square out of each corner. This process is time consuming, produces a lot of waste, and is not as precise as I’d like, so I’ve solicited a quote from a local laser cutting business.
When I was preparing the details for the quote, I was hoping the business might be able to give me a rough price per square inch cut. Instead, they provided me with a quote for the largest piece of fabric that would fit on their cutting bed (which I suppose amounts to about the same thing). At this point I began to wonder (not for the first time) if there is a way to calculate how many crosses I can cut from different sized sheets of material. I know how many crosses fit in the 40*40cm quilts that I’ve made previously, but what if the fabric I want to cut doesn’t divide neatly into 40*40cm squares? What if my fabric isn’t even square shaped? Is it possible to come up with a formula for crosses per square inch (or at least a measurement that was less unwieldy than 40*40cm)?
I decided to try and work it out.
Minimum Unit of Repetition
I began by looking for the smallest segment of the overall pattern that will tessellate when repeated. There’s probably a mathematical term for it, but I’ve been using ‘minimum unit of repetition’ or MUR. My thought was that if I know how many crosses are in a MUR and I know how many MURs fit into a given piece of fabric, then I can calculate how many crosses I can make from fabric of any size.
After some trial and error I identified this five inch segment as the MUR:
At first glance it seems as though each MUR contains two crosses:
But, it’s not quite that simple. You see, when two MURs are combined they produce an additional cross at the joining line between them:
When four MURs are combined in a square they produce an extra cross at each adjoining line, andan additional cross at the vertex (the intersection where adjoining lines meet).
At this point I had all the information I needed to work out my formula (two crosses per MUR, plus one cross for each line connecting two MURs, and an additional cross for each vertex joining four MURs) and I no longer needed to work directly with the cross pattern. Instead, I drew up a simple grid with the rough dimensions of the laser cutting bed (to the nearest multiple of five inches). Then, I proceeded to add up the relevant MURs, lines, and vertices.
The MUR Grid
The grid above is six MURs high and nine MURs wide and fits a cutting bed with dimensions a little larger than 30*45inches.
Calculation #1: 6*9 = 54 MURs
Calculation: #2 2*54 = 108 crosses
If we take 𝑥 to designate the number of columns in our grid and 𝑦 to designate the number of rows, we can begin to work out a formula that will work for any grid made up of MURs.
Algebra: The number of MURs in a grid is equal to 𝑥*𝑦, but we’re looking for the number of crosses. There are two crosses in every MUR so the first part of our algebraic formula is 2𝑥𝑦.
Horizontal Adjoining Lines
The grid has nine columns with five adjoining lines in each column (one less than the number of rows).
Calculation: 9*5 = 45 lines.
Algebra: The number of horizontal adjoining lines in a grid is equal to 𝑥(𝑦 – 1).
Vertical Adjoining Lines
The grid has six rows with eight adjoining lines per row (one less than the number of columns).
Calculation: 6*8 = 48 lines.
Algebra: The number of vertical adjoining lines in a grid is equal to 𝑦(𝑥 – 1).
Vertices are the points between rows and columns where adjoining lines meet. If you look at the image above it appears as if eight red lines (one less than the number of columns) intersect with five blue lines (one less than the number of rows) creating forty green vertices. This turns out to be an easy way to calculate the number of vertices in a rectangle.
Calculation: 8*5 = 40.
Algebra: The number of vertices in a grid is equal to (𝑥 – 1)(𝑦 – 1).
If we add up the crosses from the MURs, lines, and vertices in our 6*9 grid we get:
Calculation: 108 + 45 + 48 + 40 = 241.
Now, let’s put our formula together, then plug in our 𝑥 and 𝑦 values and see if we get the same number!
Algebra: 2𝑥𝑦 + 𝑥(𝑦 -1) +𝑦(𝑥-1) + (𝑥-1)(𝑦 -1)
Expansion: 2𝑥𝑦 + 𝑥𝑦 – 𝑥 + 𝑥𝑦 – 𝑦 + 𝑥y – 𝑥 – 𝑦 + 1
Formula: 5𝑥𝑦 – 2𝑥 – 2𝑦 + 1
Testing the Formula
Substitution: 5*9*6 – 2*9 – 2*6 + 1
Calculation: 270 – 18 – 12 + 1 = 241