Turning Sequences into Blankets

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Suppose you want to make a blanket (or placemat, or wall hanging,…) and you want it to be, you know, mathy. One way is to pick your favorite sequence of positive integers and use that sequence to create the blanket.

For example, suppose you decide to start with the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, … (ignoring the leading 0). Start by knitting/crocheting/quilting a 1×1 square. Here “1” might refer to a specific size (1 cm, for example) or it might refer to the number of stitches or rows that are crocheted/knitted.

Onto this initial square add on another 1×1 square, then a 2×2 square, a 3×3 square, a 5×5 square, an 8×8 square, etc. There are different ways that you could do this.

For example, you could use a spiral pattern,which gives a finished product that looks like the diagram that accompanies pictures of the Golden Spiral. (In the first diagram the numbers refer to the order in which the squares are added, rather than the size of the squares):

Alternately, you can start in one corner and add squares outward:

This is all well and good, but what if you don’t want to use the Fibonacci Sequence? This method can be adapted to any other sequence of positive integers, but the pieces added each time won’t necessarily be squares. They’ll be rectangles where one side is set to align with the previous pieces (so that you don’t have overhang) and the other side is determined by the number in the sequence.

For example, suppose you want to use the digits of π: 3 1 4 1 5 9 2 6 5 …

Start with a 3×3 square. The next number is 1, so add on a rectangle with width 1 (the height will be 3 to align with the initial square). Then add a rectangle with height 4 (the width will also be 4=3+1 to align with the two previous pieces). Then add a rectangle with width 1, then a rectangle with height 5, etc.

As with the Fibonacci Sequence, you can add rectangles in a spiral pattern (where the numbers again refer to the order in which rectangles are added)

or build outward from a corner.

One of the benefits of working with the decimal expansion of your favorite irrational number is that the numbers are between 1 and 9, so the rectangles don’t get as big as with the Fibonacci Sequence (and they don’t always grow in both dimensions). In all of these cases, you can continue until the blanket is the size that you’d like.

If choosing a sequence is too much pressure, you can use a Random Number Generator to determine the size of the rectangles. Indeed, this exactly what was done for the piece “Chance Log Cabin” (on the mostly-knitting blog January One), which is what inspired this post. Thanks January One!