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Artsy Math: Squaring Square Patterns

a Square made of many other sized squares and colored with purples, teals, peaches and pinks.

This is the second in a series of posts that blend art and math. Jump in and choose your medium to indulge in creative expression. Many have played with this through the years; a list of references is at the bottom. Thanks to Steve Heller for pinging me to play with squarable numbers and add them to inquiries.link along with some variations.

Warm-up

Grab a sheet of paper and a ruler. Make a square.

A plain white square with dark border.

Now, split this square into squares. You might do something like this with four squares:

A square that has been split into 2x2 squares

or these with 9 or 16 squares:

A square split into 3x3 squares

a square split into 4x4 squares

Now try to draw a square split into squares that are not all the same size by changing one of the smaller squares above. This time feel free to add some color to your squares.

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Ok, what did you make? How many squares does it have? Here are a couple of ideas:

One with 7:

A square that has been split into a 2x2, but then the lower right corner is split into 2x2. The coloring is peaches and pinks and purples.

You can also try joining squares together. Take the 3x3 grid; we can join four squares to change 9 into 6.

A square that has been split into 3x3, but then the top 2x2 squares were joined to make one square so there are six in total. One larger square surrounded by smaller on the bottom and right. The colors are peaches, pinks, and purple.

Making Patterns - a primer

Now, let's think about the sequences we can form and art we can make with the ones constructed.

{1, 4, 9, 16, 25, ...}

A single peach square.

A square split 3x3 colored with peaches, teals, purples, and pinks.

A square split 5x5 with peaches, pinks, teals, and purples.

A square split 6x6 colored peach, purple, teal, and pink

Now, what if you devise your own pattern? Create a rule for what comes next. For example, let's look at a pattern where we take a square that has been split into equal squares (like above), and then we join any middle squares to make a frame:

a square is framed by other squares of progressively larger sizes where the first has 9, then the next has 13 with the middle square having a side of 2, and then one with 17 squares that has a square in the middle with a side length of 3.

This would give us a sequence:

{9, 13, 17, ...

What comes next?

A large square framed by 20 squares

Did you say 21? How can we write this pattern? Here are a couple of ways:

  • 5 + 4n
  • (n + 2)2 - n2 + 1

Check to see if the next one fits how you wrote it - do you get 25?

A large square framed by 24 squares

Activity: Making Patterns - your turn

Now it's time to make some art. You can do this through folding paper, watercolors, quilting, or whatever medium suits you.

  • Come up with your method to construct a pattern by dividing and/or joining squares.
  • Decide if you will paint just one or many from the pattern.
  • Contemplate what the sequence might look like as it goes on. Does it have an end?
  • Are there rules to your pattern for how to color it? How many colors do you need so no two adjacent regions share a color?

Here is a gallery of example compositions. How were they made? What came before and what comes next?

an L frame of a 5x5 grid is on the bottom right that then frames in another smaller 5x5 and so on to the upper left corner creating an almost optical illusion of stairs going to the upper left. colors are purple, orange, pinks and teals.

a pattern where blocks of 2x3 are placed diagonal in a square and then recursive into the spaces that are not 2x3 or 3x2

a square of squares where smaller ones are more towards the upper left, but a lot of them are different sizes, but there seems to be a pattern in a pattern.

Spiraling L shapes made of squares get smaller inward with teal, peach, pink and purples.

L-shaped checker patterns frame more L-shaped patterns in a fractal-like pattern.

a square split 3x3 where each square is then split into the square number of its location from 1 to 9 and then shaded with a diagonal gradient of colors from peach to purple to teal

More to Contemplate

  • Are there other sequences that might start the same, but could go a different way?
  • What if you start with something other than one square? Like a 4x4?
  • Play around with "what ifs"
    • What if you change where you divide the squares?
    • What if you add a step to your construction method (like split, join, then split again)?
  • Look up your sequence by typing the numbers into OEIS to see if it is there.
  • Write your sequence as a formula.
    • For example: {1,4,7,10,13,...} can be expressed as 3n − 2 where n is the position in the sequence.
  • Is there a sequence where a square is divided into squares that all have different sizes?
  • Are there numbers that you can't make by dissecting squares into squares?

Example Art:

watercolor on wood of windows in windows with a diagonal teal with the rest blue.

A 13x13 quilt with 11 pieces made of floral and autumn fabric.

Resources and extensions

  • Wikipedia has some variations, including Mrs. Perkins' Quilt
    • A few who have played with this: Henry Dudeney (1917), John Conway (1964), Martin Gardner (1966), Antonick & Finkel, NYT Numberplay (2013), Math for Love lesson by Finkel (2017), and many others.
    • Fun note: to do this post, I went down quite a rabbit hole for how to piece the quilt together.
  • Origami papers are a nice tool to play with for this activity - you can cut, fold, and place papers to think about arrangements and dissections.
  • There is a tool called Squarrels on Inquiries to explore this. (Note: squarrel is the name because squirrels are fun, and the words square-able, squareable, and squarable all look funny.) This tool has hidden word surprises - type: "fish", "tree", "oregon", "all", "banana".
  • Do the same thing, but with rectangles
    • What rectangles can be tiled with squares of all different sizes?
    • Are there patterns that you can make when the outer shape is a rectangle that you can't with a square?
    • There is a tool here for this.
  • Do the same thing, but with triangles - here is a tool.
  • Try this with L-shapes as well (L into squares, L's into L's)

an L split into squares