To launch our explorations we noticed and wondered about several images:
This is the first one:
What do you notice? What do you wonder?
What we noticed:
- 6 right triangles composed of 14 square units across and 14 square units down.
- There are different shapes across the tops of the triangles. Some are the same and some are different.
- Each “triangle” has a key with four patterns made of three squares. Three of them have black-black-black as one of the patterns. The others have black-black-white.
- The second one seems like the easiest.
What we wondered:
- How do you get the shapes into the pattern underneath?
- Is there an overall pattern to producing each of the individual patterns using the “key”?
- How do the clusters of 3 fit to make a large triangle? I think the number in the big triangles is a multiple of 3. Does that mean I can make the big triangle out of little L-shapes?
- Does the key have to have 4 elements? Could there be more?
A second image helped us expand on our noticings and wonderings:
What we noticed:
- It says “rule” instead of “key”.
- There’s less black and more white as you read the rule from left to right.
- There are little and big triangles.
- The arch-looking pieces above the big triangle could be formed from using two tiles from the rule.
- Tile 4 and tile 7 are very similar.
What we wondered:
- Do they have to tesselate or is there blank background space showing? Are all the gray spaces “element gray” or “background gray”?
- What’s going on as we read the rule from left to right? There are different numbers of dark and light squares in the elements, so it’s not just about position.
- Under what circumstances could you get a big black triangle in the middle?
- Do the four squares together always have to be three on top and one underneath? Could they be flipped or rotated?
- Is there a pattern in which you have to use the tiles?
- Where do you start? Does the pattern grow from a particular spot? Does the rule decide what comes next?
A third set of images clarified the picture further:
What we notice:
- The rules have different 1’s and 0’s.
- The 0’s and 1’s correspond to whether there is a black or white tile as the stalk of the T. This seems to be true regardless of the top row.
- 11110 is 30 in binary. (But I don’t know what that has to do with the pattern.)
- 110110 is 54 in binary.
What we wonder:
- Is there a pattern happening in the top row?
- Is there a connection between the successive elements in each rule?
- Does this have to do with binary numbers? (It’s 0’s and 1’s.)
- Why are there different colors of black tiles on top of the T’s? (They are slowly filling in right to left.)
We wanted to understand how the image was generated from the key, but there were still some missing pieces, so we built one together using Polypad. To figure out the color of each box, we looked at the three boxes above it and used the rule to tell us the color. For example, the next empty box in the picture will be blue because the rule tells us that when the three boxes above are maroon-blue-blue, the box below is blue. (Try it yourself at Polypad!)
After that, we explored individually and together using another tool that generates the patterns automatically: https://discover-automata.vercel.app/
We explored some patterns that were generated from a single black cell and different rules and some other different settings. Here are some of our observations:
Some rules that have only 4 elements in their “keys” make triangles. Here are the ones we found:
Rule 6:
Rule 9:
Rule 14:
Here are some rules we liked that have 8 elements in their “keys”:
Rule 62:
Rule 30 (This one we noted seemed “disordered inside, but still a triangle”):
We called rules 126 and 129 “opposite rules” that generated “opposite shapes”:
There’s lots more to explore in these beautiful patterns. What questions do you have?
