Exploring possibilities
What questions do these initial observations suggest?
Perhaps it might be illuminating to think about the ‘seed’.
Can the ‘seed’ be just any arrangement of dots and linesegments?
Suppose we start by exploring these three ‘seeds’:
We already know that we can make a Cretan labyrinth by joining up, in a particular way, lines and dots of the first ‘seed’ on the left.
I have ‘made up’ the other two ‘seeds’. What happens with them?
We could join points like this…
...and like this:
Have we created labyrinths?
The pattern obtained using the middle ‘seed’ is a labyrinth, but the other one isn’t.
However, if you experiment, you will find that other ways of joining the points of the middle seed do not create labyrinths!
An interesting line of enquiry would be to make up other ‘seeds’, and experiment with various ways of joining their points. Maybe you can discover common properties of successful ‘seeds’, and what has to be true about successful systems for joining the points?
Or you might first investigate further the new labyrinth that we have managed to create:
It has only five levels – fewer than the Cretan labyrinth.
Having earlier seen several superficially differentlooking versions of the Cretan labyrinth, you might think of bending and stretching the new labyrinth to look at other topologically equivalent versions of it.
After playing around with this idea for a while, you might go one step further – before bending and stretching the lines, cut into the labyrinth by splitting down the middle of part of one side of the path just beyond the entrance. We can bend and stretch this labyrinth into a ‘rectangular’ shape:
The ‘rectangular’ shape that we have ‘chanced upon’ gives rise to a promisinglooking new line of enquiry – if you do a similar thing to the Cretan labyrinth you may begin to see a link between this fivelevel labyrinth and the Cretan labyrinth:
Seeing this labyrinth...
...might encourage you to persevere further with this line of enquiry.
Questions have arisen that we haven't yet answered. And there's much about labyrinths still to find out.
Here is a more structured class activity  designed by Tony Phillips, Professor of Mathematics at the State University of New York  in which students are shown criteria that help them design their own labyrinths.
Wanting to ‘get to the bottom of’ ideas and situations that appear to have little or no practical significance has driven mathematical enquiry for hundreds of years. And sometimes, perhaps many years later, the findings of apparently ‘recreational’ mathematical enquiries have proved to be profoundly useful in scientific discovery.
