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Secondary Magazine - Issue 64: Focus on


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Created on 01 July 2010 by ncetm_administrator
Updated on 20 July 2010 by ncetm_administrator

 

Secondary Magazine Issue 64
 

Focus on...labyrinths

Philosophy is written in that great book which ever lies before our eyes – I mean the universe. This book is written in mathematical language and its characters are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.
Galileo Galilei, from Il Saggiatore, published in 1623

orange circles

A conventional definition
When we try to apply mathematical language to the study of labyrinths we need to be clear about exactly what we are using the word ‘labyrinth’ to describe.

History suggests that we define a labyrinth as a single path from the outside ‘world’ that intertwines with itself, until eventually it reaches a dead-end, which is the ‘centre’ of the labyrinth.

If you travel through a labyrinth you never have to choose between possible routes. When you reach the centre there is only one way back – along the path that you have already trodden. 

It is helpful to regard labyrinths as special mazes.
 

green circles

Origin
Thinking about, and making, labyrinths are very ancient human activities.

People have assumed that the legendary labyrinth in which Theseus killed the Minotaur is the ‘circular’ labyrinth depicted on a 3rd century BC tetradrachm coin found in Crete. The coin also bears the inscription ΚΝΩΣΙΩΝ (Cnossus) thus linking it to the Palace of Knossos, which is traditionally associated with the Minotaur legend. This particular labyrinth is therefore known as the Cretan labyrinth.

cretan labyrinth at Tintagel in Cornwall photogrpah by Simon GarbuttMap of Jericho in 14c Farhi Bible by Elisha ben Avraham CrescasYou can see it on the cliff face at Rocky Valley at Tintagel in Cornwall, where it was cut into the rock – possibly during the Bronze Age.

It appears in a 14th century Farhi Bible
as a plan of the walls of Jericho.
 


Labyrinth on the portico of the cathedral of San Martino at Lucca, Tuscany, Italy. Photograph by Beatrix used under the Creative Commons Attribution-Share Alike 2.5 Italy license. It also appears on the portico of the Duomo di San Martino at Lucca in Tuscany.

The Latin inscription to the right of the labyrinth has been translated as ‘This is the labyrinth built by Dedalus of Crete; all who entered therein were lost, save Theseus, thanks to Ariadne's thread’.

Minotaur in the centre of a labyrinth from a 16th century gem, image source MAFFEI, P. A. "Gemmae Antiche," 1709, Pt. IV, pl. 31.

The Minotaur is represented at the centre of a different labyrinth that is carved on a 16th century gem that is now in the Medici Collection in the Palazzo Strozzi in Florence.
 

purple circles

The same or different?
cretan labyrinth from the Nordisk familjebok
In most representations of the Cretan labyrinth, the path ‘sides’ are drawn. The path itself is then the space between the lines – as in this image that was first published between 1876 and 1899 in one of the 20 volumes of the first edition of the Swedish encyclopedia Nordisk familjebok...

 
...and as in these other depictions of the Cretan labyrinth…

cretan labyrinths from left to right:: version of the ancient "Cretan" type of labyrinth contained in a circular disk, by AnonMoos; version of the ancient "Cretan" type of labyrinth, shown with the only curved lines being concentric semicircular arcs around the center of the whole labyrinth, by AnonMoos;  version of the ancient "Cretan" type of labyrinth, made with straight lines only, by AnonMoos; diagram of a Caerdroia labyrinth  by Simon Garbutt; version of the ancient "Cretan" type of labyrinth, drawn only with concentric circular arcs and radial lines, by AnonMoos; classical "Cretan" type of labyrinth in pentagonal form, by AnonMoos.

All seven images above are topologically equivalent drawings of the Cretan labyrinth – what others could your students design?

These are also labyrinths, but they are not topologically equivalent to each other. Is either one equivalent to the Cretan labyrinth?

medieval labyrinth by Sebastián Asegurado, and radial labyrinth

This maze is not a labyrinth because there is not one unique way to journey from the outside to the centre with the traveller passing just once along every part of the maze.

Labirinto labyrinth redrawn by McSush
 

blue circles

‘Recipes’ for drawing labyrinths
To draw a labyrinth you can follow instructions found in books or on the Internet, such as those in this YouTube video, or in this one.

Would your students enjoy making an enormous labyrinth as these did?

You are instructed to start with a ‘nucleus’ or ‘seed’, consisting of an arrangement of short line-segments and ‘dots’. You then proceed to join up, systematically, end-points of the line-segments and dots, linking a dot, or the end-point of a line, on one side of the ‘seed’ to the nearest available one on the other side. For example, to draw a Cretan labyrinth you are usually told to start with a particular ‘seed’ – shown at the top left below - and draw eight connecting lines like this:

Drawing a Cretan labyrinth from a seed and eight connecting lines

Why this method?
We can copy procedures like this without really understanding them. When we ask ourselves why they ‘work’, and begin to explore possibilities, we start to act mathematically. If we notice properties – that perhaps some examples share and that others don’t – we may be able to make some conjectures, that we can test and then try to explain.
 

red circles

Some initial observations
We see that the pattern that we’ve drawn is a labyrinth because there is just one route from the outside to the ‘centre’ that takes us just once over every part of the space between the lines:

Diagrams showing one route from the outside to the centre

There are four end-points - at what were originally the dots of the 'seed'.

There are four end points at what were originally the dots of the 'seed'.

So there must be two line-segments – which are curved in this diagram – we might think of them as two pieces of elastic string.

We can also see a series of ‘levels’, shown here in different colours. If we regard the outside as a level, there are nine levels – we could number them from 0 to 8:

Diagram showing nine levels within the labyrinth
 

orange circles

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 line-segments?

Suppose we start by exploring these three ‘seeds’:

Diagram showing three labyrinth 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…

We could join points from the seeds

...and like this:

And like this

Have we created labyrinths?

Have we created labyrinths. The pattern obtained using the middle 'seed' is a labyrinth, but the other one isn't

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:

Further investigation shows that the new labyrinth has five levels

It has only five levels – fewer than the Cretan labyrinth.

Having earlier seen several superficially different-looking 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:

Bending and stretching the lines into a rectangular shape

The ‘rectangular’ shape that we have ‘chanced upon’ gives rise to a promising-looking new line of enquiry – if you do a similar thing to the Cretan labyrinth you may begin to see a link between this five-level labyrinth and the Cretan labyrinth:

Bending the Cretan labyrinth into a rectangular shape

Seeing this labyrinth...

Triple spiral labyrinth by Anonmoos

...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.

 
 
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