Monthly Archives: February 2014

Designing A Maze

(Information found: Design Your Own Maze)

Designing a puzzle maze
Here is a fool-proof way to make a branching puzzle maze.

Take some squared paper and draw out a rectangle with an odd number of squares on each side. This design has 19 squares on each side, but it could be a rectangle. When drawing the maze, use a soft pencil, as you’ll be rubbing lots out.

Fill in alternate lines and columns like the picture on the right, making a waffle pattern. All the white squares will end up as part of the paths. All the dark blue squares will be walls. Medium blue squares will be either paths or wall. You don’t have to use colour, of course, but shade the medium blue squares very lightly, as you will be rubbing some of them out.

Designing a unicursal maze

This type of maze may seem strange if you haven’t met it before, as you don’t ‘solve’ it. However, when you walk it, you constantly twist and turn, getting closer and further from the centre, until finally you arrive there. You walk continuously, without pausing to make choices or back-track. Children love to run through these mazes! A unicursal maze can also make an attractive pattern.

Perhaps surprisingly, it is hard to design a good unicursal maze. When you make a unicursal maze, then the main path must cover the whole area. You can’t fill in spare areas with dead-ends as you do with puzzle mazes. So I suggest that you start by looking at existing patterns to see how they work. The Cretan design has its own trick for drawing its maze; look at the Cretan webpage to see what it is. The Roman and Chartres designs have to be copied. This isn’t too hard, and it teaches you something about how these mazes are made. Let’s try to copy a Chartres maze. It’s easiest to design the paths rather than the walls of the maze.

Chartres
This is what we’re copying.
Chartres
The Chartres maze is based on concentric circles.
Chartres
Rub out from top to bottom and side to side.
Chartres
Now start from the top and join the lines as in the original.
Chartres
Now on the left. While you have the same sort of join, they are not in the same order.
Chartres
The right part has a different order again.
Chartres
And the bottom. This is the most complicated.
For three of the arms, there are two types of joins: double backs, and rejoin the circle.
Designing maze Designing maze
At the bottom, while there are double backs, there are also more complicated double backs, overlapping the others.
But by concentrating on just one part you should be able to reproduce it.

The Roman maze is based on concentric squares rather than concentric circles. See how to reproduce one on the Roman maze webpage.

Once you have copied some designs, you can try your own! First choose your shape. You’re not restricted to circles or shares. To save you time, here are some. (Right click, then click on Save As to save to your computer. Use any Paint program to make the maze.)

Concentric circles Concentric squares Concentric hexagons
Concentric pentagons Concentric stars Concentric octagons

Once you’ve chosen a shape, rub out where you’re going to make the doubling back, etc. Then join up the paths similar to the Chartres or Roman. You can bend a path back on its tracks by joining it to the next shape. You can join it to a path further in. You can continue in the same direction on the same circuit (in which case you don’t need to rub it out!) Or you can continue in the same direction, but jig inwards to the next circuit (although this is rare).

Designing maze Designing maze Designing maze Designing maze Designing maze

However, if you join paths at random, you will probably find that you don’t have a unicursal maze, but several distinct paths. You may not be able to get to the centre at all! One way to check this is to colour in the path (if using a computer, try the Paint tool). This makes it obvious whether you have a single path or not (see right). Remember that you will need an entrance, and an end (probably in the middle). Start designing maze

So now look again at the existing patterns again to see how they work. The Roman patterns are easy, since you only need to get one quarter right, then repeat it. The Chartres pattern is trickier, since you swing from one quarter to another and back again. This makes it more interesting to walk, but harder to design. Remember that you can change shape quite easily, so you can make a circular Roman design, or a square or octagonal Chartres design (or even a hexagonal one with a few changes).

Once you start designing mazes, you will start to appreciate what the problems are, and how existing mazes solve them. Even if you never construct a maze of your own, designing mazes gives you far more insight into their nature. You feel closer to the original designers of the traditional mazes. You can imagine a Roman, or medieval monk scratching in the earth, then saying “Bother, that won’t work!” Like them, you will muse “Perhaps a Greek Key will get me out of there” or “Essentially I need a wiggly spiral…” But however you do it, and whatever your success is, by the end you will get through a lot of paper and rubber, or have an aching mouse hand!

If you want a more hands-on approach, you can mark the pathway in its final location with a long piece of rope. This way you can play around with it until it looks right, and you can guarantee that you have the correct pattern which does get to the centre, and the paths don’t cross. This method allows you to design a more fluid, less structured maze.

You may wish to use a design from this site. There are modern designs which are copyright, and you should approach the designer before copying them, but you can copy any ancient design, or one of my designs, without asking permission. However, it’s much more fun to design your own. If anyone wants to download the pictures or text on this website, and use the result on a computer or in print form, feel free to do so. I’d love to hear about any of your maze designing efforts, but it’s not compulsory!

 

Constructing a flat maze

Most unicursal mazes are flat mazes, or almost flat. This means that the ‘walls’ are not really proper walls or hedges. One strange side effect of this is that you can get the paths and walls muddled up. For example, some turf mazes have the grass as the path, and some as the walls. Flat mazes emphasise the pattern. Hedge mazes are in fact rather dull to look at; all you see is a hedge. Flat mazes are usually much easier to maintain. Another advantage of a flat maze is that once you get to the centre, or if you get bored with it, you can just walk straight out. This does mean that you can’t really get lost in a flat maze, and at all times you can see what you’re doing. They’re still fun, though!

All paper mazes are flat mazes, of course, and you can have other small mazes. These are sometimes called finger mazes, as you travel through them using a finger. A lot of the earliest mazes seem to be like this, treated as designs rather than big enough to walk through. I heard a nice idea, to have a large maze, and then when you get to the centre, there’s a notice board with a small finger maze! Even if you have a large maze, it doesn’t have to be on the ground or floor. You can have one on a wall, or even the ceiling. You can also have fun with using different materials; how about a patchwork quilt!

If you want a maze on the ground, then you can use Roman mosaic mazes as a model. You can use mosaic tiles, but you can use any pathing material (contact your local gardening shop or builders for suggestions). Make sure that you have two contrasting colours, one for the path and one for the walls.

Turf mazes were made originally by lifting the grass turf in strips to reveal the ground underneath. I suspect that this was often chalk. Now, the paths (or walls!) are usually emphasised with paths made of brick or other material, which needs less maintenance. Make the paths lower than the grass, and then you can mow over them easier. However, there are other mazes that you can make with grass. You can mow a path through grass to create the paths. Or you can get armfuls of grass cuttings and pile them up as walls. Both will tend to get trodden down or kicked, so this sort of maze is strictly temporary!

In Scandinavia, there are Cretan mazes marked out with pebbles. This is suitable for Cretan mazes, since you mark out the walls rather than the path, so you walk between the pebbles rather than on them. If you make any other type of maze, make sure you mark the walls rather than the path. The pebbles might get disturbed quite easily by people’s feet, so perhaps you could set the pebbles in concrete, or at least, dig them into the earth a bit.

Even if you don’t want a high hedge maze, you can still have a garden maze. You could make the walls into flower beds, leaving the paths as grass or pathing material. Then you can plant up the flower beds as you wish. I’ve heard of a lavender maze, which is a lovely idea! Or you can make tiny hedges, made of herbs perhaps, or box. This is getting back to the idea of a parterre or knot garden, which was the origin of garden mazes.

You may get inspiration from other places. How about a water maze as in Bristol? That may sound too ambitious, but you can make a wonderful temporary maze on a beach; see right, one of the few mazes that I’ve ever made in the real world. If you scrape a channel in the sand, then when the tide comes in, you have an instant water maze. It’s very quick, and if you make a mistake, it’s easy to correct, and the sea wipes it all clean again. Beachmaze

If you have a flat, or fairly flat, maze, you may like to think of something to put in the middle, such as a sun-dial, or a tree, or statue, or bench, or even a small finger maze!

If that all seems like too much work, then you can do what I do, and keep your mazes on computer! Or follow the intriguing idea of using beads to guide you though an imaginary maze.

What Escher might have done?

There are many ways in which a tiling can be coloured systematically. the most satisfactory colourings, both from an artistic and a mathematical point of view are those which are proper and perfect. all Escher’s colourings, with one or two possible exceptions, have both these properties which we shall now describe.

A colouring of a tiling is proper if no two adjacent tiles have the same colour. This condition is familiar in the colouring of maps, where adjacent countries are usually coloured differently. A colouring of a tiling is perfect if every symmetry of the tiling is associated with a colour symmetry. H.S.M. Coxeter explains, in his contribution to this volume, the concept of a colour symmetry. Here we shall confine ourselves to giving some examples. The simplest is that of the chequerboard colouring with two colours. This can be applied to any tiling in which each tile has four adjacents, and four tiles meet at each vertex. Such a colouring is clearly proper. To see that it is perfect we need only observe that every symmetry of the tiling either leaves each tile the same colour or inter changes the two colours, and so yields a colour symmetry of tiling. Most, but not all, Escher’s colourings of tilings using two colours are of this kind.

A perfect colouring with three colours; every symmetry of the tiling is associated with a permutation of the colours and so with a colour symmetry.A perfect colouring of this same tiling by two  colours is also possible, namely the chequerboard colouring mentioned above.

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It is easy to use After Effects as a virtual camera that will simulate complex navigation across a flat piece of artwork in a way that mirrors the animation stand “moves” used in traditional filmmaking. All the techniques talked about in the preceding chapter on kinestasis and collage animation can be achieved faster, cheaper, and with more accuracy by using After Effects by referring to familiar film technology.

Pans. When shooting film, you create pans by moving the camera or the animation stand frame to give the feeling of moving across still image. To create a pan in After Effects, you simply move artwork left, right, up or down in the program’s composition window. You tell the computer where you want your pan to start and stop by setting key frames at your start and finish points.

Zooming. In film, zooming in is achieved by changing the distance, frame by frame, between the camera and your artwork. In After Effects, you can simulate a zoom by scaling or resizing your artwork over time.

Fades. Traditionally, in-camera fades are made by slowly adjusting the amount of light coming through the lens aperture. To fade out (to black) you close the lens down over, say, 24 successive frames. To fade in (from black) on the new piece of artwork, that process is reversed. In After Effects you can fade in and fade out by manipulating key frames. Think of key frames as markers of precise moments when a selected image will be manipulated in one way or another.  When making fades, a first key frame marks the moment when the computer will begin to adjust the opacity levels. If an image is set at an opacity of 0 percent, then it is not opaque at all. In other words, you can see through it. If an image has an opacity of 100 percent, then it is fully opaque and cannot be seen through at all. It follows, then, that if you cent opacity all the way to 100 percent opacity, the resulting effect will be that of watching an image go from being perfectly transparent (clear) to being completely opaque (solid) in the amount of time you have designated by choosing two key frames. In After Effects, 0 percent to 100 percent opacity appears in fade-up from black – because the black background (a default setting) will show through your image when it is not fully opaque. To the viewer’s eye this looks exactly like a fade-up from black in traditional filmmaking.

Laybourne, K., 1998. The Animation Book. New York: Three Rivers Press