Decades ago, back in school, we used to doodle a labyrinth. I thought it would make a cool painting. A bit stark and monochromatic but intriguing, nevertheless. All I needed to do was to figure out a way to draw a geometrically defined version of it.
After a bit of analysis, I figured out the various arcs to be drawn. This was done in a CAD program (TurboCAD) and saved as SVG. Then I used GIMP to rasterise it. This allows me to scale it to any resolution. I chose a 3200 x 3200 pixel image. This was reproduced on an 88 cm acrylic on glass print from the Canvas Art Factory.
The image was drawn on paper the old fashioned way and the result was broken up into a series of circular arcs. These were then drawn in a CAD program. A couple of dabs of colour were added to an otherwise black and white image. The whole thing was fitted into a large circle and roughly centred.
A number of such labyrinths have been drawn through history. Looking at just the unicursal (single path from start to end) versions, the one above comes closest to the Cretan pattern. When we drew it as kids, we called that the Chakravyuh. This is a reference to Indian mythology.
In Indian mythology, the Chakravyuh refers to a military formation where soldiers stand in the lines described by this figure. Lord Krishna was said to have described how to attack such a formation to his pregnant sister Subhadra. As he was describing how to enter the Chakravyuh, he could hear someone respond indicating that they understood. He realised this was the baby in her stomach. As his nephew, he was predicted to be his nemesis. So Lord Krishna stopped short and the boy, Abhimanyu, grew up knowing only how to enter the Chakravyuh but did not know how to get out of it.
At a tender age, during the Mahabharat war, Abhimanyu volunteered to enter the Chakravyuh as he was one of the few who knew how. He was depending on the rest of his cohort to enter with him, but they failed to. He continued on, reaching the centre but was trapped inside and killed. While the actual pattern has a few versions, in the image above, the purple arrow indicates the entry point. Once you enter the pattern, there is a zigzag path that eventually ends up at the centre at the yellow circle. There are no alternative paths short of turning back. Nor is there any part of it that is not traversed.
When drawing by hand, we started with a cross, four quarter circles and four dots as shown below. The rest was done by drawing curves starting and ending with this initial diagram.
The image can be drawn using only simple lines and half or quarter circles. Most of the curves are parallel i.e. have the same centre. The breakup of the lines into parts is shown below. Curves with a common centre is shown in the same colour along with the centre. Besides the four corners of the start of the image, only one additional centre is required. This centre draws all the curves in the top half.
Once the logic of drawing the curves was sorted out, it occurred to me. Are there bigger labyrinths to be drawn the same way?
It looked like there are but there is actually a simpler version that can be drawn. Let us call this C(0). An animation of it being drawn is shown below. This also illustrates the curves that have to be drawn.
The next one up is C(1). This is the curve we drew originally. An animation of it being drawn is shown below.
Let us take it one step further. The next one up is C(2). An animation of it being drawn is shown below.
There is no reason to believe higher order curves don't exist and have the same properties of a unicursal path. The javascript code to draw the three curves is independent and quite messy. It would be an interesting exercise to unify it and provide just the curve order as an input. There are two straight lines. All curves are quarter or half circles and they all can be drawn using the same 5 centres - the four corners of the initial start and an additional point.
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