Rotational Patterns

 How to Create Rotational Patterns


There are many different ways of creating geometric patterns, ( I am using patterns in a very loose sense here) employing various different techniques such as mirroring, arraying, rotation etc. Nevertheless, I intuitively feel that there must be some still unknown ways of creating geometrical patterns. M.C.Esher, for instance, popularized the concept of tessellations through his magnificent wood prints with superb artistic touch. That was before the time of PC. And now that computers are ubiquitous part of our lives, some of the laborious job of creating patterns can be done on a computer at a touch of a button (well almost!). I would like to introduce one such way to create patterns  possible only through the sheer computing power of PC.

The followings are the method of constructing a new type of geometrical patterns:

There are many elementary shapes that we can use as a seeding element. Here, however, an octagon is selected to act as a seed. Just to make the final result more unpredictable and exciting, a square is added inside it as shown below.

The figure shows that there are 2 possible pivot points where rotation can take place. This is shown in the figure below.

We are going to array both the octagon and square around pivot1 and pivot2 eight times( this number can be selected arbitrarily. Here I have chosen eight simply because it is the number of sides in an octagon). This will produce eight copies of the octagon and the square around the selected pivot.  

Now,1 is the resulting array around the pivot1 and 2 around the pivot2. Notice the complete difference in their overall configurations. This illustrates that the location of pivot is an important aspect in rotational patterns.

Upon inspection, several candidates emerge from a few possible nodes as a pivot.

All these dots show possible pivot points for arraying and they are capable of producing a unique pattern of their own. For the purpose of demonstration, only around the pivots with teal colour will the arraying process be applied.

Every one of the patterns produced around the each pivot does have unique individual hallmark of their own. Pattern 1 arrayed around pivot A generates pattern A-1, and pattern 1arrayed around pivotBgenerates patternB-1 and so on.

Finally the process can go on infinitely, but I find that more than 4 level of arraying will jumble up the whole thing rather than creating visually meaningful patterns. The following are some of patterns generated in the following level.