Sunday, December 21, 2014

Circles as Numerical Sequences



Mathematics is a universal language, and our best way to communicate with those who are very different from us. We see increasingly complex mathematical designs appear in crop fields worldwide every year - in grass, wheat, barley, corn, snow and ice (I call them all “crop circles” for simplicity’s sake). Even the seemly simple ones hide a treasure of mathematics if you dig below the surface. These crop circle designs have an underlying order and beauty that even the mathematically challenged can appreciate. 

It is my belief that these beautiful designs encode data, because they contain blocks of objects that repeat, and there are rules about how the blocks can be assembled. This basically describes how language works, so it’s probable that these crop circle designs are using a geometric-based language. I think of it as a geometric object-oriented programming language that may be self-executing.

Let me clarify here that I am only talking about non-people-made (NPM) crop circle designs. The designs made by people don’t follow the same geometric rules, and have problems with proportion, scaling, and placement of the designs with respect to tram lines. Some of you may be asking – if they’re not made by people, then who? My answer to that is - an intelligent being who knows more than I. 

The Concept of Number

With the idea that math is universal, let’s start with something basic like the concept of number. In the book Contact by Carl Sagan, a radio transmitter on earth picked up a non-terrestrial signal with a series of beeps and spaces that communicated intelligent knowledge of prime numbers. Suppose we wanted to do something similar, except instead of using sound we wanted to use a visual medium to transmit the first 7 prime numbers.   

The best option for communicating numbers using geometric figures would depend on how many spatial dimensions you have available to communicate, and from which dimension the figures would be viewed from.

1-D

In one dimension (1-D), the best option to visually represent number is a sequence of lines and spaces. A line of length 2 would be followed by a space, then a line of length 3, followed by a space, etc… Note this would need to be viewed from a 2-D perspective in order to decode the numbers. 
Linear 1-D Representation of Prime Numbers

2-D

In two dimensions (2-D), the best option to visually represent numbers is a sequence of circles, placed concentrically or within a 2-D coordinate system. Radial lines on a polar coordinate system, triangles or squares also provide other options, but circles are superior because of their simplicity. Lines alone pose difficulty because of their 1-D nature. The line would need to be thick enough to be seen, but then a second dimension is introduced as well as a second number indicating the width. Using equilateral triangles or squares to represent pure numbers is also possible, but practically speaking, circles are still required to construct a perfect 60 or 90 degree angle.

Circles are the best 2-D choice to represent number because:
  • A circle is defined by one number – its radius
  •  A circle consists of one continuous line
  •  All points on the circle are the same distance from the center, which makes a circle resilient to distortion
  • Circles are the easiest geometric figure to construct
The simplest, most elegant way to represent prime numbers in 2-D is with a sequence of concentric circles with radii (or diameters) measurements equal to the prime numbers. Does this look familiar to anyone?

3-D

In three dimensions (3-D), the best option to visually represent numbers is a sequence of spheres, placed concentrically or within a 3-D coordinate system. It would be impossible, however, for a being living in 3-D to measure the relative sizes of nested spheres, so this type of encoding is not ideal for us here in 3-space.

Conclusion

The circle is the best 2-D geometric representation of pure number, and the best visual representation of numbers for those of us living in 3 dimensions.

Now we have a starting point for our crop circle language. The next logical step in decoding is to look at the sequences actually being generated by the designs in our crops. Then comes the tricky task of nailing down the positions of the design elements using a 2-D (polar?) coordinate system.
There is much work to be done… Are you on board? Are you ready?

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