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phillip1882, on 12 September 2011 - 03:42 PM, said:

That's the fun and challenge of trying to think of things in terms which weren't or aren't sensible from our current perspective. Mathematically, we can talk about a complex number of points or sides. For instance, we might talk about a given polygon having imaginary sides, sides that are only on the imaginary axis and in the complex plane. Under such a view, we might see the real part as being the count of the number of sides in the real plane. Using Gaussian integers would make good sense in such a system. Similarly, we might talk about counts of points on the real and imaginary axes in the complex plane.

phillip1882, on 09 September 2011 - 04:53 AM, said:

phillip1882, on 09 September 2011 - 10:03 AM, said:

phillip1882, on 12 September 2011 - 10:49 AM, said:

Is there a geometric interpretation for your definition of complex polygonal numbers? I suspect taking pascal's triangle and splitting it complex-wise might give an idea. Perhaps unrelated, I've been looking at imaginary powers of e for inspiration about complex polygonals. There's something compelling to me about the way a triangle can tend towards becoming a circle.

Turtle, on 07 September 2011 - 03:48 PM, said:

Unfortunately, I can not list the "first few" as the Gaussian integers inherit the same property as the complex numbers. They can not be ordered in general. We can represent them using Argand diagrams. Every integer is a Gaussian integer. Not every Gaussian integer is an integer. For instance, 1 is a Gaussian integer and an integer. 1+i is a Gaussian integer but not an integer.
The general set of Gaussian primes includes Gaussian integers which are not integers and can not belong to the set of natural numbers such as 1+i.
I'm not simply adding imaginary numbers to describe what we're seeing. I'm interested in seeing the more general non-polygonals, the set of non-polygonal Gaussian Integers.
The diagrams above are depicting inclusion relationships. The Gaussian integers include the integers as a subset, the integers include the natural numbers as a subset, the natural numbers include the (natural) polygonal and non-polygonal numbers as a subset.
The Gaussian integers are to the integers what the Complex plane is to the real numbers. The Complex plane includes the Gaussian Integers as a prop (strict) subset.
For all z, in rectangular form, where Z is the set of integers, .

For the subset Where the primes and the Gaussian primes intersect, all the numbers of the form 4k+3 are elements of the natural Gaussian primes. Think of this like the tip of the iceberg. You're seeing a much larger set poking through into our universe of natural numbers. Where the primes and the Gaussian primes do not intersect are composite numbers, which includes a subset of the prime. "Elements" we thought were irreducible, non-partitionable, yet they can be split with complex solutions. The algebraic closure brought by i allows for polygon solutions which were not possible before; therefore, we should expect members which are polygonal in a more general sense than the natural polygonal numbers: complex polygonal numbers.

Here's the updated Venn diagram as promised. Check it for errors or clarifications, please.

phillip1882, on 03 September 2011 - 04:28 AM, said:

At this time, I'm not entirely certain how to go about generalizing the set of non-figuratives to include complex and imaginary cases. What I do know is that the natural non-figurates are strictly a subset of the complex non-figurates.
If I can find a reasonably priced copy of Coxeter's work on regular complex polygons, I might actually construct a formalism. Frankly though from what I've gathered, I don't understand complex polygons and the idea of complex figurate numbers stumps me pretty quickly. For the natural figurates, we can think of the number as representing a number of dots which when arrayed geometrically construct a regular polygon of s-sides. Hopefully as I tidy up the algebra and trigonometry of Gaussian integers, it will all make more sense. What is and is not a non-figurate may change under the Gaussian integers similar to what happens to the natural primes.
I think the best start would be working with interpreting n and s geometrically. The complex polygons link I posted earlier is probably a good place to begin with that.