### Table – #1.

### Table – #2.

### Notes

In the drop-down menus of “polygonal spirals” & “polar Spirals”

other Ulam-like spirals are illustrated, and it is worth examining

to see if they have any ‘utility’, meaning if they reveal

interesting rows of prime numbers.

To look for such rows the tetragonal spiral was chosen

because of all the polygonal spirals it is the closest to the

standard Ulam spiral, see in Fig’s. -#1 & -#2.

The tetrapolar spiral in Fig. -#1 reveals

interesting prime number rows.

2 prominent rows are immediately discernible.

The larger one is color-coded blue and

its numbers are also listed in Table -#1,

marked by the pointers.

The smaller one is color-coded red and

its numbers are also listed in Table -#2,

marked by the pointers.

Following the numbers of the blue row in table -#1

from its highest number: “1597”

to its lowest number: “79”

the row remains completely ordered, algebraically.

But in he spiral that row becomes increasingly chaotic

each time it crosses over a sector-line.

Following the numbers of the red row in table -#2

from its highest number: “631”

to its lowest number: “19”

the row remains completely ordered, algebraically.

But in the spiral that row becomes increasingly chaotic

each time it crosses over a sector-line.

But:

when these rows are located in a tetrapolar spiral

they remain entirely ordered because

the rows are not disturbed by the sector-lines

any time they cross over them.

See this in Fig. -#1. of:

Utility (2).

Therefore:

the tetragonal spiral is a competent tool only for

displaying those rows that do not

cross over any of the sector-lines.

Interestingly:

when these 2 rows are plotted in the standard Ulam-spiral

they become completely chaotic from the start.

Thus:

the tetrapolar spiral generates rows that

the standard Ulam-spiral can’t..

### Conclusion

The tetragonal spiral -and other polygonal spirals- aren’t

competent tools for finding rows that cross over any of the sector-lines

because these lines disturb the geometry of

the rows when they cross over these lines.

But:

among all the other rows there may be

many that are lengthy and of interest.

Fig. -#2 is shown as a reminder that similar rows can be fund

when searching thru spirals composed of composite adjacents.

A super-computer enabled research program of

maximally extensive polygonal spirals

should produce many interesting results indeed.