The Ulam Spiral

… and many other spirals 
inspired by it …

Ulam-spiral-for-6-folds-Det1

Det. -1.

Spiral populated by:
type – (1,1) 6-folds
type – (1,0) 6-folds
type – (0,1) 6-folds
type – (0,0) 6-folds

All the 6-folds

All spirals this page: 
Spiral entries to be
multiplied by “6”

Ulam-spiral-for-6-folds-Det2

Det. -2.

Spiral populated by:
type – (1,1) 6-folds
type – (1,0) 6-folds
type – (0,1) 6-folds

All the 6-folds
having 1- or 2
prime adjacents. 

Ulam-spiral-for-6-folds-Det3

Det. -3.

Spiral populated by:
type – (0,0) 6-folds

All the 6-folds
having no
prime adjacents. 

Ulam-spiral-for-6-folds-Det4

Det. -4.

Spiral populated by:
type – (1,1) 6-folds
type – (1,0) 6-folds

All the 6-folds
having a lower
prime adjacent. 

Ulam-spiral-for-6-folds-Det5

Det. -5.

Spiral populated by:
type – (1,0) 6-folds

All the 6-folds
having only a lower
prime adjacent. 

Ulam-spiral-for-6-folds-Det6

Det. -6.

Spiral populated by:
type – (1,1) 6-folds
type – (0,1) 6-folds

All the 6-folds
having an upper
prime adjacent. 

Ulam-spiral-for-6-folds-Det7

Det. -7.

Spiral populated by:
type – (0,1) 6-folds

All the 6-folds
having only an upper
prime adjacent. 

Ulam-spiral-for-6-folds-Det8

Det. -8.

Spiral populated by:
type – (1,1) 6-folds

All the 6-folds
having a lower- & upper
prime adjacent. 

Findings.

Det. -1:
This detail gives an efficient and immediate
overview of the interdistribution among each other of
the 4 types of 6-folds.

Det. -2:
In this detail a significant number of
horizontal- & vertical rows of 6-folds can be seen,
but these rows come in 2 kinds, as follows.
1):
Rows running on straight sections of the spiral,
in the manner of a train running on its tracks.
Such rows are ‘unremarkable’ because they simply
represent sections of the numbers-line from “0” to “∞”,
showing just the 6-folds in these sections.
•Such rows can be defined as “on-track” rows and are to be
regarded as unremarkable in any spiral.
2):
Rows running across the parallel straight sections of the spiral.
Such rows are ‘remarkable’, because they consist of
6-folds that are far away from each other on the numbers line.
•Such rows can be defined as “across-track” rows and are to be
regarded as remarkable in any spiral.

Due to the crowded population of the 6-folds
remarkable rows can emerge by chance, to some extent.
The determination of the situation -chance or not- might require a
computerized analysis of a more extensive spiral.

Det’s. -4, -5, -6, -7:
The situation in these details is much the same as it is with detail -2.

Det. -8:
The population in this detail might well be to sparse
-even in a more extensive spiral with computer analysis-
to present a meaningful display of rows of 6-folds.

Det. -3:
This detail presents the strongest appearance of
across-tracks horizontal- & vertical rows of 6-folds,
forming an array of interest and
it seems to be be a good candidate for
further computerized investigation of a more extensive spiral,
and maybe to find out what forms these rows.