When observing the spiral in Preview”,

*with the 6-folds turned off,

*with only the composite adjacents shown,

one will notice a surprising -and highly noticeable-

degree of vertical symmetry in the left sector of the spiral.

This same symmetry also exists

when only the prime adjacents are shown but

that symmetry then consists only of

the voids caused by the absence of

the composite adjacents.

Arrays consisting of voids are hard to distinguish,

especially when surrounded by other voids.

Since these voids are caused by the absence of

the composite adjacents that have their place there,

these voids become “visible” by

looking at these composite adjacents.

Thus, by looking at

an Ulam type spiral consisting of composite adjacents

one attains a deeper understanding of the structure of

the Ulam spiral consisting of prime adjacents.

People trying to understand the Ulam spiral and its mysteries

might want to take note of this.

This page will analize that sector featuring

the symmetry alluded to above.

## Fig. -2.

(As in Preview, but here with spiral-origin at 0.)

– with additional numbers shown

beyond extent of spiral

at left side.

Showing **composite** numbers

adjacent to multiples of 6.

The numbers array between sector-lines -3 & -4

is strongly suggestive of symmetry around

the indicated symmetry-axis

exhibited by most of the numbers in that area.

These numbers are located

within- **& under** the sides of

the “grouping square” shown in Fig. -1.

**62** numbers are located

within- **& under** the edges of that square.

Of these:

**49** numbers are located in an arrangement that is

symmetrical about the vertical line going thru

the top- & bottom corners of the square.

These numbers are highlighted by red circular backgrounds.**5** of these numbers are located on the symmetry-axis &

there are **22** pairs of numbers, with each pair consisting of

numbers located symmetrically about the symmetry-axis.

The numbers in the symmetry-pairs and on the symmetry-axis

are shown by themselves in the take-out below.

They constitute **79.03…. %** of all the numbers located

within- & under the edges of the square.

## Fig. -3.

(As in Preview, but here with spiral-origin at 0.)

– with additional numbers shown

beyond extent of spiral

at left side.

Showing ** Prime** numbers

adjacent to multiples of 6.

**59** numbers are located

within- **and under** the edges of that square.

Of these:

**46** numbers are located in an arrangement that is

symmetrical about the vertical line going thru

the top- & bottom corners of the square.

These numbers are highlighted by red circular backgrounds.**4** of these numbers are located on the symmetry-axis &

there are **21** pairs of numbers, with each pair consisting of

numbers located symmetrically about the symmetry-axis.

The numbers in the symmetry-pairs and on the symmetry-axis

are shown by themselves in the take-out below.

They constitute **77.96…. %** of all the numbers located

within- & under the sides of the square.

## Fig. -4.

(Fig. -1 from above, partial.)

The square-size is fortuitous because

by **all 4** of its sides running **over** the rows of numbers

it is of a size such that it can most conveniently “tile”

the entire area located between sector lines -3 & -4,

this for drawing comparisons between

various areas one might wish to compare.

Such tiling can establish a “grid” system

for the entire area between sector lines -3 & -4,

with the origin of the system located at 5 and

its axes starting there.

The tile shown is shown again in Fig. -5

and it can be seen to consist of

16 smaller minimal-sized “unit tiles”

arranged in a 4 x 4 pattern.

Thus it can be defined as a 4 x 4 tile.

Note that the symmetry axis shown here

runs thru crossing-numbers & the isolates

and these only.