The Ulam Spiral

… and many other spirals¬†
inspired by it …

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.

findings-fig-1-111219

Fig. -1.
(As in Preview but here with spiral-origin at 0.)

– with additional numbers shown
beyond extent of spiral
at left side.

Showing composite numbers & prime numbers
adjacent to multiples of 6.

findings-fig-2-111219

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.

findings-fig-3-111219
findings-fig-4-111219

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.

findings-fig-5-111219
findings-fig-6-111219

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.

findings-fig-7-111219

Fig. -5.