The Ulam Spiral

… and many other spirals¬†
inspired by it …


Fig. -2.


Fig. -3.


Fig. -1.


Fig. -4.

The Figure at left represents the right sector of
the spiral shown on the previous page,
facts discussed on this page are typical for all the sectors.
The Figure shows the 6-folds and all the adjacents,
prime- & composite-.

In the Figure,
a checkerboard pattern is delineated and
that checkerboard pattern is shown again
by itself in Fig’s. -2 & -3.

In Fig. -2 the checkerboard squares are
shown with only the 6-folds,
in Fig. -3 the adjacents, both prime- & composite-
have been added to the 6-folds.

The squares in Fig. -2 each contain an
identically shaped triad of numbers,
for convenience they can be called “triads”.
Each of these triads is configured as a
sharp 90° angle, and can be visualized as an
arrowhead pointing to the direction of larger numbers.

From seeing Fig’s. -2 & -3 one can deduce the following:
the 90° angle configuration of each triad enforces that
5 of its adjacents are located in a
corner-to-corner x-shaped array within the square and with
1 remaining adjacent abutted against
the outside of the square, isolated from the others.
For convenience, adjacents such as the latter
can be called the “isolates”.

Also to be deduced:
the squares are corner-contiguous, therefore
the corner-to-corner x-shaped arrays of adjacents within them
are also corner-contiguous.
Thus  the endless-, uninterrupted rows of adjacents
oriented in both diagonal directions are formed,
as seen in Fig. -4.


From the foregoing it becomes obvious that:

  1. The diagonal orientation of the rows of adjacents
    is caused by the 90° angular shape of the triads,
    and caused by that only.
  2. The endless, uninterupted extent of these lines is
    caused by the corner-contiguous arrangement of
    all these checkerboard squares
    -squares filling out the sector towards infinity-
    and caused by that only.

These rows start at the sector-lines,
or adjacent to these lines,
or nearby these sector lines, at their far side.

Fig. -4 shows that the array of adjacents is
regularly sprinkled with
“crossings”, circled in blue &
isolates, circled in red.
These items are paired in such manner that
in each pair
the isolate-number is larger by 2
then the crossing-number. 

Note that in the figure above
the typical periphery of the void spaces is
identical to the
typical periphery of the fillled spaces.