# The Ulam Spiral

## Fig. -1.

In Euclidean geometric space no polygon
can have less than 3 angles.
In spherical geometry a polygon
can have as little as 2 angles.

In Fig. -1:

At (a)
a miniature globe is shown and on it
2 straight lines are shown,
starting at the North Pole and
ending at the South Pole.
These lines enclose
2 90° angles, 1 at each pole.
Thus these lines form
a polygon having 2 angles,
known as a “Digon”.
It also poses the question:
is there such an entity as
a “digonal spiral”?

At (b)
a slice contained between
these 2 lines & the polar centerline
has been cut away leaving
2 perfectly flat surfaces 90° apart.

At (c)
the 2 spokes of the putative digonal spiral are
shown on the flat surfaces, 1 from each corner,
with their ends abutted against the center and
stretched around it to abut against each other.

At (d)
the 2 webs are added to the spokes.

At (e)
the 90° cavity -and the digon-
have been opened up to 180°,
resulting in a digon with
2 180° angles &
2 enclosed coplanar surfaces.

At (f)
the digonal spiral has been rotated to
horizontalize the spokes.

Due to (e) & (f) the digonal spiral
can now be presented in plan view,
as shown in Fig. -2.

Digon.

## Fig. -3.

Digon
construction
diagrammatic.

Webs
-color-coded orange-
are composed as
triangular numbers.

Items equidistant
from center
-color coded brown-
to be identical.
(Refers only to items
color-coded orange .)
typ.

## Fig. -4.

Numbering system &
spiral system.
Spiral origin is at 0.

Given the method by which it is generated
this spiral can be called the “Diporal Spiral”.
There are other similarly generated polar spirals.