The Ulam Spiral

… and many other spirals¬†
inspired by it …

Digonal-spiral-fig1-2

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.

Digonal-spiral-fig2

Fig. -2.

Digon.

Fig. -3.

Digon
construction
diagrammatic.

Digonal-spiral-fig3

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.

 

Digonal-spiral-fig4

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

To see these polar spirals click on “Polar Spirals”.