   # The Ulam Spiral

### … and many other spirals inspired by it …    ## Det. -1.

Spiral populated by:
type – (1,1) 6-folds
type – (1,0) 6-folds
type – (0,1) 6-folds
type – (0,0) 6-folds

All the 6-folds

Spiral entries to be
multiplied by “6”.

## Findings.

The highest number populating the spiral above is 1599,
see upper left-hand corner of spiral.
Since the lowest number populating the spiral is “0”,
given that the lowest 6-fold is “0”,
The quantity of numbers populating the spiral is 1600.

Quantity of numbers of each type of 6-fold:
type – (1,1): 199 numbers, = 12.4375% of all numbers. (1/8 – part of all numbers.)
type – (1,0): 401 numbers, = 25.0625% of all numbers. (1/4 + part of all numbers.)
type – (0,1): 384 numbers, = 24.0000% of all numbers. (1/4 – part of all numbers.)
type – (0,0): 616 numbers, = 38.5000% of all numbers. (3/8 + part of all numbers.)

The pie chart below shows
comparative population sizes
of the 4 6-fold types. Comparative population-sizes
of the 4 6-fold types
for a spiral populated by 1600 numbers.

Whatever governs the numbers-array in a spiral,
including all the “across-track” numbers-rows
shown in the previous 3 pages,
must must also determine the comparative population-size
of each of the 4 types of 6-folds.

The pie chart is revealing but of limited use because
it only reveals comparative population-sizes specific for
a spiral populated by 1600 numbers.
As the quantity of the numbers populating the spiral changes,
slightly or significantly, the pie chart also changes, accordingly.

An informative presentation would be one wherein a
computer generates a constantly updated pie chart of
a spiral whose population is rapidly increasing,
number by number,
from a trivial single number
to an extensive population of numbers,
thereby presenting an instructive video.