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                                       Triplets   

The opening scene shows  the basic Cone, i.e.,  three circles
fitting between their common external tangents. An internal
tangent between the top, and the bottom circles is also drawn in.
That tangent crosses the central axis of the Cone at the Internal
Centre of Similitude between the two (top and bottom) circles.
                        What is the Cone Good for?  
The Cone's 36° angle makes it ideally suitable for inclusion
in  a regular five-pointed star. The Cone's tip coincides with
one of the star's tips, but, where should the star's centre be?

             Theory Itself as the Best Possible Solution 

For the answer we have to look to the general theory itself.
In other words, what we see in the image from the standpoint
of geometry - that's what it is in the artist's mind. Clearly,
this particular solution is the least chaotic & the most
deliberately significant case amid all the possibilities.
Once we start trusting our impression that the ancient artist
has geometrical theory in mind, all we have to do is follow
the general theory, and fill in the missing pieces.
This particular case is fairly simple. In the Cone, theory
sees three symmetrical circles, and their External Centre
of Similitude (similarity). So, where is the complementary
Internal Centre of Similitude?
It can only exist between non-overlapping circles - which
leaves only the top and the bottom circles. Their internal
centre of similitude becomes the centre of our experimental
five-pointed star.

          Justification For Drawing the Inner Circle 

Five-pointed stars are constructed in a circle, hence it is
customary to draw such stars in their original circle. Since
this circle passes through the star's outer points, we draw a
circle around the inner points, as well, as the complementary
circle in this position.

          Cone's Middle Circle as a Catalyst  

Drawing the inner circle makes manifest that this circle,
and the Cone's Middle circle are almost twins, sizewise.
A closer look reveals that the radii of these circles also
fit the star's arm five-times to a small +/- margin of error.
This fact prompts us to draw a circle, whose radius equals
exactly one-fifth (1/5) of the star's arm,  i.e., the Unit-circle. 
Thus, the Unit-circle is one of three types of nearly identical
circles - the Triplets - existing on any given five-pointed star.

           A Theoretically Logical Conclusion 

Living up to its name, the Unit-circle maps the Cone's sides
with a surprising outcome. The act of mapping creates a direct
visual solution to the puzzle of the Cone's original construction.
The Unit-circle turns out to have been the Construction-circle.

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