The K-circle is part of the balanced layout of Athena from head to toe, as we just saw. Next, it forms the Cone with two more circles:

The external tangents, which demonstrate that the three circles are symmetrical, and the circles form the Cone.

            The Cone's middle circle

The Cone's middle circle belongs to a group of circles apparently reiterated throughout the engraving.  Based on this frequency of appearance, the group appears standardized, and could be based on a unit circle. Measuring, or mapping the Cone with it came naturally.  (diag.below)
Draw five unit circles spaced by the radius along each arm of the Cone. Also, draw the unit circle to be concentric with the K-circle. It will be a perfect tangent to the two nearest unit circles on the Cone. This caps the Cone, and makes it perfectly logical! 

The fourth circle on the left, and the third circle on the right support the square's horizontal diagonal!
A circle from the center of the Cone's capping circle, drawn as a tangent to the Cone's sides is the K-circle!
The K-circle intersects the horizontal diagonal of the Square both at its center, and the top corner! Thus, it is possible to next reconstruct the Square,  the Torso Lens, etc.

Since measuring out 5 unit-circle radii along the Cone's sides is is the key part to its solution, does this indicate that the hypothetical star to be imposed upon the 36 degree Cone should have arms 5 unit-unit circle radii long? 

Problems of perception The new Cone-star (S-star) creates some special visual effects. Drawing the circumcircle of the Cone-star concentrically with one of the Torso circles, shows that the two radii differ by one pixel only. (diag. above) The Cone's circumcircle also seems to be a perfect tangent to one side (north-east) of the Main  Square.  At the time of the experiment, I was quite perplexed by this. Did the construction of the Square from the Cone maybe duplicate the Cone's circumcircle on the Torso?  Note that when drafting by hand, a  similar question arises with the so called Circle-Triplets, shown in the diagram below. All three seem close to being identical in size.  But, are they identical? _ My curiosity went unsatisfied until I could recreate the position in CAD, on a $2,000, 10 MHz pc :)    The Circle-Triplets Take a circle, which passes through the inside  points (corners) of our 5-pointed star.This circle (dubbed Inner-circle) on our experimental Cone-star seems to be just as big as Unit-circle, but also just as big as the Middle-circle of the Cone. On paper, the radii of both the inner and the middle circles fit the Cone-star's arm a near perfect  five-times, once a hair short - and once a hair too long. In the diagram above, the three CAD produced Triplet circles were made concentric, and then highly magnified. Yet, despite their magnification, these three circles look like a single circle. But, the following is true:                          If the Unit circle's radius  =  [1]                     then the Inner circle's radius  =  [1.0040..]                         radius of the Middle-circle = [0.9975..] One mathematical consequence of this unit system: The side of a pentagon inscribed into the 'inner'-circle measures         1.1180339887... -   the square root of 5 divided by 2 The PHI-ratio equals the same square root of  5 divided by 2   +  0.5                1.11 80339887... + 0.5  =  1.6180339887.. Now, measuring of the Cone by the unit-circles makes perfectly good sense.

The size and position of the K-circle on the Cone
The K-circle was generated from the capping circle's center as a tangent to the Cone's sides.
But, this center-point may be found by  an even quicker method - the upper intersection of the third row of  the mapping circles on the central axis marks the mid-point between the K-circle's center, and the tip of the Cone. 

The size and position of the Middle-circle on the Cone
It is also derived from the unit circles describing the Cone: the fourth row of  these circles (whose centers are 3 radii from the tip of the Cone) intersects at the center of the Middle circle (diag. below). 

The size and position of the Small-circle on the Cone

It is meant to fit between: 
                                         a) the internal tangents
                                         b) the external tangents (Cone's sides). 

The size and position of the Square with respect to the Cone

The Cone, the Mother-star & Geometry

The Cone's concept turns out utterly methodical, and scientific. Neither Euclid nor Pyrthagoras could improve upon it. Our caveman thinks, as if he were a professor of mathematics. 

I had come across some diagrams looking like the Cone, in a collegiate handbook on geometry. The diagrams dealt with the topic of similarity of  circles: That's when I learned that theory calls the Cone's sides "common  external tangents - and the Cone's tip the "external center of similitude.
Next, the book revealed that there are internal counterparts to these:

1) the Internal Tangents between the Cone's top, and small circles.
2) the Internal Center of Similitude, at the point S.
 

In the illustration above, we use parallel diameters of two circles to find their centers of similitude, or symmetry: A line between opposed ends of the diameters intersects the line of circle-centers at the internal center of symmetry "S" _  An internal tangent is drawn from S to both circles.
A line between the correspondent ends of the diameters intersects the line of circle-centers at the external center of symmetry "V". An external tangent is drawn from V.
The Ancients had indicated their knowledge of external tangent construction by symbolizing the Cone. But, could they have indicated their knowledge of the complementary internal tangent construction?
Well, our own construction of internal tangents between the Cone's top and bottom circles results in a spectacular effect..

In this situation, the four numbered points produce six different lines.  However, all these straight lines give an illusion of being one and the same line (see diag. above). The effect is as beautiful, as it is unlikely. The indication is that the Ancients knew, and played with, the idea of internal tangents.

Now, the Cone's external center of symmetry sets one tip of its star. To be consistent,  the Ancients should have used the Cone's internal center  of symmetry for the center of its 5-pointed star. - This would explain  the smallest circle of the three on the Cone, as wedged between the  internal and external tangents. The internal tangent was first drawn as a tangent to the K-circle from the 5-pointed star's center, then it was extended, and then the small circle was drawn.  

Go to animation on the Triplets