Transformative Structure,

same model, with noteworthy profiles,

a visual study.

Right > Referring to a 3d CAD modeling instance, of the first picture, 3 views appear identical, in an axonometric projection, (meaning no perspective). This is similar to a view seen through a telescopic lens. Are 3 identical- orthogonal profiles significant for one and the same model? Is this repetition a signature of a special curve?

Right> second picture down is a perspective view, (40 degree view angle). (This is similar to many ordinary views in photography lenses). These first two projections visually feature the twist factor of the equilateral tetrahedron. What in 2d appears like a box or parallelogram with a fulfilling 'X' , is in fact a 3d tetrahedron projection. The 'dog biscuit' shape is this same curve which, in the next picture, projects a near perfect circle profile. I wonder if the word triploid might befit this cuve.

Right> third picture down is an axonometric projection. In all these 3d projections, both 3d models keep the same position relative each other. The fact that the triploid curve now projects, as a nearly perfect circle is what grabs me. This projection coincides with the pyramidal- plan overview aspect of the tetrahedron.

Right> these last two perspective projections display the transformative profile of this particular triploid curve, (refering to casual observations of the general shape). When many of these triploids are linked, as into a mesh or built into a devider wall, then many of these transformations become visible all at once. See art links below.

I'm not completely versed in the human history of the tetrahedron, but it's manifold significance as an important, geometric, building block was academically and industrially popularized by Bucky Fuller. That the tetrahedron "molds" a good variety of curves which have occupied my research interests for years, qualifies it as critically significant for my geometric work.

Therefore I am thankful to the Synergetics list of Kirby Urner, where through the stimulating dialog, I learned about the enigma of the tetrahedron and those most interesting books: Synergetics I &II (now on the web). Sites dedicated to all of the names in this paragraph can easily be searched on the web.

I have found other digital artist/ geometric modelers who are also interested in this triploid, giving other names to it.

I found a very similar occurrence of this triploid curve, long before I became aware of this equilateral- tetrahedral, version. The fact that a cube, sub divided into 8 equal cube parts, also molds a close triploid curve intrigues me deeply. Click below to see triploids as 'warped discs'. All the following art work is based on 'triploids" which were molded in this 4 cube mold.......

Alpha mask art derived from triploids on a geodesic. (104 K jpeg)

By molds I mean: in a 3d CAD program, parts of one object are used as snap points to produce a second object. (This is CAD jargon which compares with fastening a flexible batten or wire to particular points on another physical object) .

In the tetrahedron, snapping on the mid points of certain edges produced this triploid curve. An initial study of this curve can also be made as follows using paper and scissors.

Cut out one whole and one half circle and draw radii with pen , lines from circle- centers, to define all the quarter 'pies'. Cut one of these 'diameter lines' from the whole circle edge to circle center, (no further). Bend (or partly fold) cutouts along these lines, alternately, so that the cut spreads somewhat open. Then tape the half cut circle in this opening and bend, (that is: partly fold), along radii lines so that all the folds alternate.

Though the folds are sharper, the paper model is meaningfully related. Closer replication can be achieved with stretchable paper media and circumferential trimming.

When this paper model is tossed randomly into the air, the folds seem to balance the fall so that it almost always lands evenly on three points.

More updates are planned, please visit again.

The triploid structure is the basis for the yellow background on many pages of this site. The background "tiles" pictures of many triploids viewed side by side with a super wide angle lens , (done digitally in CAD).

Click here to see a list of other enersearch pages

boa1@pivot.net