One of the earlier things I wanted to do, a desire born probably around the age of ten years old but never abandoned, was to be able to sculpt stone.  This wish arose both from an even older almost intrinsic love of stone, it’s strength and truthfulness and beauty, as well as the things I began to observe as being wrought from stone.  That summer of 1972 we had moved to a new town (West Bay Shore) and one of the things one does (or did) was get a library card.  The first two series of books I took out were high class picture books on both the history of World War II and on the works of great artists and sculptors. Rodin and Michelangelo were very impressive and so was the idea that one could represent the imagination in stone.  Nigh on formative.

Recently my nephew was visiting and I wanted to demonstrate the power of an angle grinder with a kutz-all disc on it, so I took a piece of alabaster I had lying around, put it in the wood vice, vrrm, vrrm, and a very impressive hollow appeared almost instantly.  I was struck by how fast that was, as in the sculpting of stone, from my limited experience, the gross removal of material is the cumbersome part.  I made a note that I’d make something out of this particular block just to try a few things.

I call this post a detour because I had actually gone to the workshop to make the first shaping of the oaken elephant I’ve been planning.  The alabaster was in the vice that I was to put the oak block in.  ADHD.  How could I remove the alabaster without working on it?  I could not..  Speaking briefly of that elephant, you might have to click on the pictures below, but you should certainly be able to see both side and front views.

Anyway, so the elephant had to get in line.  This alabaster block, I drew a circle on it, and beneath that a triangle, and all I really first thought was to use various tools to flesh out (subtractively) those shapes.  The angle grinder makes very short work of alabaster – I think for an outside project though, as opposed to in a closed workshop where the air quality rapidly deteriorates and smoke alarms quickly get set off.  Also it’s noisy to the point of obscuring thought.  Usually I listen to music and think as I make things.  In the future I’d save the use of the angle grinder for cases where a lot of material wants to be removed quickly and it’s outdoors.

After I unplugged the fire alarm I mused as the dust cleared, and finally, thinking Cyclops really, I sketched an eye in the circle.  When I brought the one-eyed circle triangle rough upstairs that night my younger daughter said, oh, it’s an Illuminati.  I did not know that there were such things (not counting ancient secret societies and/or conspiracy theories) as creatures but decided to run with it.  Really my purpose was mostly to discover how certain tools, it didn’t matter here what I made so much as the observations while making.

I figured if you take the nose and mouth away from a cyclops, just leave the eye, that can be an Illumati.  For the purposes of explaining how such creatures survive, their exceedingly sensitive and specialized eye absorbs the energy of the light it beholds and transforms that into electricity.  That powers their immense cognitive apparatus (far more efficient than ours, as the in their model the thing perceived actually creates the energy for thought) AND the little energy they devote to their specific physical incarnation.  An Illuminati with hair as rich as this one

probably took a very long time to get that way.  But let’s leave the ponderous bits aside for a moment.

A standard flat and round file/rasp combo was very quick to cut this stone (I think it’s a Moh’s hardness rating of 2), as was a farriers rasp I had on hand.  I had a Dremel tool with a pea-size kutz-all ball bit that was almost as easy to use as a magic marker, as far as making lines.  What surprises me most though was how a simple orbital sander with grits ranging from 60 to 220 readily put silky polish on the stone.  A stone like this really shames the reputation of stone as being harder than wood, because it’s not.  Nonetheless, at a hardness like this I can’t see how anyone can say that tools limit their ability to shape stone.  The files, rasps and, and sandpaper were perfectly adequate.  An awl could have done that the Dremel did.  Because of this experience I really have to open my thinking to the idea that the real limit is more conception than execution, that sure, you could think of things so wispy that the stone could not hold the form, stallions rearing on two feet as well would not work, but most things will be fine.

So back then, for a moment, to this imaginary creature.  I kept wondering how it would advantaged or disadvantaged by having only one eye.  No stereopsis (fancy for binocular vision) as least real time anyway, as models could be made by memory and traversal.  It’s the immediacy though, which of course leads the wandering mind, a la Flatland, to the mind of a fly – if flies have minds as we understand them, and whether the idea of a two dimensional reduction of the multidimensional world would seem such a non-starter, as only the barest hint of the underlying reality is so portrayed.  But enough of these conjectures.  I think the singularity of perspective, at least in the immediate sense, maps as a scary quality.  Maybe cyclopes needed to move about more, to obtain the multiple angles we take for granted.  Maybe this need to act to obtain perspective made them wiser, recognizing that a single point of view was not enough; maybe it made them more circumspect, less inclined to think that to see was to know.

What a Cyclops knows, he knows by heart.  Such has been this alabaster detour.

Nonagons and nonsense, nonagonography, non-sequiturs, nonesuch, none of that now, not for the faint of heart…  This is a long post and would that it were just about nonagons there would be the weak chance that some overall sense could be made of it. It is more the case here nonagons are just being used, exploited even, to discover what properties may adhere to irregular but equi-angluar polygons.  Henceforth in this post such irregular but equi-angular polygons will be referred to as IBE polygons.

This started simply enough, a plastic cup as pencil holder on my desk taunting me that it needed to be restored to the proper destiny of plastic cups and that a suitable wooden substitute be created in it’s place.  Not having a lathe I immediately began to consider options in regular polygons and quickly settled on nine as being nicely divisible into 360, being non-standard (no pun), as affording sufficient creative license.

The next question that seemed a natural follow-on was ‘Why regular?’.  The answer was half-practical, so that the angles could be known and cut, but some variance, of the side lengths perhaps, as a rectangle is a stretched square, an IBE square if you will, seemed interesting.  This led to thoughts about whether IBE polygons could be made with an odd number of sides (yes), a prime number of sides (yes) and what constraints describe the construction of IBE polygons.  Just as a matter of assurance, completely irregular polygons were never considered.

Using the fact that 40 degrees is the central angle for a regular nonagon pie slice it stood to reason that 20 degrees would be a proper bevel to create the 70 degree wall side angle needed to create a nonagon.  I cut an array of such beveled lengths to test simple ideas.

Main learning here was that IBE polygons were a subtler reality than initially perceived, that they bore an intriguing similarity to the way crystals grow in nature, and that I could describe almost nothing about the constraints that pertain to them.

I talked to a few of my general sages and did some browsing/research.  I came upon several interesting things, among them Naploeon theorem – http://mathworld.wolfram.com/NapoleonsTheorem.html and the extensibility of Viviani’s theorem (http://en.wikipedia.org/wiki/Viviani’s_theorem) to IBE polygons.  I came upon a nice applet that lets one deform regular polygons into IBE polygons.  http://www.cut-the-knot.org/Curriculum/Geometry/EquiangularPoly.shtml.  The java on this page was out of date but running it proved satisfactory.  I was able here to become sure of both that there were more possibilities of irregularity than I could readily organize into intuitive classes and that yet that I am certain that some collection of line lengths are excluded.  It might be that the answer of what is in the set is expressed as range of ratios for the members, but that seems unlikely – something about prime factoring is tempting but likely false – in short, cluelessness.

Many experiments were conducted

The IBE nonagon pencil holder was built

But a satisfactory answer on the question of what criteria describe, for any IBE polygon of N sides, ratio ranges of valid segments is unanswered.  In the case of regular polygons we can express the area in terms of the number of sides N (assume 1 as the side length).  For IBE polygons that 1 must be substituted with some sort of complex ration expression dictated by the segments.

I’ll return to this unless the committee on coherent expression and useful theories weighs in.  I believe that there’s a body of knowledge out there that has terms for the majority of what I’m grappling at and has at least a portion of the insight I am seeking.  If anyone has a  hint please share.  I’m thinking I’ll do some rough looking at whatever theories describe the growth of crystal formations.  They’d have to have language that addresses some of this.

thanks,

(June 24th – update)

No theoretical progress but the blades being set did yield an almost involuntary nona-cone/wooden tepee, spire thing.

One thing too about doing anything nine-sided and getting to use the nona- prefix, is that given the conventional non- as not, a lot of possible humor arises.  A nonaspire, for example, aspires not.