-must be gorgeous...
-must smack of Gothic architecture, only better...
and the work flow I have developed over years of having scads of ideas that seem awesome for an hour or so is essentially: start off at an effectively arbitrary point that seems most interesting to work on, immediately assign arbitrary constraints on what I want which will be the anchor points for the obsessing I do later on to stall progress, discover that said constraints require lots of math, google around a while for math information of the type I think I need with the "optional" step of getting progressively more irritated as I either find my leads being unrelated despite sounding exactly like what I'm looking for from google, find that my leads ramble incomprehensibly across a Rorschach of bush-round-beating mostly made up of poorly described examples, or find I can't tell whether my lead tells me what I want to know usually due to a sort of incomprehensible rambling across a Rorschach of bush-round-beating mostly made up of poorly described examples...then I apply what I learn, then I rinse and repeat, or realize it's 7am, tell myself "just another hour" realize it's 11am, and...AHEM. Moving on.
A
feature seen in many Gothic cathedrals, one that I rather like, are the pillars carved to look as though they are several pillars clustered together.The design I have been making began with the idea of a cluster of four such pillars, though each pillar made up of three sub pillars, each of which would be the same circumference, and tangent to one another. Rising from the floors, each pillar cluster would diverge, while from each diverging pillar cluster the three sub pillars diverge as well, arching and intersecting to form the vaulting.
In a floor plan, the bases of such pillars can translate into a trio of circles, each of which have the same radius, and each of which is tangent to the two others.
For some reason I also decided to make design decisions based on the Fibonacci sequence. By that reckoning I decided the outermost edge of the outermost sub pillar of a sub pillar cluster should be exactly two meters from the center of the four clustered pillars.
Opening autocad, I began trying to draft the three circles first, and I realized I did not know how to do that. I had decided the outermost edge of a pillar must be two meters from the center of the cluster.
Once upon a time I was a child with a handful of dowel sticks. A handful of two dowel sticks, assuming you know the diameter of each, is the only handful you can know the exact diameter of a hole they bot
h will fit through at once without doing a bit of convoluted trigonometry. A handful of three dowels is the only handful you could easily keep held such that they would fit within a perfectly circular hole. Try clutching a handful of four dowels and see if you can keep their innermost edges equidistant. When you figure out why you have been trying to do that for upwards of 45 minutes tell me, and I'll say that's why I did so when I was a kid once upon a time with a handful of dowels...I'll make something up for why I did it while not a kid...If you draw one circle that is tangent to the origin, then draw another circle the same radius also tangent to the origin, the only arrangement of the two you'll find them not overlapping one another somehow is if they are offset 180 degrees from one another. If you then add a third circle of the same radius there is no arrangement where it may be tangent to the others, and to the origin.
In essence I knew the radius of the circle that encloses all four pillars. That was the only known I had. The dimensions of all the other circles in fact depend upon that one known quantity, so I began looking for a quantity within the unknowns that I could daisy chain into a set of relationships by which the one could be derived from the other.
One set of relationships I did know of as having a well established set of equations for finding all the dimensions involved if you have only one of them is the equations for polygons that have n sides. In autocad it's not so great a chore to draft a triangle, and use it to draft a trio of circles that are tangent to one another. I just need to draw the triangle, draw a circle centered on a triangle's point, and expand it to the midpoint of that triangle's side. Of course knowing that doesn't tell me directly how big the triangle needs to be for the circles to fit inside of a circle of known radius.
From the set of equations for finding dimensions of an n-sided polygon I derived which unknowns I needed to stick together to equate them directly to the one known I had. The length of a side, 'a', of a polygon with n sides can be found if you know the radius of the circle that encloses the polygon, 'R'. a = 2*R*sin( pi/n ).
The radius of a circle tangent to it's two neighbors, each of which is the same radius, is just half of a. Since the circles are centered upon the points of the polygon, the smallest circle that encloses the lot of them is the length of their radii away from their centers. So, the smallest circle to enclose said circles would have a radius of R + ( a/2 ). Substituting using the aforementioned formula, it is R + R*sin( pi/n ).
Crackers. So now, to find the R and find the a, I set the radius of the enclosing circle to 1, and solve for R. R = 1 / ( 1 + sin(pi/n) ). So there's the unit circle of the issue. Now it just remains to multiply by the radius I had decided on and derive from that the radius of the four pillars, then from that radius the radii of the three sub pillars making each up.
R1 = 2*( 1/(1+sin(pi/4)))
= 4-2*sqrt(2)
~ 1.17157287525380990239
a1/2 = 2 * ( 2 - sqrt(2) ) * sin( pi/4)
= 2*sqrt(2)-2
~ 0.82842712474619009760
R2 = ( 2 * sqrt(2) - 2 ) * ( 1/(1+sin(pi/3)))
= (4*sqrt(2)-4)/(2+sqrt(3))
~ 0.44395275812755717173
a2/2 = ((4*sqrt(2)-4)/(2+sqrt(3)))*sin(pi/3)
= 6-6*sqrt(2)-4*sqrt(3)+4*sqrt(6)
~ 0.38447436661863292586
...of course the problem with this solution is it does not find the dimensions such that the sub pillars actually touch those of the other pillars. There's gaps. I shall need to see about those now...
