#Non Euclydina Geometry
12 messages in this thread
I have never really examined a 3DS GSPHERE to notice that the faces
varied. In that case, you'll have to use an automated algorithm.
I'll develope some algorithms, and if I find one I'm happy with I'll
post it here.
James – An AutoCAD user on this forum pointed out that the model I'm
after has to fall in the catagory of regular polyhedrons, of which
there are a limited number. The tetrahedron (4 sides), dodecahedron
(12 sides) and icosahedron (20 sides) are 3 examples. I'm pretty sure
there are fewer than 10 of these objects, where all sides are the
same. So, maybe that will assist you in your algorithm
ax-ploration…
I looked in a great illustrated dictionary I have today, and under
"polyhedron", it actually had pictures of Tetrahedron, (4 triangles),
Hexahedron, (6 squares… 'sounds like a "cube" to me <g>),
Octahedron, (8 triangles), Dodecahedron, (12 pentagons), and
Icosahedron, (20 triangles). It went on to say…"Each of these five
regular polyhedrons has identical regular faces and identical angles.
There are no other shapes with these properties."
So it looks like this might be where the quest ends, although I
imagine it might be possible to "fudge" a bit with slightly different
size cones or endpoints to end up with more. Anyway, an Icosahedron
doesn't look too hard; I'll try to come up with something this
weekend, if you think it's busy enough.
Isn't it weird that there are only 5 of these objects? Something
integrally related to the curvature of space, probably, or the number
of orders of unity beyond square root of -1 (there are 6, 9 if you
count 1, -1 and the square root of -1…Hm… does this relate to
Jimi Hendrix's tune "If 6 Was 9"? <G>) or the number of angels that
can dance on the head of a pin. I *do* think 20 will be plenty of
cones. I'll have to look at it, but I think d' quest has, indeed,
ended with the Icosahedron. Thank you so much, John, for pursuing this
train of thought with/for me!
It would appear that you beat us to the solution singlehandedly with
a dictionary! Can AutoCAD create all of these regular polyhedrons?
If so, David's problem becomes only a matter of geometric
construction.
Also, I recall David mentioning that a Dodecahedron, (12 pentagons)
was not sufficiently "busy". Using the Icosahedron (20 triangles),
would help, but the packed nature of the construction might
contradict the desired effect. If the radii of the circles were
reduced so that there was more space between the cones, (hence, they
would not be tangent, but the general construction principles would
be the same) this might visually enhance the image.
I discovered that constructing an icosahedron and inserting cone bases
in the triangles leaves a hole in the center of every circle of five
cones, which is probably not the desired effect, but, since this hole
is in the shape of a pentagon, another slightly smaller cone fits
perfectly tangently in there, with its peak at the center of the
polyhedron, just like all the other cones. This gives a total of 32
cones, all touching their neighbor cones, with all these points being
tangent to the center. I've sent one via email to David; 'hope he can
use it.
AutoCAD doesn't automatically create these polyhedrons, but it does
automatically create polygons, so it was a simple matter using
geometry to create the final objects. The biggest hassle was later
defining separate object layers and nomenclature, so each cone would
be separate in 3DS; I also included separate base faces for all the
cones in case he needed those too. If anyone is interested, I'll
upload both the dodecahedron and icosahedron, both made of cones;
they're actually pretty small files, and were fun to make.
John,
>> If anyone is interested, I'll upload both the dodecahedron and icosahedron
<<
Please upload the files of which you speak… I'd like to check them out.
Thanks,
Mark
I would definitely be interested in seeing what they look like. For
some reason I am having a hard time picturing how to build them let
alone what they actually look like. Could make for some interesting
lisp routine to write in AutoCAD. (spoken like a true nerd I'm
afraid).
Yann,
Tried to upload the files today…got a message saying the libraries are full.
I'll try on a daily basis 'til it works.
John
Hey, BTW…were you around for that discussion of the 3D AME spring that was
showed in Cadalyst or Cadence in the R12/Mac review? Any idea how that was
done??
Thanks for trying to upload the file. Let me know when you get it up there.
I'm afraid I wasn't around for the 3d AME spring dicussion, nor have I seen the
r12MAC review in CADence or CADalyst. Is there some thing I should know about?
Springs can be created with AME, it's just that it's a hard manual process.
I've thought about writing a lisp routine to do it automatically. Unfortunately
I haven't had any time to do it nor do I have an actual use for that myself. I
would like to write one eventually. The math could be interesting to figure
out.
Yann Bertaud – VPG, Autodesk.
Yann,
The discussion on the AME spring was on the ACAD forum a few weeks back. On
page 107 of the Nov. 93 Cadence is a screen shot of a 3d spring created with
AME on the Mac. Everyone wanted to know how it was done, but no conclusions
were reached. The author said that drawing the basic entities on a Quadra 800
was faster than on a 486/50. I just thought you might know something about its
construction we didn't; it's an interesting problem…'was just curious. I
personally don't need one for anything right now.
John
Yeah, James, the dodecahedroid (new word! new word!) was not quite
what I had in mind. Actually, "busy" was not the right word. I'm
after a spherical overall impression and 12 cones was too obviously
not spherical, that's all. 20 is pretty *&&^%$% nice, tho'!