Yes, I realize many people don’t know what I’m talking about and even among those who do, interest is probably pretty slim, but this has been bugging me for about 30 years and I really want to know whether it is possible to create a non-periodic tiling of the surface of a sphere.
For those of you not yet fascinated by tesellations, a periodic tiling is one where if you made the pattern out of wires you could shift it over and it would still line up. An example of a periodic tiling of a spherical surface might be a geodesic dome in which the surface is entirely tiled with triangles. Non-periodic tilings can be generated if you start (for example) with two types of rhombuses, one with an angle of 72 degrees and one with an angle of 36 degrees.
The distinction between Periodic/non-periodic tilings only make sense on an infinite surface like a plane. I’m not sure how you would even define the terms on a finite surface.
I’m not sure what you mean. I don’t think you can tile a sphere with a single regular shape, or with two or three or four shapes. I’m certain that you can tile with a whole bunch of shapes – the usual grid you get when drawing latitude and longitude lines will tesselate a sphere, for instance. Variations on this theme will work, as well (replace some straight lines with wavy lines, or with teo lines with an angle between them. I suspect that this isn’t what you have in mind, though. I’ve never seen a tesselation of a sphere that didn’t involve such a bipolar pattern, though.
That said, I can easily see how to “tesselate” a sphere by inscribing a tetrahedron, then drawing a shape at each vertex, then working my way out from there and repeating the pattern around each vertex. Eventually you reach the boundary with the other patterns. You can repeat this with an inscribed square, octahedron, dodecahedrom, or icosahedron. Heck, you can do it with any inscribed polyhedron. In that way, you’ll end up with a tesselation that re-uses the same shapes several times on the surface of the sphere. You just end up with odd shapes of different sizes at the “grain boundaries”.
But I don’t know if that’s what you’re talking about.
I thought the typical geodesic dome (if continued to complete the sphere underground, so to speak) was “tiled” with regular pentagons and regular hexagons. But I guess this would be periodic and you want non-periodic?
Wow. Four responses in under two hours. Is this cool, or what?
OK, so any geodesic (polygon inscribed on a sphere) will be a periodic tiling, such as tetrahedron, icosahedron, geodesic of 2- or higher frequency. The pattern can be “slipped” so it matches up again, usually by shifting a vertex around which there are five regions to the next such vertex.
Yes, the distinction between periodic and non-periodic tilings is meaningful on a sphere but I’m still wondering if you can - for example - tile a sphere with a slight variant of the Penrose rhomboids, or the dart-bowtie duo.
My interest, by the way, is partly because I want to create non-periodic tensegrity forms, if anyone’s interested in those creatures.
It’s not clear to me either what would be meant by a non-periodic tiling of a sphere. Simply by the nature of a sphere, it would have to have a period of one revolution of the sphere. I would imagine you might have a tiling that had no smaller period, but it would certainly have the one revolution period.
I have no earthly idea what a tensegrity form is, but if I wanted to generate a sphere tiling that didn’t look like some regular mesh, here’s what I’d do:
First, use a pseudorandom number generator to generate triplets in the range [-1,1], and renormalize each triplet to lie on the surface of the sphere. Be sure to check that at least one value in the triplet is non-zero, and throw out any duplicated surface points.
If you really care about avoiding clustering, throw out all generated points (before renormalization) that fall outside the unit sphere, or that fall inside a 1/2 unit sphere. This will give a uniform sampling density, and avoid precision problems due to all values geing too close to 0.
Also, if you would like a somewhat regular surface, simulate a repulsive force between surface points that falls off as the square of the distance between them. Start small, to avoid any precision errors, and snap the moved points back to the surface of the sphere. Depending upon how precisely you converge to a steady-state, this can also get rid of clustering.
Finally, do a Delauney triangulation of the points to find a good surface.
This WAG is based upon a paper by Greg Turk, Retiling Polygonal Surfaces, in the SIGGRAPH '92 Proceedings.
The best sources I’ve found for this sort of thing are the 3 books by Magnus J. Wenninger:
Polyhedron Models
Spherical Models
Dual Models
If you want a truly “random” design, just draw irregular shapes all over a balloon, dividing and connecting them at random. But, of course, it’ll still repeat every revolution (assuming that we’re limited to 3 dimensions).
According to David MacCaulay in his current PBS series Building Big, the geodesic dome, as designed by Buckminster Fuller, is composed of triangles; maybe you’re thinking of a soccer ball, whose skin is made of pentagons and hexagons.
Punoqllads wrote:
Again from Building Big, a tensegrity dome is basically a circus tent with the poles cut off in mid-air and joined to the others with diagonal cables. The Georgia Dome in Atlanta is an example. The concept was orginally credited to Fuller, but it was actually one of his students who came up with it.
The roof of the Georgia Dome is really quite amazing – the fabric roof weighs just 68 pounds, covering about 400,000 square feet, but will support the weight of a fully loaded pickup truck. If you’re ever in Atlanta, check it out.
Draw a scalene triangle on your sphere somewhere: That’s your first tile. The rest of the sphere is your second tile. Unless you’re counting the 360[sup]o[/sup] rotation as a periodicity, this will give you an aperiodic tiling. If you do count the 360[sup]o[/sup] rotation, then it’s trivially impossible.
And by the way, the name of the soccer-ball shape (also the Buckminster Fullerene shape) is the equilateral truncated icosahedron.
[QUOTE]
*Originally posted by 3waygeek *
**don willard wrote:
68 lbs of teflon coated fiberglass covering 395,000 sq ft ? Seems too light to have any significant strength. I know this quote is from the website you referenced but… 68 lbs?! 9.068 acres of area protected by a material that weighs 7.5 lbs per acre. Truly amazing if this is so.
