## Is it possible to aperiodically tile a sphere?

For the discussion of math. Duh.

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Shinju
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### Is it possible to aperiodically tile a sphere?

I'm currently doing a Ceramics project, making wire-frame versions of the various different polyhedra, I started with the Platonic solids and then moved onto the Archimedean ones and so on. It occurred to me that another perspective on what I was doing was that I was tiling a sphere.

Now I've always been fascinated with Penrose tiling, there's something very beautiful about a five fold symmetry, and in art having things almost but not quite symmetrical always works really well. So it occurred to me what would be really interesting would be something like the polyhedra I'm working on, but with a quasi symmetry rather than the full symmetry I'm currently using.

So is there a penrose tiling on a sphere? Are there other aperiodic or non periodic tilings on the sphere (I'd prefer something with a five-fold or other odd numbered symmetry as they are generally more aesthetically pleasing)?

I'm also very interested in the Maths side of it too, so any more general information on Penrose tilings will be greatly received. I'm conversant on the construction method, but I've yet to actually see a proof of the tiles aperiodicity. I'd also be interested in hearing about any other aperiodic or nonperiodic tilings of the surfaces of other three dimensional objects, such as cylinders or toruses (tori?).

Mike_Bson
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### Re: Is it possible to aperiodically tile a sphere?

Very interesting question. Well, for one, you'd want the two polls to have the shape that is in ''the middle'' of the tessellation.

PM 2Ring
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### Re: Is it possible to aperiodically tile a sphere?

A finite tiling of the sphere can't be aperiodic, as it has to wrap onto the sphere. The usual planar Penrose tilings ought to be infinite to be truly aperiodic, but if we use tiles of diminishing size, then we can fit an infinite number onto a finite sphere. One way to do this is to use the Riemann sphere.

Here's one such tiling I made a few years ago, using POV-Ray.

You'll notice that the tiling "peters out" before it gets to the South Pole. Sorry aboout that. I don't think it's possible to cover the Riemann sphere totally with an aperiodic tiling with a finite number of computations, but I guess there may be a way to do it using deBruijn techniques.

As a comparison, here are top & bottom views of a simple square tiling of the Riemann sphere.

Here's a flat Penrose tiling of mine on the POV-Ray site.

See this thread and this one for more of my flat Penrose tilings.

Sadly, the POV-Ray newsgroup database seems to have become mangled, and quite a few of my messages & images are displaced or missing.

You can see some more of my Riemann sphere renderings & other tilings rendered in POV-Ray at a friend's blog:
http://newtonexcelbach.wordpress.com/2008/06/30/pythagoras-penrose-and-pov-ray/, including this one, which uses two overlapping Penrose tilings.

Shinju
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### Re: Is it possible to aperiodically tile a sphere?

>>>A finite tiling of the sphere can't be aperiodic, as it has to wrap onto the sphere. The usual planar Penrose tilings ought to be infinite to be truly aperiodic,

Sorry, but I need to ask, what's the proof of this? I mean it might not seem likely but then neither is the existence of an aperiodic tiling of the plane in the first place?

Consider that there are trivial non periodic tilings of the sphere, ones where there's just no quasisymmetry at all - 3 random circumferences of the sphere will divide the surface up in a manner where there is no period, its just that then you dont have any repeating tiles. Its trivial if we allow at least one unique tile - however I'm interested in less trivial solutions.

Any suggestions for the other questions I had?

gmalivuk
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### Re: Is it possible to aperiodically tile a sphere?

I think you're understanding aperiodicity differently. If it has to go on a sphere, then going around any circumference is going to bring you back to where you started, and thus you will repeat the same set of tiles.
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Shinju
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### Re: Is it possible to aperiodically tile a sphere?

I don't think you're correct about that. It sounds like your imagining that the surface of the sphere could be unwrapped in some smooth transform to be laid out in a regular shape which would then tessellate, but I don't think there is such a transform.

The definition on Wiki for periodic tiling is for the plane and is as follows :- tilings which remain invariant after being shifted by a translation. I think for the surface of a sphere we can define it similarly to be tilings which remain invariant after being shifted by a rotation by an angle < 2pi (After all we wouldn't consider the standard penrose tiling to be periodic because if you rotated it 360 degrees it looks the same).

PM 2Ring
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### Re: Is it possible to aperiodically tile a sphere?

Ok, but I'd prefer to call that an irregular tiling, rather than aperiodic. There are such tilings for spheres and ellipsoids, but I can't recall the details at present. I read about them years ago, in an article on the guy who designed this:
Spoiler:

And you're right, you can't "peel" off the surface of a sphere or torus into a (Euclidean) plane surface without stretching it, although that's obviously easy to do with a cylinder.

Shinju
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### Re: Is it possible to aperiodically tile a sphere?

I dont seem to get a picture there. Link broken?

PM 2Ring
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### Re: Is it possible to aperiodically tile a sphere?

Shinju wrote:I dont seem to get a picture there. Link broken?

Hmm. That's odd. Try this page, from the site of the designer of the giant easter egg himself.

Link removed due to now being broken. - gmal

Cleverbeans
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### Re: Is it possible to aperiodically tile a sphere?

Or you can just Google "Vegreville pysanka" there are loads of images available. I'm somewhat stunned it managed to show up in this forum, even more so in the math subforum but apparently all the teasing I gave to Vegrevillians over their "famous egg" in the past was unjustified.
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antonfire
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### Re: Is it possible to aperiodically tile a sphere?

It's not hard to come up with examples where all the tiles are the same. For instance, take a tetrahedron and cut each face in two along some height. You can do it so that the resulting shape has no symmetries (for instance by having three of the heights meet at a vertex).

Tilings with no symmetries are easy since you can achieve them by mucking with something locally.

The Penrose tiling is interesting because it's aperiodic but still somehow ordered. For instance, every finite chunk of it reappears in infinitely many places. It's also related by a substitution to a scaled-up version of itself. Neither of these things make sense when you talk about tilings on a sphere.
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Shinju
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### Re: Is it possible to aperiodically tile a sphere?

>>>The Penrose tiling is interesting because it's aperiodic but still somehow ordered. For instance, every finite chunk of it reappears in infinitely many places.

Yes thats exactly the sort of thing I'm after, something quasi symmetric. I'm not after something made from just random shapes.

>>>It's also related by a substitution to a scaled-up version of itself. Neither of these things make sense when you talk about tilings on a sphere.

I appreciate that there may be differences between what I am attempting to find and a proper Penrose tiling, but I haven't yet been given any proof no such aperiodic tiling exists. I think what its looking like, is that either so far Mathematicians have not found any tiling (i.e. no one has looked for them) or that no one on the forums here has heard of any being found. The scaling up problem I think is the most likely direction to take in proving no such tiling can exist, but on its own it isn't a sufficient condition, since while it may be the method used on the flat plane to create aperiodic tilings, that doesn't imply it is the only method possible. As we've seen uninteresting aperiod tilings are possible, so it still seems to me to be possible that interesting aperiodic tilings with areas of local symmetry should exist too so I'll continue looking.

gmalivuk
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### Re: Is it possible to aperiodically tile a sphere?

Shinju wrote:
Neither of these things make sense when you talk about tilings on a sphere.
I appreciate that there may be differences between what I am attempting to find and a proper Penrose tiling, but I haven't yet been given any proof no such aperiodic tiling exists.
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Shinju
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### Re: Is it possible to aperiodically tile a sphere?

I don't really have the mathematical knowledge to be as precise as I'd need to be. I was hoping someone here could translate what I've written here in english into mathological combobulations.

I'm looking for an aperiodic tiling of the sphere (for periods <2pi, aperiodic as opposed to non periodic) and I'd like it to display certain levels of symmetry, just as the aperiodic tilings of the plane do, such as the Penrose tilings, I can't tell you precisely what property that is (although I see the word quasi-symmetry bandied round a lot) since I don't have the mathematical basis for understanding that.

I don't really get the (what seems like) resistance to the topic as it would seem to be quite an interesting one, at least it does to me.

Shinju
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### Re: Is it possible to aperiodically tile a sphere?

Maybe if I go the opposite way, and go simpler and less specific it might help.

I saw a pretty picture on the web, it looked like this.

I wanted to make one, however after contacting the author and talking to people here I found out it wasn't real.

I still want to make something similar, but something that is real.

gmalivuk
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### Re: Is it possible to aperiodically tile a sphere?

Shinju wrote:I don't really get the (what seems like) resistance to the topic as it would seem to be quite an interesting one, at least it does to me.
There is not resistance to the topic. There is simply a lack of responses because it remains unclear what properties you want your tiling to have, since the usual ones that make Penrose tilings so interesting can't be applied to a sphere.
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### Re: Is it possible to aperiodically tile a sphere?

I saw a pretty picture on the web, it looked like this.
Nice. But from the way the tiles on the periphery are shrinking, it does look a lot like a view from above of a Riemann sphere or other stereographic projection to me. FWIW, I've done some sphere tilings with that chunky tiled look, but I prefer the smoother look of the one I posted earlier.

Using stereographic projection, you can project any planar tiling onto a sphere, but you will always get shrinkage. Stereographic projection is nice because it preserves angles, but obviously it doesn't preserve areas. But because it preserves angles all such tilings are still ultimately Euclidean, and to tile a sphere's surface properly, you need to use a geometry with positive curvature.

Have you had a look at the work of Jos Leys? He does amazing things with hyperbolic tesselations, but I think he might have done some work on the sphere as well. Also check out the official MC Escher page - Escher did a few spherical patterns, and his hard-core fans have continued the tradition. But I don't know if these guys have done the kind of thing you're looking for.
Last edited by PM 2Ring on Tue Aug 03, 2010 6:06 pm UTC, edited 1 time in total.

Shinju
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### Re: Is it possible to aperiodically tile a sphere?

I absolutely adore Escher. Hyperbolic geometry and the Poincare disc model are something I have planned for the future but I haven't come up with something I actually want to create quite yet, its gently mulling itself over in the back of my mind.

As for your choice of texture, yes I absolutely agree, particularly for ray traced objects, that was a very clunky looking texture on the one I showed but it was one of only two depictions of a penrose type sphere I found on the net, your's is a lot nicer - I particularly liked the pastel shaded one. Mind you I've got quite different limitations when it comes to glazing ceramics, lol.

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### Re: Is it possible to aperiodically tile a sphere?

Thanks for the compliment, Shinju. I haven't been in the mood for a while to do much ray-tracing, I really ought to get back into it before I forget how.

Good luck with your ceramics work. Maybe you should post photos of some of your geometric pieces on this forum.

I'd love to see some of my creations in solid form. Hopefully, that will happen in the not-too-distant future when one of my friends gets a 3D printer.

birds for the mind
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### Re: Is it possible to aperiodically tile a sphere?

Hello,

I am trying to find possible ways to cover double curvature surfaces with aperiodic tiling (penrose ones)..
do you think it's doable?..
I would like to cover a surface with aperiodic tilings. More specifically, I would like to start with double curvature surfaces and then experiment with non-orientable ones..

gar37bic
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### Re: Is it possible to aperiodically tile a sphere?

Oddly enough, I landed here after googling 'aperiodic tiling of a sphere'. I hadn't really thought it through - I was just taking an analogy into the third dimension. So we are starting from an analogy to the tiling of the plane, merely trying to find something interesting.

So, having read this and some other things, I've figured out the following. Regular tessellations of the surface of a sphere with a single tile shape are equivalent to composing one of the Platonic solids. It's been proved that this is all there are. There are a number of tessellations involving two tile shapes, such as the 'soccer ball' combining hexagons and pentagons. I refer the reader to http://en.wikipedia.org/wiki/Platonic_solid#Tessellations and also the articles on polyhedron, polytope and related topics.

So I think the question comes down to whether there is a non-Platonic polyhedron - a polyhedron composed of a limited number of tiles, that do not decompose into an equivalent of one of the Platonic solids. At first glance (IANAM) it seems that this is impossible. However, at one time aperiodic tilings of the plane were thought to be impossible so this idea can't be dismissed out of hand.

In order to take this farther, we have to recognize differences between the plane and the sphere. Aperiodicity in the planar case relates to translational symmetry and (in most cases) rotational symmetry. So in the spherical case, we might pose the question as follows: Assume the 'skin' of the tesselated sphere can be shifted around arbitrarily. Then we define aperiodicity as meaning that no sliding or rotating of the skin around will allow the tesselation to match up over all of the equivalent tiles of any of the Platonic solids simultaneously. In other words, the tessellation can not be mapped onto a Platonic tiling. One might call it a 'non-Platonic' tiling.

This is not the same thing as aperiodicity in the plane but it could be interesting. It seems unlikely that this is possible given the Platonic solids proof - I don't see how any such tiling can irreduceable to the Platonic case - but it is well beyond my meager skills in math to even compose a reasonable approach to the question.

One interesting (i.e. "hand-waving") approach would be to do something analogous to the discovery that Penrose tilings were related to a projection of from a five-dimensional structure:
In 1981, De Bruijn explained a method to construct Penrose tilings[21] from five families of parallel lines as well as a "cut and project method", in which Penrose tilings are obtained as two-dimensional projections from a five-dimensional cubic structure. In this approach, the Penrose tiling is viewed as a set of points, its vertices, while the tiles are geometrical shapes obtained by connecting vertices with edges.
- (http://en.wikipedia.org/wiki/Penrose_tiling)

So maybe projecting some higher-dimensional construct onto the sphere might have some interesting results. Unfortunately, adding dimensions (plane to sphere) is adding constraints in this case, not removing them. Higher dimensional spaces have fewer equivalents to the Platonic solids, except the four-dimensional 24-cell. All higher dimensions than four have only three convex regular polytopes. So this approach also seems unlikely to be of assistance.

Yesila
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### Re: Is it possible to aperiodically tile a sphere?

Shinju wrote:
I wanted to make one, however after contacting the author and talking to people here I found out it wasn't real.

I still want to make something similar, but something that is real.

You can cheat. The sphere will undoubtedly be either hanging or sitting on a stand? either way part of the sphere will be obscured and you can put your "cheating" tiles there.

Deveno
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### Re: Is it possible to aperiodically tile a sphere?

i think this can be done. my idea is this: create an "asymmetric" tiling that has a regular border "near the equator" for a hemisphere.

then create another "similar" tiling for the other hemisphere, and glue together. this ought to get around the "shrinkage problem".

EDIT: idea #2: create two approximately circular "polar caps", and treat the remainder of the sphere as a cylinder. to avoid scaling problems, expand each horizontal strip between the "caps" with a tile cluster in the middle.

tetrahedron
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### Re: Is it possible to aperiodically tile a sphere?

Hi folks. Just ran across this forum thread. I've been studying alternative periodic table depictions for some years now. Back in 1979 I realized, while still a young chemistry undergrad, that the periodic table could be thought of as a distorted tetrahedron, if you extended the atomic numbers to 120 and cut out the 4 orbital blocks and stacked them in order vertically. Much later, after finding that someone else had independently rediscovered my solution (see Valery Tsimmerman's Perfect Periodic Table site) I worked to try to find a solution to what I regard as failures of the model. In 2009 I came up with a system of skew rhombi of close-packed spheres (each such rhombus representing a square number of spheres, equivalent to 2 equal-length periods as found in the Left-Step Periodic Table of Charles Janet (from 1928), which has the s-block elements on the right. This table organizes orbitals in terms of their order of actual introduction into the elements rather than 'surface' chemical behavior (that is, with the 'noble' gases on the right). When the skew rhombi are stacked, like taco shells of increasing size or like Russian dolls, they form ever larger tetrahedra with no gaps. I only learned this past year that I'd been anticipated by at least 6 years by Russian physician Dmitri Weise (see esp. figs 4 and 8 at Weise's pythagorean approach in chemistry page, which you can find on Google) .

Still, the model fails to maintain the continuity of Mendeleev's Line, the numerical continuity of all the atomic numbers, in order. In the past year I've perfected 8 different models (apparently covering all the logical possibilities) that manage to keep the line continuous. The basic pattern is that of fig.6 at the same site, but for co-centered tetrahedra of 20 and 100 spheres (that is, of 4 periods each: 2+2 and 8+8 for 20, and 18+18 and 32+32 for 64). Each period acts like a Penrose tile to some extent. Translations on the surface won't create periodicity (don't confuse chemical periodicity with tiling periodicity!), for all tiles higher than s-block-only (2 spheres) the tiles are self-similar but of growing sizes. Some chirality is involved as well, as well as different labeling orders, leading to the 8 possibilities (4 if you ignore simple mirror-image tetrahedra). I don't know if these models exhibit quasicrystalline patterning.

Anyway this thread has lain dormant for some time- does anyone here know whether professionals (or even amateurs) have toyed with tiling patterns like these have been investigated over the surface of a tetrahedron? Thanks.

Jess Tauber

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### Re: Is it possible to aperiodically tile a sphere?

By the way, it turns out the both the electronic and nuclear shell systems get mathematical motivation from Pascal's Triangle. For the electronic system, every other s2 element's atomic number is every other tetrahedral number (thus the tetrahedral mapping). For the simple harmonic oscillator model of the nuclear shell system, for spherical nuclei the super-stable 'magic' magic numbers are exactly doubled tetrahedral numbers: 2,8,20,40,70,112,168.... This is due to the fact that the intervening shells are doubled triangular number in size:
1s=2
1p=6
1d2s=12
1f2p=20
1g2d3s=30
1h2f3p=42
1i2g3d4s=56
1j2h3f4p=72

Note that the orbitals are all segregated for positive or negative parity in these 'period analogues'.

For ellipsoidally deformed nuclei, the values of the numerator and denominator of the oscillator ratio (which measures the relative extents of the matter wave in the polar and equatorial directions, respectively), standing in as a measure of deformation (versus the usual 'deformation parameters beta, delta, and epsilon) determine how these doubled triangular number intervals are used to generate new magic numbers.

For the sphere (the default ellipsoid), with oscillator ratio (OR) 1:1, we have ONE doubled triangular number interval between every single magic, and each interval is used only ONCE. But for a prolate ellipsoidal nucleus of OR 2:1, though there is still one doubled triangular number interval between each magic, each interval is used TWICE to generate a magic. SO 2,2,6,6,12,12,20,20,30,30.... giving magics 2,4,10,16,28,40,60,80,110,140, which match published sources.

OR 3:1 has each interval used THRICE to generate magics, again giving the same sequence of magics as published sources. Thus the OR's numerator MULTIPLIES the use of the intervals.

For an oblate harmonic oscillator nucleus of OR 1:2, each doubled triangular interval is used just once, but now its between not every single magic, but between every SECOND. And for 1:3, between every THIRD and so on. The denominator of the OR DIVIDES the system, just as the numerator MULTIPLIES.

A professional mathematical physicist informed me that quantum harmonic oscillators ALWAYS deliver numbers of stable states whose values are terms in Pascal Triangle diagonals. Which diagonal the terms come from depends completely on the physical dimensionality of the system. The outer rows of Pascal's Triangle are all 1's, so just a single unchanging state, for a 1 dimensional system. Then the natural numbers are used for a two dimensinoal system. The triangular numbers pattern a 3 dimensional system, and we double for spin.

There are other regularities in the harmonic oscillator model. For example it is found that the total shell energy remains conserved over deformation, even though the individual components of these shells change energies in regular ways. For the shell energy to remain constant means all the changes (which involve different numbers of particles as well) have to be coordinated in some way.

The more realistic spin-orbit model of nuclear shells retains a number of the Pascal Triangle motivations. For example, for spheres the new magic numbers are 2,*6,14,28,50,82,126,184... Intervals here are all doubled triangular PLUS TWO. But this hides the way these are actually constructed. All these new magics increase the harmonic oscillator magics by monotonically increasing amounts, due to the so-called 'intruder levels', which are orbital partials of highest spin lowered in energy by the spin-orbit effect sufficiently to join the structure of the preceding shell.

It turns out that the SIZES of these intruders are exactly those needed to increase the shell size from the harmonic oscillator's doubled triangular number values TO THE VERY NEXT HIGHER doubled triangular number values. Thus 1f2p (20 nucleons) adds 1g9/2 (10 nucleons) giving 30 total. Then 1g2d3s (30) adds 1h11/2 (12) to give 42 total. And so on.

Furthermore, the depths of intrusion (how far the intruder level dips from its original harmonic oscillator shell position to the spin-orbit position) is always in terms of doubled triangular numbers.

For 1g9/2 depth is 2. For 1h11/2 depth is 6. 1i13/2 has depth 12. And 1j15/2 has dept 20. For neutrons anyway. There's some ambiguity for protons in one shell, where depth 20 is found where 12 is expected.

And conservation of shell energy seems to be preserved over deformation in the spin-orbit model as well, though harder to tell due to poorly made illustrations I've used (Nilsson diagrams). It will be interesting to see whether the numerator and denominator of the OR in the spin-orbit model determine new magic numbers as they do for the harmonic oscillator.

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### Re: Is it possible to aperiodically tile a sphere?

cyanyoshi
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### Re: Is it possible to aperiodically tile a sphere?

"The higher level of attention needed to understand the geometry could distract a driver's view away from the road for longer than necessary which could therefore increase the risk of an incident."

I prefer to read this statement as drivers being so enthralled by the mathematical beauty of a geometrically accurate football sign that they forget how to drive, rather than drivers being distracted by a new sign that they aren't familiar with.

Eebster the Great
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### Re: Is it possible to aperiodically tile a sphere?

I like that story. Of course, it's a pretty tiny thing, but the government felt compelled to provide multiple arguments for preserving the current sign anyway. I understand the idea that changing signs without a good functional reason can be dangerous, but the government's other argument actually contradicts this (and is correct): nobody would really notice the change at all. But since Matt isn't advocating replacing existing signs and just that whenever they put new signs up, they use a more accurate picture, I'm all for it. It is a pretty strange error.

HES
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### Re: Is it possible to aperiodically tile a sphere?

Eebster the Great wrote:But since Matt isn't advocating replacing existing signs and just that whenever they put new signs up, they use a more accurate picture, I'm all for it.

This would, however, require an amendment to the Traffic Signs Regulations and General Directions 2016, which would have to be voted through parliament. Give it another 14 years perhaps?

Also, the line at the end isn't correct - at 100,000 signatures, the petition will be considered for debate in Parliament - and rejected.

(It did provide some amusement within our Highways department, though)
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elasto
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### Re: Is it possible to aperiodically tile a sphere?

Why on earth are parliament micromanaging rather than delegating the exact image allowed on a traffic sign? (Yet parliament isn't allowed any management over any of the details of Brexit...?)

Our country is so dysfunctional sometimes...

HES
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### Re: Is it possible to aperiodically tile a sphere?

We have strictly defined symbols to ensure consistency across the country, which in turn enhances readability. It's a lot of effort for the payoff, but British road signs are generally able to be read faster than those elsewhere. Same reason we use mixed-case rather than all-caps, so that you can recognise the word-shape of your destination rather than having to read the whole thing.

Uh, we seem to be veering a little off topic...
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### Re: Is it possible to aperiodically tile a sphere?

HES wrote:We have strictly defined symbols to ensure consistency across the country, which in turn enhances readability.

You can have strictly defined symbols country-wide without MPs being the ones to define them. Parliament doesn't have to have a vote every time NICE adds or removes a treatment from the NHS, no reason why they should have to if the Highways Agency wants to make a change.

Uh, we seem to be veering a little off topic...

Meh