Sphere with fewest unique faces

[David Russell]

All (particulary Taff!) I'm trying to get my head around a combined problem:  First I'm trying to construct a high order sphere with the fewest unique faces. The standard geodesic has more and more faces as the order increases. The truncated icosahedron seems to be the thing, but I'm having trouble finding a model with high order. I thought I found a Taff model, but then I couldn't find it again once I got Sketchup working.  Are there any better choices for the model? The Deltoid reticulation looks like the same faces, but I haven't gone through all the math yet.  The order I'm trying to reach is a sphere 20-40 meters in diameter, where no face is larger than one meter.  Once I've got that, I need an accurate projection to make each face about 100mm thick. Luckily this does not have to be exact.  I've been reading and reading, and I keep getting lost, there's so much!.  Can someone point me to the right data? Thanks!

[David Russell]

Just an addendum, the panels will be 3D constructed so they do not have to be flat. I'm hoping that makes a difference, but it could complicate the work in Sketchup...

[Dick Fischbeck]

Or the fewest parts will unlimited faces.

On Thu, Oct 23, 2014 at 7:20 PM, David Russell <da...@firelinx.com> wrote:

All (particulary Taff!) I'm trying to get my head around a combined problem:  First I'm trying to construct a high order sphere with the fewest unique faces. The standard geodesic has more and more faces as the order increases. The truncated icosahedron seems to be the thing, but I'm having trouble finding a model with high order. I thought I found a Taff model, but then I couldn't find it again once I got Sketchup working.  Are there any better choices for the model? The Deltoid reticulation looks like the same faces, but I haven't gone through all the math yet.  The order I'm trying to reach is a sphere 20-40 meters in diameter, where no face is larger than one meter.  Once I've got that, I need an accurate projection to make each face about 100mm thick. Luckily this does not have to be exact.  I've been reading and reading, and I keep getting lost, there's so much!.  Can someone point me to the right data? Thanks!

-- 

[David Russell]

Dick: I think you get it.  We will be building molds for the panels, so the fewest number of molds is the key.  Where can I find larger order numbers for the buckyball type?

On Thursday, October 23, 2014 7:20:24 PM UTC-4, David Russell wrote:

[TaffGoch]

Tall order -- likely impossible.

For a 20 meter dome, the circumference is about 62 meters. Divide that by your 1-meter panel criterion -- that's ~60 panels. Here's a depiction of a class-I dome, with 60 triangles around the perimeter:

This particular version hasn't been "optimized," and has 30 different triangles. With judicious "massaging," the count can be reduced, but (just estimating) probably not less than 12, or so.

Off-hand, I can't recall any ingenious solutions. If there have been any, it would have been well-celebrated and, therefore, pretty well-publicized and known (and often employed.)

Temcor's triacontahedron subdivision, championed by D.L. Richter, produces the likeliest low count (that being 12, in the above example.)

I'd love to hear about, and impart, a satisfactory solution, but, in 30 years, haven't come across such.

-Taff

[David Russell]

Taff: Thanks for jumping in. I knew I was getting into uncharted waters. I got the idea from the Hex-Pent discussions that the hexagons were all the same, I take it that's not correct.  Does it have at least fewer unique faces?

On Thursday, October 23, 2014 7:20:24 PM UTC-4, David Russell wrote:

[David Russell]

Let me outline my thinking. I'm not trying to find a perfect solution, just the best one I can.  If I understand what I've been reading so far (and that's far from a foregone conclusion) The icosahedron has 12 equal faces. The pentagonal Hexacontahedron has sixty identical pentagons. The Hexakis Icosahedron has 120 identical triangles (polyhedra.org).  Now is the problem that these look like they're exact but are actually not?  I found a quote that, "Unfortunately, it is a well-known group theoretical result that there are no completely regular point distributions on the sphere for N > 20." -- Max Tegmark  If we take that as the upper bound, and had an N=20 regular point distribution on a sphere, and we didn't have to push the vertices out towards the sphere because we can make curved panels, is there a polyhedral shape that can map into that distribution? it should be just the dual of the distribution points (?). If I'm wrong on those points, then the next best thing is to find the polyhedron with the fewest number of unique faces I can.  It may well be (N<=20) I have to give up my dome size or max panel size restrictions, we'll deal with that separately.

On Thursday, October 23, 2014 7:20:24 PM UTC-4, David Russell wrote:

[Ken G. Brown]

One way to cover a hemisphere with a single unique spherical triangle is to use long skinny gore-shapes going from the zenith to the equator. Think silo roofs. You can have as many as you want. 

Maybe that idea will help your quest in some way. 

The 120 triangles you mentioned are half right and half left handed so are not all identical. 

Ken G. Brown,

from my iPhone

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[David Russell]

Still adding more...  for example in http://www.emis.de/journals/EM/expmath/volumes/12/12.2/pp199_209.pdf  the coloring of figure 17 would lead me to believe that the sphere could be constructed with only 3 unique faces with optimized Coulomb energy points.  Taff: If you find this as interesting a problem as I do, but don't want to thrash it out in the forum, shoot an email to da...@firelinx.com and we can report back when we have an answer.  Do you know of a calculator for the hex-pent dome where we could compare the number of faces to this same basic geodesic size?

On Thursday, October 23, 2014 7:20:24 PM UTC-4, David Russell wrote:

[David Russell]

kbrown:  You are right about the triangles being left and right, I missed that at first glance.  But you see the benefit -- we now have 120 faces with only two unique sizes. The problem comes in where I have to restrict the size of one panel.  I could make a regular icosahedron as big as a football stadium if I could have any face size and just have one unique face. It's getting a big dome out of a minimum number of small faces that's the interesting part of the problem.  Thanks for helping though, you never know when it might spark something else.

On Thursday, October 23, 2014 7:20:24 PM UTC-4, David Russell wrote:

[norm...@gmail.com]

If you check out the John Sumner dome I made (https://groups.google.com/forum/#!searchin/geodesichelp/youtube/geodesichelp/9resE0V706k/j-JWslIj-kYJ) you'll see the left and right triangles are the same size merely flipped.

If you use spherical trig it is possible to do it with one "size"

[Ken G. Brown]

My vote is to keep discussion on the list, you never know when someone else may have something to add. 

Ken G. Brown,

from my iPhone

--

[Adrian Rossiter]

Hi David

On Fri, 24 Oct 2014, David Russell wrote:
> the next best thing is to find the polyhedron with the fewest number of
> unique faces I can. It may well be (N<=20) I have to give up my dome size
> or max panel size restrictions, we'll deal with that separately.

There are also non-polyhedral tilings, where the vertex of one tile
lies on the edge of another, if these are of interest

http://cs.stmarys.ca/~dawson/images4.html

Adrian.
--
Adrian Rossiter
adr...@antiprism.com
http://antiprism.com/adrian

[David Russell]

Norm: That was an interesting discussion, have you continued to look at scaling up the total number of faces? How far can you go before you have more than two triangles?  This is heading the way I need, though, especially looking at Taff's addition to your article, "Domical Structure" it looks like there's 4 unique faces for 60. That's still not as good as the pentagonal Hexacontahedron with 60 of one face, but that's as far as it can go. What happens now as we push for more faces with this model?

On Thursday, October 23, 2014 7:20:24 PM UTC-4, David Russell wrote:

[norm...@gmail.com]

The 60 panel one is one piece flipped too.  I haven't explored scaling up the total number of faces...I tend to go the opposite way and try to reduce the total number of faces.

[norm...@gmail.com]

This is a good link too.  thanks for posting Adrian.

[David Russell]

Adrien: Thanks for that, I hadn't even gone down that particular rabbit hole yet.  It is not obvious to me how these tilings can be expanded to larger spheres, i.e. more tiles of smaller size to make a bigger sphere.  Anyone care to chime in on that?  Does anyone have experience with counting the unique faces on the floret tessellations? These seem to be pretty high order with very few unique faces.  One of Taff's examples attached here.

On Thursday, October 23, 2014 7:20:24 PM UTC-4, David Russell wrote:

[norm...@gmail.com]

Have you considered Lobel Frame type structures?

[David Russell]

Norm: No hadn't seen that one yet.  I can see how you could build big structures with just the triangles -- not domes, but some kind of structure.  It doesn't look like there's any usable space inside, though. In the larger structures they again play with squares and pentagons so you're back to the same geometry we already have. Thanks, though. Learn something new every damn day.

On Thursday, October 23, 2014 7:20:24 PM UTC-4, David Russell wrote:

[David Russell]

Norm's last comment got me thinking:  Perhaps we should widen the net.  I don't care if the end structure is a dome, or a torus, or a cube.  I just want to build the structure with the fewest number of unique panels.  Granted, we all know the dome is the strongest structure of the type enclosing the most area, but a torus should be pretty interesting. Sure we could build a big cube from a lot of little squares, but it would not be stable. I found a paper on geodesics of torus (torii?) that led me to believe that as long as you followed the geodesic -- which spirals around the torus -- you could tesselate it with pentagons or perhaps pen-hex again.  I found this paper that claimed tessellation with triangles and regular hexagons  http://forums.wolfram.com/mathgroup/archive/1999/Nov/msg00113.html

There's got to be a "sweet spot" here somewhere that gets us at least the fewest number of unique faces for a given size.

On Thursday, October 23, 2014 7:20:24 PM UTC-4, David Russell wrote:

[norm...@gmail.com]

I'm surprised Dick didn't mention his randome.  Although it is built with vertexes instead of faces, they are the same size and shape and with a mold you could pump them out fast.

Your only limit would be how small of an angular defect you could produce with the mold.  For example if each piece had an angular defect of 1 degree then you'd have 360 pieces for a half-sphere.

[Dick Fischbeck]

Hey Norm!

I did, in a way. I wrote:

"Or the fewest parts (with) unlimited (number of) faces."

On Thu, Oct 23, 2014 at 7:20 PM, David Russell <da...@firelinx.com> wrote:

All (particulary Taff!) I'm trying to get my head around a combined problem:  First I'm trying to construct a high order sphere with the fewest unique faces. The standard geodesic has more and more faces as the order increases. The truncated icosahedron seems to be the thing, but I'm having trouble finding a model with high order. I thought I found a Taff model, but then I couldn't find it again once I got Sketchup working.  Are there any better choices for the model? The Deltoid reticulation looks like the same faces, but I haven't gone through all the math yet.  The order I'm trying to reach is a sphere 20-40 meters in diameter, where no face is larger than one meter.  Once I've got that, I need an accurate projection to make each face about 100mm thick. Luckily this does not have to be exact.  I've been reading and reading, and I keep getting lost, there's so much!.  Can someone point me to the right data? Thanks!

[David Russell]

Sorry Dick, I didn't get what you were saying. I haven't seen that structure before. Are these small plastic panels? Glued together?  My first thought is for a small emergency shelter, which seems to be the case, it should work fine, but for a larger structure with significant mass wouldn't the pieces tend to tear apart (assuming finite strength of the adhesive). How does it get to be watertight?  Illuminate us more, if you would, on how it's formed and put together.

On Thursday, October 23, 2014 7:20:24 PM UTC-4, David Russell wrote:

[Dick Fischbeck]

A dome shell within a dome shell. The shell is quite thin but the two shells are bridged. A single vertex elements throughout, unlimited (theoretically) frequency. The elements can be stamped out of aluminum, for example. Flat elements with a built-in vertex can be made out of flat sheet stock. Other materials and other methods work, too.

http://www.google.com/patents/US7389612

Craig Chamberlain used spherical caps, which are compoundly curved. 

http://www.google.com/patents/US4270320

Bucky's plydome fits your requirements as well. More to chew on.

http://www.google.com/patents/US2905113

On Thursday, October 23, 2014 8:22:12 PM UTC-4, Dick Fischbeck wrote:

Or the fewest parts with unlimited faces.

[Dick Fischbeck]

Hi David

Enjoyed your Mars blog. 

Overlapping vertex elements, made out of sheet material, fasteners (through-bolts) waterproofed with o-rings or other self-sealing washers, I like aluminum. 0.040" is nice. I am not an engineer, but I do understand something of forces. Others here have heard my spiel before. The patent explains a lot and I am happy to fill in the gaps. (no pun intended)

Bucky named these structures polyvertexia.

Edges form between any two elements. Faces form between any three elements.

https://www.flickr.com/photos/8152259@N03/6167562967/

Dick

Freedom, Maine

[Dick Fischbeck]

Here's a sharp image of that model of round playing cards. 2004. Bridges Kansas.

http://www.cs.berkeley.edu/~sequin/ART/BRIDGES2004/domes/DSCF0095b.html

On Fri, Oct 24, 2014 at 1:05 PM, <norm...@gmail.com> wrote:

[TaffGoch]

On Fri, Oct 24, 2014 at 6:01 AM, David Russell wrote:

"...I got the idea from the Hex-Pent discussions that the hexagons were all the same, I take it that's not correct.  Does it have at least fewer unique faces?"

David,

Hex/pent tessellations employ hexagons that only APPEAR to be identical. As in triangular tessellations, the COUNT of different hexagons is comparable. Generally, the hexagons are neither equilateral, nor regular. For examples, see...

https://3dwarehouse.sketchup.com/model.html?id=86cf0d924a4e77ebcce427aa8e13641c

___________________________________

On the topic of spherical distribution, using charge-repulsion separation of nodes, you may find this java exploration of interest:

http://thomson.phy.syr.edu/thomsonapplet.php

This java app will also depict the "dual," composed of pent/hex faces (but will not provide cartesian coordinates for the dual nodes. That function is limited to the triangular tessellation, only.)

Again, various different faces are produced, triangular or polygonal.

-Taff

[David Russell]

Dick: Hey, I've been googled! Thanks for that, the Mars One blog has been up since summer, but I don't get much feedback. The funny thing is last August I got a call from Fox Network, saying they were starting a new reality show named Utopia. It puts 15 people on their own for a year.  They said they were up to their eyeballs in the homeless, the jobless, and the clueless. They had read the blog and asked me to apply for the show. They needed someone who at least had thought about these scenarios.   I did, and we discussed it for a while, but if you've seen the show you can see in the end they went another way. The people you want to send on a long term isolation mission are NOT the ones that will make good reality TV.   I'll be taking a closer look at your structure.

Taff:  Thanks for clearing that up for me, I haven't found a lot of discussion that talks at all about the faces themselves.  Is the same true of the floret tessellations? 

daver

[Dick Fischbeck]

Off topic: In the news this week.

http://news.nationalgeographic.com/news/2014/10/141015-mars-simulation-mission-space-psychology/

Back to dome construction, edge lengths can be the same ala Clinton/Goldberg. Unless a "perfect" sphere is import to you...

http://en.wikipedia.org/wiki/Goldberg_polyhedron

http://www.domerama.com/wp-content/uploads/2012/08/ClintonEqualEdge.pdf

[Dick Fischbeck]

Our friend George Hart added this video just last year. Worth a few minutes.

https://www.simonsfoundation.org/multimedia/mathematical-impressions-goldberg-polyhedra/

On Sat, Oct 25, 2014 at 2:17 PM, David Russell 

[David Russell]

Ok, guys, I have to go back and challenge my initial assumptions.  Viewing these geometries without doing the math gives you the impression that the faces are equal. Let me start at the beginning, and see if we can work through this logically:  First step, I believe that I'm correct that the regular icosahedron  has all faces equal (by definition), and the truncated icosahedron has two unique faces that are all equal.  Can someone confirm for me that I've got that much right?  Similarly, spherical tiling of the icosahedra (2,3,5) all of the faces should be identical (albeit flipped)?

[Dick Fischbeck]

Tim Tyler weighed in years ago on Goldberg polyhedrons here:

http://www.hexdome.com/essays/greenhouse_conjectures/index.php

[Dick Fischbeck]

More from Boston Tim (joke), he moved and married, but he is Brit:

http://hexdome.com/introduction/index.php

http://timtyler.org/

Tim can be reached at   

t...@tt1.org

Classic Tim:   https://www.youtube.com/watch?v=kH31AcOmSjs

[David Russell]

Dick: THANKS! it's starting to make sense now. If I look at the 7,0 Goldberg Polyhedra in the Wiki article, am I correct that the shading indicates 1 type of face for the pentagon, and 2 different types for the hexagons? That would be fine. Also, you are correct that I don't care if they are perfectly spherical.  The wiki article also said, "Icosahedral symmetry ensures that the pentagons are always regular, although many of the hexagons may not be."  Which I take to mean the two different hexagons.  I assume that as the frequency increases the number of distinct hex faces will as well (as Taff alluded), but it looks like it would not be as bad as the triangular tessellation.  I'm also encouraged by the other notes that suggest we could get the hexes to be the same.  Are there any more sources/conjecture there? 

[Dick Fischbeck]

In my inexpert opinion, hexagons and pentagons with very nearly equal edges will make great spherical shingles. The distances between vertexes are not the same but who cares. Ask yourself how Nature builds her shells. Check Ernst Haeckel's book,  http://commons.wikimedia.org/wiki/Kunstformen_der_Natur

[Dick Fischbeck]

Plus Jack Bruce died today so all bets are off!

On Sat, Oct 25, 2014 at 5:26 PM, David Russell <da...@firelinx.com> wrote:

[David Russell]

This was new and interesting...

https://www.sciencenews.org/article/goldberg-variations-new-shapes-molecular-cages

[Gerry in Quebec]

Great link, Adrian. Lovely collection of spherical tilings. I'm really familiar with the first ("an interesting edge-to-edge tiling") but I hadn't seen the others. Thanks.

- Gerry

[Dick Fischbeck]

Some New Tilings of the Sphere with Congruent Triangles

http://archive.bridgesmathart.org/2005/bridges2005-489.pdf

[David Russell]

Gents:  To get back on track, here's where I lose the mental image: Attached are Taff's Goldberg-Clinton Critter and 3V icosaheron.  Taff states that the 3V has two unique triangles. My question is that as the frequency increases, what is the relationship to the number of unique triangles?  In the Goldberg-Clinton model, does the equal edge lengths imply equal faces, or are they distorted so that there are more unique faces at the cost of equal edge lengths?  Finally, can the great circle icosahedra be cut into smaller triangles with more great circles?

[Gerry in Quebec]

Dave,

The attached table (jpg image) gives a partial answer to your first question.

- Gerry in Quebec

[Adrian Rossiter]

[Resending]

Hi Gerry

On Sun, 26 Oct 2014, Gerry in Quebec wrote:
>> http://cs.stmarys.ca/~dawson/images4.html

> Great link, Adrian. Lovely collection of spherical tilings. I'm really
> familiar with the first ("an interesting edge-to-edge tiling") but I hadn't
> seen the others. Thanks.

You might also know the scalenohedron, which comes from crossing
the rhombi of that tiling with (gold) edges vertically rather
than horizonatally

https://www.google.com/images?q=scalenohedron

You can also raise pyramids on the skew rhombi to make a
polyhedron with all congruent triangles. I call this
polyhedron a subscalenohedron in Antiprism.

The corresponding spherical tilings for these two forms are
included in the "Semilune and quarterlune tilings" section
of the tilings page.

[Adrian Rossiter]

Hi Gerry

On Sun, 26 Oct 2014, Gerry in Quebec wrote:
>> http://cs.stmarys.ca/~dawson/images4.html

> Great link, Adrian. Lovely collection of spherical tilings. I'm really
> familiar with the first ("an interesting edge-to-edge tiling") but I hadn't
> seen the others. Thanks.

You might also know the scalenohedron, which comes from crossing
the rhombi of that tiling with (gold) edges vertically rather
than horizonatally

https://www.google.com/images?q=scalenohedron

You can also raise pyramids on the skew rhombi to make a
polyhedron with all congruent triangles. I call this
polyhedron a subscalenohedron in Antiprism.

The corresponding spherical tilings for these two forms are
included in the "Semilune and quarterlune tilings" section
of the tilings page.

[Gerry in Quebec]

David,
Your second question was: "In the Goldberg-Clinton model, does the equal edge lengths imply equal faces, or are they distorted so that there are more unique faces at the cost of equal edge lengths?"  I'm not quite sure what you're referring to in the second part of the question, but maybe I can answer the first bit and distinguish between a cage and a polyhedron.

In Joe Clinton's versions of the Goldberg tessellations of the sphere, there aren't actually any flat faces (polygons), with a few exceptions. So, the structures are "wireframes", or "molecular cages" for biologists and chemists. Technically, they aren't polyhedra.

But, as you know, it was demonstrated recently by UCLA's Stan Schein and James Gayed (and described in the article you mentioned earlier) that there are planar versions of these equilateral structures.... and these are therefore genuine polyhedra. But the vertices aren't all equidistant from the centre. Back in 2010, Taff Goch posted SketchUp models of two of the simplest of these polyhedra, in Google's 3D Warehouse. He made these models in the context of a thread on Yahoo's Dome Living discussion group (now called Dome Times), about the Clinton Conjecture. This, of course, was a few years before the paper by Schein and Gayed.

The fact that all the edges of these polyhedra are the same length does not imply equal or congruent faces. For example, in an equilateral Goldberg polyhedron with 12 pentagons and 470 hexagons, there are eight types of hexagons, only one of which is regular. An image of Taff's model of that polyhedron (I 4,4) can be seen here:

https://groups.google.com/forum/#!searchin/geodesichelp/Goldberg/geodesichelp/hI5vfZ_ZJhY/bGXdQc08qT0J

Cheers,

- Gerry in Quebec

On Tuesday, October 28, 2014 8:42:28 AM UTC-4, David Russell wrote:

[norm...@gmail.com]

To answer the third question.  Yes you can divide the spherical surface by more great circles to get small triangles.

[Adrian Rossiter]

On Tue, 28 Oct 2014, Adrian Rossiter wrote:
> You might also know the scalenohedron, which comes from crossing
> the rhombi of that tiling with (gold) edges vertically rather
> than horizonatally
>
> https://www.google.com/images?q=scalenohedron
>
> You can also raise pyramids on the skew rhombi to make a
> polyhedron with all congruent triangles. I call this
> polyhedron a subscalenohedron in Antiprism.
>
> The corresponding spherical tilings for these two forms are
> included in the "Semilune and quarterlune tilings" section
> of the tilings page.

http://cs.stmarys.ca/~dawson/images4.html

Just to note, the scalenohedron can be made from crossing spherical
kites, and not just rhombi.

Also, the models that are based on diamonds can all be twisted
like "A twisted edge-to-edge tiling", not just the 'gyroelongated
dipyramid' which is mentioned in that section.

I have included images of the twisted models made by crossing
the rhombus diagonals each way, and by raising a pyramid on
the rhombus (it is also possible to invert the apex point, but
this corresponds to a tiling that overlaps on the sphere).

[Gerry in Quebec]

Just to follow up on Norm's reply, here's a link to an earlier Geodesic Help conversation about great-circle domes. In this case, the examples are based on a patent by engineer John (Jake) Jacoby from Idaho, USA.

https://groups.google.com/forum/#!searchin/geodesichelp/Jacoby/geodesichelp/MDkUW9P30Lc/ZtNTSx-AD4cJ

- Gerry in Quebec 

On Wednesday, October 29, 2014 8:47:04 AM UTC-4, norm...@gmail.com wrote:

[Adrian Rossiter]

Hi Gerry

On Wed, 29 Oct 2014, Gerry in Quebec wrote:
> Just to follow up on Norm's reply, here's a link to an earlier Geodesic
> Help conversation about great-circle domes. In this case, the examples are
> based on a patent by engineer John (Jake) Jacoby from Idaho, USA.
>
> https://groups.google.com/forum/#!searchin/geodesichelp/Jacoby/geodesichelp/MDkUW9P30Lc/ZtNTSx-AD4cJ

I have just had a quick look and have written Python 3 script to make
these domes. The script generates a planar section of a tiling of a
specified frequency, raises it to a specified height, and projects
it onto the unit sphere by gnomonic projection. Besides the triangle
tiling, I also included a square (produces non-planar squares) and
crossed square tiling.

To run the program, copy both python files into the same directory and
run jacoby_dome.py with Python 3.

Examples:

Help
python3 jacoby_dome.py -h

Frequency 5 at height 4, triangles
python3 jacoby_dome.py 5 4 | antiview

Frequency 10 at height 4, triangles
python3 jacoby_dome.py 10 4 | antiview

Frequency 5 at height 2, triangles
python3 jacoby_dome.py 5 2 | antiview

Frequency 5 at height 5, squares
python3 jacoby_dome.py -t s 5 4 | antiview

Frequency 5 at height 5, crossed squares
python3 jacoby_dome.py -t x 5 4 | antiview

[Adrian Rossiter]

On Thu, 30 Oct 2014, Adrian Rossiter wrote:
> tiling, I also included a square (produces non-planar squares) and
> crossed square tiling.

The crossed square version could be used to make an "octagonal"
dome. If the number of squares on the tiling edge are divisible
by three the corners can be truncated from the tiling.

[Gerry in Quebec]

Hi Adrian,

Thanks for the Python files & images of the Jacoby domes. The gnomonic projection seems like a much simpler and more elegant way to generate these domes than my tedious trig-based system.

- Gerry