Some Tessellations of the Sphere by Triangles

[Christopher Jones]

My web page at https://chrisjones.id.au/Spherical%20Tessellations/

describes what I believe to be a new method of projecting a subdivided polyhedron on to a sphere. Under this projection, all vertices lie on parallels of latitude which are small circles (except for the equator) and hence not geodesic. 

This makes it very easy to slice the sphere at any latitude containing vertices, to get a dome which can sit on the ground on a flat circular polygon.

In such a structure, presumably all of the struts on these small circles will be in tension, and all the others in compression.

Christopher Jones

[luke Orlowski]

Looks like Gnomic projection. I used this method to create egg shaped domes.

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[Christopher Jones]

I don’t think so. I am projecting a polyhedron on to a sphere. A gnomic projection projects a sphere on to a plane. 

To view this discussion visit https://groups.google.com/d/msgid/geodesichelp/CAMhX8B9E-bFBNy12xXc9GkpZKHY_%2BZLMCr3SYaW3X4XxSs8wnA%40mail.gmail.com.

[Bryan L]

Hello Christopher,

have you researched the Krusche method? It has one or more parallel lesser circles away from the equator.

There has been plenty written about it over the years in this forum with plenty of examples.

To view this discussion visit https://groups.google.com/d/msgid/geodesichelp/CABS3LeGgnuPRhHD4tTwp%2BRFSevEET%3D%3D%2BBheaa3Xm0OOKS1YadA%40mail.gmail.com.

[luke Orlowski]

That's also true, I was refering to method used by engineers to build "Zeiss Model I"  specially Walter Bauersfeld that needed a perfectly smooth, lightweight hemispherical ceiling to act as a projection screen.

To build it, he took a regular 20-sided geometric solid (an icosahedron) and used the concept of a gnomonic projection to map it outward from the center onto a sphere. He subdivided the flat triangles of the icosahedron into smaller networks and projected those lines straight out onto the sphere's surface.

To view this discussion visit https://groups.google.com/d/msgid/geodesichelp/CABWq%3Di6v-Q7xqSwJmqGGFMjsF8cHk1mVNKbSY%2B8QX3Y9gTsu9Q%40mail.gmail.com.

[Christopher Jones]

Bryan,

I have not been able to find a full description of the Kruschke geometry, but as you say, it only applies to a few cirlcles near the equator, and for this reason, perhaps, is sometimes called a "debasement" of the geodesic dome.

My method applies consistently over the whole sphere.

[Christopher Jones]

This sounds like the usual projection of an icosahedron from its centre on to a sphere, typically called an icosphere, and corresponding to the spherical projection on this page of mine:-

https://chrisjones.id.au/Spherical%20Tessellations/projected16.html

[Bryan L]

The Kruschke method allows lesser circles for all latitudes (and their symmetric equivalents) up to V5. For frequencies greater than V5, there can only be up to V - 1 lesser circles - so all but one can be lesser circles.

Full symmetry is maintained throughout the surface and all vertices have equal radius. 

To view this discussion visit https://groups.google.com/d/msgid/geodesichelp/85e603e6-13bc-4608-ab97-a2019e2435b8n%40googlegroups.com.

[Christopher Jones]

Bryan

ChatGPT was unable to provide me with a Python script to model the Kruschke method for a dome with 16 subdivisions. It said a full historical chord-factor table implementation needed the actual Fuller/Kruschke tabulated factors, including chord factors, and it did not currently have a reliable frequency-16 historical Fuller/Kruschke chord-factor table to embed. Do you have access to any software that can do this?

[Bryan L]

Here is an image of a V16 Kruschke I created in SketchUp quite some years ago.

I would have to go back and revisit whatever method I used - but it was purely geometric as far as I can recall.

I think I made a mistake saying there could be up to n - 1 lesser circles with this method. It may be that for some higher n, the possible number reduces.

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[Levente Likhanecz]

hi Chris. the Kruschke method not just the equator plus/minus 1 plane.
the lesser circle above equator is distributed all around the sphere following the plentifold symmetry of icosahedron.
its a good van with 5V 4 planes:

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[Levente Likhanecz]

Hi Bryan, it would be interesting to understand geometrical solution for more than 1 lesser circle.
As i remember for 2 lesser circles we made already some iterative math approach.
(to have every vertices on unit radius). that was the 5V with 4 planes. 
The Dome — Fuller Dome

these are the 4 lesser circles/planes (2+2)

i recall also Taffgoch'es work. he made 1 more, a 3+3 lesser circle version, just by his hand-geometric-iteration method. it was a higher frequency dome with 6 planes.
but it had some cheat.

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[Gerry in Quebec]

Dear Christopher, Lev and others,

Christopher, your two versions of DP5 (pentagonal arrangements) remind me of the geometry of the external geodesic grid of the Montreal Biosphere, formerly the US pavilion at Expo '67. In that dome by Fuller & Sadao, the upper hemisphere, comprising 24 rows of triangles, uses two subdivision methods similar to the first of the two DP5s. The lower section of the dome has 12 rows of triangles following lesser circles, similar to your second DP5.

 Here's a bare-bones png depiction of the various subdivision methods used for the Montreal dome.

 

Also check out the lesser circle domes designed by American architect and engineeer  T.C. Howard via his Charter Spheres company during the 1970s (maybe 1960s to 1990s?). A good Internet search word to find that stuff might be Synergetics... and you should also be able to find mentions of those Charter domes in this Geodesic Help Group using that keyword.

 

- Gerry in Québec

 

 

[Bryan L]

Hi Lev,

I am wishing I had kept notes on all this stuff because the method I used escapes me now...

I can remember up to V6 I was able to achieve it reasonably easy in sketchup - I think by intersecting planes that were established from some given vertices that were defined by symmetry. For V7 I remember I had to use a vertex on the rear of the sphere as a reference point to establish a plane. For the even frequencies, I was able to use the previous one - as in the V4 vertices can be used as a base for the V8. I also remember that there was a way to calculate the radius of the plane of a lesser circle but I can't find any spreadsheet with those calculations - I suspect I just did it in a calculator...

To view this discussion visit https://groups.google.com/d/msgid/geodesichelp/CAOkvEDm-W7bLTuko%3DO01xiSF6fmkjEjDKvVmy8aWFyhhH6p7nQ%40mail.gmail.com.

[Christopher Jones]

I have updated my web page at https://chrisjones.id.au/Spherical%20Tessellations/

It now ilinks to an additional page where the the models are coloured with a distinct color for triangles of each distinct shape, as suggested by Chris Kitrick.

[Bryan L]

Christopher, I don't understand from your webpage what the method is or how it works. Can you elaborate?

And two questions: 

Are all the vertices equidistant from the origin?

and

Do all the vertices follow the typical symmetries associated with for example the icosahedron?

You mention not finding a list of chord factors for the Krusche method but I don't see any chord factors with any of your models. Not that chord factors aid in the recreation of a model - a list of vertices' coordinates within the Schwartz triangle is sufficient if a model has full icosahedral symmetry. 

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[Christopher Jones]

Bryan,

The two main features of my method are

1. Subdivision of the Schwarz triangles is performed using a triangular mesh applied to those triangles in a 2D equirectangular projection, which is then inverse projected back to a sphere.
2. Schwarz triangles with one vertex at a pole are expanded laterally in the 2D equirectangular projection while subdividing with the triangular mesh, before  inverse projection back to a sphere.

All of the vertices at the same height (z value) are equidistant from the z axis.

The vertices do not follow the typical symmetries associated with for example the icosahedron. 

This is clear from the new page where the models are coloured with a distinct color for triangles of each distinct shape.

Models produced by my method look very different from the typical spherical projections on adjacent rows.

I don't use any chord factors. I only mentioned these because they were brought up in connection with the Kruschke method.

Chris

[Christopher Jones]

Gerry

I can't see any resemblence of either DP5 models to the Montreal Biosphere.

The lower section of 8 light blue rows below the equator does resemble the 8 roes below the equator in my k5A5 model.

Not sure why you distinguish the bottom 8 light blue rows from the bottom 2 brown ones.

It looks like you are the author of that diagram. Can you tell me how you sourced the information about the yellow and purple sections?

Is there any 3D model (.obj or .glb) of the Montreal Biosphere which can be downloaded?

Chris

[Gerry in Quebec]

Hi Chris,

From the bottom of your post up....

 - I have not posted my CAD model of the Montreal Biosphere, only a little collection of screenshots in this discussion group nine years ago.

https://groups.google.com/g/geodesichelp/c/fIuS_UfEP6c/m/j4Cm1sylAQAJ

 -  To compile the coordinates of the yellow and purple sections, and identify the subdivision methods, I worked backwards from Fuller and Sadao's architectural drawings. The owner/moderator of this group, David Price (aka TaffGoch), did some SketchUp models many years ago based on photos, and these were a good starting point for the subsequent modeling I did in Excel and Antiprism about 12 years ago.

 - I erred in referring to 12 rows of isosceles triangles below the equator. You are right, there are 10 rows, eight immediately under the equator and then another two rows at the base. I don't know exactly why the bottom two rows are different, but that is the arrangement specified in Fuller & Sadao's architectural drawings.

 -  Right you are.... Your DP5 models have little in common with the Montreal Biosphere except for the lesser circle rows of triangles below the equator.

 Cheers,

- Gerry in Québec

[Christopher Jones]

Gerry

I found your 2014 pdf which gives more details of the geometry of the dome: Montreal-Biosphere-datasheet-April-22-2014.pdf.

I also found TaffGoch's 3D model at 3D Warehouse. His accompanying text said 

"The Biosphère, as built, after truncating to a 3/4 dome • Geometry, from the "equator" down, is modified to provide struts parallel to the ground.

This modification, or debasement, of the true geodesic configuration, allowed the 3/4 dome to "sit flat" on the foundation."

That would be those 10 rows in your diagram.

It is possible that the top one of those rows which meets the equator does not have all oblique struts of equal length, as their equal horizontal strut lengths at the equator would not match the strut lengths of the radially projected polygon above at the equator.

The important observation that you have brought to light is that the central section of my projection method (for knAn models) is not a novelty. It is not very difficult to get the strut lengths for a stack of antiprisms with varying sizes of the top and bottom polygons.

The novelty of my method lies in the continuation of the horizontal regular polygons, with decreasing number of edges, into the top and bottom caps, and their seamless joining to the central section.

Chris

[Christopher Jones]

I have just updated the pages which display the models to provide a link in the first column of the tables to open each model in the Online 3D Viewer, 

which has a larger display in a full browser window so that you can see them better. You can also download the models from this viewer.

On Friday, 12 June 2026 at 14:51:09 UTC+10 Christopher Jones wrote: