Do you really need all those flat planes in your dome?
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Christopher Jones
5:57 AM (11 hours ago) 5:57 AM
to Geodesic Help Group
On page 8 of his Dome Cookbook of Geodesic Geometry, David Kruschke took issue with Domebook 1 and Domebook 2 by Pacific Domes, as they claimed that some domes did not have a level bottom course. His frequency 3 and 4 frequency dome subdivisions address and solve this problem.
His method is overkill, however, if all you want is a 3/8 or 5/8 cutoff of a 4 frequency dome. You only take advantage of 2 out of 3 of the horizontal cutoff planes. The only contribution of the 6 other flat planes resulting from triangular symmetry is to reduce the number of different chord factors.
With help from ChatGPT I have devised a method of subdivision which gives two specified horizontal flat planes but avoids other unnecessary constraints, resulting in a dome with a perhaps more pleasing arrangement of triangular faces. It is also applicable to any frequency of subdivision, whereas the Kruschke method is problematical for frequencies greater than 5.
Here is an example 3D model showing the top 15 spherical triangles of a dome with frequency 15 subdivision. It shows the arcs joining vertices instead of the struts.
https://3dviewer.net/#model=https://chrisjones.id.au/Spherical%20Tessellations/icosahedral_dome_v15_s3_gnomonic_piecewise.glb
The red arcs are great circles subdivided by radial projections of the linear subdivision of the icosahedral triangular edges.
The green arcs are small circles subdivided by a gnomonic construction based on blue coordinate families.
The blue arcs are continuous great-circle segments from side to green row, across green row, and from green row to side.
His method is overkill, however, if all you want is a 3/8 or 5/8 cutoff of a 4 frequency dome. You only take advantage of 2 out of 3 of the horizontal cutoff planes. The only contribution of the 6 other flat planes resulting from triangular symmetry is to reduce the number of different chord factors.
With help from ChatGPT I have devised a method of subdivision which gives two specified horizontal flat planes but avoids other unnecessary constraints, resulting in a dome with a perhaps more pleasing arrangement of triangular faces. It is also applicable to any frequency of subdivision, whereas the Kruschke method is problematical for frequencies greater than 5.
Here is an example 3D model showing the top 15 spherical triangles of a dome with frequency 15 subdivision. It shows the arcs joining vertices instead of the struts.
https://3dviewer.net/#model=https://chrisjones.id.au/Spherical%20Tessellations/icosahedral_dome_v15_s3_gnomonic_piecewise.glb
The red arcs are great circles subdivided by radial projections of the linear subdivision of the icosahedral triangular edges.
The green arcs are small circles subdivided by a gnomonic construction based on blue coordinate families.
The blue arcs are continuous great-circle segments from side to green row, across green row, and from green row to side.
Robert Clark
1:02 PM (4 hours ago) 1:02 PM
to Geodesic Help Group
Chris,
I'd be interested to know in what way you utilized ChatGPT to reach a solution? That could be really useful.
Robert
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