Flatness of icosphere layers near the equator

[Christopher Jones]

When an icosahedron is radially projected on to a sphere so that two opposite vertices are at the north and south poles, an even subdivision of its triangular faces will result in a flat polygonal row of edges at the equator. I have just noticed that an odd subdivision  v >= 5 results in two nearly flat rows on either side of the equator. Not sure whether this is common knowledge.

With flatness defined as zmax-zmin as a percentage of the radius of the sphere:-

v       Flatness
5 0.691225%
7 0.510883%
9 0.358848%
11    0.295662%
13         0.238142%
15         0.206908%

v              0.02705529238%/v approximately

This could be helpful in creating a dome slightly higher than a hemisphere without needing special geometry at the base such as Krueschke subdivision.

Chris Jones

[Levente Likhanecz]

the 5V has an iteration resulted 4 flat (lesser circles) version:

this one absolutely flat (all 6 sets) and all vertices on sphere radius.
Kruschke 5V 9strut multilevel flat truncation symmetric unit chord lengths 0.159901114924194 red 0.187373294396636 blue 0.236567227622598 gray 0.256694505885747 orange 0.240079981119830 green 0.271356209578508 purple 0.258519251299602 cyklamen 0.262082087735626 brown 0.247747374895913 light blue

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[Ashok Mathur]

Dear Levent

My request is to satisfy my curiosity; what is the deviation from flatness when instead of struts of length 0.25669 and 0.2585 you use an average length of these two? Similarly substitute 0.24774 and 0.24007 with their average.

Regards

Ashok

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[Levente Likhanecz]

i'm afraid i can not carry out those things simply. there was a solver excel file (i've lost) that brought the light blue struts (the central quilateral triangle) in level with some other strut on another icosahedral triangle. Chris Kitrick explained in a lengthy thread.
so i dont have it automated. or parametrized.
i can only painstakingly reconstruct by hand, from the strut lengts.

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[Levente Likhanecz]

Class I Icosahedron 7V Truncations {3,5+}(7,0)

here Chris (Kitrick) explains about 3,4,5,7V Kruschke stuff

[Levente Likhanecz]

Release icosahedron_classi_truncatons · ckitrick/icosa_classI_truncation

here there is some c source code from Chris doing the slicing

[Ashok Mathur]

Dear Levent

Do not do this by hand. Gerry at Qbec is the master to request for help with Excel solver functions.

Can you do this Gerry?

Regards

Ashok

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[Robert Clark]

I took this as a puzzle challenge. I only attempted to solve for the 1/3 and the 2/3 5V domes with flat bases. I was also able to eliminate scalene triangles so that they are all now isosceles panels. The strut count has been reduced from 9 to 7. Added bonus is that all vertices remain on the same spherical surface. I hadn't thought I'd be able to do that. Arriving at the solution using SolidWorks is like doing a Sudoku puzzle. I am solving a logic problem, not a math problem. SolidWorks can be very sensitive when so many constraints are being applied. I had to incrementally creep in on a solution without blowing up the model, which happened many times. Anyway, it was an interesting way to fill a rainy Saturday!

-Robert

.15998348 yellow
.18746919 red
.23666780 light blue
.24762965 orange
.25018969 pink
.26194799 light green
.28418078 dark blue

[Ashok Mathur]

Looks very easy to assemble even though there is a great deal of strut length variation. , similar size struts are kept together giving nearly equilateral triangles.

Ashok

Regards

Ashok

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