Re: sphere volumes

[Dick Fischbeck]

If a 100 frequency sphere has a volume x, what frequency sphere has a volume of 2x?

[Ashok Mathur]

Dear Dick,

To frame a proper challenge, we need to make some rules. For instance, a hundred frequency dome may have a large number of struts, say over 150 of them. Obviously, all of them can not be used in the dome with double the volume.

So you need to say what single strut length needs to be present in both domes.

One rule could be that the longest strut in the single volume dome should be the longest strut in the other frequency dome.

Or the rule could be formulated that the weighted average length should be the same in both domes.

So please formulate a suitable rule.

Regards

Ashok

On Fri, Apr 22, 2022 at 9:24 AM Dick Fischbeck <dick.fi...@gmail.com> wrote:

If a 100 frequency sphere has a volume x, what frequency sphere has a volume of 2x?

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[Dick Fischbeck]

"Or the rule could be formulated that the weighted average length should be the same in both domes."

Sounds good.

On Fri, Apr 22, 2022 at 10:17 AM Ashok Mathur <ashokch...@gmail.com> wrote:

Dear Dick,

To frame a proper challenge, we need to make some rules. For instance, a hundred frequency dome may have a large number of struts, say over 150 of them. Obviously, all of them can not be used in the dome with double the volume.

So you need to say what single strut length needs to be present in both domes.

One rule could be that the longest strut in the single volume dome should be the longest strut in the other frequency dome.

So please formulate a suitable rule.

Regards

Ashok

[Dick Fischbeck]

Hi Ashok

"If a 100 frequency sphere has a volume x, what frequency sphere has a volume of 2x?"

Since volume equals 20F^3, we can say a 100 F sphere has a volume of 20,000,000. Double this to get 40,000,000. We get F=125.99 for a sphere of twice the volume.

Thoughts?

Dick

On Fri, Apr 22, 2022 at 10:17 AM Ashok Mathur <ashokch...@gmail.com> wrote:

To view this discussion on the web visit https://groups.google.com/d/msgid/geodesichelp/CAD-a3OigneWVeHmaJRonKXOuskEi%2BAi%3Di2AfgGaqhPq4cBaa%2Bg%40mail.gmail.com.

[Dick Fischbeck]

Then a sphere with 3 times the volume will have a frequency of 144.22.

A sphere of 4 times the volume is f=158.74. And so on.

[Dick Fischbeck]

Just playing around. These are the frequency of sphere volumes that increase in steps, each one n+1.

100

126

144

159

171

182

191

200

208

215

222

229

235

241

247

252

257

262

266

271

278

280

284

288

292

296

300

[Ashok Mathur]

Dear Dick

I have tried to formulate a solution in this manner but have not succeeded in finding a solution.

1.       What is the decreasing rate at which the weighted mean average of strut lengths at increasing frequencies happens in Geodesic domes? Can one device a general formula for it?

2.       Except for sub-division method, attributed to Duncan Stuart, which generate the same number of struts as the frequency of sub-division, other sub-division generate a varying number of struts different from the frequency. I assume that we are following Class 1 Method 1 as used by Hugh Kenner for answering the question.

3.       Most dome calculators work on the basis that the radius of the sphere is 1 (i.e. the volume does not vary with the frequency) and then proceed to find out strut lengths.

4.       We want to do this first and then reverse the process.

5.       We do that by calculating the weighted average length of the struts (WAS). (I am simplifying my working below by not taking the Weighted Average but simply taking the arithmetic average for convenience.) For instance, in 2v division, the two strut lengths (Dome calculator data) are 0.618 and 0.546 with an average length of 0.582.

6.       We then set this WAS to 1 by taking WAS reciprocal which comes to 1.72 and which yields a volume of 5 that is almost 6 times the previous volume.

7.       Such calculations based on Dessert Dome data are given in the attached sheet for frequencies up to 6. They show that with higher and higher frequencies, the increase in volume slows down. The 6V sphere encloses only 1.9 times the volume of 5 V sphere.

8.       To minimize the number of calculations for a 100 frequency sphere, start with a 110 frequency sphere. Then do a binary search. If 110 V sphere’s volume is already over 2 times of 100V sphere, drop down to 105 V sphere. Otherwise jump to 115 V sphere. Soon you will get the right frequency closest to your answer.

9.       Spreadsheet enclosed. Please do point flaws in my thinking.

 

Regards

Ashok

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[Dick Fischbeck]

Ashok- To me, it is easier to keep strut length constant and have the volume increase with frequency.

[Dick Fischbeck]

Dividing a sphere into many pyramids connected to the center allows us to relate the sphere's surface area and volume.

[Dick Fischbeck]

cite- http://blog.zacharyabel.com/tag/spheres/

[Dick Fischbeck]

Hi Ashok

I think xyz and ivm are incompatible in thought even though there is a way to translate one to the other.

Either I can imagine a continuous surface or I can't. Mutually exclusive. Do you know what I mean?

School is almost out for summer! Long year, it's been.

Cheers

Dick

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