# The Golden Solid Angle – Yet Again!!

Another hobby of mine is “experimental mathematics” – I think I may have mentioned (ages ago) my Prime number hunting server.
Well way back in June 2007 (just before actually) I discovered something that seems to have been overlooked for centuries – The Golden Solid Angle.
Everybody knows the basics, the 1-D Golden Ratio, the 2-D Golden Angle (equal to 2*Pi/Golden Ratio^2) and there it seems to have come to a halt for a few centuries. Why hasn’t anybody (until moi) taken this up one further dimension to form the Golden Solid Angle? Or have they? Nobody has got back to me on this and I’ve not seen it in any book or publication on the subject. So why not?
To find the Golden Solid Angle (in steradians) is pretty straightforward. Draw a sphere of radius r with the total surface area of the sphere being 1 + Golden Ratio. Then the solid angle subtended by the unity area part of the sphere is the Golden Solid Angle which is given by 4*Pi/Golden Ratio^2.
I have just sent this off to Wolfram Mathematica to see if they will publish it. Way back in 2007 I sent it off to the Mathematical Gazette and they answered – “Just because it is a new discovery in mathematics doesn’t make it necessarily publishable” which after a lifetime in science with over 120 refereed research journal publications to my name – was a new one on me.
Let’s see if the Stephen Wolfram outfit are a little more enlightened – or not
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# Fibonacci-Primes

Here are the first 200 Fibonacci numbers, and the list below them shows you whether they are Prime (True) or not (False).

{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, \
2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, \
317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, \
14930352, 24157817, 39088169, 63245986, 102334155, 165580141, \
267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, \
4807526976, 7778742049, 12586269025, 20365011074, 32951280099, \
53316291173, 86267571272, 139583862445, 225851433717, 365435296162, \
591286729879, 956722026041, 1548008755920, 2504730781961, \
4052739537881, 6557470319842, 10610209857723, 17167680177565, \
27777890035288, 44945570212853, 72723460248141, 117669030460994, \
190392490709135, 308061521170129, 498454011879264, 806515533049393, \
1304969544928657, 2111485077978050, 3416454622906707, \
5527939700884757, 8944394323791464, 14472334024676221, \
23416728348467685, 37889062373143906, 61305790721611591, \
99194853094755497, 160500643816367088, 259695496911122585, \
420196140727489673, 679891637638612258, 1100087778366101931, \
1779979416004714189, 2880067194370816120, 4660046610375530309, \
7540113804746346429, 12200160415121876738, 19740274219868223167, \
31940434634990099905, 51680708854858323072, 83621143489848422977, \
135301852344706746049, 218922995834555169026, 354224848179261915075, \
573147844013817084101, 927372692193078999176, 1500520536206896083277, \
2427893228399975082453, 3928413764606871165730, \
6356306993006846248183, 10284720757613717413913, \
16641027750620563662096, 26925748508234281076009, \
43566776258854844738105, 70492524767089125814114, \
114059301025943970552219, 184551825793033096366333, \
298611126818977066918552, 483162952612010163284885, \
781774079430987230203437, 1264937032042997393488322, \
2046711111473984623691759, 3311648143516982017180081, \
5358359254990966640871840, 8670007398507948658051921, \
14028366653498915298923761, 22698374052006863956975682, \
36726740705505779255899443, 59425114757512643212875125, \
96151855463018422468774568, 155576970220531065681649693, \
251728825683549488150424261, 407305795904080553832073954, \
659034621587630041982498215, 1066340417491710595814572169, \
1725375039079340637797070384, 2791715456571051233611642553, \
4517090495650391871408712937, 7308805952221443105020355490, \
11825896447871834976429068427, 19134702400093278081449423917, \
30960598847965113057878492344, 50095301248058391139327916261, \
81055900096023504197206408605, 131151201344081895336534324866, \
212207101440105399533740733471, 343358302784187294870275058337, \
555565404224292694404015791808, 898923707008479989274290850145, \
1454489111232772683678306641953, 2353412818241252672952597492098, \
3807901929474025356630904134051, 6161314747715278029583501626149, \
9969216677189303386214405760200, 16130531424904581415797907386349, \
26099748102093884802012313146549, 42230279526998466217810220532898, \
68330027629092351019822533679447, 110560307156090817237632754212345, \
178890334785183168257455287891792, 289450641941273985495088042104137, \
468340976726457153752543329995929, 757791618667731139247631372100066, \
1226132595394188293000174702095995, \
1983924214061919432247806074196061, \
3210056809456107725247980776292056, \
5193981023518027157495786850488117, \
8404037832974134882743767626780173, \
13598018856492162040239554477268290, \
22002056689466296922983322104048463, \
35600075545958458963222876581316753, \
57602132235424755886206198685365216, \
93202207781383214849429075266681969, \
150804340016807970735635273952047185, \
244006547798191185585064349218729154, \
394810887814999156320699623170776339, \
638817435613190341905763972389505493, \
1033628323428189498226463595560281832, \
1672445759041379840132227567949787325, \
2706074082469569338358691163510069157, \
4378519841510949178490918731459856482, \
7084593923980518516849609894969925639, \
11463113765491467695340528626429782121, \
18547707689471986212190138521399707760, \
30010821454963453907530667147829489881, \
48558529144435440119720805669229197641, \
78569350599398894027251472817058687522, \
127127879743834334146972278486287885163, \
205697230343233228174223751303346572685, \
332825110087067562321196029789634457848, \
538522340430300790495419781092981030533, \
871347450517368352816615810882615488381, \
1409869790947669143312035591975596518914, \
2281217241465037496128651402858212007295, \
3691087032412706639440686994833808526209, \
5972304273877744135569338397692020533504, \
9663391306290450775010025392525829059713, \
15635695580168194910579363790217849593217, \
25299086886458645685589389182743678652930, \
40934782466626840596168752972961528246147, \
66233869353085486281758142155705206899077, \
107168651819712326877926895128666735145224, \
173402521172797813159685037284371942044301, \
280571172992510140037611932413038677189525}

{False, False, True, True, True, False, True, False, False, False, \
True, False, True, False, False, False, True, False, False, False, \
False, False, True, False, False, False, False, False, True, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, True, False, False, False, True, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, True, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
True, False, False, False, False, False, True, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False, \
False, False, False, False, False, False, False, False, False, False}

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# The Sunflower Seed-Head Yet Again

Having spent the whole day on the Golden Ratio, the Fibonacci sequence, the Golden (Planar) Angle, and Phyllotaxis – here is a Mathematica produced piccie of the Sunflower seed-head pattern using 2,000 “seeds”.

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# Still Even More on the Golden Ratio

The well-known Fibonacci sequence formed by summing consecutive terms looks like:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89……….

And a well known property of this sequence is that if you take the ratio of consecutive terms you get closer and closer to the Golden Ratio the higher up the sequence you go. So if we divide 89 by 55 we get 1.61818181…….

We can create a Fibonacci series incorporating the variable x in the following way:

1, x, (1+x), (1+2x), (2+3x), (3+5x), (5+8x), (8+13x)…

Now a remarkable property of this sequence is that once again, if we take the ratio of consective terms in the sequence, we get the Golden Ratio – Phi – provided x=Phi, which I think is a remarkable result.

What this means is that looking at the sequence with Phi replacing x we get:

1, Phi, (1 + Phi), (1 + 2*Phi), (2 + 3*Phi), (3 + 5*Phi)…

And if taking the ratio of consective terms gives us Phi, this means that:

(1 + Phi) = Phi^2

(1 + 2*Phi) = Phi^3

(2 + 3*Phi) = Phi^4

The first result comes directly from the basic definition of Phi itself, namely that (1 + Phi)/Phi = Phi

Taking a deeper look at the above sequence I was able to come up with a general result, namely:

Phi^n + 2*Phi^(n+1) = Phi^(n+3)

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Here is a 3-D view of comet Garradd. You can view the stereograph by:

1. Using the Cross-Eyed method.
2. Using 3D prism glasses (berezin.com/3d/3dprism.htm)
3. Using Brian May’s OWL Stereoscope (londonstereo.com)

I took the image of the comet using a Hyperstar III on a Celestron C11 SCT and an M25C OSC CCD camera from Starlight Xpress. The Dr. Brian May created the stereograph from the CCD image.

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# Greg’s “3” Asterism in Leo – in 3D

Here is a 3-D view of Greg’s “3” asterism which can be found in the constellation Leo. You can view the stereograph by:

1. Using the Cross-Eyed method.
2. Using 3D prism glasses (berezin.com/3d/3dprism.htm)
3. Using Brian May’s OWL Stereoscope (londonstereo.com)

I took the image of the asterism using the Sky90 refractors and the M26C OSC CCDs on the MiniWASP array at the New Forest Observatory. Dr. Brian May created the stereograph from my Deep-Sky image.

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# Cabbage White Butterfly Eggs in 3D

Here is a 3-D view of the eggs of the Cabbage White butterfly. You can view the stereograph by:

1. Using the Cross-Eyed method.
2. Using 3D prism glasses (berezin.com/3d/3dprism.htm)
3. Using Brian May’s OWL Stereoscope (londonstereo.com)

I took the images of the Cabbage White butterfly eggs using a Canon 5D MkII DSLR and a research trinocular microscope. The 2 images are focus-stacked using the Helicon Focus software, and Dr. Brian May created the stereograph.

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# 3D VIEW OF DNA

Here is a 3-D view of 2 and a half periods of the DNA molecule. You can view the stereograph by:

1. Using the Cross-Eyed method.
2. Using 3D prism glasses (berezin.com/3d/3dprism.htm)
3. Using Brian May’s OWL Stereoscope (londonstereo.com)

I put together the 3D model from a Cochrane’s of Oxford kit, and Dr. Brian May created the stereograph.

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# 6-Minute Space Station Pass 02/12/2022

It’s been a loooong time since I’ve done one of these – so I’ve got a bit out of practice. But this was tonight’s Space Station pass at 5:37 p.m. I got part of a nice long 6-minute pass and you see it disappears off the top of the screen where there is the glow from a very bright Moon. I will do better next time

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# Star reduced Horsehead nebula and Belt Stars of Orion

The Horsehead nebula and the Belt Stars of Orion region. This is a 2-frame mosaic using a single Sky90 and a single M25C OSC CCD. H-alpha and OIII data was also included. Imaging time around 4-hours (or more) per frame. This version is with the star reduction app that works with Russ Croman’s StarXTerminator.

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