The Golden Solid Angle – Yet Again!!!

Posted on December 30, 2022 by Greg Parker
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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The Original Heart Nebula After Star Reduction

Posted on December 28, 2022 by Greg Parker
Heart_Nebula_StarXT_Forums

This is the original Parker/Carboni Heart nebula taken with a single Sky90 and M25C OSC CCD after star reduction has been applied.

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The Original M31 with Star Reduction

Posted on December 28, 2022 by Greg Parker
M31_StarXT_Forums

Well this is very interesting! Applying star reduction to the original Parker/Carboni M31 image, taken with a single Sky90 and M25C OSC CCD – brings out all the blue (hot young stars) which was not visible with all the stars there.

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Climate Change

Posted on December 28, 2022 by Greg Parker
Climate change? What climate change? Where’s your proof???
You aren’t going to believe this one.
So I suffer from Hayfever – big deal – why are you talking about Hayfever at the end of December??
I know Hayfever symptoms (I’ve had it for over 60 years) and I felt I had Hayfever at the BEGINNING of December. Surely not possible. However, talking to my wife, she said she had seen the Catkins out (Hazel and the first indicator that I’m going to get Hayfever) since the beginning of the month. I went out for a walk and sure enough all the Hazel trees down the lane were covered in Catkins!!!
10 years ago I got thoroughly pissed off when the Catkins came out at the beginning of February and kicked off my Hayfever.
5 years ago I got even more pissed off when the first Catkins opened up at the beginning of January.
And now in 2022 we have Catkins opening up at the beginning of December.
Congratulations you really have f***ed this Planet up good and proper.
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The Sadr Region in Cygnus

Posted on December 27, 2022 by Greg Parker
CombineSadr_20subs_10mins_28subs_15mins_Forums

Nice clear night last night and I managed to grab 20 x 10-minute subs with the Canon 200mm prime lenses and the ASI 2600MC Pro CMOS cameras on the Sadr region. A bit perverse imaging Cygnus at the end of December maybe, but this isn’t the first time. I added last night’s data to 28 x 15-minute subs taken using the same kit on the same region, and the resulting composite is shown above. North is to the left in this image. Star reduction was also applied to the composite image.

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Star Reduced Sadr Region With the 200mm Lenses

Posted on December 25, 2022 by Greg Parker
Combine_200mm_Sadr_StarXT_Forums

A bit of star reduction on the wide field Sadr region works wonders.

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Altair Region with Star Reduction

Posted on December 23, 2022 by Greg Parker
200mm_Altair_Forums

Surprisingly, even a region which is predominantly stars can benefit from a little star reduction.

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Fibonacci – Primes

Posted on December 22, 2022 by Greg Parker

Here are the first 200 Fibonacci numbers – and the list below shows whether they are Prime (True) or not (False). Courtesy of Mathematica.

{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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Sunflower Seed-Head Again

Posted on December 21, 2022 by Greg Parker

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”.

Sunflower_2000_Forums

 

 

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Yet Another Update on the Golden Ratio

Posted on December 21, 2022 by Greg Parker

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 sequence 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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