I have been putting together DSS2 red and blue channel data (with a synthesised green channel courtesy of Noel Carboni’s Astronomy Actions) and got to this point before the computer started to give up with the mass of data.  This is the Tulip nebula region in Cygnus with the Northern Cross (open cluster) and “Red-Necked Emu” (open cluster) to the left.  The ultra red star near the top of the image is S-type star AA Cygni.

See plenty of new fractal renders on my Flickr Mathematical Structures page.

It will not have been mentioned before in this blog, but I like certain aspects of pure mathematics as much as I like deep-sky imaging.  I think most people will have heard of the Golden-Section, or the Golden-Ratio, and how it can be obtained by dividing a straight line up into two sections one of length unity, and the other of length tau or 1.61803398…  What is less well-known is that if you wrap the line round into a circle, so the circle perimeter is divided into lengths of unity and 1.618, then then angle subtended by the unity length of the perimeter at the centre of the circle is 137.507 degrees – or the Golden Angle.

That’s where the story seems to have been left, for a very long time, but I have to wonder, why?  We started with a line (one-dimension), then moved to a circle (two-dimensions), where’s the spherical case (3-dimensions)?  I did a long search a couple of years back and couldn’t find anything on this.  So I wrote a paper on “The Golden Solid Angle” for the Mathematical Gazette, which was in fact turned down as “although the result was new, just having a new result is not necessarily having something worthy of publication” – well that’s a new one for me!  So wishing to stake my claim as the discoverer of the Golden Solid Angle (sent to the Mathematical Gazette on Thursday 14th June 2007) here’s the thing explained for the first time below.

Divide the surface of a sphere into two regions, one of surface area unity, and the other of surface area 1.618…  The surface area of unity will subtend a solid angle gamma at the centre of the sphere.  By noting the total solid angle about a point is 4Pi Steradians, we can derive the following equation for gamma:

(4Pi – gamma)/gamma = 4Pi/(4Pi – gamma)

Giving a quadratic in gamma which can be solved in the usual way to give:

gamma = 1.52786Pi Steradians or 15757.2 square degrees.

Question is, does anyone out there know where the Golden Solid Angle, gamma, makes an appearance in the Natural world (or basically, anywhere)?  If you do then please let me know ASAP

Since writing the above I have found that using Phi to represent the Golden Ratio we have:

Golden Angle = 2 Pi / Phi squared

Golden Solid Angle = 4 Pi / Phi squared

Nice!!

Simple – but I really like this one 🙂

I will put most of my fractal renders on Flickr and will now only post the most impressive here.

Two more fractal renders from the amazing Incendia software.

A new Incendia fractal composite.

I have been playing with fractal rendering recently, but I don’t think any of my attempts come close to Daniel White’s 3D-Mandelbulb.  You can read about how Daniel dragged the Mandelbulb out from hiding here, it is a fascinating story.

If, during those boring moments where you idly pressed numbers into an electronic calculator and randomly hit function buttons, you played extensively with the square root function – you might have seen something interesting.  Provided that the integer you keyed in was not itself a perfect square, then the result of pressing the square root button always seemed to produce an irrational number – how odd is that then?  I came across a proof that the square root of 2 is irrational – arrived at by a totally different route from the norm.  The normal route is the centuries old reductio ad absurdum.  This “new” route I found not only beautiful, but extremely powerful in that it proved that the square root of any integer that is not itself a perfect square will always yield an irrational.  Here’s how it goes.

We start off in the conventional way by assuming that the square root of 2 IS rational and may be given as m/n where m & n are both integer.

1)  √2 = m/n

2)  m² = 2 n²

That much we have seen before, but now comes the really clever bit.  Factor m & n into their unique prime factors:

3)  m₁ m₁ m₂ m₂ m₃ m₃ … m_r m_r = 2 n₁ n₁ n₂ n₂ n₃ n₃ … n_r n_r

Now here’s the problem.  The prime number 2 will appear an even number of times on the left hand side of the equation (if it appears at all) and an odd number of times on the right.  Since the decomposition into primes is unique the prime number 2 cannot appear an even number of times on one side of the equality and an odd number of times on the other.  So the square root of 2 cannot be written in the form m/n with m & n integer.

A)  Although the above was shown to be true for root 2, the same argument of course would hold true for any prime number.  The square root of any prime number is an irrational.

B)  Replace 2 with any integer that is not a perfect square.  Now that integer may be decomposed into its prime factors, and, if the number is not a perfect square, then once again we will have an odd number of primes on one side of the equality and an even number of primes on the other.  The square root of any integer that is not a perfect square is also an irrational.

C)  Finally, replace 2 with any perfect square.  Now the perfect square will decompose into pairs of primes and this will give the possibility of even numbers of primes on both sides of the inequality and therefore the possibility of a rational solution.