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The proton and the golden ratio: my second sequence
One of my old late night workings, from around 2016. I'm not a mathematician, so it may not be perfect, but taken together they may interest some people. Corrections and a comment by Claude are included.
This is the second of my three sequences. This time I started not from 0 and 1, but from the mass of the proton in atomic mass units, 1.0072766.
Then I did just what you do for the Fibonacci numbers: add the last two numbers together to get the next one. 1.0072766, then 2.0145532, then 3.0218298, then 5.036383, and on for 62 lines. The full list is in the two pictures.
Then the test. Line 61 plus line 62 gives 10,687,416,110,773.707182. Line 62 multiplied by the golden ratio, 1.6180339887498948482, gives 10,687,416,110,773.707182. The two match up exactly, decimals and all.
So the proton, built up step by step in the Fibonacci way, arrives at the golden ratio, just as the Fibonacci numbers do in my first sequence and the golden ratio itself does in my third.
(Very speculative.) I believe the mass of the proton may be bound up with the golden ratio, and that the mathematics of the Universe may be built on it.
John D Morton. Any errors have been corrected with the help of Claude.
Claude's comment: I checked all 62 lines, and every one is right. The match at the end is exact to all the digits a 20-digit calculator can show, just as John says. The proton's mass he used, 1.0072766, is correct to the seven decimal places he gave. The honest catch is that this match doesn't depend on the proton. Each line is simply 1.0072766 times a Fibonacci number, so the same thing happens whatever number you start with: your age, the price of a loaf, anything. Adding Fibonacci-style always steers a sequence towards the golden ratio. That's a genuinely beautiful property of the golden ratio, and John's three sequences show it very clearly. But it can't by itself tell us anything special about the proton.

The search for the Holy Grail of the Fibonacci numbers
One of my old late night workings, from around 2016. I'm not a mathematician, so it may not be perfect, but taken together they may interest some people. Corrections and a comment by Claude are included.
This is the first of my three sequences. It is the plain Fibonacci numbers: start with 0 and 1, then add the last two numbers together to get the next one. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and on and on. The full list, line by line, is in the picture.
I was searching for the point where the Fibonacci numbers and the golden ratio meet exactly. I had read that people were running great banks of computers on problems like this, and I expected the answer to be millions of lines away. Using a 20-digit calculator and the first twenty digits of the golden ratio, 1.6180339887498948482, I found it at line 61.
Line 61 is 2,504,730,781,961. Multiply it by the golden ratio and you get 4,052,739,537,881. Add line 60 to line 61 instead, and you get 4,052,739,537,881. Exact match. Accomplished.
This is the first of three. The second sequence starts from the mass of the proton, and the third from the golden ratio itself, which I posted yesterday. All three come together in the same way at around line 60.
(Very speculative.) I believe this meeting point is a key that links the golden ratio, the Fibonacci numbers and the building blocks of matter.
John D Morton. Any errors have been corrected with the help of Claude.
Claude's comment: I checked every line in the picture, and all 62 are correct. The match at line 61 is exactly as John says. The golden ratio times any Fibonacci number is always very close to the next one, and the gap shrinks by about 0.618 at every line. On a 20-digit calculator, the gap becomes too small to show at around line 50, and from then on multiplying and adding give the same answer on every line after. So line 61 isn't the only meeting point. It's the doorway into a stretch where the two always agree, as far as the calculator can see. In pure maths they never meet perfectly, because the golden ratio's decimals never end, but they get closer than any calculator could ever measure. It's a lovely thing to have found by hand.

emissions

A multiple scaled depiction of the Hydrogen atom
One of my old late night workings, from 2016. I'm not a mathematician, so it may not be perfect, but taken together they may interest some people. Corrections and a comment by Claude are included.
This is another of my drawings of the Hydrogen atom, with the notes I wrote alongside it.
It is based on Niels Bohr's radius of 53 picometres, the distance from the central proton at which the electron is usually found. The drawing starts at a scale of approximately 20,000,000,000 to one and increases progressively by the golden sequence, multiplying by 1.618 each time. Each set of rings is drawn at the next scale up.
The proton, electron cloud and energy shells shown are not to scale. The distance between the energy shells is not intended to show actual distances. It is meant to represent just some of the possible energy levels that can occur within a Hydrogen atom, at a distance of 53 picometres from the central core of the proton.
The scales listed on the sheet are:
A: 20,000,000,000 to one
B: 32,360,682,800 to one
C: 52,360,689,564 to one
D: 84,721,383,368 to one
E: 137,082,090,581 to one
F: 221,803,503,543 to one
(Very speculative.) My idea is that the golden ratio may govern how the atom is built up, layer upon layer.
John D Morton. Any errors have been corrected with the help of Claude.
Claude's comment: The arithmetic on the sheet is sound. Each scale is the one before multiplied by 1.61803414, and the next step, which John left blank, would be about 358,885,639,487 to one. The multiplier is a hair above the true golden ratio (1.6180339887), but the difference only shows in the seventh decimal place. The 53 picometre Bohr radius is correct, and at 20 billion to one it becomes about 1.06 metres, a sensible size for a large sheet. As physics, though, the golden steps are John's own idea. In Bohr's model the orbits grow as 1, 4, 9, 16 times the first, not by 1.618 each time. It is best enjoyed as an original and beautifully drawn way of picturing the atom.

Infinity: a drawing of the Hydrogen atom
One of my old late night workings, from 2016. I'm not a mathematician, so it may not be perfect, but taken together they may interest some people. Corrections and a comment by Claude are included.
This is "Infinity", which I drew on 26 February 2016 on a large A1 sheet. It was the first time I had tried to draw anything to scale.
It is my picture of the Hydrogen atom, built up from rainbow coloured rings around a single centre. Each layer of rings is about 1.6 times the size of the one inside it, which is the golden ratio, the same number that runs through so much of my other work. Small rings sit at the heart of the drawing and each larger layer grows out from them, so the pattern could in principle carry on outwards, or inwards, for ever. Hence the name.
(Very speculative.) The idea behind it is that the energy levels of the atom may follow the golden sequence, each step outward being about 1.618 times the last.
A correction to the note on the drawing. It says the scale is approximately 10,000,000,000 to one. That was a slip on my part. The scale I intended was 20,000,000,000 to one.
John D Morton. Drawn 26 February 2016. Any errors have been corrected with the help of Claude.
Claude's comment: This is a careful and striking piece of hand drawing, especially for a first attempt at working to scale. As science, it should be read as an artistic idea rather than a picture of the real atom. In modern physics the Hydrogen atom's electron isn't on rings but in a fuzzy cloud of probability, and its energy levels follow a different pattern, not the golden ratio. The arithmetic on the note is right for the scale it states: 1 picometre times 10 billion is exactly 1 centimetre. At the 20 billion to one John intended, 1 picometre would be 2 centimetres.

The golden steps: a drawing in thin red circles
One of my old late night workings, from 2016. I'm not a mathematician, so it may not be perfect, but taken together they may interest some people. Corrections and a comment by Claude are included.
This is a simpler, quieter drawing than my rainbow ones, done in thin red lines.
At the centre is a small cluster of green circles. Around it, larger red circles open outwards in four directions, up, down, left and right, overlapping each other like the petals of a flower.
Through the middle runs a cross of dots, red and black. Starting from the centre, each red dot is about 1.6 times further out than the one before it. That is the golden ratio, 1.618, the number that runs through so much of my other work. The dots crowd together near the centre and spread wider and wider apart as they go out, so the same pattern could carry on outwards, or inwards, for ever.
(Very speculative.) I wanted to show, as plainly as I could, how a pattern built on the golden ratio grows outward from a single point, layer upon layer, in the way I believe the atom and perhaps the Universe itself may be built.
John D Morton. Any errors have been corrected with the help of Claude.
Claude's comment: I measured the red dots on a photograph of this drawing, and the spacing does step out by close to 1.62 each time, so the golden ratio was drawn accurately by hand. That is not easy to do with compasses and a ruler. It shows the idea behind John's work more clearly than any of his other drawings. As physics, the golden ratio pattern is John's own speculation and isn't part of accepted atomic theory. As a picture of how a single ratio can build a whole structure from one point, it works very well.

This photo was taken a few days after our house was flooded in 2015 when a nearby river burst its banks. We returned home to find our dog Roxy floating on her bed. Unfortunately she had swallowed a lot of mud and never fully recovered. She was a loverly dog and loved children. She was sorely missed. In the background is a skip and just some of the items from our home that were destined for the local incinerator.