I spent a serious amount of time on this when I discovered it, a couple of years after it was published.
Firmly believing that "variations on a theme is the crux of creativity", I studied what happens to the same construct in other numbering bases. Well, not many: based on the first theorem, only unary, binary and ternary were interesting.
I should unearth the report I wrote back then -- although I have no idea where a digital copy might be stored.
> What is the reason why there are 92 elements in this mathematical method
Those elements are necessary and sufficient for closure in exactly the weird formal automaton that the informal look-and-say sequence describes, i.e. once you have a graph and transition-states spelled out this much, there's just nowhere else for sequences to transition to. This would maybe be an unsatisfying answer except.. it's surprising at first that it's even finite, or for that matter that you can even analyze this kind of question in the first place.
They proved it originally by simplifying the cases that had to be examined, then enumerating stuff manually. All of which was probably so ugly and error-prone that it's no wonder several versions of the proof were "lost" ;) But later confirmed independently with code in various ways.. see the thing I linked elsewhere in thread. Or here's some work using haskell: https://www.cs.cmu.edu/~kw/pubs/conway.pdf
So besides graphs and transitions, number of elements is ultimately implied by the number base and the sequence rules (like reading left to right). Someone tried roman numerals but I can't find details. Ternary gives 24 elements apparently: https://arxiv.org/pdf/2405.11103
92 or 94 or whatever does not match the number of elements on the periodic table.
There are only 83 elements that are stable enough to persist during the lifetime of a planet, so they are frequently called primordial elements.
Most of the content of the Earth of these 83 elements (from H to Bi, subtracting Tc & Pm and adding Th & U) is inherited from the primitive Solar System, from before the formation of the Earth.
If you want to consider besides the long-lived chemical elements also the short-lived chemical elements, then you must choose some threshold for the lifetime, and the number of elements will depend on the chosen threshold.
Among the unstable elements, chemists usually forget to count the element with Z=0, i.e. neutron, which is nonetheless a distinct chemical element, belonging to the group of noble gases, which contains 7 noble gases, 5 that are stable and 2 that are unstable, i.e. neutron & radon. Hydrogen with Z=1 is the second chemical element, not the first, when also counting the unstable chemical elements.
In cosmic events like supernova explosions or neutron star collisions, all chemical elements up to Z=100 (fermium) are created. So for a very short time after the explosion there are around 101 chemical elements.
Of these elements formed in an explosion/collision some are very short-lived and decay before reaching other stellar systems. Nevertheless, some still persist after many millions of years. Thus the primitive Solar System, before the formation of the planets, contained a few more elements than today, the most abundant being plutonium (due to its half-life of over 80 million years for 244Pu). For some time after its formation the Earth still contained significant quantities of plutonium, but almost all of it has decayed until today, though there have been claims that extremely small quantities of primordial plutonium have been detected.
The category of "transuranic" elements is not defined by an intrinsic physical or chemical property. It is defined by the current age of the Earth. From Z=90 to Z=100, the half-life decreases very steeply and all the "transuranic" elements happen to have half-lifes much smaller than the age of the Earth. Had we lived on a much younger Earth, where plutonium was still abundant, we would have talked about "trans-plutonic" elements. On a planet formed even more quickly after the generation of its matter, the heaviest still abundant chemical element would have been even farther towards Z=100.
On Earth, besides the 83 primordial elements, there exist also 7 unstable elements (Po to Ac & Pa), which are produced continuously by the decay of thorium and uranium, which compensates their continuous decay. Counting these will result in 90 elements that compose the Earth in non-negligible amounts. So you can count 83 elements or 90 elements as the components of Earth, but not 92.
A few other elements are produced naturally in negligible amounts by the spontaneous fission of uranium and by neutron capture events following such fission events.
The elements with Z>100 can be produced only in collisions between nuclei accelerated to very high energies. Such nuclei exist in the cosmic radiation, but the amount of heavy nuclei is extremely small in the cosmic radiation, so the chances of collision between such heavy nuclei are even smaller. The amount of superheavy chemical elements produced artificially by humans, even if it is very small, exceeds by much the amount of such elements produced by natural causes.
> So you can count 83 elements or 90 elements as the components of Earth, but not 92.
Not that it matters.. but actually we might hit 92 since 2 new alkaline earth metals are works in progress: ununennium and unbinilium. Fingers crossed for that island of stability
Oh, what memories!
I spent a serious amount of time on this when I discovered it, a couple of years after it was published.
Firmly believing that "variations on a theme is the crux of creativity", I studied what happens to the same construct in other numbering bases. Well, not many: based on the first theorem, only unary, binary and ternary were interesting.
I should unearth the report I wrote back then -- although I have no idea where a digital copy might be stored.
Oh hey, I tried to drum up some interest in a recent submission too. A 2024 connection to automata theory here: https://arxiv.org/abs/2409.20341 And there's code too: https://github.com/AleksandrStorozhenko/ConwayTransducer
I used to refer to this as "MUP", for "Most Useless Problem". I wonder if there has ever been any use for it, anywhere.
What is the reason why there are 92 elements in this mathematical method, and why does it match the number of elements on the periodic table?
> What is the reason why there are 92 elements in this mathematical method
Those elements are necessary and sufficient for closure in exactly the weird formal automaton that the informal look-and-say sequence describes, i.e. once you have a graph and transition-states spelled out this much, there's just nowhere else for sequences to transition to. This would maybe be an unsatisfying answer except.. it's surprising at first that it's even finite, or for that matter that you can even analyze this kind of question in the first place.
They proved it originally by simplifying the cases that had to be examined, then enumerating stuff manually. All of which was probably so ugly and error-prone that it's no wonder several versions of the proof were "lost" ;) But later confirmed independently with code in various ways.. see the thing I linked elsewhere in thread. Or here's some work using haskell: https://www.cs.cmu.edu/~kw/pubs/conway.pdf
So besides graphs and transitions, number of elements is ultimately implied by the number base and the sequence rules (like reading left to right). Someone tried roman numerals but I can't find details. Ternary gives 24 elements apparently: https://arxiv.org/pdf/2405.11103
At about 6:00 into this Numberphile video ‘Look-and-Say Numbers’ [1] Conway says that there’s ‘No connection whatsoever’.
Myself, I find this highly suspicious. The pattern is the pattern. There are no coincidences. Or maybe there are, who knows.
[1] https://youtube.com/watch?v=ea7lJkEhytA
92 or 94 or whatever does not match the number of elements on the periodic table.
There are only 83 elements that are stable enough to persist during the lifetime of a planet, so they are frequently called primordial elements.
Most of the content of the Earth of these 83 elements (from H to Bi, subtracting Tc & Pm and adding Th & U) is inherited from the primitive Solar System, from before the formation of the Earth.
If you want to consider besides the long-lived chemical elements also the short-lived chemical elements, then you must choose some threshold for the lifetime, and the number of elements will depend on the chosen threshold.
Among the unstable elements, chemists usually forget to count the element with Z=0, i.e. neutron, which is nonetheless a distinct chemical element, belonging to the group of noble gases, which contains 7 noble gases, 5 that are stable and 2 that are unstable, i.e. neutron & radon. Hydrogen with Z=1 is the second chemical element, not the first, when also counting the unstable chemical elements.
In cosmic events like supernova explosions or neutron star collisions, all chemical elements up to Z=100 (fermium) are created. So for a very short time after the explosion there are around 101 chemical elements.
Of these elements formed in an explosion/collision some are very short-lived and decay before reaching other stellar systems. Nevertheless, some still persist after many millions of years. Thus the primitive Solar System, before the formation of the planets, contained a few more elements than today, the most abundant being plutonium (due to its half-life of over 80 million years for 244Pu). For some time after its formation the Earth still contained significant quantities of plutonium, but almost all of it has decayed until today, though there have been claims that extremely small quantities of primordial plutonium have been detected.
The category of "transuranic" elements is not defined by an intrinsic physical or chemical property. It is defined by the current age of the Earth. From Z=90 to Z=100, the half-life decreases very steeply and all the "transuranic" elements happen to have half-lifes much smaller than the age of the Earth. Had we lived on a much younger Earth, where plutonium was still abundant, we would have talked about "trans-plutonic" elements. On a planet formed even more quickly after the generation of its matter, the heaviest still abundant chemical element would have been even farther towards Z=100.
On Earth, besides the 83 primordial elements, there exist also 7 unstable elements (Po to Ac & Pa), which are produced continuously by the decay of thorium and uranium, which compensates their continuous decay. Counting these will result in 90 elements that compose the Earth in non-negligible amounts. So you can count 83 elements or 90 elements as the components of Earth, but not 92.
A few other elements are produced naturally in negligible amounts by the spontaneous fission of uranium and by neutron capture events following such fission events.
The elements with Z>100 can be produced only in collisions between nuclei accelerated to very high energies. Such nuclei exist in the cosmic radiation, but the amount of heavy nuclei is extremely small in the cosmic radiation, so the chances of collision between such heavy nuclei are even smaller. The amount of superheavy chemical elements produced artificially by humans, even if it is very small, exceeds by much the amount of such elements produced by natural causes.
> So you can count 83 elements or 90 elements as the components of Earth, but not 92.
Not that it matters.. but actually we might hit 92 since 2 new alkaline earth metals are works in progress: ununennium and unbinilium. Fingers crossed for that island of stability
https://en.wikipedia.org/wiki/Unbinilium https://en.wikipedia.org/wiki/Ununennium