Enjoy forums? Start your own community for free.
zIFBoards - Free Forum Hosting
Welcome to Dozensonline. We hope you enjoy your visit.
You're currently viewing our forum as a guest. This means you are limited to certain areas of the board and there are some features you can't use. If you join our community, you'll be able to access member-only sections, and use many member-only features such as customizing your profile, and sending personal messages. Registration is simple, fast, and completely free. (You will be asked to confirm your email address before we sign you on.)
Join our community!
If you're already a member please log in to your account to access all of our features:

Name:   Password:


Forum Rules RULES~ "General Chit-chat"

"OffTopic" does not mean "fill this forum with random spam, warez, pornography, insults or general bile". Please respect each other and be polite. This forum is for anything which does not fit into the other forums, for general random banter, fun and games, and anything else which one fancies.

 

 Double Sharp's Alternate Earths, showcasing various bases
Double sharp
Posted: Dec 13 2017, 03:22 PM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



This has been an idea I have been talking about at various times this year (January and December 2017!), so here's the relevant stuff quoted from these two threads.

Here are the January posts from the first thread:
QUOTE (Double sharp @ Jan 5 2017, 08:45 AM)
QUOTE
(Placeholder to make the text big)

I'm not sure a tetradecimal culture would feel the need to include fifths in their auxiliary base and use 420 or 840. Their priorities in this range might well be the same as ours: find a nice, round number that approximates the number of days in a year. This is why I am so very tempted to locate "Heptadactyl Earth" at around 1.24 AU from the Sun and keep its rotation rate the same (I'd like to keep the circadian rhythms at about 24 hours to make the time units match up), just so that the year would be 504 days long! (It would still be at about the edge of the habitable zone, if much colder and harsher than the real Earth, just like tetradecimal is harsher than decimal but not unusable.)

I haven't really thought about what to do to the other planets. It's interesting to note that if you scale everything by about 5/4 this way, Venus ends up at about 0.9 AU from the Sun, which is also within the habitable zone, getting a year of about 312 days. So maybe there would be a pentadactyl decimalist culture here, and it would use 360 or 300 for degrees (if the latter, it's probably going to get decimalised later). That would be very interesting when they both invent telescopes and see what the other civilisation has been doing on their own planet! But if we want to show the tetradecimal-decimal difference purely, we may not need to have any of this, because we will be reminded of everything familiar in decimal by making it unfamiliar but analogous in tetradecimal. So I've not made up my mind on whether I want to have something like that or not.

QUOTE (Double sharp @ Jan 5 2017, 09:04 AM)
QUOTE
(Placeholder to make the text big)
I think I mentioned some of it on the thread for base 504, but I think my "Heptadactyl Earth" culture would keep time as follows.

Consider Herschel's improved Gregorian calendar, whose rules are: a year divisible by 4 is a leap year, unless it's divisible by 100 in which case it isn't, unless it's also divisible by 400 in which case it is again, unless it's also divisible by 4000 in which case it isn't. This makes for a year length of 365.24225 days, which is so close to the real one that by the time this goes wrong the year length would probably have changed enough.

The obvious path to making "Heptadactyl Earth" both similar and weirdly alien at once is to port this exactly. Instead of 969 out of 4000 years being leap years, we'd have dad{e} years out of 4000{e} years being leap years. So the rule would be: a year divisible by 4 is a leap year, unless it's divisible by 100{e} in which case it isn't, unless it's also divisible by 400{e} in which case it is again, unless it's also divisible by 4000{e} in which case it isn't. (Though if we're interested in how the base came about they wouldn't have figured out such a rule yet.) So if we make the year length exceed 504 by about the same amount, we'd have 509.24608 days in a year (approximately), with 2701 years in every 10976 being leap years. This results in a distance of 1.248 AU from the Sun. Incidentally, since the actual Earth has an average temperature of 288 K, and we are assuming "Heptadactyl Earth" to be similar, its average surface temperature would be about 258 K, multiplying 288 by sqrt(1/1.248) - coincidentally, about -15 degrees Celsius instead of +15. So I imagine we need more of a greenhouse effect to push most of the surface above zero, but I would still like it to be significantly colder and harsher.

(It is incidentally true that since a tetradecimal millenium is much longer than a decimal one the criticism of the last addition is sounder, so perhaps the rule would simply be db{e} out of 100{e} years being leap years. It makes essentially no difference, giving 509.24617 days instead and the distance is still 1.248 AU to three decimal places, the fourth being a zero.)

{e} (default tetradecimal)
So now we could basically port the Gregorian calendar over, except that instead of months of 28, 29, 30, and 31 days, they would be 2c, 2d, 30, and 31 days! (But the weeks are probably still 7 days long, for the heavenly lights, since none of that has changed.)

Then each day (still the same length) could be subdivided into 1a hours (of the same length), each hour in 60 minutes, and each minute into 60 seconds, which would each be about half the length of an actual Earth second. (So their Babylonian-equivalent civilisation would be quattuoroctogesimal instead of sexagesimal, perhaps combining duodecimal and tetradecimal tendencies.)

One could similarly derive systems of measure like SI for this world (not seeking to improve on it, but rather imitating its historical development), and I think it would show very well how much the perceived convenience of ten as a number is related to that being how many fingers we have and hence the base of our numeration system. The reason why I would suggest doing it for tetradecimal is because 10=2*7 just as a=2*5, so many things look just familiar enough to fall in a sort of uncanny valley of numeration.

(Of course, not everything would be ported over verbatim. For one, life expectancy should remain largely constant if these are really like humans. The commonly taken figure of seventy - threescore and ten - would become about fifty of these longer years, as it obviously must because of the swapping of seven for five everywhere. But tetradecimalists would probably round it to 37, the square of seven, with lucky individuals reaching 40 as the eighty or fourscore analogue, and the age of majority would probably be rounded to 10, which is about nineteen and a half. Indeed, you can essentially double these tetradecimal decades and read them as decimal to convert them to Earth years. No one would reach 70: already passing 50 would be remarkable and passing 60 extremely rare.)

QUOTE (Oschkar @ Jan 5 2017, 06:25 PM)
QUOTE
(Placeholder to make the text big)
I'm not sure about this. If you scale Saturn by a factor of 509.24608:365.242215, it will be on average 12.582 AU from the Sun. That's already a quarter of the way to our universe's Uranus. Will it still be visible, or would it be more convenient to have a Vulcan inside the orbit of Mercury?

QUOTE (Double sharp @ Jan 6 2017, 03:17 AM)
QUOTE
(Placeholder to make the text big)
{a}

I didn't bother to work it out properly, but if Uranus is still sort-of visible to the naked eye (if undistinguished), Saturn will definitely still be visible out there. The danger (and the reason I really should work it out properly) is that if it is too low in magnitude, it may be visible and be thought of as a star.

Anyway, H_Saturn = -9.7 according to this website. According to Wikipedia, d_BS = 12.582 AU, d_BO = 12.582 - 1.248 = 11.334 AU (a naughty average), d_OS = 1.248 AU, and since it is a superior planet we can ignore phase and set p(χ) = 2/3. So m_Saturn = -9.7 + 2.5 log_10 (3/2 * (12.582)^2 * (11.334)^2 * (1.248)^-4) = 0.549. So, not a problem. happy.gif

The slight issue I find with the scaled-up Venus is that 0.9 AU may still be close in enough to cause a runaway greenhouse effect. Still, it may not be so great a loss. After all, if I wanted to illustrate the differences starkly (and show how the choice of 10 is so related to the base of our number system), it seems to me that the best way to go is to bring a decimalist from actual Earth into a portal to "Heptadactyl Earth".

(And I think most of the differences between my and Oschkar's imaginary worlds are because they seem to be intended for very different purposes. Mine seems to be mostly to illustrate a point about the similarity of ten and fourteen as bases, whereas his probably has more depth than that and does not require this sort of "convergent evolution".)

QUOTE (Oschkar @ Jan 6 2017, 03:26 AM)
QUOTE (Double sharp @ Jan 6 2017, 03:17 AM)
After all, if I wanted to illustrate the differences starkly (and show how the choice of 10 is so related to the base of our number system), it seems to me that the best way to go is to bring a decimalist from actual Earth into a portal to "Heptadactyl Earth".

Actually, let’s create more alternate universes where mammals evolved to have four and six digits as well, and set them in hyperplanes parallel to each other.

QUOTE (Double sharp @ Jan 6 2017, 04:12 AM)
QUOTE
(Placeholder to make the text big)
{e}

I usually use the tetradecimal setting to make a point about rounding. My favourite pair for this sort of thing is 37 and 38: right next to each other, and the two different universes would argue vehemently about which is rounder. And then instead of racism you'd have basism (I think Kode suggested this) and the opportunity for yet another copy of Romeo and Juliet (I think Kode suggested this too), except that because of the base annotations I am very tempted to make them Alfonso and Estrella.

The tetradactyl and hexadactyl universes may not illustrate the point quite as well because I think the mentality of users of a base that isn't an even semiprime is a little too different. So I actually think it might be better for the decimalists to first observe the practices of the tetradecimalists before encountering the duodecimalists and octalists, to sort of ease into it (since I think it's pretty clear that 10 is not optimal as a base). We also wouldn't quite be able to port it directly, because it's not clear why a duodecimal or octal civilisation would go for anything other than strict powers of the base: it's difficult to have auxiliary bases when your base already has an optimal prime factorisation for its size.

And after a while of constructing this heptadactyl universe I am starting to think of 7 as a half and 37 as a quarter, which I never did quite get to for octal. Maybe it's because octal surrogacy is less clear (they tend to overlap because 8 is a prime power), but I think it's also because 7 and 37{a} are not so common as numbers and can take on these roles without causing a mental gear shift and confusion.

EDIT: I can already sort of imagine what would happen in the simplest scenario with just pentadactyls and heptadactyls. After the initial culture-shock and squabbling and obsessively checking everyone's numbers of fingers and toes, both the decimal and tetradecimal universes would suddenly have to work together for some contrived reason, a moral would be delivered, and everyone would be happy forever, at least until somebody interprets 100 as 72 and crashes a space probe. The most plausible solution to that I can imagine is to mandate that when two bases are involved, people communicate with the other party's native base, so that no one's giving quantities that aren't understood. But that won't work for more than two bases.

(There followed some stuff that was more on the number bases than the worldbuilding.)
QUOTE (Double sharp @ Jan 6 2017, 06:32 AM)
QUOTE
(Placeholder to make the text big)
{e}

(This is a separate post because it focuses on the world-building elements rather than the more purely radicological elements.)

The main reason why I'm not sure how to go forward with tetradactyl and hexadactyl universes is because of how my heptadactyl universe got started in earnest. It came about because I noticed that 280 makes a good tetradecimal auxiliary base, and that it mimics 360{a} perfectly. Making the year longer and the planet colder then seems to match the "feel" of tetradecimal as familiar, yet colder and harsher.

I'm not sure how I could match the feel of octal or duodecimal, which tend to withdraw inwards to their own prime factors and alienate themselves more significantly from anything else, even though those are not so useful (except for 3 in octal which would be accommodated). Even further, I'm not sure how to start, as 4 and 6 are not primes and multiplying 2^3 * 3^2 by either of them will just give two or three gross. Also, "Pentadactyl Earth" is conservatively very close to the inner edge of the habitable zone, and I couldn't make it significantly more comfortable and warmer for "Hexadactyl Earth" without running a high risk of causing a runaway greenhouse effect.

So I'm not exactly sure how to start for the octal and duodecimal universes, following the theme of the tetradecimal one.

QUOTE (Oschkar @ Jan 6 2017, 06:13 PM)
{a}

QUOTE (Double sharp @ Jan 6 2017, 06:32 AM)
I'm not sure how I could match the feel of octal or duodecimal, which tend to withdraw inwards to their own prime factors and alienate themselves more significantly from anything else, even though those are not so useful (except for 3 in octal which would be accommodated). Even further, I'm not sure how to start, as 4 and 6 are not primes and multiplying 2^3 * 3^2 by either of them will just give two or three gross.


Is there a problem with the year being about two or three gross days long? Notice that in octal, two gross is 440{8}, still with two significant digits. As for duodecimal, 300{c} is probably a reasonable year length, only about a day longer than my conworld’s year.

QUOTE
Also, "Pentadactyl Earth" is conservatively very close to the inner edge of the habitable zone, and I couldn't make it significantly more comfortable and warmer for "Hexadactyl Earth" without running a high risk of causing a runaway greenhouse effect.


Wouldn't it be "Tetradactyl Earth" that you would need to worry about? That would have to be only 0.864 AU from the Sun, over halfway to Venus. Its average surface temperature would be 310 K, within the human body temperature range. It's evident that at least the more equatorial parts of Tetradactyl Earth will probably be uninhabitable to humans, but could it still be manageable by lowering the amount of greenhouse gases naturally found in the atmosphere?

As for Hexadactyl Earth, to have an orbital period of 305.2b29{c} days, it would have to orbit at 1.127 AU from the Sun, and it would have an average surface temperature of 271 K, barely below freezing. That seems easier to terraform than either the octal or tetradecimal worlds.

QUOTE (Double sharp @ Jan 7 2017, 04:30 AM)
QUOTE
(Placeholder to make the text big)
I wanted to go for the feel of the base, where twelve is by far less resistive than fourteen (and indeed has even more leverage than ten, though the amount of resistance is matched), but on reflection it would probably be easier and more convenient to go by magnitude. So indeed I would only need to worry about the octal world, and using 440{8} and 300{c} as year lengths. The Wikipedia article on the runaway greenhouse effect suggests a lower limit of 0.84 AU, so the octal world would just squeak through. (Senary and hexadecimal worlds would be outside the habitable zone, which matches what I think of those bases for humanity.)

QUOTE (Oschkar @ Jan 7 2017, 05:10 AM)
QUOTE
(Placeholder to make the text big)
A study by A. Zsom, S. Seager, J. de Wit and V. Stamenković suggests that the inner boundary of the habitable zone can be moved further inward by lowering the relative humidity of the planet. A mean relative humidity between 31.25% and 43.75% could be reasonable and would place the octal world unambiguously beyond the lower limit of the habitable zone. It would be a desert world, probably reflecting how "bone dry" the prime factorization of 8 actually is...

QUOTE (Double sharp @ Jan 7 2017, 05:37 AM)
QUOTE
(Placeholder to make the text big)
I find those percentages amusing in context, because I'm mentally reading them as 24%{8} and 34%{8}!

I'm not so sure octal is "bone dry". Thanks to Icarus' usage the term makes me think of large primeflanks like 60 and 240, whereas eight is next to 32 and 7, and its square is next to 5*13. In fact, those are the exact same primes that tetradecimal can test for, except that tetradecimal can't test for 9 (but I don't think that's terribly important). The stubborn sticking to twos alone mean that it is the only human-scale base in which one can test immediately for an arbitrary power of two.

I've mostly given up on comparing {8, 10, 12, 14} to see which is "best", but I do think that all four have their own distinct and interesting ways in which they treat numbers, if I may indulge in a bit of personification. Maybe this is because my initial reaction to 0.333... as a child was not that it was ugly, but that it was cool, and that 0.142857... was cooler. I found 1/13 and especially 1/17 less cool because I couldn't remember their periods (I remember 1/13 = 0.076923... now).

I should also note that we could remove most of the conditionals from the decimal leap-year rule by changing it to the following: every 4th year is a leap year, but every 128th is not. This is very accurate and works even better in octal!

Anyway, the paper also says "By maximizing the cooling effect of clouds on an Earth-like planet, the inner edge could be located as close as 0.87 AU around the Sun based on 1D models with clouds". If we added an entire extra 8-day week to the octal year, then it would indeed be at 0.870 AU, with a year of 296.2378 days. It is somewhat cheating, though.

QUOTE (Oschkar @ Jan 7 2017, 06:00 AM)
QUOTE (Double sharp @ Jan 7 2017, 05:37 AM)
I find those percentages amusing in context, because I'm mentally reading them as 24%{8} and 34%{8}!

I did that on purpose, but was too lazy to type all the BBCode for the subscripts, so I wrote them in decimal instead.
QUOTE
I'm not so sure octal is "bone dry". Thanks to Icarus' usage the term makes me think of large primeflanks like 60 and 240, whereas eight is next to 32 and 7, and its square is next to 5*13. In fact, those are the exact same primes that tetradecimal can test for, except that tetradecimal can't test for 9 (but I don't think that's terribly important).

That’s why I said the prime factorization, not the base itself. By itself, 8 is just a power of 2; it’s its neighbours 7, 9 and 65 that give it the transparency that a natural-scale base deserves.
QUOTE
Anyway, the paper also says "By maximizing the cooling effect of clouds on an Earth-like planet, the inner edge could be located as close as 0.87 AU around the Sun based on 1D models with clouds". If we added an entire extra 8-day week to the octal year, then it would indeed be at 0.870 AU, with a year of 296.2378 days. It is somewhat cheating, though.

Well, we could assume both a larger than average cloud covering and a slightly drier atmosphere, both effects strengthen the negative feedback loop against the greenhouse effect.* The post-industrial octalists have to be much more careful with their emissions, though... I guess 445.1716{8} = 293.237792{a} days in a year isn’t that bad.

* EDIT: Just realized that this sentence is probably an oxymoron. Maybe a thinner but more widespread cloud covering? I don’t know how to reconcile both effects...

QUOTE (Double sharp @ Jan 7 2017, 06:28 AM)
QUOTE
(Placeholder to make the text big)
Meanwhile, the tetradecimalists are probably welcoming global warming with open arms!

The tetradactyls could certainly well be more conscious about climate change, since the danger of causing a runaway greenhouse effect and becoming like Venus for them would be very real. I just hope they don't start with a problem, since the trouble with a larger-than-average cloud covering is that clouds should also have a warming effect. So while 310 K definitely seems fine (it is 22 K higher than "Pentadactyl Earth", so while the tropics are definitely too hot, the more polar regions would be okay), it assumes that conditions are similar to normal Earth, which may not be a reasonable assumption in this case.
[...]

(The remainder of that post was again on the bases, instead of the worldbuilding.)
QUOTE (Double sharp @ Jan 7 2017, 06:46 AM)
QUOTE
(Placeholder to make the text big)
Going by the octal leap-year rule (a year divisible by 4 is a leap year, but one divisible by 200{8} is not), 31 years out of 128 are leap years, resulting in a year length of 293.2421875 days (or should I say 445.174{8}?). Then the proverbial lifespan of seventy years would be octal 127{8}. Unfortunately the conversion factor of 5/4 does not work out so nicely in octal as that of 5/7 works out in tetradecimal (the latter works well because, as I mentioned, 49 and 50 are round numbers in the two different bases that are right next to each other, so 50{e} easily becomes 37{e}). Porting the Gregorian leap-year rule directly instead makes very little difference.

I would port the Gregorian leap-year rule to duodecimal by default, so that b8b{c} out of 4000{c} years are leap years. This gives a year length of 437.24465 days, giving a distance from the Sun of 1.127 AU. Again we get not-so-nice numbers because seventy years become 4a{c}. So, thanks to a numerical coincidence, the tetradecimal world actually seems to work out more similarly to the decimal one than any of the others. But this is perhaps not surprising because fourteen is essentially ten's big sister among number bases: both are even semiprimes where the other prime factor is included is medium-large. (I think I've previously called 2 the queen of the primes and 3 the princess. Following that, 5 and 7 would be important court figures, 11 and 13 might be occasionally called-upon retainers, and the higher primes - the first tier of them being 17, 19, 23, 29, and 31 - would essentially merge into the general populace, following a binary classification.)

QUOTE (Oschkar @ Jan 7 2017, 07:32 AM)
QUOTE (Double sharp @ Jan 7 2017, 06:28 AM)
I just hope they don't start with a problem, since the trouble with a larger-than-average cloud covering is that clouds should also have a warming effect. So while 310 K definitely seems fine (it is 22 K higher than "Pentadactyl Earth", so while the tropics are definitely too hot, the more polar regions would be okay), it assumes that conditions are similar to normal Earth, which may not be a reasonable assumption in this case.

It’s still 22 K... If this graph is any indication, even at the Palaeocene-Eocene Thermal Maximum, the average temperature on Earth was only about 15 K warmer than it is now...

Another possibility would be to change the mass of the Sun slightly in each universe to ensure a more habitable temperature on each universe’s Earth. I’m working out the exact parameters right now.

Or maybe... The Sun has a mass characteristic of a G5V star, but it is considered a G2V star because of its temperature and luminosity, which are higher than the average for stars of its mass. To a first approximation, the planetary equilibrium temperature varies linearly with the temperature of its star. We need to bring it down 22 K, or a ratio of 310:288. If we reduce the temperature of the Sun by the same ratio, it would descend from 5778 K to 5367 K. We could replace the Sun by a G9V star slightly more massive than typical, preserving the same mass, and thus, the same semi-major axis for the same orbital period. The duodecimal and tetradecimal worlds should probably sustain their temperature by the greenhouse effect, though.

QUOTE (Double sharp @ Jan 7 2017, 10:57 AM)
QUOTE
(Placeholder to make the text big)
[...]
I agree that the duodecimal and tetradecimal worlds are probably fine as they are, while the octal world needs a little help. One thing I have certainly learned from this conworlding endeavour is that the Earth is actually quite close to the inner edge of the habitable zone, so that while the worlds for {10, 12, 14} work out fine without additional changes, the world for {8} doesn't. I agree that you could terraform the duodecimal and tetradecimal worlds to be very much like the decimal ones in temperature too, though there is probably a limit to how much you can do with CO2 before it freezes out, so that while "Hexadactyl Earth" might reach Earthlike temperatures I think "Heptadactyl Earth" would still have to be significantly colder (but I'm almost certain that it can still be pushed above the freezing point of water). Checking the higher natural-scale reveals that {16} ends up at 1.36 AU, and is essentially on the edge of causing the CO2 to freeze out of the atmosphere. Well, it seems that the habitable zone is about as small as the human-scale range! happy.gif

But I absolutely love the thought of an octal world just at the innermost edge of the habitable zone, and having to worry about turning into Venus, either from their own efforts, or because their sun is getting warmer! Some of the papers I've read on this suggest that Venus could have been habitable 1 billion years ago, if it were a land planet. So if the octalists managed to evolve 1 billion years faster through contrived coincidence, they might realise this and desperately try to save themselves and their home world from being doomed!

QUOTE (Oschkar @ Jan 7 2017, 07:30 PM)
QUOTE
(Placeholder to make the text big)
Let’s try to contrive coincidences in each planet’s atmosphere and in the luminosity of each Sun so that the average temperature difference between adjacent worlds is about 4 K. The tetradecimal world will be stuck in a permanent ice age, possibly approaching some scientists’ depiction of a "snowball Earth", and the duodecimal world would look a little like the average of the Last Glacial Maximum (map), with Beringia still being exposed. The octal world would look mostly like this.

If the names are any indication, we’re dealing with each world’s Spain. In Ray and Adams’ Ice Age map, Iberia is mostly steppe, with tundra surrounding the main mountain ranges (the Cantabrian, Iberian and Central Systems in the north, and the Baetic System in the south). The Pyrenees would be permanently frozen over. In Dubia, Iberia is an island, separated from continental Europe by a strait running through the Midi. Its climate I can only assume to be the equivalent of Köppen’s Cfb, though because of the higher temperatures, it would be more similar to the present day humid subtropical climates (Köppen’s Cfa).

QUOTE (Double sharp @ Jan 8 2017, 06:48 AM)
QUOTE
(Placeholder to make the text big)
{e}

Well, to be fair, not everything can be the same as normal Earth. The ideal form of this conworld would kill it, because just for starters, anything relying on the number of fingers people have would be different.

Just considering music as an example, vocal music would not have to be any different, but instrumental music would. If pianos look the same, I can't imagine that composers wouldn't exploit the possibility of extremely rapid scales played with the fingering 1234567 instead of 1231234, or the possibility of wider stretches and tone clusters for a wider heptadactyl hand. (Already you can exploit the differences in tone colour on a standard keyboard; chromatic scales and figuration such as C-E-D-F-E-G-F-A can be played with three, four, or even all five fingers, and the more fingers you use, the more overlap and speed there will be, but the less clear the sound.)

The bases people would use may also be a bit different. We would still have outliers like {6, 11} in the decimal world arising from ritual counting systems or body-tally systems. Even if {10} is the dominant base, {7} would be pretty widespread as a substrate, and there may well be special words for the powers of 7 (especially 37 and 1a7); there would also probably be some {20}. The increase difficulty of 10 as a base over a seems to indicate the possibility of binary counting leading to some octal or hexadecimal, or even through finger binary. (Hexadecimal could also arise from counting fourteen fingers and two feet, and be used through doubling and halving via finger binary as a proxy; the sixteen easily manoeuvred bits make a four-digit hexadecimal integer, and you can use one of the bits as a sign bit. Octal could come from positional finger-counting, using one hand for eights and the other for units.) Counting phalanges would give us octodecimal (not counting the thumb) or vigesimal (counting it), and the latter could easily become decimal!

The funny thing about this universe is that to illustrate the arbitrariness of the choice of a over 10, c, or 8, everything must be very similar; but it cannot be exactly the same in everything but the radix, because the choice of dominant radix influences not just what other radices you see, but some other skills involving manipulation of digits!

QUOTE (Double sharp @ Jan 9 2017, 01:52 AM)
QUOTE
(Placeholder to make the text big)
{e} (doing this consistently has now become a personally imposed challenge to think like a heptadactyl for this thread)

I was originally worried there might even be a little problem maintaining the heptadactyly in the tetradecimal world. If I try to make it start that much earlier, from the tetrapods, the trouble is that 7 is such a large number and I'm not entirely sure what evolutionary advantage there is to keep having seven digits on each appendage as opposed to five. After all, many of the descendants of these pentadactyl tetrapods lost some of their digits, and there were in fact early heptadactyl tetrapods in our universe (and there were some earlier ones with eight and some later ones with six.)

Thus I also toyed with another scenario, where the point of divergence was pushed much later, so that heptadactyly (of the very rare sort that actually looks normal when you first see it) only became a trait through random mutation and got selected for due to early hominids' strange preferences. But it would be really rare to have two complete extra functional digits. Maybe this isn't so bad: for cats, 5 is supposed to be normal, but anything between 4 to 7 isn't that uncommon. But then you can't really guarantee that it always shows up on all four limbs. Admittedly for the worldbuilding we would only need the hands to be heptadactyl; heptadactyly of the feet might be a rare trait that would only really be societally interesting for reaching the perfect number 20 instead of the highly composite 1a. (Yes, I admit that this idea came straight from you telling me about constructed Almean octodecimality; although this would seem to give the hilariously strange result of most people looking fairly normal until you look at their hands, with a significant minority wearing rather strangely shaped shoes. It also gives our beloved tetravigesimal!) The octal and duodecimal worlds seemed safer since 4 and 6 are next to 5 while 7 is not.

But after a lot of reading, it seems that the genes for developing the number of digits are also connected to developing much more important parts of the body, so the answer's simple. Once 7 got decided on, it would stay consistently and lead to 10 and 20 in the absence of pressure against it, because it would be really rather risky to change it.

P.S. Here are two links discussing the question, concluding that having 16 digits in total might be optimised.

QUOTE (Double sharp @ Jan 9 2017, 12:54 PM)
QUOTE
(Placeholder to make the text big)
{a}

I've done some further research, mostly for the tetradecimal world where all of this started.

If that is at about 1.25 AU, Heptadactyl Earth will receive about 64% the insolation of Pentadactyl Earth. So it would be worse than the last Ice Age at about 30 Kelvins lower (just as I calculated roughly, which is a good sign because someone's redone the calculations independently), but not enough to cause a Snowball Earth: that would need us to go 50 Kelvins lower. (A hexadecimal world would seem to start making the carbon dioxide freeze out of the atmosphere and cause a positive-feedback cooling loop.) The equatorial regions would be habitable even without an additional greenhouse effect, but people would need to master more technology before expanding further northwards and southwards if that was the case. This would seem to require so much divergence in the language-development history we know because of the much smaller inhabited area, so that it really would be unbelievable if the researchers from each world could understand those from the other.

We also want there to still be 12 months, but now each of them becomes about 42 days each. This seems to require the Moon to be at four-thirds its current orbiting distance. Now, tidal strength follows an inverse-cube law. If we set the solar tidal force on our Earth to 1, then the lunar tidal force on our Earth is 2.1. So the solar tidal force on Heptadactyl Earth is (5/4)^(1/3) = 0.93, and the lunar tidal force is 2.1 * (4/3)^(1/3) = 2.1 * 0.908 = 1.9. It doesn't actually change very much.

Still, the fact that I would like them to meet and notice how similar everything is seems to force everything to have been the same, except that everyone would have seven fingers on each hand and seven toes on each foot instead of five. This would probably change nothing except the numeral base used. This may realistically affect some of the names for the pre-existing numbers even before we reach ten, but it would be more amusing to keep those constant.

Also in that thread were some philological notes, mostly from Oschkar, about how to adapt Indo-European number names to non-decimal bases, creating a naturalistic sort of nomenclature for octal, dozenal, and tetradecimal. But I will quote those highlights in a separate thread (presumably to be titled "Oschkar's Indo-European Number Names", or something like that).
Top
Double sharp
Posted: Dec 13 2017, 03:27 PM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



And here are the posts from December, originally in this thread:
QUOTE (Double sharp @ Dec 9 2017, 04:42 PM)
QUOTE
(Placeholder to make the text big)
Oschkar, you might like this "salvation" of the octal conworld where I still imagine this Xing metrology is in use: slow-rotating planets can be nearer to their host stars before boiling over like Venus, because thick cloud layers get localised on the side facing the Sun, raising the planetary albedo. To quote Wikipedia on terraforming Venus, "A terraformed Venus with the current slow rotation but a more Earth-like atmosphere would result in a global climate similar to warmer versions of Alaska or Siberia with "day" and "night" periods each about 58 days long. The "day" would resemble a short summer with warm humid climate, a heavy overcast sky and ample rainfall. The "night" would resemble a short winter with quite cold temperature and snowfall. There might be periods with more temperate climate and clear weather at sunrise and sunset." Worldbuilding Stack Exchange has some interesting ideas about a Venus-like day-night cycle.

Our octal planet could be there; the year could still be two gross five Earth-days, with "days" and "nights" just over six dozen one days long. Presumably, given how long this schedule is, you could handwave things and say that our tetradactyls using this Xing metrology got used to this as a vaguely "seasonal" cycle (perhaps dividing things into eights or sixteens, distinguishing parts of each season), with an awkward extra day to be sliced off somehow. (For reference; this hypothetical octal planet is 0.864 AU from its Sun, not quite as close as Venus. Its average surface temperature should be around 288 / sqrt(0.864), or 310 K; around 37 degrees Celsius, if not for the slow rotation leading to more clouds and a higher albedo, thus presumably reducing it back to an Earthlike average temperature - since a 400{8}-day day instead of 1-day one reduces the temperature by 20 K on average.)

(BTW, a senary world would have to be at 0.716 AU, pretty much like Venus - at which even slow rotation may not be enough to save you. Oh well. Anyway we have already had enough of this theme. Similarly the hexadecimal world at 1.363 AU is almost at Mars' perihelion, making bases 8 and 14 good places to stop.)

All right, now we've gone and made the octal conworld distinct enough that I'm not sure how to make the dozenal (1.128 AU, average temperature 271 K if all else is equal) and tetradecimal conworlds (1.248 AU, average temperature 258 K if all else is equal) fit this. If the octal conworld is going to resemble a terraformed Venus, then I would want the tetradecimal one to resemble a terraformed Mars, except that I'm not sure where that leaves the dozenal one. And all else being equal both of these are going to look like "Snowball Earths", so they need to have that much more greenhouse gases in their atmospheres. Still, I'm not sure what to do to make things as weird as the long seasonal days and nights that would be on the octal world, which is actually a nice depiction of how the mentality of a prime power base is quite different from that of a diprime like decimal, and how octal's mentality is basically "the sun can shine on 2 as a prime and nothing else". Presumably that means the tetradecimal world has a "normal" day-night cycle like ours, but something interestingly different should happen on the dozenal world.

QUOTE (Double sharp @ Dec 11 2017, 07:20 AM)
QUOTE
(Placeholder to make the text big)
On Our Multibase Multiverses

Actually, this and the other 4 K-difference ideas Oschkar had run into one particular problem: no one is going to believe, given all these differences, that the language is going to be similar. I suppose we could still have recognisable humans without straining suspension of disbelief, but already the world is going to be quite different-looking and the idea that language families expanded the same way is absurd. Together with the cultural differences the new temperature regimes and odd day-night cycle here would imply (and we've been trying this for a while, because if you don't change things by desertification or slow rotation the octal world quickly becomes veneraformed happy.gif), I think the more plausible result is going to have a history and culture that is miles different from what happened in our world.

Kode's Dozenal Indo-Europeans

I suspect that all these different metrologies and Indo-European-style octal, dozenal, and tetradecimal numerals actually come from a different sort of "alternate history" Kode, you, and I have been considering. It's not really an alternative universe because it comes from ours with a single point of divergence. Since rare bases that aren't {(5), 10, (20)} have occurred multiple times in world history, with one (60) even becoming the main base of an early civilisation, it is not much of a stretch to posit as our point of divergence that Indo-European speakers had settled on eight as their base instead of ten, perhaps collapsing out the old words for "nine" and "ten" in favour of pure octal constructions, and perhaps retaining them (I prefer the latter idea, because it gets rid of the "eleven and twelve" problem in Germanic). So it would be the case of one otherwise rare base (8) gaining momentum because it gets picked up by a culture that quickly becomes preeminent.

For the dozenal and tetradecimal worlds, just replace 8 with 12 or 14, and "extend" the digits instead of collapsing them down. I don't even want to mess with dactyly for this because if it isn't a single point of departure this far back, then too much can change, especially when so many things involve digital manipulation. For this, I can believe that everything is the same except for the base. Mind you, you would get a "coexistence" scenario, since I would have trouble believing that Proto-Semitic and Proto-Sino-Tibetan (which you'd need to get all the major world languages under one roof - anything else will get influenced a lot by these anyway) independently made the same innovation in base. So it is still rather interesting, even though these hypothetical worlds are essentially only around for making a plausible pseudo-Indo-European nomenclature. (If you want a pseudo-Semitic or anything else nomenclature, simply posit that that family was the one to make the innovation, come up with names for the new digits, and see what comes out. Although this step does mean that it's easier to come up with pseudo-octal nomenclatures than for the other two bases, since there is then no need to construct transdecimal names.)

The DS-Oschkar Multiverse

My "multi-base multiverse" is actually a little different. Indeed it is inspired greatly by anirudhk's number-base version of Gulliver's Travels, especially in its desire to reflect how the base feels like. Which is why I have been trying to strike a middle ground between the other alternate-history and making things so different that it stops being an "octal conworld" and starts being a "conworld which just happens to have octal conlangs".

Now I'm not sure how to handle the language bit. For one thing, practically no one is going to be interested in a linguistic thriller. Even Tolkien's unfinished Notion Club Papers skirts around this issue by having much of the decipherment work being already done before the start of the narrative, and leaving a substantial appendix for the conlang nerds. I suppose we could learn from that and LOTR and save a discussion of the differences in "roundness" and so forth that would be necessitated by a base of 8, 12, or 14 instead of 10 to a sort of appendix separate from the main narrative. Only, because we are here for those differences, we are essentially writing the appendices with at most only the very briefest outlines of a story, with nothing more than perhaps the intrigue (because, as Kode and Don discussed, of course humans will switch from racism to basism), and some technobabble to get people back to their home universes.

A More Detailed Treatment of the Octal Conworld

We could develop some aspects of those conworlds, and make mention of what civilisations existed and what base they were using (there should be a lot of hexadecimal on the octal world, and I suspect a good bit of senary and septenary on the dozenal and tetradecimal worlds, maybe a bit of tetravigesimal and octovigesimal as well). To save the language thing, I suspect you'd have to presuppose that a large bunch of humans from the pentadactyl world (ours) got sucked earlier into that other world, but earlier in time; they then jump-started some things early thanks to their knowledge, and their language became predominant, except that they consciously changed it to fit with everyone counting in another base. Since Oschkar, based on the names, suggested that we were looking at each world's Spain, I suppose we need a good bunch of Romans to pass through and provide a jump to civilisation on the tetradactyl planet, and then yield recognisable Romance descendants over a wide swath of the planet. They would not be exactly like modern Spanish, French, Portuguese, Italian, Romanian, etc., but we should have a reasonable number of alien Romance languages around, and asking for one to be mutually intelligible should do it.

EDIT: Except that only names that already existed among the Romans could be taken over in such a scenario. Which rules out some of the ones Oschkar and I have already been using. Oops. I think I will just pretend that those are simply illustrative ones from a Kode-like universe where everything but the base and dactyly really is the same and that they don't have anything to do with this fixed-Venus very different octal planet. Otherwise you need the Romans to come after they've adopted Christianity and it becomes exactly like zompist's Elenicoi. Though I like his idea to save things: as he wrote here, "Out of the infinity of possible universes, those with planets similar to ours are easier to get access to. (Perhaps there is another effect which makes it harder to get to near-duplicate earths.)"

Of course there were already tetradactyl humans around with advanced civilisations; but we can posit the pentadactyl Romans arriving in what was previously a backwater area, making the first multiversal Empire. happy.gif (And this presents an opportunity to come up with Almean-style mixed-dactyls.)

We may need to mess around more with the rest of the system. For one thing, the octal solar system, thanks to everything being compressed in our initial approach, is going to have a visible Uranus (I haven't bothered to calculate it, but Uranus is already on the edge of visibility in our world). And with such a slow rotation (thankfully it doesn't have to be retrograde as well), it seems likely that there is no moon on the octal conworld. There may have been one before, but it should have turned into a giant meteor long ago. (Hey, maybe that was what killed its dinosaur-equivalents! tongue.gif)

In fact, maybe these are not alternative universes at all, and the portal to "Tetradactyl Earth" just leads us to a faraway planet orbiting a solar twin. Whatever happened to the portal anyway after the Romans went through? Presumably it closes and opens temporarily and only goes one way, or else there is no story. So it can't plausibly be the same one; a new one had to have opened somewhere along the line. Now deliberately walking into one-way portals, with no clue of what lies behind, is not something that smart people do. So this needs to be contrived as something of an accident, presumably with enough advanced technology on the other side to figure out what has been going on and send them back. I mean, an octal-hexadecimal society could be expected to figure out information technology significantly sooner. Given Pascal's decimal mechanical calculator and Leibniz's use of binary (not to mention the I Ching), it strikes me as plausible.

On The Lack of Details About the Dozenal and Tetradecimal Worlds

And I still haven't worked out what exactly is going on with the hexadactyl and heptadactyl worlds, other than noting that they should be kind of like habitable Mars-analogues. Except that Mars' problem is really just the small size, and the greenhouse effect should be able to fix things enough. Which is quite fine and not very alien for the tetradecimal world, which is really another instance of a diprime; it should just be more extreme, which could be done by giving it a larger tilt. Dozenal is more unkind to numbers "off the grid" and generally feels more introverted, though less "my way or the highway" than octal; I'm not sure how to represent that. Nor how these extra two worlds come into the picture without a bewildering selection of portals, all of which just so happen to lead to similar, habitable planets. And how the languages were made to work out: evidently the portals decided to share.

To be fair, I am still not quite happy about how the dozenal and tetradecimal nomenclatures look. The adjectival derivations of the new transdecimal digits are fine with "seanen, freen, owlen, sown", but "friendred" for the gross sounds to me a bit too whimsical, so we might need a different derivation for the name of the dozen. So I may give that a bit of a pass first. It's evident that the challenges in making a Venus-style planet habitable require more interesting solutions than making a Mars-style one habitable (just dump in lots of CO2), so it looks like the octal world will end up being my focus for now.

On the Presence of Two Different Conversations Here

And now I realise we are having two entirely different conversations here. Please carry on while I decide on where to reposition this Xing-metrology-using octal society. Perhaps we should split out these posts and the ones on the "Base Fortnight" thread into a new thread on my and Oschkar's multibase multiverse idea. Except that both are pretty closely related to all the other discussion there, along the lines of Piotr talking about senary vs dozenal, being vehemently against tetradecimal, or finding a home for this octal metrology, so that a split will likely not help matters one bit.

Oh well. I think that we can set atmospheric pressure as 1 atm there too and make the tetradactyl planet a true Earth twin. The 8-vigil cycle should probably just be considered as their circadian rhythm, now with precious little to do with the astronomical cycle of two days per year. And there are probably no months, so this is more of a concession to practice here and not part of the system there. And so is calling it the "xing" metrology; maybe the Chinese is meant to represent an unknown other culture there.

QUOTE (Double sharp @ Dec 11 2017, 03:41 PM)
QUOTE
(Placeholder to make the text big)
I think I've decided how this ought to go:

This octal metrology does not really need to exist in an octal con-culture, any more than Primel needs to exist in a dozenal con-culture. It is just one component of the octal "lens", the octal way of looking at things, applied to measurement. Nonetheless, it could certainly exist in any octal culture on an Earth facsimile, and the idea could be applied to any other planet (replace "day" with whatever that is for your planet, "gravity" with whatever that is, and "water" with whatever solvent life uses there). When we speak of hypothetical cultures using bases like {6, 8, 10, 12, 14, 16, 18, 20} based on their number of fingers (and toes), it is with this sort of idea, where everything but dactyly (and hence the base) ends up the same, and we're not really considering much else besides the bases.

The octal conworld I've been derailing this thread about (though to excuse myself I might say that it's my thread and it's not without cause) is a little different. Its octalness is just one of many things about it, and while it did intend to evoke the feel of 8 as a base, its focus is not just that, and it can do other weird things to save its analogies with the decimal situation that don't help demonstrate the direct analogies of roundness (e.g. 4 vs 5).

Now we can turn back to dimensionless quantities.

QUOTE (Double sharp @ Dec 12 2017, 08:25 AM)
QUOTE
(Placeholder to make the text big)
I think I figured out what to do for the dozenal world. Because a planet further from its sun needs more CO2 in the atmosphere, so that the thick atmosphere equalises global temperatures more effectively, it can have crazier tilts without having its hemispheres alternately being frozen wastelands and boiling infernoes. For an Earth-like planet tilted 90 degrees at 1.4 AU, the conclusion of a Kasting-coauthored study is that (source; original paper has DOI 10.1006/icar.1997.5759):
QUOTE
its climate would be positively balmy -- the equator would be 11 deg C (52 deg F), and the poles would never rise above 46 deg C (115 deg F) or fall below 3 deg C (37 deg F). Earth would have no ice anywhere on its surface, except on some of its highest mountains.

Which seems pretty reasonable for the feel of dozenal: everything is consistently nice, and you just need to avoid 5 and 7, while 2 and 3 are treated amazingly well. So I would think that the classification I would follow here is to group bases as either having (1) one prime factor or (2) more than one prime factor, and splitting each category as (a) squarefree or (b ) squareful. Thus, up to 30, we have:

1a: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 (the primes)
1b: 4, 8, 9, 16, 25, 27 (the prime powers)
2a: 6, 10, 14, 15, 21, 22, 26, 30 (other squarefree composites)
2b: 12, 18, 20, 24, 28 (other squareful composites)

Now it is true that the numbers don't quite work out, but you can play around with the luminosity of the host star. The octal star can stand to be more luminous than ours (since the slow-rotation was meant to save Venus, and the octal planet isn't really that close), and the dozenal star needs to correspondingly be less luminous. (EDIT: Although that may indeed be unnecessary; the octal planet is also protected by being farther from its sun, while the dozenal planet is already within the "outer habitable zone" from 1.1 to 1.4 AU. Since enough CO2 gets spewed out in the carbonate-silicate cycle to make things stay above freezing, the general idea of a world with an equator at around ~10 degrees Celsius and the poles fluctuating from the forties to just above freezing should stay accurate.)

The octal and decimal worlds end up with greater temperature extremes with snowfalls (the decimal world being less extreme), while the dozenal world is indeed very balmy. But while 8 and 12 treat their primes better they are also much harsher to their coprimes than 10 (or indeed 14 and 16), so I've given them strange day-night cycles.

Anyway, the distances from their host stars for the worlds for {6, 8, 10, 12, 14, 16, 18, 20} should still be {0.72, 0.86, 1.00, 1.13, 1.25, 1.36, 1.47, 1.58} AU. Since the CO2 clouds appear to have a warming effect still, it would strike me that the worlds shouldn't look too different outside those for bases {8, 10, 12}; 18 should look like 12, and 14 like 10, even though 6 (coprimes are small), 16 (prime power with good neighbour), and 20 (abundant with good neighbour) are harder to classify. So indeed I think {8, 10, 12} is all you really need for a first tour: for 14, while it is a good study, it is mostly as a comparison to 10, and it's just a matter of degree rather than kind.

EDIT: Just realised that the high partial pressure of CO2 needed for this trick (at 1.4 AU for Earth, atmospheric pressure needs to be 3.12 bars and the partial pressure of CO2 2.12 bars; 1 bar may still work, but it's still too much for us) is going to kill the octal and decimal visitors to the dozenal world in short order. I'll need to look up the original paper and do some calculations, but it seems like deep enough oceans could also make things work, equalising temperatures via the water instead of the air.

Second EDIT: Though at 1.12 AU, you only need about 0.1 bars of CO2 to do the trick, and that is mostly okay if you have a lot more N2, O2, and H2O in the atmosphere. We can certainly live in a 1% CO2 atmosphere, so it mostly becomes a matter of adapting to 10 bars of atmospheric pressure, and the unfortunate side effect that at those partial pressures oxygen toxicity is problematic. We just can't seem to get this right, can we?

QUOTE (Double sharp @ Dec 12 2017, 03:30 PM)
QUOTE (Double sharp @ Dec 12 2017, 08:25 AM)
EDIT: Just realised that the high partial pressure of CO2 needed for this trick (at 1.4 AU for Earth, atmospheric pressure needs to be 3.12 bars and the partial pressure of CO2 2.12 bars; 1 bar may still work, but it's still too much for us) is going to kill the octal and decimal visitors to the dozenal world in short order. I'll need to look up the original paper and do some calculations, but it seems like deep enough oceans could also make things work, equalising temperatures via the water instead of the air.

To be fair, I think that making some aspects of "alienness" step in, despite the humanoids' almost-believably-human-looking nature, ought to work. I cannot really bring myself to believe that everything looks the same. For one thing, now that I've made the octal world always have days and nights lasting months, and the dozenal world do that except at the equinoxes, surely there would be some adaptations to it, perhaps with some sort of freewheeling circadian cycle, or perhaps indeed just hibernation through the night (though in both cases the night doesn't seem that bad, so I don't want to go that route).

I don't really see a way in which the dozenal world with this sort of high CO2 level is going to work out unless our dozenal humans have kept the ability to photosynthesise, which I have to admit is really cool. But then while they can certainly come over to the octal and decimal worlds, the natives of those worlds can't go there without breathing masks, although they can probably adapt over time to the higher 3-atm-like pressure. Given that, perhaps just like Treisaran's tour, the first step is a more unplanned one from decimal to a non-decimal, non-dozenal base (14 in his case, 8 for mine), and the second is to dozenal. (After all, dozenal doesn't really assert itself as superior unless you think about it, while the benefits of octal, which are nice if a bit incomplete, are easier to reduce to short soundbites.) That way the decimal natives can accidentally get to the octal world and come out not too badly off, unlike if they went to the dozenal world first and got carbon dioxide poisoning.

EDIT: The trouble with photosynthesis alone as a food source for humans is that we'd need to be see-through to get enough surface area, and stationary because otherwise not enough energy is available. But I suppose we could have some mechanism that filters out CO2 from the atmosphere when the dozenalists breathe it.

Now I'll continue talking about this here, now that I've made a dedicated thread for it.
Top
Double sharp
Posted: Dec 14 2017, 04:36 AM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



I think the carbonate-silicate cycle is really the problem with our mental conception of the tetradecimal world. We'd think that it would be much colder and be more extreme, being further from the Sun, but actually past about 1.1 AU, the cycle creates enough carbon dioxide in the atmosphere (0.06 atm and up!) that it's not going to freeze at all except on the high mountains, and even tipping the planet over to 90 degrees like Uranus isn't going to make it uninhabitable (except for sweltering polar summers). (I think we can just make the Sun a bit more luminous to make 1.12 AU act like 1.1 AU would. In fact it might not even need to be that much more luminous; G1V is already a bit too much.)

Indeed it would feel exactly like the "everything is nice" feeling of dozenal except that the high CO2 partial pressure will kill you. Unless we dilute it to 1%, resulting in a 6-atm mixture, at which nitrogen narcosis is significant (though not yet lethal). A mixture of 0.06 atm CO2, 0.5 atm O2, and 5.44 atm N2 ought to be survivable (it goes without saying that I am not a medical expert), with the little problem that visitors are going to have significant nitrogen narcosis. But I guess that's a step up from death.

With such interesting extremes on the octal world, it may not even need an axial tilt, because climate is determined by time of day and season; hence you get tropical weather in the daytime, polar weather in the night, and temperate weather at sunrise and sunset. And on the dozenal world indeed the weather is going to be devastating with six times the atmosphere and we should do something about that.

I think we can vaguely handwavingly say that the point of the high CO2 content, past the greenhouse-gas issue, is that the poles don't cool down quickly as the atmosphere is quite opaque in IR. That accounts for 70% of the small latitude temperature gradient at 1.4 AU. The remaining 30% comes from the increased thickness of the 3-atm atmosphere. And we ought to be able to do something like that with the atmosphere at 1.12 AU, which should be somewhere between 1 and 3 atm; I'll handwavingly say 2 atm. Since the paper notes that 1.1 AU is enough to reach this kind of climate, with an amazingly small temperature range at a sensible 23.5-degree obliquity (low 1.5 degrees Celsius, high 23.4 degrees Celsius in the 1.4-AU model, but the paper says that things are similar for 1.1 through 1.4 AU), the problem is only how animals adapt to the high 0.06-atm (3%) CO2 levels. And in fact they might do all right, since the strict physical limit is more like 6%; at 3% most people can function for a month continuously without problems other than moderate respiratory stimulation; even exercise doesn't result in an exaggerated response, though I would not recommend it. And 2 atm is definitely more reasonable than 6 atm; indeed if it really is 3 atm like the 1.4-AU case, then things are even better as CO2 is only 2% instead of 3% now.

Unless my handwaving is missing something, I think I've fixed the dozenal world: our octal and decimal natives had better not stay outside for months at a time, but short exposures to the air should be all right, and they can always calibrate things indoors to make themselves happy. The weather would be spectacular, but not too bad: it won't snow except on high ground, where things really would get cooler significantly faster (see this on worldbuilding reddit for some of the more spectacular consequences).

It's only that the tetradecimal plan is kind of scotched by natural feedback cycles, unless it is so far out that CO2 condenses out of the atmosphere and cools things down again. Or maybe it is just that its sun is an even more luminous G0V star, so that it is what you'd expect before 1.1 AU even at 1.25 AU? (The tetradecimal world is certainly going to be stuck in a perpetual Ice Age, though a habitable one, which makes sense.) While F stars would also work (temperature list), they also have a bad habit of flooding the planet in UV radiation.

P.S. Because of the weird day-night cycles on the octal and dozenal worlds, I was trying to figure out for a while how "photosynthetic hibernation" would work for plants, until I belatedly realised that I was reinventing deciduous trees. biggrin.gif
Top
Double sharp
Posted: Dec 14 2017, 10:32 AM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



One interesting prospect of the slow rotation on the octal world is that the day truly does become a very fundamental reality for its tetradactyl inhabitants! Because the day-night cycle is more like the seasonal cycle on our planet, whereas there is no counterpart to our day-night cycle, circadian rhythms ought to be relatively freewheeling. After all, the sunlight's not going to run out for a while!

As is rather obvious (and if it isn't, here's a description), there is now a large deviation between rotation period and day length, because there is one extra rotation coming from the revolution of the planet around its host star. While 366 vs 365 is no big deal for Earth, for Mercury and Venus, as well as this octal planet, it is assuredly a big deal! I'd like to have 72-Earth-day days and nights (and thus 144-day nychthemera) that annoyingly don't really fit the approximately 293-day year; it keeps slipping out a little more each time, becoming about the same every 29 cycles because 5 * 29 = 145 (now where have I seen those numbers before?), although it's not exact and there's still something not quite right. (I know it's a bit more than 293, and I'd scale up 144 by the same factor to avoid a nice but hard-to-explain coincidence.)

When I'm not on my phone I'll work it out properly, but the way I'm seeing it, circadian rhythms would be freewheeling, not having to adapt to much of a light cycle; thus detailed keeping would presumably stick to negative octal powers of the immensely long day, with calendars serving for most of the longer cycles (I could see four phases for the four quasi-seasons, like Kode's Phasic metrology). Things would be standardised somewhat in the more modern era, though a return to the natural freewheeling cycle would definitely be advocated. So something like how some Arctic animals only show circadian rhythms in spring and autumn when there are such cycles, except that there are none on the octal world (there are some on the dozenal one), so that there never really are any circadian rhythms either. That would imply some very interesting timekeeping schedules there!

EDIT: Okay, it should be 144.1190273 and 293.2421875 respectively instead of 144 and 293. That's 2.034722222 days per year we want, so 3.034722222 rotations, or a 96.62900458-day rotation period.
Top
Double sharp
Posted: Dec 15 2017, 10:05 AM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



(Notes for the octal nomenclature: the multiples of eight are "eight, twaught, thright, foraight, fivaight, sixaight, sevenaight, thaured", with final "-aight" reduced to "it". Still wondering what 512 ends up as, on the assumption that it ended up similarly to the reconstruction of the Germanic and Balto-Slavic decimal word as *tusdḱomti given on Wikipedia.)

{8}

A few interesting thoughts about this: if the day length were to be an exact submultiple of the year length, then the stars you would see would vary by longitude as well as latitude, because you'd never see those that always happen to fall during the day. But it isn't exact, so they slowly drift, and a determined observer wishing to see any star from his homeland need only wait. After just over thraight cycles, or about two decimal decades, the cycle repeats, and presumably complete star charts would have been produced this way in antiquity. Of course, in the modern octal world, you head to a network of observatories positioned so that you always have some in the dark.

A rather freewheeling circadian cycle would have worked throughout most of history, with generally segmented sleep patterns, but with the rise of civilisations a cycle of a negative power of two of a day would have been adopted. (A season is something like half a day here! With the cross-quarter days being more marked, we really could get divisions of eight!) With a tiny axial tilt (I'm envisioning something like Mercury), the "day half" and "night half" are pretty much always the same duration. So we would presumably count in half-days, and in between dates would be numbered from 00 to 77. Why two digits? Because one thauredth of half a day is approximately 1.1 of an Earth day; it's just over twenty-seven hours. Which I would handwavingly say to be not too far from the tetradactyls' native circadian-like freewheeling cycle, and then go on to divide it octally six times to give a unit of about 0.37 seconds. So time keeping would be done as a single octal number: your number of half-days from some event, a radix point, followed by two octal digits for a date, and then four or six octal digits for a normal clock. (Stopwatches would of course keep going to more digits.) Something like 3740.531662 would be a perfectly normal-looking time.

There's no good reason for time zones here; the sun rises and sets so slowly that deviations on the order of Earth hours are trivial, so you don't get the "Beijing time or Xinjiang time" problem that occurs when you try that on our planet! At the same time, though, the thought of the entire world going on the same schedule boggles the mind; then everything completely shuts down everywhere and starts back up! A more realistic approach is a sort of "shift" idea; you'd have something like four shifts staggered throughout the day, so nothing really stops. Instead the stopping happens in a sort of eight-day week; presumably the last day of the week is when it happens (staggered to the first day for those providing services on that last day, of course). (Hurray for the three-day resting 7-0-1 period spanning two weeks!)

There is however a good reason for "date zones": of course they are not really dates, but Earthlings can think of them that way, especially those living in the Arctic and Antarctic circles. I would see them as proceeding in intervals of biosciadays: your date zone theoretically goes from -40 to +40, depending on when the sunrise and sunset happen for you (at some reference meridian, of course, they happen at 00). If you go across the planet, the time zone remains the same, but instead of sunrise happening on even days and sunset on odds (or vice versa), sunrise happens on odd days and sunset on evens (or vice versa).

Okay, this terminology is getting too confusing and Earth-biased and I have got to fix it somehow. I'd use "nychthemeron" for the ~220-Earth-day period, and "day" and "night" only for those halves. "Cycle" could mean the 1.1-Earth-day period instead. (Yes, in technical literature, to avoid the ambiguity inherent in the word "day", English borrows the Ancient Greek νυχθήμερον instead of resorting to its Germanic heritage - presumably because the components of the Dutch etmaal are entirely obsolete with those meanings in English today.)
Top
Oschkar
Posted: Dec 20 2017, 05:37 AM


Dozens Disciple


Group: Members
Posts: 575
Member No.: 623
Joined: 19-November 11



QUOTE (Double sharp @ Dec 15 2017, 10:05 AM)
Still wondering what 512 ends up as, on the assumption that it ended up similarly to the reconstruction of the Germanic and Balto-Slavic decimal word as *tusdḱomti given on Wikipedia.)

I’ll first normalize this to *túh₂sdḱomtih₂. I don’t believe that there ever was a *dḱomt in this word (the Germanic, Slavic and Baltic evidence all point toward a feminine active participle *tuh₂séntih₂ ~ *tuh₂sn̥tyéh₂s), but I’ll play along with it to see what comes out of it.

PIE *túh₂s-h₃ḱtōw-ih₂ > PGmc *þūhstōwī > OE þūstēow > ModE thoustew /ˈθaʊstjuː/
Top
Double sharp
Posted: Dec 20 2017, 08:49 AM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



{8}

Wonderful. So eight eights make a thaured and eight thaureds make a thoustew. (What does this construction give us in dozenal and tetradecimal, BTW?)

I'm not sure I believe this etymology either, but in such cases I'm willing to take practically anything to preserve distinctiveness. Because if such a word doesn't come from the base, then it probably won't come out any different, and we crash another space probe.

This even shows up in how I deal with the first octade in your scheme. I'm not happy with letting "11" and "12" be read as "eleven" and "twelve" at all, and "13, 14, 15, 16, 17" as "thireight, foureight, fiveight, sixeight, seveneight" read rather close to "30, 40, 50, 60, 70" as "thraight, foraight, fivaight, sixaight, sevenaight". In order to stop people arguing over whether some dearly departed rich relative meant to write an "a" or an "e" there, and because even stressing them differently ("fouréight" vs "fóraight") doesn't help us for "thireight" (never mind that contrastive stress while counting will produce "thíreight, fóureight, fíveight, síxeight, séveneight" anyway), I suspect that octal English might go for the route of regularising the teens to "eight-one, eight-two, eight-three, eight-four, eight-five, eight-six, eight-seven". Perhaps "nine" and "ten", or rather the roots that led to them, could be posited to have existed as roots and have been preserved in languages whenever the otherwise resulting constructions end up in confusion.

We'd have to come up with something completely different for 1000 in Latin, so that it doesn't end up with exactly the decimal word again; presumably your stoirent is its French descendant. But I think octal would really prefer a four-grouping, which would allow hexadecimal compatibility since 10^4 = 20^3 (together with sanstre for 400 in French, which is full of vigesimal influence here, and should be full of hexadecimal influence there). I'm just not sure how we can come up with something that isn't "myriad" for 10^4.

Meanwhile we have only really solved this problem for the Indo-Eurosphere; the obstacle I keep running into trying to do it for the Sinosphere is the tendency to give graphic etymologies instead.

And I realise this has turned into the nomenclature thread that I should start instead. But you can certainly reply here if you get to it before I get time to collect everything; I'll just collect this material there too in quotes.
Top
Kodegadulo
Posted: Dec 21 2017, 12:13 PM


Obsessive poster


Group: Moderators
Posts: 4,184
Member No.: 606
Joined: 10-September 11



Just a (hopefully) honorable mention: "checker" for eight squared; "(check)mate" for eight cubed. A bit tricky to give it an alt-PIE etymological history though, I admit...
Top
wendy.krieger
Posted: Dec 21 2017, 02:25 PM


Dozens Demigod


Group: Members
Posts: 2,432
Member No.: 655
Joined: 11-July 12



Kode has never played fairy chess.

Great chess is played on a board that is ten deep and twelve wide.

My own forays into this area have been into multi-dimensional worlds. You propose a model that 7 is one circle surrounded by six equal circles, and that the corresponding number in 4d would be a sphere surrounded by 12 spheres.

The point where Double-sharp and I come to the conclusion the bases become unrulily difficult (ie 30-36 area), moves up into the order of 120. (I don't know what possessed me to say 5. The room was under attack at the time. Thanks to those pointing it out!)

In 4d, you have time zones and season-zones. All of the daylight periods are 12 hours, but the seasons are caused by the inclination of the sun. This corresponds to some circle of 47 degrees diameter on the lattitude circle, which means the sun could be 47 degrees lower in the sky when it's winter.

But where in 3d, you can think of the season-zones as a double-ended pointer, where the north is 6 months behind the south, in 4d, you get the full circle: christmas in spring, indeed!
Top
Double sharp
Posted: Dec 21 2017, 03:52 PM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



Surely we can surround a sphere by 12 spheres of equal size in 3D, not 5?

I think that despite all those fairy chess variants, some of which are incidentally very good, the standard chessboard in the Western world will always be thought of as 8 by 8, just as the standard Chinese one is 9 by 10 and the standard Japanese one is 9 by 9. (Though you might be interested in the older and once more popular 12 by 12 variant of Japanese chess.)
Top
Kodegadulo
Posted: Dec 21 2017, 09:36 PM


Obsessive poster


Group: Moderators
Posts: 4,184
Member No.: 606
Joined: 10-September 11



Oh I've heard of fairy chess, but I was speaking in English, not Faerie. smile.gif I also understand that the Japanese don't call their nine by nine game "chess", but rather shōgi (将棋), and the Chinese call their nine by ten game xiàngqí (象棋). So if we ever need words for sixzy-nine and sevenzy-six, we're covered. smile.gif

But it appears that Wendy wasn't claiming that the kissing number in four dimensions is five (we all know it's twenzy), she seems to be accusing someone else in the conversation of making that claim. I can't imagine where she would get that idea. But as usual, Wendy's logic and diction is as clear as mud, and her context-setting is non-existent, so there is no way to tell what she was saying.
Top
Double sharp
Posted: Dec 22 2017, 03:21 AM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



As for the context: she originally wrote "...the corresponding number in 4d would be a sphere surrounded by 5 spheres" [sic], and I pointed out that the kissing number in 3D was in fact 12, not 5, upon which she edited the post and added:
QUOTE (wendy.krieger @ Dec 21 2017, 03:52 PM)
(I don't know what possessed me to say 5. The room was under attack at the time. Thanks to those pointing it out!)
Top
Kodegadulo
Posted: Dec 22 2017, 04:55 AM


Obsessive poster


Group: Moderators
Posts: 4,184
Member No.: 606
Joined: 10-September 11



QUOTE (wendy.krieger @ Dec 21 2017, 02:25 PM)
You propose a model that 7 is one circle surrounded by six equal circles, and that the corresponding number in 4d would be a sphere surrounded by 12 spheres.

The kissing number:
in TWO dimensions ... is SIX
in THREE dimensions ... is ONE DOZEN
in FOUR dimensions ... is TWO DOZEN

Can you FINALLY get that straight? Aren't you supposedly the self-declared expert on "higher space"?

You can pack a DOZEN three-dimensional SPHERES around a SPHERE
You can pack TWO DOZEN four-dimensional HYPERSPHERES around a HYPERSPHERE

Got it? Or are four dimensional hyperspace aliens attacking your room again?
Top
Double sharp
Posted: Dec 22 2017, 05:50 AM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



Since she is extrapolating from 3D to 4D, she probably is making the assumption that since the kissing number in 2D (one less than 3D) is 6, close to the subitisation limit that makes bases less than 6 (and their products just less than 36) rather difficult to use, the fact that the kissing number in 3D (one less than 4D) is 12 should double that for the human-scale bases, and quadruple it for mixed bases.

Running with this hypothesis, our four-dimensional people would find that dozenal feels too small, like senary; hexadecimal would probably be closer to the start of their "sweet spot" for "human-scale-like" bases. Tetravigesimal would probably be ideal; trigesimal might be a little large and is not very deep, despite being full of regulars. So they'd think of {16, 20, 24, 28} closer to the way we think of {8, 10, 12, 14}; {18, 30} would also assert themselves as possibilities. 32 and 36 may prove a bit large, with 40 and 42 beyond the limit.

The prospect of having a race of 4D aliens for which Oschkar's "higher natural-scale" becomes the human scale is very interesting. Admittedly, I am not sure I believe this, because the factors militating against large bases are also rather mathematical. What this seems to do to a base like 24 is to make the divisors easy as they ought to have been, and ameliorate the neutral digits too so that they feel like decimal 6 instead of splitting into many cycles (since we can't hold that long a line in our short-term memory). That's wonderful, but the problem is that there are still lots of totatives, double that of dozenal, of which only three are at all helpful (1, 5, and n). If we guess that they'd find each totative row about as annoying as dozenal 5 or 7, it still remains that they have to worry about five opaque totatives instead of two, even if the length isn't a factor. And that's not getting into all the differently scrambled sequences in the neutral digits; each one is easy on its own, but now you suddenly have a lot.

So it seems to me that simply increasing the mnemonic powers of these 4D people, to raise their short-term memory limit to "13 plus or minus 2", is not really enough to "fix" 24 outside the divisors. I think it's more likely that they'd use 18 or 20 as their base, maybe 16. In order to get 24 to work, I think you'd need to increase their mnemonic powers even further than just a better short-term memory, because of the quadratically expanding tables. It's possible that I'm being a bit too pessimistic, but without increasing memory quadratically as well, we seem to be in the situation where the number of opaque totatives alone is enough to cause problems.

And I also am not sure how the kissing number can be linked to this other than coincidentally. The proof that the kissing number in 2D is 6 is very trivial; the one that the kissing number in 3D is 12 is absolutely not, as is the one that the kissing number in 4D is 24.
Top
wendy.krieger
Posted: Dec 22 2017, 09:46 AM


Dozens Demigod


Group: Members
Posts: 2,432
Member No.: 655
Joined: 11-July 12



The kissing numbers are established as exact for 2, 4, 8, and 24 dimensions. The value of 3d has been shown by exhaustion that 13 spheres can not touch the core. Thise of 5, 6, and 7 correspond to the root-lattices of E5, E6, E7. But there are non-lattice spheres that have the same kissing number and packing efficiency known for these dimensions too.

The reason i chose 7 and 13 as numbers representing visual smallness, in 3d and 4d, is because the field-of-vision is N-1 (ie 2d, 3d), and these represent the central core plus the kissing spheres. The rest of the stuff is done on a logrithmic scale.

The reason that i make distinction between up and down dimensions, is that some things make sense only if counted down from solid. For example, a surface divides space, and is thus a 'down' dimension. We have in 4d, a surface is three-dimensional. The edge of a sword, is not 1d, but two-down dimensions, since the solid (N), is cut (ie -1), by the sweep (ie -1) of the blade, and thus must be N-2 dimensions.

Regarding asking for a pound at the deli, yes i tried it a few years back, and no, they did not understand it. But i live with people who are much my senior, and they still talk in pounds and inches. Ironmongery and builders still understand inches and feet, but they use metrics as well. So they say it's '800 wide' and one is left bamboozled about what 800 millimetres might mean.
Top
Double sharp
Posted: Dec 22 2017, 11:19 AM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



QUOTE (wendy.krieger @ Dec 22 2017, 09:46 AM)
Regarding asking for a pound at the deli, yes i tried it a few years back, and no, they did not understand it. But i live with people who are much my senior, and they still talk in pounds and inches. Ironmongery and builders still understand inches and feet, but they use metrics as well. So they say it's '800 wide' and one is left bamboozled about what 800 millimetres might mean.

Surely this is instead in response to Harold at your thread?
QUOTE (Harold @ Dec 22 2017, 01:24 AM)
I would be curious as to what type of service she would get in the market if she went to the deli counter and asked for quantities in pounds.  If the attendant is young to middle aged, they would have no clue what she is asking for.  Someone older might know but would be highly confused and would have forgotten decades how to convert to grams so it can be weighed on the scale.  Then there the people in the queue behind her who are getting pissed because she is holding up the queue.
Top
wendy.krieger
Posted: Dec 23 2017, 01:26 PM


Dozens Demigod


Group: Members
Posts: 2,432
Member No.: 655
Joined: 11-July 12



You did not make this sort of comment, when Kode cross-posted his comment on your thread onto mine! Here it was an accident that i made that comment here. Not a delibarate decision.
Top
Double sharp
Posted: Dec 23 2017, 01:39 PM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



QUOTE (wendy.krieger @ Dec 23 2017, 01:26 PM)
You did not make this sort of comment, when Kode cross-posted his comment on your thread onto mine!  Here it was an accident that i made that comment here.  Not a delibarate decision.

Why should I need to make a comment on his actions, when he said upfront that he was deliberately making the comments there, and when I agree with him that they are off-topic on my thread? Surely in such a case there is no need to query whether he meant to reply where he did.

And speaking of going off-topic on my threads, why are we engaging in this here?
Top
icarus
Posted: Dec 23 2017, 02:20 PM


Dozens Demigod


Group: Admin
Posts: 1,913
Member No.: 50
Joined: 11-April 06



Let's stay on topic folks. Merry Christmas Wendy.
Top
« Next Oldest | General Chit-chat | Next Newest »
zIFBoards - Free Forum Hosting
Create your own social network with a free forum.
Learn More · Register Now

Topic Options



Hosted for free by zIFBoards* (Terms of Use: Updated 2/10/2010) | Powered by Invision Power Board v1.3 Final © 2003 IPS, Inc.
Page creation time: 0.0574 seconds · Archive