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An uncle who designed jet engines at General Electric said one advantage GE had over European counterparts was the greater diversity in sheet metal gauges/thicknesses used to build the engines; they could maintain required strength throughout an engine while paring down engine weight. He attributed the better mix to the English system encouraging divisibility by 2/3/4/6 rather than 2/2.5/5/10 at various scales.
Ideally this kind of thing should be roughly on a log scale, along the lines of https://en.wikipedia.org/wiki/Preferred_number, e.g. https://en.wikipedia.org/wiki/Renard_series https://en.wikipedia.org/wiki/E-series_of_preferred_numbers
You can pick a coarser or finer scale depending on the particular need.
For a base twelve world, I would recommend using binary logarithms written in base twelve (“dublogs”) whenever possible; 0.1 on such a scale matches one semitone in a 12-note-per-octave equal-tempered musical scale.
> Ideally this kind of thing should be roughly on a log scale, along the lines of https://en.wikipedia.org/wiki/Preferred_number, e.g. https://en.wikipedia.org/wiki/Renard_series https://en.wikipedia.org/wiki/E-series_of_preferred_numbers
Traditionally various "gauge" units have been used (e.g. AWG[1]), which are more or less logarithmic. Not quite as elegant as Renard/E series, but still craftsmen and engineers of the past were not completely oblivious of the utility of such scales.
> https://en.wikipedia.org/wiki/E-series_of_preferred_numbers
One interesting thing about E-series is how it includes rounding in its definition. If we take E192 as an example, naively thinking the 186th value should be 10^(185/192) = 9.19479 which rounds to 9.19. But instead it actually is 9.09*10^(1/192) = 9.19967 which rounds to 9.20 (!). I haven't been able to figure out how you would come to that figure without needing first to calculate all the preceding 185 numbers.
edit: I just noticed that the E-series tolerances do not actually match the series in the upper end; for example for 2% tolerance, E48 actually leaves gaps in the range and would be more suitable for 2.5% tolerance. For 2% tolerance hypothetical E60 series would make more sense (and kinda also mirror E6 ~ 20% series). Same applies for E96/1% and E192/0.5% where E120 and E240 (or 1.25% and 0.625%) should have been used instead respectively. I suppose it doesn't matter for practical engineering, but it does tarnish the elegance a bit.
I'm honestly not sure what you're saying, but it sounds compelling!
I'm saying that if redesigned today for our current society (either metric or imperial) sheet metal gauges should probably use something like preferred numbers for widths.
In a hypothetical base twelve world, they could instead use a log scale (think decibels) explicitly, and one nice option is to use binary logarithms, since then twelfths correspond approximately to nice duodecimal fractions. This is the basis of the Western musical scale.
Stock car racers would pare weight down by dipping the sheet metal parts in acid to thin them out. The weight came off, and the car was still "stock".
> the English system encouraging divisibility by 2/3/4/6
I'm not sure that the Old English system used sixths much. Eighths for sure though.
Fun fact: The NYSE and NASDAQ switched from fractional dollar quoting to decimal dollar quoting in the 2002-2003 time frame. The reason was to narrow the bid/ask spread, as 1/64 of a dollar rounds to 2 cents.
The side effect was that prices could no longer be represented exactly as standard IEEE 754 floating point numbers and all calculations now have some small amount of error. Or you could do decimal math and be 50x slower. Take your pick!
They should have just reduced the minimum bid to 1/128th of a dollar!
The most interesting argument I've heard from the dozenal society is that the decimal system is inherently selfish. There is a reason that cases of beer come in packs of 6, 12, 18, 24, 30 instead of 5, 10, 15, etc. It is much more likely you will be able to divide them equally amongst a group of arbitrary size.
That argument sounds like a stretch. I would’ve thought those unit counts were instead used due to resulting in a convenient, space-efficient rectangular ‘slab’ size when packaged together: 6 pack = 2x3, 12 pack = 3x4, 18 pack = 3x6, and so on.
And an argument could therefore be made that decimal is more capitalist since it is more likely to result in a franctional remainder. And I would imagine that would end up in the coffers of seller.
Personally, I endeavor to cut my pizzas into 60 slices.
Isn't that their entire argument?
Well it isn't usually couched in terms of morality.
Only marginally related, but I spent one of my summers at PROMYS[1] doing research into which complex numbers could be successfully used as bases. As different as the various common integer bases "feel" in terms of hand-computation, there's nothing like cranking out conversions of fractions to base 1+i to make you realize that there's a whole wacky universe of number representations out there. I thought it was a really cool field of study, and I remember wishing there were more readily apparent applications. Anyone?
Similarly, see Pavel Grinfeld's argument [2] advocating the vector basis L = {1, x, x²-1/3} w.r.t. Legendre polynomials.
[1] https://en.wikipedia.org/wiki/Pavel_Grinfeld
[2] MathTheBeautiful: Why {1,x,x²} Is a Terrible Basis [video] https://www.youtube.com/watch?v=pYoGYQOXqTk&index=14&list=PL....
Donald Knuth developed the "quarter-imaginary number system" (based on 2i) for a high school talent search. It can represent every complex number with just the digits 0,1,2 and 3, without the use of a sign character.
Cool, a fellow PROMYS alum! The only application I can think of is to use the patterns to generate fractals graphics e.g. http://demonstrations.wolfram.com/NumberSystemsUsingAComplex...
I recommend base 5/2 + sqrt(3)i/2. You can write points in the plane using 7 possible 'digits' at each place. http://fractalcurves.com/Root7.html
I stumbled onto these guys a few years back when I was going nuts about base 18 (which you can do on a standard 5x2 abacus!). I think they have their own symbols for digits > 9 instead of using 'a' and 'b' in the hexadecimal fashion.
Non-standard radixes are pretty fun, and it makes you think about nebulous questions like "what does ten mean", and if you happen to carry that train of thought way past the station, "what does it mean to be a number?"
I learned base 12 arithmetic in gradeschool and the textbook used T and E for 10, 11.
I'm pretty sure this was before people started using hex notation for computing and logic.
For a fictional treatment of converting a society to base 12, Leo Frankowski's High Tech Knight series is a lighthearted and fun read (starts with The Crosstime Engineer).
[I didn't like the books past the original 4, YMMV]
I wonder how they feel about base e?
I can't tell if this was meant as a joke or not