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My Love Affair with Dozens (1972) [pdf] (dozenal.org)
50 points by dalke 25 days ago | hide | past | favorite | 46 comments

I enjoy dozenal. I also enjoy weightlifting. An interesting property of dozenal is how compatible it is with American barbell weights. I haven't heard anyone mention this before (who knows why. the intersection of mathematicians and powerlifters?)

American weights are normally a 45lb barbell, 45lb plates, and then additional smaller plates. The large plates are just called "plates" and counted on one side. 1 plate -> 135 (45*2+45bar), 2 plates -> 225, 3 plates -> 315, 4 plates -> 405. These seemingly weird numbers become "round" when you've been lifting them a while.

What about jumps smaller than that? It gets a little weird. 45 doesn't exactly break up nicely. You might have 35s, definitely 25s, maybe 15s (especially olympic lifters) then 10s, 5s, and 2.5s (called "twos"). "A plate, a twenty-five and a two" is 190. Then you get into "washers". Maybe a 1.25. Or if you have a few pairs of 0.75s and 0.5s you can make any integer.

This is not ideal. You need to have more different types of plates than is most necessary. The math is summing up lots of odd numbers.

Enter dozenal plates. I'll put a d at the front for dozenal measurements. A dozenal plate would be d40, or 48lbs. Pretty darn close to the 45s. The bar is also d40. So a bar with a plate on each side is... d100. Well that's nice. 2 plates -> d180, 3 plates -> d240, 4 plates -> d300. And the smaller plates: d20, d10, d6, d3. If those jumps of 6 total pounds are too big, you could add d1.6, and d0.9 washers. All perfectly split.

That's a neat trick. Of course some of the number weirdness comes from that fact that standard kg plates are translated approximately into lbs in the US. So your 45lb plate is actually a 20kg plate and your 25lb plate is based on a 10kg plate. Although 10kg is actually around 22lb. I'm not sure whether the same colour plates genuinely weigh slightly differently in the US vs the rest of the world.

They do weigh at their marked weight (nowadays at least). But you’re right they’re just slightly modified kg plates.

Not much of a powerlifter, but my working squat was > 2x bodyweight for a while.

I find it easier to lift weights in metric. It helps that I'm EU based, so my plates are in KG.

One plate is 20kg, and the bar is also 20kg Jumps smaller than that, 10kg, 5kg, 2.5kg, 1.25 kg. I've never needed to get down to washers, 1.25 kg jump is small enough to make even with 'small muscle' exercises (wrist curls)

Powers of two all the way down, sums are easy.

The trick is to do sets of 12. Sets of 10 are too easy. So the dozenal still comes in :)

I find lifting in metric way easier. I can do about twice as much!

I don't understand the benefit - either system can accurately represent any integer number of pounds, right? And as you said, after a while the "weird" numbers become normalized after a little while. And it seems like you need the same number of different smaller-weight plates either way (45/25/10/5/2.5) vs d(40/20/10/6/3). If you want the plates to be integer multiples of each other, why not go with the already-widely-adopted kilogram weights?

Also, with d(6/3/1.6/0.9), you're still summing up odd numbers.

If you do the math all the time it’s easy, but most don’t.

You have one more plate in dozenal before you have to get fractional. And 6 and 3 aren’t weird numbers in dozenal. They’re like a half and quarter, respectively. 6+6=10.

Ah, that makes some sense, thanks.

One of my favorite language Youtubers, jan Misali, has an interesting video on Dozenal and "Seximal" (base 6) number systems.


Very good, but I think all these alternate base systems need entirely different digits. People are too used to thinking of '10' as ten.

Also has a followup at https://www.youtube.com/watch?v=wXeX_XKSNlc including a comparison of various bases in representing reciprocals (eg 1/3 = 0b.010101 = 0s0.2 = 0d0.333333 = 0z0.4 = 0x0.555555).

jan Misali makes excellent content! Most of what I know about linguistics I learned from him!

I feel strongly that base 6 or base 11 would make more sense than base 5 or base 10. yes, I have 10 fingers, but I can also represent 0 with no fingers, giving me 11 states.

11 is awkward, but using one hand for the ones place and one hand for the sixes place works quite well.

That’s because you’re doing it wrong. Go full binary and you can count to 31 with one hand, 1023 with both.

Binary finger counting does not line up well with the manual dexterity of the human hand. What does line up well is base-12 counting by touching each of 12 phalanges in your hand with the thumb in order.

and this is surprisingly easy to learn.

Start with the pinky, and 4 becomes an interesting number

4 works fine with thumb = 1 too!

But clearly you are right, the pinky is by nature the smallest digit. :)

<Hand raised with 9> Rock on!

11 is terrible for a numeric base, because it's prime. It doesn't evenly divide into anything until 121. 12 is nice because it's a highly composite number. 6 is as well, but kinda on the small side, and the extra 2 in 12 relative to 6 would be nice.

16 hours since this post was made, and not one Spinal Tap reference. I'm getting old.

How often do you count with your fingers?

For the Tolkien nerds, Elvish numerals use base 12.


That system is base 10, although like a lot of Western languagea it has remnants of when it was in a different base.

I once saw a carpenters square that was a foot long on each side, but one side was divided into quarter inches and the other divided into thirds of an inch.

I was told it was used to make 3/4/5 right triangles, which was kinda fascinating.

Why would 3/4/5 right triangles be useful for carpenters? 30-60-90 are definitely useful though.

In layout, to determine a right angle longer than your framing square. We measure multiples of 3/4/5 on perpendicular walls to check their square, and then measure diagonally corner to corner to check again.

Not parent, but I was also confused by the tool -- if the square (which is by definition 90 degrees where the two sides joined) is x and y long on the two sides -- then what's the purpose of being able to make a square from the 3/4/5 rule that's limited to the x and y lengths?

I'm having trouble explaining more succinctly why this seems like a totally pointless feature of a woodworking square - so it's possible I've misunderstood how it could be used to extend beyond the length of its arms.

I just meant for a larger project we lay aside the square, and measure out 3/4/5 on the ground or whatever. So you're gonna dig the foundations of a house, use a string line and measure out 30 feet and 40 feet. If they're 50 feet apart, you're good.

Oh I understand your question now, yes I agree.

To make sure your 90 is really a 90. Easy and quick to check.

You can do that with the square itself.

You can check the corner is square with a square. Checking the walls and the room can be done with a string.

a^2 (9) + b^2 (16) = c^2 (25)

Without addressing the benefits of using base-12, the supposedly simplifying arithmetic example is atrociously awful:

>The clerks then described a typical problem, getting the cubic content of a package measuring 2'6" x 3'6" x 4'2"

Duodecimals (or any positional system) are a bad representation of these numbers. The author proceeds with a very tedious computation of 2,6 x 3,6 in base 12.

Instead, you can do it all in your head:

2'6" x 3'6" x 4'2" = (5/2)" x (7/2) " x (4 1/6)" = 35 x (1 1/24) ft^3 = 35 + 35/24 ft^3 = 36 11/24 ft^3

There is more to say for the "duodecimal" musical notation, where each note lives on its own line (abolishing the sharps and flats).

This notation is called the piano roll, as it was already in use in mechanical player pianos. It's not great for writing with a pencil, but excellent for screens and editing - which is why most producers would go straight to the piano roll.

It's not, however, connected to base-12 in any way.

76 cm x 107 cm x 127 cm, should be about a cubic meter, give or take (if it was -25 cm on one side and +25 cm on another, it'd have been a 1x1x1 cube).

I thought for a second it wouldn't work like that, but it does!

You are increasing the length of one of the sides by 33%, but decreasing by 25% to get to roughly-a-cube. The product of ratios is 3/4 * 5/4 = 15/16, or about 6% underestimate.

Of course, this is already pretty close, and the third side being 107 and not 100 corrects for that.

In the end, that's because (1-x)(1+x) = 1-x^2, so if if the amount we're doing this with is under 20%, our error is under %4 (i.e. pretty good for rough estimates).

I worked with a member of the Dozenal Society of America. He taught me that pandas have 6 digits on each paw, which is why the panda is their mascot and why almost all panda merchandise is wrong.

For a moment there, I held my breath, thinking the dozen is somehow built into the structure of Pandas, the Python library

Not mentioned is that you can count to 12 on one hand using your thumb to point to finger joints/segments and still have a zero state. So you can actually pretty compactly represent up to 144 on your hands!

12 is a Highly Composite Number, which means it has a lot of factors: 1, 2, 3, 4, 6. In fact, it is a Superior Highly Composite Number. I have heard this makes it extra useful for fractions. --- https://en.wikipedia.org/wiki/12_(number)#Mathematical_prope...

Base 8 is ideal. Easy to count: multiplication table is almost half that of decimal, numbers have just a bit more digits than devimal, very easy to convert to binary.

And we can count using fingers! Just discard thumbs! They are not standard anyway.

Base 8 is pretty naff IMHO.

8's prime factors are 2×2×2, so the only nice patterns it can represent involve twos, fours and eights; but we can already do that using binary, so it's rather a waste of factors. Base 8 is useful as a 'compact notation' for binary (hence why it's easy to convert); along with base 16, base 24, base 32, base 64, etc. Even then, I prefer hexadecimal over octal.

If we change one of octal's factors to a 3 we get base 2×2×3 = 12, which can represent patterns involving twos, threes, fours, sixes, eights and twelves. That's a lot more 'bang for the buck' than octal.

Dozenal is still 'wasteful' by having two factors of 2. If we get rid of one we get seximal, which can represent patterns involving twos, threes, fours and sixes. In that sense, dozenal almost a 'compact notation' for seximal, but not as easy to convert as octal<->binary; we get easy conversion for base 36, base 216, etc. but they seem unwieldy.

The next meaningful jump up from seximal/dozenal is base 2×3×5 = 30, which also seems unwieldy. Dropping the factor of 3 gives us decimal that we all know and hate ;)

> And we can count using fingers! Just discard thumbs! They are not standard anyway.

We can count dozenal on one hand, using the finger joints. Our non-standard thumbs can be useful for pointing at the joints.

> so it's rather a waste of factors

Not really. Multiplication by two is by far the most common thing that happens in life (when cells divide, they divide by 2, not by 3). So easy multiplication by 2 AND 4 is far more important than the occasional by 3, which is no better, really, than in the decimal system.

I wonder if the original Atlantic article is available somewhere for reading, it would be interesting from historical curiosity point of view.

One of my favorite moments in literature: "... and nothing more was said - may God forgive me - of duodecimal functions."

Are there any algorithms anyone knows that are more efficient/ can only be achieved in a given base?

i much prefer seximal/heximal.

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