
The fifth hyperfactorial: 5⁵×4⁴×3³×2²×1¹=86400000 milliseconds is exactly 1 day - slbenfica
https://twitter.com/fermatslibrary/status/996736533511266304
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simonvc
It was useful for sailors to be able to divide the day into shifts, so our day
is the most divisible number of hours long.

1 x 2 x 3 x 4

That way you could have half days, quarter days, or third of days.

An hour is divided into 60 minutes: 3 x 4 x 5

The word "second" means a "second division by 3 x 4 x 5"

~~~
jlebar
This article (from downthread) suggests that 60 comes from the Sumerians, who
had a base-60 number system, and 24 comes from the Egyptians, who had a
base-12 number system. Although it agrees that 12 and 60 are conveniently
divisible in the way you describe, I'm not finding any evidence to connect
this specifically to sailors, or even to people dividing up shifts. Rather it
seems likely that the civilization's numeric base was chosen first, and then
it was applied to timekeeping.

[https://www.scientificamerican.com/article/experts-time-
divi...](https://www.scientificamerican.com/article/experts-time-division-
days-hours-minutes/)

~~~
kungtotte
Shipboard life was one of the first large scale applications of precise
timekeeping. You need it for navigation and for organising shifts and jobs
(though I think navigation is the major player here).

For most people at that time there was no need for even hourly timekeeping,
much less minutes or seconds. You woke up in the morning and ate your
breakfast, milking the cows took however long it took, then you gathered the
eggs from the hens, probably around that time you had lunch, then depending on
the season you'd do whatever was needed in the fields until the sun started
setting and then you ate supper.

But with navigation you had to know how far to sail in any given direction to
not get knocked off your course.

I'm certain that the use of 60 and 12 base systems came from the Sumerians and
Egyptians, but their application for timekeeping is probably influenced
heavily by sailors.

~~~
derefr
It seems like even without precise knowledge of absolute time, precise
knowledge of intervals would be useful; you could evaluate a time-trial to
know if you’re improving as a footman/chariot rider; you could evaluate the
maximum speed of a horse; you could measure height by dropping things off of
the sides; etc.

So I would expect that, even if we didn’t have hours, we probably had second
and minutes fairly early on.

~~~
gowld
You don't need any particular units for that, though. Any periodic oscillator
you have at hand would work. (And with an analog clock you don't need to name
the subdivisions, just look at the clock). You only need standard units with
nice divisions for mass-coordinated scheduling.

~~~
derefr
Or mass-coordinated record-keeping for something like the Olympics. How do you
know whether this year’s best runner beat last year’s best runner?

~~~
ansible
> _Or mass-coordinated record-keeping for something like the Olympics. How do
> you know whether this year’s best runner beat last year’s best runner?_

That sort of timekeeping a modern affectation. In the ancient past, it was
sufficient to just be the fastest in whatever was the most recent contest. If
you wanted to see if this year's best runner could beat last year's best, then
you set up a race between them...

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drivers99
24 hr per day, 60 min per hour, 60 s per min, 1000 ms per s

    
    
        24   : 2 3 4
        60   :   3 4 5
        60   :   3 4 5
        1000 : 2   4 5
                     5
                     5
               -------
       count:  2 3 4 5

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adamnemecek
Babylonian number system was based on 60 (which is why we have 60 minutes in
an hour and 360 degrees). 60 is dope because it has divisors
2,3,4,5,6,10,12,15,20,30 which is a lot of divisors.

Bee tee dubs Babylonians were like way ahead of their time math-wise. They
were aware of Fourier for example.

[https://en.wikipedia.org/wiki/Babylonian_mathematics](https://en.wikipedia.org/wiki/Babylonian_mathematics)

~~~
mikekij
Bonus points for combining "dope" with mathematics.

~~~
205guy
But quickly negated by "bee tee dubs" which is a cryptic mannerism that is
actually longer than its original ("by the way" via btw). It's like whiskey
tango foxtrot, just without the cool spoken alphabet.

~~~
205guy
In reply to ad-hominem's dead comment: word.

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test6554
The fifth hyperfactorial: 5⁵×4⁴×3³×2²×1¹=86400000 milliseconds is exactly _24
hours_. 24 hours is approximately 1 earth day.

~~~
vertexFarm
It would be very suspicious if our day was such an even number of
milliseconds. It isn't, but still pretty cool! Things like this are what leap
years are for, after all.

Or does that count? Is that caused by a partial day in the revolution of the
earth around the sun or a failure of a day to fit into exactly 24 hours? Or
both? Are we talking solar or sidereal days?

~~~
bwbw223
How so? Hours, minutes, seconds, and milliseconds are all man-made units. It’s
the failure of 24 hours not fitting exactly into a day, not the inverse.

Leap years have nothing to do with milliseconds in a day- it’s days that fail
to fit into a year; days and years are defined by the orbit and rotation of
the earth. Leap seconds, however, are another story...

~~~
vertexFarm
I know they are man-made units, but I don't think the men who made the second
could measure the length of a day and split it into 86,400,000 units
accurately.

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mxwsn
Unfortunately, the definition of one second as 9,192,631,770 energy
transitions of the Cesium atom factorizes in a less pretty way:

2 x 3^2 x 5 x 7^2 x 47 x 44351

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gowld
Previous discussion from last time the same OP made a tweet with the same
content.

[https://news.ycombinator.com/item?id=15888591](https://news.ycombinator.com/item?id=15888591)

~~~
dmix
5 months for an exact repost by the same person getting a similar amount of
upvotes/comments. That's a new one for me...

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oytis
It's not a pure coincidence. Coefficients used for time measurements (12, 24,
60) were deliberately chosen to have as many (small) integer divisors as
possible.

~~~
mywittyname
I was going to make a comment to this effect -- I think this is by design.

This is a logical alternative to our metric system. We use powers of ten for
increasing units because we operate in base-10. Older civilizations created
larger units by multiplying a smaller unit by either an existing member of the
set, or the next largest factor. So you end up with a sequence like 2, 6
(3x2), 12 (3x2x2), 60 (5x3x2x2), 360 (6x5x3x2x2), 2520 (7x360), etc.

In effect, this is akin to saying a kilogram is 10^3 grams. It's novel to use
because we were not taught to think that way. I bet a Babylonian would find
this tweet to be kind of obvious.

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TheRealPomax
Ehhhhh, sort of? But more: "surprisingly close to one day", because if you
want to do correct time-keeping, a real day (or rather, a sidereal day, the
time in which the earth makes one full rotation wrt "fixed" stars) is
currently 86164090.7 milliseconds long.

You could also look at the solar day (the time it takes the earth to rotate
such that the sun appears in the same place), in which case a day is actually
a little longer than 86400000ms.

~~~
vignesh_m
That's basically because our definition of second is no longer based on the
rotation of the earth. If it was it would be exactly that.

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Xophmeister
An equally uninteresting coincidence is that 10! seconds is 6 weeks

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beagle3
And a related mnemonic: The number of seconds per day, in modern programming
language notation, is 864e2 (That's 8-6-4-2 with an 'e' inserted before the
last digit).

I generally prefer to have a SECS_PER_DAY constant, or write (24 * 60 * 60) to
make the value clear, but when code golfing, and as a mnemonic, I remember
that SECS_PER_DAY=86400=864e2

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JoeAltmaier
Well, a civil day anyway. A solar day is 86400002

~~~
jvolkman
off by 2

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mattdeboard
4 hard problems in computer science

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jvkersch
Hence, a very poor approximation to pi is given by 365 × 5⁵ × 4⁴ × 3³ × 2² ×
1¹ / 10^10.

~~~
OskarS
Why would "the number of milliseconds in a year divided by 10 billion" have
anything to do with pi?

~~~
Shadow6363
I feel like I was just nerd sniped, but this drove me crazy until I thought
about it for a bit.

If we treat Earth's orbit as a perfect circle, then the number of milliseconds
in a year would be its circumference. To get to pi then, we just need to
divide that by its diameter or 2*its radius. In addition, we have the
circumference in ms so we want to convert that into a distance or the radius
into ms so we need the speed the Earth is rotating around the sun.

The average radius of the Earth to the Sun is 149,600,000 km so the diameter
is 299,200,000 km. Earth's average orbital speed is 30 km/s or 0.03 km/ms.
Combining these two numbers to get ms, (299,200,000 km / 0.03 km/ms) =
9,973,333,333.333 ms, which is very nearly 10 billion.

~~~
Armisael16
This isn't a reason or an explanation, just a statement of the arithmetic.

The siblings comments to yours are right - it's nearly random chance (give or
take conservation of momentum during accretion of the solar disk into
planets).

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Aardwolf
It is a pretty great coincidence, but there are things that helped improve the
chances of this:

Numeric bases are often chosen to have many divisors. The numeric bases 60, 12
and 10 are used in time, which have many 2's, 3's and 5's as divisors.

So if you multiply them all, you get exactly such a product. The only
coincidence is how nicely the powers line up.

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logfromblammo
Utter coincidence, and it only works at all if you throw in the milliseconds
term, which makes it seem forced.

A lot of our numbering systems are inherited from early mathematics that dealt
mainly with ratios of low whole numbers. And so selecting bases with many
prime factors made the rational math easier. When your base is 60, it's easier
to divide by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30.

You might as well divide up the mean solar day into 10! = 1 * 2 * 3 * 4 * 5 *
6 * 7 * 8 * 9 * 10 = 2^8 * 3^5 * 5^2 * 7 = 3628800 chunks, which are each
1/42nd of a second. Or maybe make the 7 represent the 7 days in a week, and
use 518400 chunks per day, each 1/6th of a second. You could divide up your
time by _so many_ whole factors.

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DenisM
In related news a (typical) year is 10^2 + 11^2 + 12^2 == 365 days.

Also known as 13^2 + 14^2.

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aetherson
This isn't much of a coincidence. We have days, hours, and minutes that are
designed to be divided into 3's, 4's and 5's, and then there are lots of
factors of 10's in the fifth hyperfactorial, getting us down to milliseconds.
It's fun that it happens to work out to the hyperfactorial, but if it didn't,
it was always going to be just a couple of additional or fewer 2s, 3s, and 5s
away.

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vram22
Speaking of factorials, this post I had written a few years ago may be of
interest:

Permutation facts:

[https://jugad2.blogspot.in/2016/10/by-vasudev-ram-
nicomachus...](https://jugad2.blogspot.in/2016/10/by-vasudev-ram-nicomachus-
theorem-3d.html)

It mentions many kinds of factorials and other interesting types of numbers
too.

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aembleton
10! is the number of seconds in 7 weeks

[https://duckduckgo.com/?q=10!+-+(60*60*24*7*6)&t=canonical&i...](https://duckduckgo.com/?q=10!+-+\(60*60*24*7*6\)&t=canonical&ia=calculator)

~~~
zimpenfish
6 weeks? Unless I'm radically misunderstanding that multiplication.

~~~
aembleton
Yes, 6 weeks. Not sure why I typed 7.

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brlewis
Despite being well aware that 60 was chosen to be evenly divisible by small
numbers, I find it a really cool coincidence that the exponents fell into
place so perfectly.

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aaroninsf
Factoid:

The reason the number 108 recurs in Hinduism and Buddhism is that as the third
hyperfactorial it was esoteric knowledge discoverable by sacred geometers.

(3³×2²×1¹ = 27×4×1 = 108)

~~~
vignesh_m
Lol. Its just because its a highly divisible number. Saying basic
multiplication is "esoteric" shows how little you expect of ancient
civilisations. They estimate the circumference of the fuckin earth and you say
its "esoteric" to think 108 is holy.

~~~
jobigoud
I would say that considering a number as "holy" is esotric by definition.

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ChuckMcM
And perhaps in our lifetime the next hyper factorial (6^6 * 5^5 * 4^4 * 3^3 *
2^2 * 1^1) will be a lifetime (127.8 years (approximately)

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stretchwithme
So is 5^5 * 4^4 * 3^3 * 2^2 * 1^1 * 0^0

~~~
jobigoud
There is no consensus for what 0^0 means.

~~~
jonsen
That's why they are called emoticons, just conveying emotions ;-)

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brolover
10 factorial seconds is 42 days.

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juancn
And 10! seconds is exactly 6 weeks.

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sandov
It would be impressive if days and seconds weren't defined arbitrarily.

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partycoder
You can divide a day in any arbitrary manner. This is just a coincidence.

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jiveturkey
yeah but as other comments explain, this isn’t some numeric “property” as
such, giving insight into anything. It is by definition.

Furthermore, a day is not even a day. We need leap seconds to sync up the
solar day.

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m0skit0
A "real" day is 86164100 milliseconds (23h 56m 4.100s)

~~~
rimliu
And what makes stellar day real, and mean solar day not real? Having day
defined in terms of the star Earth is orbiting around has a benefit of having
more or less consistent time for noon. Having a day defined in terms of the
mean sun instead of the real one has a benefit of having noons at exactly 24h
apart instead of variable inter between noons of the real sun (see
[https://en.wikipedia.org/wiki/Equation_of_time](https://en.wikipedia.org/wiki/Equation_of_time)).

~~~
jonsen
_... what makes stellar day real, ... ?_

It's not really real, just side-real.

~~~
unit91
Oh man this was the pun of the week. Well done.

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mbfg
almost like the person who thought up time keeping, thought about how it would
be most useful.

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arthurcolle
signs of The Architect

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SketchySeaBeast
The Architect was off in terms of the ACTUAL time it takes for the earth to
rotate - 86,164,100 milliseconds. What a "stellar" mistake for numerical
perfection. Did The Architect screw up the math?

~~~
vignesh_m
Our "original" definition of seconds was in term of our earth's rotation so it
is exactly OPs number. We just changed our definition to a more constant
second that slightly varies from our old definition.

~~~
SketchySeaBeast
But "milli" implies the metric system's version of a second, which is defined
as per the decay of a caesium-133 atom. I doubt very much The Architect wanted
us to mix our units of measure, unless The Architect was referring to that
window of time where "milli" existed, but we hadn't yet changed the second to
a uniform length of time (as opposed to one that would be affected by the
variable rotation of the earth). If so, we are already past His golden age.

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mike_ivanov
> exactly

Which day, specifically?

[https://en.wikipedia.org/wiki/Day_length_fluctuations](https://en.wikipedia.org/wiki/Day_length_fluctuations)

