
Don’t Fall for Babylonian Trigonometry Hype - robinhouston
https://blogs.scientificamerican.com/roots-of-unity/dont-fall-for-babylonian-trigonometry-hype/
======
svachalek
I did a little digging in curiosity about the Plimpton 322 story, and
discovered that essentially Wildberger was already a proponent of rational
trigonometry and then figured out that this tablet was based on it, which is
the opposite order the news stories led me to believe.

But anyway the aspect of it that has me curious is whether it benefits 3D
graphics or anything else that depends on lots of triangle computations. I
found a paper that concludes it's more easily computed, but it seems to make
some pretty simplistic assumptions (cost of sqrt = cost of sin, all algebraic
costs = 0).

~~~
jacobolus
Using base 60 makes no sense for computer graphics.

However, doing everything possible in terms of vectors and never using angle
measure unless interfacing with humans or legacy software is a generally good
idea.

If you want to represent a rotation use a complex number of unit magnitude. If
you want to represent a 3-dimensional rotation use a unit quaternion. If you
want to store one of these using only 1 or 3 parameters, respectively, take
the stereographic projection and optionally reduce the precision of your
floating point numbers afterward.

This doesn’t imply turning everything into Wildberger’s rational trigonometry
per se, but if you need metrical comparisons then you end up developing/using
some of the same machinery.

~~~
vog
_> doing everything possible in terms of vectors and never using angle measure
unless interfacing with humans or legacy software is a generally good idea_

More generally, use scalar products instead of angles. Calculations based on
scalar products are faster, numerically more accurate, simpler, and easier
chained. Convert from/to angles only for input/output.

Even more generally, always use sane representations internally, and press
cumbersome stuff to the borders of your system.

Examples:

\- You should use unicode strings internally, and decode/encode only on I/O.

\- You should use simple, native data structures internally, and deal with XML
representations only at the boundaries.

\- And so on.

------
anateus
This is a great article touching on the several ways in which this new
Plimpton 322 hype is overblown. There's usually a paper like that comes out
every few years :)

If you're interested in learning more about Babylonian math-history I highly
recommend Jens Høyrup's _Lengths, Widths, Surfaces_.

------
Houshalter
Wildberger is a recurring subject on /r/badmathematics including this recent
story. He's almost a crank and has some very very strange views on
mathematics. He strongly objects to the use of infinity and proofs and
mathematics that involve it, which includes real numbers.

If I understand correctly, he goes even farther and claims even really big but
finite numbers don't "exist". And through this he claims to have "resolved"
the Goldbach conjecture and other strange things.

~~~
bergoid
In mathematics education, I always disliked the way inifinity is shoehorned
into the different sets of numbers and treated as "just another number", so
that you can write shorthands like: 1/∞=0, instead of the more accurate ∀ε>0,
∃c>0: x>c ⇒ 1/x<ε

Infinity is not a number. It's a limit.

Treating it as a number only confuses real understanding of it.

~~~
jacobolus
It makes sense to treat ∞ ≡ 1:0 as a “number” in the context of projective
geometry, the Riemann sphere, etc.

The quarrel Wildberger has is not with the proportion 1:0, but with the
concept of a limit.

------
chroem-
This article seems to boil down to "stop liking what I don't like." Is
Wildberger's distaste for real numbers eccentric? Yes. Is there anything
fundamentally wrong with what he is proposing? Absolutely not.

As long as rational trigonometry is logically consistent then you really can't
argue against it except on aesthetic grounds. I don't see why popsci outlets
like Scientific American should feel the need to rally the public against
rational trigonometry on emotional grounds.

~~~
Lazare
I disagree.

Mansfeld and Wildberger made a number of trivially provable false claims. I
read the SciAm article as 1) focusing on their false claims and 2) not trying
to "rally the public" against rational trigonometry at all, much less on
emotional grounds.

In particular, Mansfeld and Wildberger (especially as reported in other
outlets) are making the claim that "rational trigonometry" is _more accurate_
, and that's a concrete claim with is 1) easily verifiable and 2) wrong.

And that's not even touching on the bizarre claim that "we count in base 10,
which only has two exact fractions: 1/2, which is 0.5, and 1/5." There's so
much wrong with that, it's hard to know where to start.

~~~
chroem-
>that's a concrete claim with is 1) easily verifiable and 2) wrong

I fail to see how it's wrong. If you call Math.sin(x) it will return an
iterative numerical approximation of sine. Would you care to elaborate rather
than casually dismissing it?

> There's so much wrong with that, it's hard to know where to start.

Such as? 60 has more prime factors than 10, and therefore using it as a
numeric base will result in you encountering fewer irrational numbers. You
really can't argue against that.

~~~
jdmichal
> Such as? 60 has more prime factors than 10, and therefore using it as a
> numeric base will result in you encountering fewer irrational numbers. You
> really can't argue against that.

Actually, I can argue with that. Irrationality is a property of the number,
not the representation. An irrational number cannot be written as a
terminating "decimal" in _any_ base. And, of course, a rational number can be
written as an integer fraction in _any_ base.

[https://math.stackexchange.com/a/625481](https://math.stackexchange.com/a/625481)

EDIT: Ignoring, of course, doing silly things like saying pi is rational in
base-pi... Which I think deserves a spot in some circle of mathematical
purgatory. Also, I wonder if every number _except_ pi and its multiples are
non-terminating in base-pi...

~~~
sillysaurus3
_silly things like saying pi is rational in base-pi... Which I think deserves
a spot in some circle of mathematical purgatory_

Hmmm...

Anybody know any resources related to that? I'm interested in the properties
of base-pi.

I found this:
[https://math.stackexchange.com/a/2188852](https://math.stackexchange.com/a/2188852)

Strange bases are pretty cool. Any integer can be converted into base-sqrt(2)
by interleaving zeoes into its binary representation. E.g. 5 in base-sqrt(2)
is 010001, 21 (0b10101) is 0b0100010001, etc. Possibly useless, but neat.

Also: [https://www.quora.com/Can-we-have-a-number-system-with-
the-b...](https://www.quora.com/Can-we-have-a-number-system-with-the-base-pi)

 _Knuth considers transcendental bases (ee is a transcendental number) in The
Art of Computer Programming – in the section called Positional Number Systems.
One of the consequences of using irrational bases is unfortunately that
numbers need not have a unique representation._

EDIT: Wow, this is awesome: [https://www.quora.com/What-is-10-in-base-
pi](https://www.quora.com/What-is-10-in-base-pi)

Specifically the answer from Dave Williamson. It's ridiculously comprehensive.
He shows how to calculate it using pure math, using a calculator, by writing a
program, and demonstrates a partial proof on some of the properties.

~~~
Someone
Neither mentions base π, but you may be interested in
[https://en.wikipedia.org/wiki/Non-
integer_representation](https://en.wikipedia.org/wiki/Non-
integer_representation), which even claims applications: _" There are
applications of β-expansions in coding theory (Kautz 1965) and models of
quasicrystals (Burdik et al. 1998)."_ and
[https://en.wikipedia.org/wiki/Quater-
imaginary_base](https://en.wikipedia.org/wiki/Quater-imaginary_base).

(Surprisingly, there seems to be _less_ content on this on mathworld)

------
arjie
Has there ever been a "ancient tablet unravels secrets undiscovered" that
contributed to science and not history? I think everyone can safely eliminate
this sort of thing in a Bayesian sense.

~~~
leoc
The discovery of Pāṇinian grammar by nineteenth-century European linguistic
scholars
[https://en.wikipedia.org/wiki/P%C4%81%E1%B9%87ini#Modern_lin...](https://en.wikipedia.org/wiki/P%C4%81%E1%B9%87ini#Modern_linguistics)
probably comes close, although of course Pāṇini was never lost or unknown in
India.

------
dfboyd
The story says George Plimpton bought the tablet in 1922. George Plimpton (I
assume it's the same one) was born in 1927. I admit he was awesome enough that
Jonathan Coulton wrote a song about him, but could he buy a tablet five years
before he was born?

~~~
david927
It's his grandfather:

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

------
akyu
Glad to see Wildberger getting some recognition, even if some of it is mixed.
His rational trigonometry is an entirely rigorous and interesting piece of
mathematics even if you don't think it's particularly pragmatic.

~~~
rocqua
How does he deal with the diagonal of the square? because that needs sqrt(2).

~~~
jacobolus
He bases his metrical geometry on “quadrance” (squared distance) and “spread”
(squared sine) rather than distance and angle measure.

~~~
gus_massa
Then, how does he deal with the the pentagon? because that needs
sqrt(5+sqrt(5)) or something similar with two nested square roots.

~~~
jacobolus
To deal with pentagons you need to extend the field of rationals by an
additional element. Wildberger is kind of ambivalent about finite field
extensions.

------
tree_of_item
> but seems like a solution to a problem that doesn’t exist.

I'm pretty sure the vast majority of mathematics could have been described
this way when it was first developed. Hilarious.

~~~
_kst_
Perhaps so, but in this case the solution is one that's already well known
(that you can express trig functions exactly when you're dealing with right
triangles with integer sides).

If you want to do some computation involving a given angle, you're not likely
to have the luxury of picking the angle that makes the computation easy using
rational arithmetic. The sine of 45° is an irrational number (1/sqrt(2)).

------
IshKebab
And stop calling it base 60. It's clearly a mixed base, call it base 6-10.

Calling it base 60 makes it sound like they have 60 different digits which
would be insane.

~~~
jacobolus
They clearly treated it as a base sixty floating point system, even though
they wrote each “sexagesimal digit” using collections of two symbols one of
which represented 10 times the other. The symbol for 10 was only ever used to
represent 10*60^n. Tens were never themselves the unit.

(Though the abstract sexagesimal system evolved out of earlier concrete
systems with less standardized mixtures of physical units.)

------
svat
Although the hype is overblown, there is nothing particularly wrong with the
article (except for the hype), which is why it was accepted into _Historia
Mathematica._

A good introduction to Old Babylonian mathematics (the broader context for
this paper) is in a paper by (surprise) Knuth, called _Ancient Babylonian
Algorithms._ (You can find PDF links here:
[https://scholar.google.com/scholar?cluster=10887370978433539...](https://scholar.google.com/scholar?cluster=10887370978433539328))
It's a very nice paper, and an example of Knuth's scholarly approach: despite
not being a historian in this paper he produced the first translation of many
old Babylonian tablets into English, by comparing the published German and
French translations and then looking up Akkadian and Sumerian dictionaries to
resolve differences! (Unfortunately he misread one of the secondary sources
and later published a retraction; ignore the first column on page 672, about
sorting (everything about Inakibit-Anu.)

Now to Plimpton 322: it has, for various triples (a, b, c) satisfying a^2 +
b^2 = c^2, columns containing a, c, and either a^2/b^2 or c^2/b^2. This has
been known for decades; what's been unclear is what this table was used for.
Some have proposed this was number-theoretic (just a listing of Pythagorean
triples), but this doesn't answer why these specific triples were chosen. Some
have proposed this was trigonometric, but the concept of an angle is not
otherwise known in Babylonian mathematics.

Enter Wildberger: for many years now he's championed "rational trigonometry"
(trigonometry using only rational numbers). So instead of the sine and cosine
he proposes using their squares; instead of angles there's something called
"spread", etc. It all works reasonably fine, is not significantly harder to do
than regular trigonometry (he claims it's easier, but it's about the same),
and you get to work with only rational numbers. (This is also related to a
broader project within mathematics which rejects infinite numbers -- see
finitism, ultrafinitism.)

So the Plimpton 322's fractions of the form a^2/b^2 are exactly like the
functions of "rational trigonometry", and the authors of this paper
([http://www.sciencedirect.com/science/article/pii/S0315086017...](http://www.sciencedirect.com/science/article/pii/S0315086017300691))
make the argument that these tables were used for such a purpose. In one of
the recent discussions, a commenter here pointed out this demonstration they
(the commenter) made:
[https://teacher.desmos.com/activitybuilder/custom/59a05b5f50...](https://teacher.desmos.com/activitybuilder/custom/59a05b5f50338d34a21e61c0#preview/8b58c98f-a047-46d1-8f53-d102993b11ad)

If you ignore the "more exact fractions" (which is true under a particular
interpretation, i.e. numbers that can be written as finite sequences of
integers in the "floating-point" representation), and "more useful than
current mathematics" hype from the newspapers, the paper itself is sound and
an argument about the nature of this table that is as plausible as any other
speculation. They address previous interpretations and discussions, and have a
coherent theory of how this table could have been generated or what it could
have been used for. It even addresses the objection that Babylonians didn't
have the concept of an angle: using exactly the objections raised against the
trigonometric interpretation to say they used rational-trigonmetry-like
functions instead. :-)

The paper is a good example of one aspect of the history of mathematics, the
way arguments are adduced for speculation on topics where there's little
evidence (have you read the rhetoric in Robson's paper?), and the fun of
constructing explanations out of little data. Here it is again:
[http://www.sciencedirect.com/science/article/pii/S0315086017...](http://www.sciencedirect.com/science/article/pii/S0315086017300691)

~~~
thejynxed
To think any civilization could build structures like the Babylonians did and
not have a concept of angles is patently absurd. From what architecture they
have that still stands, there is plenty of visual evidence that they indeed
calculated for angles in some fashion.

~~~
jacobolus
They clearly had a concept of similar triangles, and a concept similar to our
notion of “slope” or “tangent” (apparently they actually used the reciprocal
of slope, i.e. length per unit depth).

However, they may not have had the concept as of ~4000 years ago of “angle
measure” per se. You’d have to ask an expert in the history of astronomy when
that concept was first developed semi-formally; my understanding is that it
was 1000–1500 years later by subsequent Mesopotamian astronomers.

Angle measure (another word for circular arclength corresponding to a
particular rotation) is actually quite a tricky thing to get right. It’s the
logarithm of the rotation (a pure bivector quantity) with the orientation
stripped away – if we call our rotation _R_ oriented in the plane of the unit
bivector _i_ , it’s _θ_ = log( _R_ )/ _i_. To go from an angle measure back to
a “complex number” (scalar + bivector) with which you can do anything useful,
you need to put the orientation back in and then exponentiate it: _R_ = exp(
_iθ_ ). In dimensions higher than 2, you need to use rotation operators via a
kind of sandwich product, because composition of rotations is not commutative.

This is not an obvious concept, didn’t really get figured out properly until
the late 19th century, and still isn’t taught very clearly to students today.

------
k_sze
What if it's just a page of math homework from Babylonian times? And not a
special trig table at all?

What if?

~~~
jacobolus
That’s one of the previous interpretations. If you read this paper you’ll find
out why these authors think that interpretation is implausible.

------
HillaryBriss
the title of this post is exactly what my father said to me as a child, every
day before i went to school.

and do you think i listened? i could kick myself. i really could!

