For the Mathematicians

[Berzerker]

https://phys.org/news/2017-08-mathematical-mystery-ancient-babylonian-clay.html

Trig just got much older...and apparently simpler

And the guy who found the tablet was the model for Indiana Jones

"Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles. It is a fascinating mathematical work that demonstrates undoubted genius.

"The tablet not only contains the world's oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.

"This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3000 years, but it has possible practical applications in surveying, computer graphics and education.

"This is a rare example of the ancient world teaching us something new," he says.

 

[warpus]

This is the important part here:

"The tablet not only contains the world's oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.

When my mom studied math in school trig tables were at the back of the book as a reference. Something like this except not quite so old:

Spoiler :

As you can see it is not 100% accurate. For the purposes of students and even engineers this was more than enough though so that's what they printed.

It seems that the Babylonians built their mathematical models in a way that lead to a way of writing trig tables using 100% exact values. For example the number 7 is 100% exact. My own terminology. Or 7/2. With trig tables you can't do that, or at least you can't do that in our mathematical system, but apparently you can in the Babylonian one.

This doesn't mean that their way of doing math was better, it just means that in this case trig function could be written in a more.. elegant way. Which is better, but you have to weigh their whole mathematical model as a whole and this is just one part of it. Either way though pretty cool, and I wonder what else they figured out that we don't know about still/yet

 

[Kyriakos]

Hm, i read the article and have some questions.

I gather this is not a theorem (proof) of the pythagorean a^2+b^2=c^2, and the tablet had been theorized in the past to just have some such pythagorean triplets, albeit large numbers. And it uses a base60 numbering system.
But: given there was no theorem on this in Babylonian times, what is the new discovery/importance about? The article mentions that the approach seems to be about ratios (of what?) instead of "angles or circles", but again where is the theorem allowing for this to be used? Afaik the first theorem (which was a theorem of ratios moreover...) is attributed to Thales, and is the one about ratios of parts of sides of triangles when created by the same line cutting them all. So where is the babylonian math basis for using such ratios? (and if it isn't using those ratios, what is it using, and with what proof? cause otherwise it was already known for a long time that this tablet had pythagorean triplets ).

 

[onejayhawk]

I took statewide prizes in Trig. It was fun.

J

 

[Manfred Belheim]

You're going off on a tangent.

 

[Ferocitus]

Hm, i read the article and have some questions.

I gather this is not a theorem (proof) of the pythagorean a^2+b^2=c^2, and the tablet had been theorized in the past to just have some such pythagorean triplets, albeit large numbers. And it uses a base60 numbering system.
But: given there was no theorem on this in Babylonian times, what is the new discovery/importance about? The article mentions that the approach seems to be about ratios (of what?) instead of "angles or circles", but again where is the theorem allowing for this to be used? Afaik the first theorem (which was a theorem of ratios moreover...) is attributed to Thales, and is the one about ratios of parts of sides of triangles when created by the same line cutting them all. So where is the babylonian math basis for using such ratios? (and if it isn't using those ratios, what is it using, and with what proof? cause otherwise it was already known for a long time that this tablet had pythagorean triplets ).

Yes, you are correct that there are no Babylonian "proofs" in the same way that Thales
or Hippocrates or others used. (By the way, Wildberger is not trying to diminish the
achievements of early Greek mathematicians. He is a great supporter of the early Greek
geometric way of thinking about many mathematical topics.)

They address more than just the Plimpton tablet. There are others as well that are
of interest. See, for example, their series of videos on Babylonian Mathematics and
the Plimpton Tablet.

It is of interest also that there are algorithmic methods described in the tablets, and
certain solutions for quadratic equations that were thought to have been "discovered"
by Arabic scholars.

Norm Wildberger has many youtube videos on "Rational Trigonometry" which is an
approach that uses the notion of "quadrances" (the squares of lengths) and "spread" in
place of angles. It is a way to avoid many (foundational) difficulties to do with square
roots and other "incomplete" infinities.
https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ

Those foundational difficulties have not disappeared with advances in mathematics
over the last century or two. They are usually either not mentioned, or hand-waved
away in undergraduate courses on real analysis and calculus.

 

[BenitoChavez]

You're going off on a tangent.

Don't be so hyperbolic

 

[warpus]

what is the new discovery/importance about?

From what I understand the discovery is pretty much that the Babylonians figured out ways to express certain trig table values as ratios. It's interesting to mathematicians because it's a different way of looking at the same thing.

 

[Manfred Belheim]

Don't be so hyperbolic

Sorry it was just a reflex.

 

[Kyriakos]

From what I understand the discovery is pretty much that the Babylonians figured out ways to express certain trig table values as ratios. It's interesting to mathematicians because it's a different way of looking at the same thing.

Any info on what sort of ratios those are? Sounds algebraic (?)

As others noted, ancient greek math often was using geometry to work with algebra-like patterns and solutions, as in using surface area of differing triangles, circles or rectangles. Archimedes used a more refined/hybrid of this (along with his lever calculations) to approximate pi (surface area of triangle tied to circle ever more approximated, iirc).

JB line there is the so-called "lever".

 

[onejayhawk]

You're going off on a tangent.

I could offer to cosine the draft copy, but that's circular logic.

J

 

[Ferocitus]

Any info on what sort of ratios those are? Sounds algebraic (?)

No, it is definitely based far more on geometry than algebra.

Distances and quadrances both measure separation in (Euclidean) space.
Quadrances are just the squares of the lengths of, for example, the sides of a triangle,.
"Spread" is the square of the sine of an angle.

Their ratios, sums, and products can be used to derive exactly the same
geometric relationships as using ordinary lengths and angles.

https://en.wikipedia.org/wiki/Rational_trigonometry#Quadrance

 

[Ferocitus]

Archimedes used a more refined/hybrid of this (along with his lever calculations) to approximate pi (surface area of triangle tied to circle ever more approximated, iirc).

Archimedes used polygons to approximate a circles. He used one polygon inside the
circle, and one (with the same number of sides) outside the circle.
Then, by increasing the number of sides (he used up to 96 sided polygons), he
constructed a sequence that led to his famous approximation: 223/71 < pi < 22/7.
Chinese mathematicians of the same time used similar techniques but, with
a couple of simple extra steps, they were able to get much better approximations.

 

[Kyriakos]

Hey, i can't follow up on what an actual mathematician says I can just note that Archimedes expanded on the use of polygons (ever increasing number of sides) approximating the periphery of a circle, first (apparently; if we go by Aristotle) proposed by some student of Plato - don't recall the name now, though, but he did use just the one polygon, from inside the circle- inscribed- increasing number of sides, whereas Archimedes used the one outside the circle as well).
A basis of all those approaches was a fundamental work on turning some (not all, of course) curved surfaces tied to circles, into rectangular surfaces. Again memory fails as to the name of the mathematician presenting that for the first time (it has been too long since i dealt with all that for some philosophy program i had been presenting!).

 

[Ferocitus]

Hey, i can't follow up on what an actual mathematician says I can just note that Archimedes expanded on the use of polygons (ever increasing number of sides) approximating the periphery of a circle, first (apparently; if we go by Aristotle) proposed by some student of Plato - don't recall the name now, though, but he did use just the one polygon, from inside the circle- inscribed- increasing number of sides, whereas Archimedes used the one outside the circle as well).

That's right. It was a very clever trick.
As I said earlier, Chinese mathematicians used some extra steps
that greatly improved the approximation. They used 96-sided polygons
as did Archimedes.

 

[uppi]

Hm, i read the article and have some questions.

I gather this is not a theorem (proof) of the pythagorean a^2+b^2=c^2, and the tablet had been theorized in the past to just have some such pythagorean triplets, albeit large numbers. And it uses a base60 numbering system.
But: given there was no theorem on this in Babylonian times, what is the new discovery/importance about? The article mentions that the approach seems to be about ratios (of what?) instead of "angles or circles", but again where is the theorem allowing for this to be used? Afaik the first theorem (which was a theorem of ratios moreover...) is attributed to Thales, and is the one about ratios of parts of sides of triangles when created by the same line cutting them all. So where is the babylonian math basis for using such ratios? (and if it isn't using those ratios, what is it using, and with what proof? cause otherwise it was already known for a long time that this tablet had pythagorean triplets ).

Maybe they used the physicists' approach to mathematics: As long as it works, we don't care whether there is any proof or whether it is actually correct

As far as I understand it, the idea is to only ever work in the space where all distances are squared and you never take the square root. So instead of talking about distance a, you are talking about the distance A = a^2. Then you can also express angles not like in Euclidean geometry, but as the ratio of the sides in a rectangular triangle, i.e., the square of the sine.
So instead of saying sin(z) = a / b, you say Z = A / B, where you take Z = (sin(z))^2 as a measure for the angle. This makes trigonometric calculations very simple: If you want to calculate B in a triangle, you could just do B = A*Z instead of b = a*sin(z). That way, you avoid having to calculate the sine and for Z you can just take rational numbers, for example 1/4 for a 30 degrees or 3/4 for 60 degrees. This means, that you do not need any irrational numbers (like sqrt(2)), which the Greeks had to struggle with in Euclidean geometry.

With this approach, the Pythagorean theorem simplifies to A + B = C. Because of that simplicity, they might not have been aware of it, because the way the system is constructed, it is trivial (essentially built-in).

If you just to trigonometry with it, the system seem simple and easy to use, but it has its drawbacks: All the values are not additive any more, so if you go the same distance two times it is not twice the distance (but 4 times). As a consequence, the triangle inequality is violated, so going first from A to B and then to C is "shorter" than going directly from A to C. In fact, this violates a lot of our intuitive understanding of distance (which the Greeks have codified in Euclidean geometry) that it should not be called distance anymore (so the modern formulation of it calls it quadrance). It is an elegant formulation of trigonometry, as long as you stay within it. But as soon as you leave trigonometry, it becomes a burden.

And I have to say, I am a bit skeptical that it is correct to infer that the Babylonians understood all this from some clay tablet.

 

[Ferocitus]

As a consequence, the triangle inequality is violated,

The triangle inequality using quadrances (squares of distances) is:
( Q3 − Q1 − Q2 )^2 ≤ 4 * Q1 * Q2.

And I have to say, I am a bit skeptical that it is correct to infer that the Babylonians understood all this from some clay tablet.

Although they focus on Plimpton 322, it is not just the content of one tablet
that shows the Old Babs used trig tables and other concepts far earlier than
previously believed.
There are other tablets that show, for example, geometric figures with
estimates of the numerical value of the square root of two.

What their "understanding" was is anyone's guess.

 

[uppi]

The triangle inequality using quadrances (squares of distances) is:
( Q3 − Q1 − Q2 )^2 ≤ 4 * Q1 * Q2.

That seems to be the triangle inequality for distances expressed in quadrances. This is something different than the (invalid) triangle inequality for quadrances. That means you cannot directly use quadrances as a metric and lose all the proofs that were made for metrics. You would have two options then: either find a metric that can be expressed in quadrances (and add a layer of complexity) or define a new metric-like concept for quadrances. I suspect that a try at the latter might end up with a less powerful concept that only goes so far - but you would need to try to find out. But even with a less powerful concept, there I wonder whether there might be problems with a lot of trigonometric calculations for which it would be sufficient and which might be more efficiently solved with rational trigonometry.

 

[Ferocitus]

That seems to be the triangle inequality for distances expressed in quadrances. This is something different than the (invalid) triangle inequality for quadrances. That means you cannot directly use quadrances as a metric and lose all the proofs that were made for metrics. You would have two options then: either find a metric that can be expressed in quadrances (and add a layer of complexity) or define a new metric-like concept for quadrances. I suspect that a try at the latter might end up with a less powerful concept that only goes so far - but you would need to try to find out. But even with a less powerful concept, there I wonder whether there might be problems with a lot of trigonometric calculations for which it would be sufficient and which might be more efficiently solved with rational trigonometry.

There are many calculations that can be simplified. Others, of course, might
not be as easy.

Wildberger has pushed the idea of rational trigonometry in many directions,
from hyperbolic geometry to algebraic topology.
He has dozens of youtube videos on the topic, as well as many that are not
devoted to rational trig.

https://www.youtube.com/user/njwildberger

I like that he is challenging mathematical foundations. Although some people try
to dismiss those difficulties out of hand, he does have some very distinguished
supporters, e.g. Fields Medal winner Vladimir Voevodsky.

Those difficulties don't interest me as such: I'm an applied mathematician and
I have your physicist's attitude - if it works, that's enough!

 

[Birdjaguar]

The Sumerians invented counting and writing so it is reasonable to expect their Babylonian descendants to be the inventors of other related things like Trig.