Wikipedia:Reference desk/Archives/Mathematics/2017 September 1

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

September 1

Compute Minor axis of of an ellpise based on knowing a perimetier and the major axis...

(For context this came up when trying to draft a sewing pattern of all things. I'm trying to add a 'stirrup' an the end of a pattern for a pair of footless dance tights.)

I have a flat ellipse (major axis parallel to X axis). The major axis is known (b) as is the perimeter of the ellipse (p). I'd like to know the length of the minor axis (a) so I can compute p/2+2a. (so that I can know how much elastic to add as a stirrup.)

This doesn't need a perfect answer so a workable approximation would be appropriate.

My understanding is that there are various ellipse approximations I could use? ShakespeareFan00 (talk) 12:08, 1 September 2017 (UTC)

There are various approximate formulas, but I can't find one that is easy to rearrange. I would use a calculator such as this one, putting in the major axis and trying out different values for the minor axis until the perimeter is sufficiently accurate. This is not an elegant solution (or even accurate) but might be sufficient for your purpose. Dbfirs 13:04, 1 September 2017 (UTC)

... later ... this is a better calculator to use because it calculates using Ramanujan's formula. The one I linked above is less accurate. Dbfirs 13:21, 1 September 2017 (UTC)

Too bad that the calculator does not allow the perimeter as input! Bo Jacoby (talk) 16:26, 4 September 2017 (UTC).

Yes, I looked for one that did, but failed to find one. How about creating one? I suppose one could implement one of the better formulas in Excel, then use "goal seek". Dbfirs 16:40, 4 September 2017 (UTC)

Converting from one set of units to another without physical comparison?

Imagine that you've established communication with an alien colony residing roughly one light-year away from Earth. You decide to teach them about our system of units - grams and meters, for example. Only problem is there is no way that you can physically show them these units so that they can compare them to their own, the "grak" and "meker" (being so far away). Luckily, you learn that they do have water on their planet, so you tell them that one cubic meter of water has a mass of one kilogram. They also happen to measure the mass of water as being one kilograk per cubic meker, and so you conclude that both systems of units are the same. Of course it's actually a meaningless comparison. Just because the ratios are the same, that doesn't mean the units are as well. Is there any sort of solution to this kind of problem? — Preceding unsigned comment added by 73.232.241.1 (talk) 23:10, 1 September 2017 (UTC)

Sure. Why not compare the mass of a single molecule of water? Or a single atom carbon-12? See mole and the Avogadro constant. Alternately, our modern SI definitions of the second, "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom", and the meter, "length of the path travelled by light in a vacuum in 1/299 792 458 seconds", should work. The kilogram is still defined by artifact, but see kilogram#Proposed future definitions. -- ToE 00:26, 2 September 2017 (UTC)

A followup question for you: Suppose you somehow had instant communication with an intelligent alien civilization so far away that you could not compare local structures. (As in, look at that nebula over there.) How would you be able to agree on the concept of left and right? -- ToE 00:37, 2 September 2017 (UTC)

That works, but only if they have the technology to measure things on an atomic scale. Suppose they were limited to much larger scale measurements. Is there an algebraic method that could be used to work it all out under those circumstances?

Follow up question seems unsolvable. If their DNA (if they even have it) had the same chirality as ours, perhaps we could use that as a frame of reference. Of course, that's assuming they could even work out which is the "head" and "tail" of the molecule! (Not sure if that's so straightforward). Edit: not to mention the problem of determining if they even have the same chirality in the first place...— Preceding unsigned comment added by 73.232.241.1 (talk) 01:16, 2 September 2017 (UTC)

One light year from Earth would be in the Oort cloud. Any alien species which could colonize there would be fully capable of measuring the mass of a molecule. For that matter, any species capable of communication over such a long distance would be, too. StuRat (talk) 01:53, 2 September 2017 (UTC)

Heh, maybe so. I was just trying to frame a mathematical problem as a sort of silly hypothetical scenario. I wonder if there's some ingenious approach that could be used here. All we have is ratios...that's not enough information to convert the units. Something else would have to go into the equation, I just can't figure out what! — Preceding unsigned comment added by 73.232.241.1 (talk) 02:24, 2 September 2017 (UTC)

Please sign you posts using four tildes (~~~~). Also, please consider creating an account. (See WP:ACCOUNT.) There are often followup answers to reference desk questions after those questions have been archived, and if you have an account you can be notified of these new answers. I hope you are faring well there in Houston. Cheers! -- ToE 02:38, 2 September 2017 (UTC)

Prior to the mid-1950s, we would not have know how to communicate left and right, but we now have parity violation. See Relative direction#Geometry of the natural environment. -- ToE 02:59, 2 September 2017 (UTC)

Interesting, so it can be done! Also, from the article on the Wu experiment:

"The results of the Wu experiment provide a way to operationally define the notion of left and right. (...) Previously, if the scientists on Earth communicate with a newly discovered planet’s scientist, and they have never met in person, it’s not possible for each group to determine unambiguously the other group’s left and right."

Funny, how ironic that I would stumble on that analogy. :) 73.232.241.1 (talk) 03:18, 2 September 2017 (UTC)

It's an interesting way to put it, but doesn't it miss something kind of obvious? Just send the message itself using a circularly polarized EM wave, and refer to the polarization. It's still a good point in principle (you could save it, for example, by supposing that there is a communication channel that sends bits only and for which you can't control anything else besides the bits sent), but it doesn't seem to be necessary for solving the problem as literally stated. --Trovatore (talk) 00:05, 3 September 2017 (UTC)

The problem stated: "Suppose you somehow had instant communication with an intelligent alien civilization so far away that you could not compare local structures". That doesn't sound compatible with EM waves. PrimeHunter (talk) 23:59, 7 September 2017 (UTC)

If you told them that a cubic meter of water had a mass of one kilogram, they would think that water on earth wasn't very dense. — Preceding unsigned comment added by 82.38.221.49 (talk) 09:17, 2 September 2017 (UTC)

Oh, right! Yeah I wouldn't last very long as Intergalactic Ambassador of Science, would I? Okay, so one kilogram per cubic decimeter... 73.232.241.1 (talk) 10:47, 2 September 2017 (UTC)

No, there isn't a purely mathematical method to go from an agreed upon reference density to a common unit of mass or length. You don't have enough information to determine a reference mass or reference volume (or length) from only knowing the ratio. A related article is our Dimensional analysis.

Excluding atomic physics, what else can be done working from properties of matter? With a common reference time (as you would likely have if you are communicating by radio), it would be easy. The triple point of water gives you a reference temperature and pressure, and you could then agree on a reference velocity as the speed of sound in some specified medium at a give temperature and pressure. Since you started with a reference time, you now have distance.

But what if you didn't even have an agreed upon reference time? Well, what do you have? From the physical properties of water, you are starting with a reference temperature (in K), a reference density (in kg/m³), a reference pressure (in PA or N/m² or kg/m/s²), and a reference speed (m/s), but I don't see how to isolate an individual unit from those references. For instance, dividing pressure by density gives you velocity squared (in m²/s²), and you could take the square root of that to get speed, which you already had. You still haven't isolated a base quantity. Tensile strength again gives you pressure. Specific strength (in pressure divided by density -- speed squared, again) is sometimes given in terms of breaking length or self support length, a distance, but that depends on an agreed upon local gravitational acceleration. (Or even just an agreed upon unit of time. "The maximum length of a rod of pure copper which can withstand (absent other forces) the reactive centrifugal force which would be exerted if it were spun at 1 Hz." And you don't have to do that actual experiment as if can be computed from the specific strength. It's not like anyone has ever actually attempted to suspend a 109 km length of spider silk!) If only we had a reference acceleration!

You may not like this any better than the atomic physics solution, but in 1798 Cavendish effectively determined the universal gravitational constant to within 1.2%, and G has units of m³/kg/s². Multiply that by density, invert, take the square root, and you have time. Voilà! -- ToE 15:44, 2 September 2017 (UTC)

I think using the universal gravitational constant as the missing link is perfectly cromulent. If the aliens know their astrophysics, referencing black holes can be handy. E.g., the minimum mass of a star that would collapse into a black hole gives you a unit of mass. -- Meni Rosenfeld (talk) 19:55, 2 September 2017 (UTC)

Thanks ToE. Yeah, I was pretty sure that was the case, just hoping in vain that I might be wrong! But yes, I suppose using atomic-scale measurements as you suggest to derive a suitable equation to work out the individual dimensions might be the only way to do this. 73.232.241.1 (talk) 23:57, 2 September 2017 (UTC)

But we've established that atomic measurements are not necessary, you can use the gravitational constant, measured with something like a Cavendish scale. -- Meni Rosenfeld (talk) 13:43, 3 September 2017 (UTC)

Sorry, I hadn't read how the experiment was actually conducted. Unbelievable that he was (effectively) able to determine G with such accuracy! Not exactly a trivial setup, but certainly proof that it can be done with less-than-advanced technology. 73.232.241.1 (talk) 18:20, 3 September 2017 (UTC)

See physical constant for a whole list of universal constants. The speed of light in a vacuum, for example, would be critical for them to know if communicating over light years. StuRat (talk) 19:29, 3 September 2017 (UTC)