Wikipedia:Reference desk/Archives/Science/2015 January 1

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January 1

Prove the Square-Cube Law

I noticed that in the wikipedia entry for this law (Square-cube law), no proof of the law is given.

I would like a proof that the ratio of the areas and volumes of 2 similar figures is the square and cube of their scale factor respectively.

Thanks. 175.156.52.140 (talk) 00:06, 1 January 2015 (UTC)

I think you can just prove that visually. Let's do area here, were we can show that the area of squares 1, 2, 3, and 4 units wide are 1, 4, 9, and 16:

*

**
**

***
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***

****
****
****
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Therefore, it varies with the square of the side length. Feel free to try any other shape besides a square. For volume, that's a bit harder to show here, but you can draw some cubes and count the number of cubic units in each. StuRat (talk) 00:26, 1 January 2015 (UTC)

What sort of "proof" of the square-cube law is the OP looking for? It's like asking for a proof of addition based on multiplication. One doesn't ask for proof of the obvious based on the obscure. μηδείς (talk) 01:54, 1 January 2015 (UTC)

Your point is perfectly reasonable, but actually the OP is correct. Mathematicians do ask for (indeed demand) proofs of the (intuitively) obvious, based on the (well-established but) obscure knowledge they possess. This is certainly one of those things, as it is intuitively clear enough that every volume must be something cubed, and every 2-d thing must be something squared, but it is a mathematical fact, and this needs to be demonstrated. IBE (talk) 06:32, 1 January 2015 (UTC)

It's been a long time since elementary geometry. Is this "law" actually a law or it is a theorem? If it's actually a law, there is no proof involved - it's taken as a given. If it's a theorem, there should be a proof somewhere. ←Baseball Bugs ^{What's up, Doc?} carrots→ 02:50, 1 January 2015 (UTC)

Actually that's a good point. Is the Square-Cube Law a mathematical theorem that can be proven rigorously with algebra and calculus (and not just purely with geometrical arguments)? Or is it a law/theory which is based on empirical evidence? If it is indeed a theorem, I haven't been able to find a formal proof anywhere by googling, hence why I'm asking the wikipedia helpdesk. — Preceding unsigned comment added by 175.156.52.140 (talk) 19:44, 1 January 2015 (UTC)

I do most of my higher math intuitively based on shapes, but it would seem trying to prove that the surface area of a cube increases by a factor of 16 when the length doubles and the volume increases by 8 is just obvious by definition. The math desk would be the place to ask for a proof proof. μηδείς (talk) 04:46, 1 January 2015 (UTC)

I assume you meant the surface area increases by a factor of 4? Surely a typo, IBE (talk) 06:35, 1 January 2015 (UTC)

Yes, that was wrong, but I meant to say the surface area of a 2x2x2 cube would increase to 24 from 6, which is a factor of four, then made the obvious guess that 24 minus eight (for some reason) was 16.

It's straight forward, and in the article, that a surface area is a square law and the volume is cubic. The proof that it is monotonic is the derivative rate of change is linear for surface area while a square law for volume. The volume will always expand faster than the surface area. --DHeyward (talk) 04:53, 1 January 2015 (UTC)

I think they want the 3-D case of one of the three 2-D proofs in [1]. An easier way might be to build a little on [2]. But honestly, how can this possibly be anything but homework? 63.228.180.122 (talk) 06:18, 1 January 2015 (UTC)

I can't say why it couldn't be homework, but I don't see why it would be either. It's a perfectly reasonable question here, but it would be a weird one to set a student. Stranger things have happened though - my chemistry teacher had to answer an exam question, "Why do you believe in the atom?" IBE (talk) 06:32, 1 January 2015 (UTC)

• So long as it's for your own interest, and so long as you know this isn't very rigorous, proceed as follows. For surface area, draw a bunch of squares of length L on the figure. Discard any leftover bits, and the area is (the number of squares) X (the area of one square). This is an approximation, because we discarded some stuff, and the "squares" aren't really square at all, if your shape is a sphere. So these "squares" are really "curvy squares". Then proceed in a sequence, with squares of size L/2, L/4, L/8 etc. Assuming the shape possesses finite curvature (ie it is not a fractal), the area of the stuff you discard will shrink to zero, and the "curvy squares" get flatter as you proceed. So it gets closer and closer to just a bunch of squares, and it approximates the true surface area. In the limit at infinity, it is exact. I call this a PGP, or "Pretty Good Proof". Proceed similarly for volumes, using cubes not squares. IBE (talk) 06:47, 1 January 2015 (UTC)

The law isn't actually correct for fractals, but then again a fractal surface would be infinite for a finite volume. Many biological structures like for instance the lungs approximate fractals to exploit this feature and scale differently within a wide range. Dmcq (talk) 12:44, 1 January 2015 (UTC)

This is a good question, and one that cannot be formally answered with elementary mathematics because the general concepts of (planar) area and of volume, simple as they are for simple geometric figures, are more complicated than they seem. The concepts are formalized by measure theory, which wasn't well developed until about one hundred years ago and is a subject which, in most universities in the United States (where a typical bachelor's degree takes four years of study), is taught to third year students in a class called Real Analysis for mathematics majors and somewhat less formally in a class called Advanced Calculus for engineering majors, and is taken only after having completed three to four semesters of regular calculus. (Physics majors often have the option of choosing between the two classes.)

Students learn in theses classes that Real numbers are stranger than they first seem, and that, just as it is possible to construct pathological functions which behave in strange ways, there are subsets of the Euclidean spaces ℝ² and ℝ³ which either don't have well defined areas or volumes, or whose areas or volumes behave unexpectedly. See Banach–Tarski paradox for an example.

The formal definition of (planar) area is touched on at Area#Formal definition and a similar definition holds for volume. Note that most of this describes the general behavior that the area (or volume) concept must obey, but that it also includes the stipulation that the area (or volume) of a rectangle (or rectangular cuboid) is what we expect; that is, a rectangle of dimensions a and b has area ab and a rectangular cuboid of dimensions a, b and c has volume abc.

Clearly the area of two similar rectangles varies by the square of their scaling factor and the volume of two similar rectangular cuboids varies by the cube of their scaling factor. Any proof that a particular subset P of ℝ³ has a given volume will use constructions which can eventually be traced back to rectangular cuboids, and so a parallel proof for the volume of a subset Q which is similar to S but scaled by a factor of s can be made using rectangular cuboids similar to those used in demonstrating the volume of S, scaled by the same factor s. Since the volume of those rectangular cuboids varies with s³, so will the volume of the subsets P and Q.

Formal treatment of surface area adds some additional complications (Surface area#Definition touches on them) but in the end the measure of surface area is related to the measure of planar area, which is itself based on the area of rectangles. And since the area of a similar rectangles varies with the square of the scaling factor, so does the surface area of similar three dimensional objects. (Where the surface area is defined, that is. It is possible to construct subsets of ℝ³ which have well defined and finite volumes but which have undefined or infinite surface area.) -- ToE 14:13, 1 January 2015 (UTC)

@ToE So, are you saying that the Square-Cube Law is a mathematical theorem that can be proven? If so, I would like to see the formal proof. Thanks. — Preceding unsigned comment added by 175.156.52.140 (talk) 20:07, 1 January 2015 (UTC)

Yes, the scaling can be proven, and I'll see what I can do for you tonight. For now, let me say that I overstated the difficulties above. I should not have given the impression that showing the scaling is hard; it isn't. It's just that volume and area (and surface area) can be defined at many different levels of rigorousness and generality, and the more advanced definitions of those concepts may be difficult to follow. What I said about measure theory is true, but it really just served to throw a red herring into the discussion. The scaling can be shown in whatever form and level that the volume and area concepts are defined. A calculus based answer should satisfy you as it would work for any object that an engineer would imagine or model. -- ToE 15:53, 2 January 2015 (UTC)

We can look at the scaling question separately with every method we use to define volume and surface area.

• Explicit formulas for common solids can be checked to verify that those given in Volume#Volume formulas vary with the cube of the linear scale and those given in Surface area#Common formulas vary with the square of the linear scale.
• Any definition for volume and area, however abstract, will eventually stipulate that the volume of a rectangular cuboid of dimensions a x b x c is abc and the area of a rectangle of dimensions a x b is ab, thus matching our expectations for simple figures. This volume (and area) scales with cube (and square) of the linear scale. The surface area of the rectangular cuboid is 2ab + 2bc + 2ac, which scales with the square of the linear scale. Thus the rectangular cuboid obeys the Square-Cube Law, but we don't use that fact directly.

We can determine the volume of solids via successive approximation by filling the solid with many, small, non-overlapping cuboids, the sum of whose volumes give a lower bound for the volume of the solid. As the cuboids are made smaller, closer approximations can be found. We also make successive approximations with a collection of cuboids which together fully contain the solid to get upper bounds for the solid's volume. If the limit of the upper and lower bounds (as the cuboids become smaller and smaller) approach the same value, then we say that the volume of that solid is defined and is equal to that limiting value. This is in essence the calculus method for determining volume. The volume of a scaled version of the solid will vary with the cube of the linear scaling because that is how the rectangular cuboids behave, and the new solid can be approximate with a scaled version of the cuboids which approximated the initial solid.

You might be tempted to do the a similar thing for determining surface area by calculating the exposed surface area of the collection of small cuboids approximating the solid from inside and from outside (ignoring that part of the unexposed surface area of any cuboid which is touching another cuboid), and doing so will give you a valid limit (for well-behaved solids) which does scale with the square of the linear scale, but this is not the surface area of the solid because the small faces of the cuboids are not parallel to the surface of the solid they are approximating. For a sphere this would give a value of 6πr² instead of the correct value of 4πr². Surface area should be rotationally invariant, but this calculation is not, with a unit cube having a value ranging from its actual surface area up to twice that figure, depending on how it is rotated with respect to the axes of the cuboids. For a more subtle problem with calculating the surface area via the limit of successive approximations, see Schwarz's paradox (or, until that article is created, see Surface Area and the Cylinder Area Paradox), where the area of an open cylinder is approximated by a series of small triangles whose vertices are on the surface of the cylinder, but the limit of their combined area as the triangles are shrunk in both height and width can be any value between the actual surface area of the cylinder and infinity, depending on how the limit is taken. The lesson here is that surface area is a more complicated concept and it must be dealt with rigorously.

• All is straight forward for a polyhedron, as its volume can be subdivided into a finite number of tetrahedrons and its faces can be subdivided into a finite number of triangles, with the volume and surface area of the polyhedron being the sum of the volumes of tetrahedrons and the sum of the areas of the triangles. A scaled version of the polyhedron can be subdivided into scaled versions of the original tetrahedrons and triangles, and since their volumes and areas scale with the cube and square of the scaling factor, so does the total volume and surface are of the polyhedron.

• Learning to calculate the volume and surface areas of more complicated solids is a large part of a class in Calculus and Analytic Geometry. Volume is done as Dragons flight showed below, but in case you are not comfortable with volume integrals, I will use a characteristic function (or more specifically, an indicator function) and integrate over all space in order to make the change of variable more obvious.

In order to determine the volume of U ∈ ℝ, define the indicator function 1_U as:

$\mathbf {1} _{U}(x,y,z):={\begin{cases}1&{\text{if }}(x,y,z)\in U,\\0&{\text{if }}(x,y,z)\notin U.\end{cases}}$

Then the volume of U is:

$v(U)=\iiint {1}_{U}(x,y,z)\,dx\,dy\,dz$

as long as that integral exists. Note that this integrates over all space, summing up volume of space when inside U and doing nothing outside of U.

Define the scaled up version of U as αU := {(x,y,z) | (x/α,y/α,z/α) ∈ U}. The indicator function 1_αU(x,y,z) = 1_U(x/α,y/α,z/α) because a point is in αU if and only if the scaled back position of that point is in U. So the volume of αU is

$v(\alpha U)=\iiint {1}_{\alpha U}(x,y,z)\,dx\,dy\,dz$

$=\iiint {1}_{U}(x/\alpha ,y/\alpha ,z/\alpha )\,dx\,dy\,dz$

This suggests the variable substitution (u,v,w) = (x/α,y/α,z/α), which give us dx = αdu, and so on. Thus:

$v(\alpha U)=\iiint {1}_{U}(u,v,w)\,\alpha du\,\alpha dv\,\alpha dw$

$={\alpha }^{3}\iiint {1}_{U}(u,v,w)\,du\,dv\,dw={\alpha }^{3}v(U)$.

So there we have volumes calculate via calculus scaling with the cube of the linear scaling factor.

Surface area is a bit more involved. Dragons flight showed the surface integral directly, but I will give a more explicit calculation which makes the change in variable more obvious.

We can calculate the area of a piecewise smooth surface by breaking it into finitely many smooth pieces which can be defined via a height function over planar support. For a given piece S, let z = f(x,y) for (x,y) ∈ T be the height of S above the x-y plane (with your coordinate axes determined to best work with this piece), where T is the area of the x-y plane under S. We learn in calculus that the surface area of S is

$A(S)=\iint \limits _{T}{\sqrt {\left(f_{x}(x,y)\right)^{2}+\left(f_{y}(x,y)\right)^{2}+1}}\,\,dx\,dy$

f_x is the partial derivative of f with respect to x, and it gives the slope of the surface (above the point it is evaluated at) with respect to a line parallel to the x-axis. If a small (essentially flat) part of the the surface over a small area of the x-y plane is parallel to the plane, then its area is equal to the small area on the plane, but if it is tilted, then its area a greater by a factor of √(f_x² + f_y² + 1). Note that this is equal to 1 when f_x = f_y = 0, is equal to √2 when f_x = 1 and f_y = 0 (or vice versa), and can become very large when either or both of f_x and f_y become large. The usual way this is derived is by stating that the small surface directly over dxdy is a parallelogram defined by the two vectors dx(1, 0, f_x) and dy(0, 1, f_y) whose cross product is dxdy(-f_x, -f_y, 1), and the area of that parallel is the magnitude of the cross product which is dxdy√(f_x² + f_y² + 1), our integrand.

Now consider our scaled up piece αS of the scaled up surface, with the region under αS being αT := {(x,y) | (x/α,y/α) ∈ T} and a height function z = g(x,y) := αf(x/α,y/α). The chain rule gives us g_x(x,y) = (∂/∂x)αf(x/α,y/α) = αf_x/α(x/α,y/α)(1/α) = f_x/α(x/α,y/α), and similarly g_y(x,y) = f_y/α(x/α,y/α), which makes sense because scaling hasn't changed the slope of congruent points of the surface. So, the surface area of αS is

$A(\alpha S)=\iint \limits _{\alpha T}{\sqrt {\left(g_{x}(x,y)\right)^{2}+\left(g_{y}(x,y)\right)^{2}+1}}\,\,dx\,dy$

$=\iint \limits _{\alpha T}{\sqrt {\left(f_{x/\alpha }(x/\alpha ,y/\alpha )\right)^{2}+\left(f_{y/\alpha }(x/\alpha ,y/\alpha )\right)^{2}+1}}\,\,dx\,dy$

which suggests the substitution (u,v) = (x/α,y/α), which give us dx = αdu, dy = αdv, and a contraction of the region of integration from αT to T, giving us

$A(\alpha S)=\iint \limits _{T}{\sqrt {\left(f_{u}(u,v)\right)^{2}+\left(f_{v}(u,v)\right)^{2}+1}}\,\,\alpha du\,\alpha dv$

$=\alpha ^{2}\iint \limits _{T}{\sqrt {\left(f_{u}(u,v)\right)^{2}+\left(f_{v}(u,v)\right)^{2}+1}}\,\,du\,dv=\alpha ^{2}A(S)$

And if the individual pieces of the surface scale with the square of the linear scale, then so does the entire surface.

I'm not sure that this gives you any more than Dragon flights answer, but I thought that offering the integrals with some innards might make them seem less like magic. -- ToE 02:33, 4 January 2015 (UTC)

If you are willing to accept calculus as a starting point then, the argument is basically some version of the following. Given a compact three dimensional object U, the volume of U in Euclidean space can be represented as:

$V=\iiint \limits _{U}1\,dx\,dy\,dz$

Further, let $\alpha \,U$ be such that $\forall (x,y,z)\in U\rightarrow (u,v,w)\equiv ({\alpha \,x},{\alpha \,y},{\alpha \,z})\in \alpha \,U$

Then the volume of $\alpha \,U$ is

$\iiint \limits _{\alpha \,U}1\,du\,dv\,dw=\iiint \limits _{U}1\,(\alpha dx)\,(\alpha dy)\,(\alpha dz)$

$=\alpha ^{3}\,\iiint \limits _{U}1\,dx\,dy\,dz=\alpha ^{3}\,V$

A similar argument assuming the area of a surface S can be described as $A=\iint \limits _{S}1\,dx\,dy$ implies that the area of $\alpha \,S$ scales as $\alpha ^{2}$

That's the gist of the argument anyway. If you want to be really detailed, one needs to dive into measure theory, Lebesgue integration, the notion of metrics, and what the hell "area" and "volume" really mean. These arguments can be really complicated if you want to be completely general, but for typical uses, an argument like the one sketched above should make the point well enough. Dragons flight (talk) 23:46, 1 January 2015 (UTC)

Thank you to ToE and Dragons flight for all the explanation and working.
A calculus-based proof definitely makes sense and is intuitive enough to understand (for someone who knows a bit of undergraduate-level calculus) so that even if one can't rigorously prove or fully-understand the formal proof of the Square-Cube Law/Theorem, one can at least see how it came about without merely only memorising how to use/apply it.

I'm afraid that the calculus based proof also hides the assumptions behind a mass of symbols. The law is not always true, it is only true with the assumptions. You have to have the right definition of area and volume. If you allow fractals then the area can go infinite for a finite volume. Basically one assumes that almost everywhere the surface looks like the plane if sufficiently magnified and small cubes are either completely filled or completely empty. Dmcq (talk) 18:36, 5 January 2015 (UTC)

Can you give an example in Euclidean space of the scaling law being false? A fractal with infinite area doesn't seem like a scaling violation per se. If you double the fractal's linear dimension then don't the area and volume scale factors of 4 and 8 still generate the correct answer except for the detail that multiplying anything by infinity is a pretty pointless thing to do? I'm not aware of any fractals with finite area or volume, where a linear scaling would produce some other result, and the fractals with infinite volumes / area would still seem to obey the scaling law, albeit in a rather trivial way. Dragons flight (talk) 19:21, 5 January 2015 (UTC)

If you use a different definition of measurement which takes the scaling into account one can assign a finite length to a fractal - but with a fractional dimension. See Fractal dimension. The square cube la does not then apply to those finite amounts. For instance if one has a base scale of 1 meter and measures a Koch snowflake one meter across with it one would get a length of about 3 meters. However it would be 4 meters when measured with a 1/3 meter scale. The dimension is log(4)/log(3) so one could say its fractal length is 3 meters^1.26... like one might say a normal length of the side of a square a square is 2 meters and its area is 4 meters². Dmcq (talk) 22:31, 5 January 2015 (UTC)

vegetable oil additive

how to mix epdm(ethylene propylene diene terpolymer),eva(Ethylene-vinyl acetate) and pmm(poly methyl methacrylate) additive with vegetable oil for lubrication49.15.140.116 (talk) 03:46, 1 January 2015 (UTC) 01-01-2015

Meta-Data question

If I want to know what Muscle relaxants cause appetite loss as a side-effect, where could I easily check it (Going through the articles\Pamphlets of all of them could be very long). Thanks for the helpers, Ben-Natan (talk) 06:38, 1 January 2015 (UTC)

Sounds like you need a copy of the Physicians' Desk Reference which any decent public library should have. Note, however, that it will probably abridge most side effects less important and less frequent than the more serious and common effects. You might have to search PubMed to confirm your findings. Are you looking for a muscle relaxant which does cause loss of appetite, or one which does not? 63.228.180.122 (talk) 08:22, 1 January 2015 (UTC)

Thanks for the detailing. I want to know which of them does indeed cause it. Ben-Natan (talk) 11:12, 1 January 2015 (UTC)

Simply googling "muscle relaxant" "appetite loss" with the quote marks will get you dozens of drugs with such affects. Happy New Yearμηδείς (talk) 16:43, 1 January 2015 (UTC)

Or of drugs that people relay to such effect. I am looking for a medical data array... Thanks anyway. Ben-Natan (talk) 21:26, 1 January 2015 (UTC)

River landscapes

What causes the typical river landscapes you see in Europe with green either side, sometimes hills. Normally this happens on very wide rivers. 176.251.149.108 (talk) 12:34, 1 January 2015 (UTC)

I'm not sure if Cycle of erosion will help. It contains additional links that may be of assistance. Bus stop (talk) 12:56, 1 January 2015 (UTC)

Fluvial, Flood plain, Valley and River phenomenon may also be useful articles. Tevildo (talk) 13:11, 1 January 2015 (UTC)

By green on either side, do you mean all the plants growing there ? Those are due to the reliable source of water. Areas with a less reliable source of water tend to be less green. Of course, plants need more than just water, but those other ingredients are typically present in Europe, with an exception in far northern reaches, where the temperature gets too cold for most plants. StuRat (talk) 16:04, 2 January 2015 (UTC)

Approximate extent of the Alpide Orogenic System

• If I understand what's being asked, continental western Europe's geography is largely dominated by mountains and plateaus, including the Meseta central of Spain, The Pyrenees, the Massif Central of France, The Alps, the Carpathian Alps, the Balkan Alps, and in areas like the North European Plain many rivers such as the Rhine still run swiftly and are highly managed in their courses. In Russian and Siberia you get plains with meandering rivers like those of the drainages of the Mississippi and the Amazon and the Orinoco river. The Spain to China Alpide belt caused by the orogeny driven by the northward movement of Africa and India against Eurasia causes this.
  

The central riparian plains of North and South America do have old plateaus like the Ozarks and the Tepui of Venezuela and Guyana, like Mount Roraima, but for the most part it is the Pacific pushing under the Americas that is the main source of mountain building, and their central planes are far vaster than the coastal planes of europe. When the rebounding Baltic Sea full rises above sea level it will have a nice meandering river system. μηδείς (talk) 02:23, 5 January 2015 (UTC)

Marlins Park

When a foul ball or errant pitch or throw toward home smacks the wall of that tank, does it cause pressure waves through the tank water that shocks the fish? 75.75.42.89 (talk) 15:02, 1 January 2015 (UTC)

There seems to have been some concerns about this while the aquariums were being built, but I'm not finding any recent sources indicating that impacts have been a problem for the fish. Deor (talk) 19:56, 1 January 2015 (UTC)

Using the night-time electricity system as a means of dealing with excess electricity generation due to renewables.

My understanding is that certain renewable methods of energy generation such as wind is somewhat problematic because it causes fluctuations in in the amount of power in the grid, I guess reflected by the number of Hz or the voltage (or both? which is it?). Some heating and cooling systems use electrical energy at night to store heat or... cold (to store cold?) which can then be used in the day when electrical energy is more expensive. I was wondering whether this system could also be used to help whomever manages the grid deal with excess of power or would this create problems with people's devices receiving power at inconvenient times (I'm figuring they'd obviously only be charged the night-time rate for consuming this excess power since it's a surplus). — Preceding unsigned comment added by 78.148.105.13 (talk) 16:54, 1 January 2015 (UTC)

• Here in the UK, the hydro electic power is like this. When the power is cheap, they pump water up to the top of the mountain and when the power is expensive, they generate hydro power. See Dinorwig Power Station --TrogWoolley (talk) 17:31, 1 January 2015 (UTC)

See Pumped-storage hydroelectricity for the general concept. As the article mentions, it's currently the mostly widely used form and may get decent efficiencies but does require a suitable location. For the wider concept see Grid energy storage which has a section and links to an article on Thermal energy storage which seems to be the closest to what the OP is referred to. Note that one of the big problems with thermal storage is that you need a big difference in temperatures for useful energy extraction, and this and other factors means you can get quite high loses, even more so if you're thinking on a longer scale than a day (which may be necessary with a number of systems, even solar will depend somewhat on cloud cover and other such things, let alone ones like wind). Using the heat for heating purposes (or unheat for unheating purposes) rather than electricity generation at least means you don't lose so much when trying to extract the energy again later, hence why when this is done it's mostly commonly for that purpose. BTW, this ultimately comes down to the issue of Load balancing (electrical power)/load levelling (also mentioned in the grid energy storage article). If I misunderstood and the OP is referring to turning on devices like storage heaters when demand is low compared to supply to storage heat (or unheat or whatever) for later direct use at times outside night rather than storing thermal energy for later electricity production, this is somewhat discussed in these articles, including perhaps the linked Energy demand management. (One obvious limitation with that idea is that it's only likely to be a small percentage.) Nil Einne (talk) 17:51, 1 January 2015 (UTC)

Power company operators from about the 1890's through the 1920's used huge banks of storage batteries in downtown business districts to smooth the demand versus supply for electricity. They could be charged during periods when demand was low and discharged during peak loads or is a supply line or generator had to be taken out of service for a short period. This was when DC power was commonly provided in downtown business districts. Pumped storage is probably much more efficient than batteries. Utilities have looked into storing energy in Superconducting magnetic energy storage as well as in battery banks with electronics to convert back and forth AC to DC. There is discussion of having a rate where the utility can buy back energy from a plugged-in electric car, which might be important when more cars are plugged in in the future. Those are "supply side" management techniques. On the demand side, as an alternative to storing and discharging power within the electrical system, operators can provide "time of use" pricing via smart meters, with the actual cost of electricity changing each hour. Via the "internet of things," one could arrange for appliances such as the dishwasher and clothes washer and dryer to run, or for the electric car to be charged at 3 am when electricity costs 1 cent per kwh rather than at 7pm when it costs 14 cents per kwh. I have actually seen negative time of use pricing, where they would literally pay a few cents for each kwh you used, and going the other way electricity priced over a dollar per khw when transmission lines and generators are inadequate to supply a peak load. There have long been businesses which freeze ice at night at low electric rates and use a heat exchanger to provide air conditioning in the day. Edison (talk) 23:42, 1 January 2015 (UTC)

The cutting edge in storage for leveling excess generation such as from wind at night is power to gas which is being built out in Europe[3][4][5] and has 22% round-trip efficiency (page 67). Compared to 85% for pumped-storage hydroelectric, that might not seem very good, but consider the storage capacity and relatively free transmission of the natural gas pipeline system. There is a pipeline network from Portugal to China, with essentially unlimited amounts of storage relative to excess generation rates, and if they run out there are a fleet of LNG tankers on land, sea, and rail to take up the slack and sell it somewhere else. Tim AFS (talk) 01:18, 2 January 2015 (UTC)

Thanks everyone for all the interesting answers and links! — Preceding unsigned comment added by 78.148.105.13 (talk) 18:57, 2 January 2015 (UTC)

Power to gas looks fascinating and the technology could be a great complement to intermittent solar and wind generation. I have asked in this Ref Desk in prior years about the possibilities of making gaseous or liquid fuel from the combination of electricity with carbon dioxide and got no indication that the technology was this far along, as in a 6mw pilot plant. I wonder about the safety and utility of simply adding hydrogen to natural gas lines. I know nat gas is not one pure chemical like methane, but wouldn't the combustion characteristic be different, so that the mixed fuel would have a different heat content, and would require a different orifice in a range burner, for instance? What would be the maximum hydrogen to be added without any adjustments by the end user? Would they get more or less heat energy per unit volume of fuel purchased? I suppose the rate charged could be adjusted to compensate for higher or lower heat content. It reminds me of US agribusiness getting legislation to require the gasoline industry to add 10 percent ethanol to gasoline, to the detriment of older car engines. Edison (talk) 20:47, 2 January 2015 (UTC)

Hydrogen has less than one third the heat per volume of natural gas. Better use of it is to make Syngas, whose cost is mostly the cost of hydrogen. Same for other gases such as ammonia, which are good for fuel and other purposes. And for liquids such as synthetic gasoline. Basically, once you've got plentiful hydrogen, you've paid most the cost of making all these more useful things. Jim.henderson (talk) 20:59, 2 January 2015 (UTC)

Analyzing data scientifically: Dealing with answers to determine preference between Coca-Cola or Pepsi, how many are gay or straight, or how many people took illegal drugs

Given a binary choice to a question, what is the best way of dealing with false results, in cases when people don't have any reason to lie, and when people do have a reason to lie? Intuitively, it makes sense not to expect any lie when answering the first question, but the second and third could lead people to choose the more main-stream answer, even if it's wrong (so the lies towards one or the other answer do not compensate). --Noopolo (talk) 20:43, 1 January 2015 (UTC)

Are you asking how to estimate what percent are lying? ←Baseball Bugs ^{What's up, Doc?} carrots→ 22:55, 1 January 2015 (UTC)

Years ago I came across a survey technique for dealing with precisely this kind of problem. Suppose we want to find out the percentage of people who cheat on their spouses. The questionnaire will ask the respondent to flip a coin in private, and answer 1 of 2 questions depending on the result of the coin flip:

1. (heads) Between Coke and Pepsi, which brand of cola do you like better?
2. (tails) Have you cheated on your spouse?

Question 1 is an innocuous question; question 2 is more sensitive, particularly if the respondent is a cheater. Question 1 provides a convenient cover to those who don't want to admit to cheating.

The estimated percentage of cheaters will be the sum of those who self-identify as such, plus the extra percentage (beyond the expected 50%) of respondents who answered question 1. --98.114.98.174 (talk) 23:07, 1 January 2015 (UTC)

Of course you then need a question 1 where the results actually are 50-50, or at least, predictable. --65.94.50.4 (talk) 01:14, 2 January 2015 (UTC)

No, you don't care about the answers to question 1, you just care about the proportion of people who answered question 1 instead of question 2, minus 50%, plus the proportion of people who answered yes to question 2. Tim AFS (talk) 01:32, 2 January 2015 (UTC)

This may be a useful additional analysis, but note it doesn't seem sound to assume everyone who doesn't want to answer question 2 is going to lie and say they got question 1 even if they got question 2. An indeterminate percentage may be truthful on getting question 2 but still lie on it. There will probably also be a small percentage who, while they would have truthfully answer question 2 as no, because they are uncomfortable with the question for some reason or just think it's a personal question that shouldn't be asked, may claim they got question 1. (It's also worth remembering there will be definition issues, people's personal definitions of cheating may vary wildly. In fact, this could be one reason people would lie about which question they got. They may feel the answer to question 2 is no, but recognise some people would claim yes for what they did. Even if you include a definitions, as shown by the Clinton case, you always run the risk of being ambigious, plus you may turn additional people off answering by such things anyway. Nil Einne (talk))

• Tim and the anonymous person posting at 04:16 are both missing the point: the subject doesn't say which question they answered, and knows they won't have to say. So we don't know how many people answered question 1; we just assume it was half of them, based on the properties of coin flips. If question 1 is a true 50-50 question and the overall answers are 60% yes, 40% no, we assume that half the people got question 2 and they answered 70% yes, 30% no. --65.94.50.4 (talk) 09:47, 2 January 2015 (UTC)

Sorry for the lack of name I used the wrong number of tildes. Anyway that wasn't stated in the original explaination. Under the alternative scenario you've now outlined, you still have the problem that it's unclear how willing people will be truthly answer to the cheating question, just because it isn't known which question the are answering to (remembering that in most cases such surveys are already anonymous with it set up such that no one is supposed to know what answer any particularly respondent gave). And you still have most of the earlier problems as well, people ignore the coin flip and answer one of the questions (most likely the Coke/Pepsi one, occasionally perhaps the cheating one for reasons such as those suggested below) because they feel like it and yet you have no idea how many have done this even if your hope is it's less than in the case when they do mention which coin flip they got. To be frank, Tim AFS/my suggestion still sounds a better way to use the coin flip, even if it also has its limitations. And my reading of 98's outline (particularly the last sentence) remains the same, i.e. that our scenario was what 98 was referring to. Are you sure your scenario has ever even been used in real research? (Admitedly I couldn't actual find details relating to what 98 appeared to refer to anyway, I would guess the details are slightly different.) Your scenario seems to run the risk of adding more uncertainty under the unproven assumption it will result in a higher percentage of people answering truthfully. (Which is not to say the scenario suggested by Tim AFS/me is perfect. While you do get more data, you're also screwing with your original data. Remembering also what I outlined in this reply but not in my initial reply, namely that some people who got the Pepsi/Coke answer may choose to answer the cheating spouse question for whatever reason.) Nil Einne (talk) 14:18, 2 January 2015 (UTC)

It was stated, actually: "flip a coin in private". --65.94.50.4 (talk) 19:58, 2 January 2015 (UTC)

I don't think anyone disputed that? At least I can't see anyone saying anything indicating they didn't know the the coin was flipped in private, definitely I didn't.

The point as me and I think Tim AFS understand the other IP, particularly given the last sentence, is the coin is flipped in private. The person then answers the question based on the coin flip. Which question the answered and their answer is known to the researcher. What isn't known is what they actually flipped, since as we all (I think) know the coin is flipped in private.

Normally you would expect there to be roughly 50/50 (for a sufficient large sample size) people answering each question if you use a fair coin. However the assumption is that you will get a higher percentage of people answering the innocous question, because they do not want to answer the cheating question (well once you do the research, this won't just be an assumption). I think a further assumption (which will ultimately always be an assumption) is that many of these would probably need to truthly answer "yes" if they were to answer the cheating question.

However as I stated, I don't think it's safe to assume all those who decided to answer the Pepsi/Coke question despite getting the cheating question would have "truthfully" answered "yes" if they were to answer the cheating question. And you also can't assume all of those who would "lie", are going to "lie" about which question they got rather than "lie" in their answer to the cheating question. And we can't even assume all those who are "lying", are "lying" "no", it's likely some are "lying" when they say "yes". Perhaps worst of all, you can't even assume everyone who is "lying" about what their coin flip are "lying" in saying they got the Pepsi/Coke question when they got the cheating one.

Nevertheless you're likely to end up with an additional data point, probably a percentage of extra people who answered the Pepsi/Coke question or percentage who didn't answer the cheating question however you want to view it, than you would expect by pure chance.

In case it isn't clear, I use the terms "lying" etc loosely hence the quotes. As I mentioned, in reality what's a lie may not always be clear in cases like this. I'm guessing, in the particular case of the coin flip, some people may not even bother to flip the coin instead choosing whichever answer they want. Whereas others may keep flipping telling themselves they screwed up (or at least do it once or twice). Or simply flip in such a way they're likely to get the answer they want. Etc etc.

I guess you could monitor this part (at a distance such they are assured you can't actually see the result) to at least force them to actually flip and ignore the result. But I'm not sure if this will help, I'm guessing too close monitoring risks making the person uncomfortable so they're more likely to lie as they may feel it isn't anonymous even if it is.

Nil Einne (talk) 14:11, 4 January 2015 (UTC)

Reporting bias is our article on the topic. DMacks (talk) 23:41, 1 January 2015 (UTC)

Excepting that it doesn't really have anything much to say. --Epipelagic (talk) 04:58, 2 January 2015 (UTC)

It's true. Great opportunity for RD to help build the encyclopedia if citeable info comes to light here. DMacks (talk) 05:06, 2 January 2015 (UTC)

Under what circumstances would someone be asked the "cheating" question? ←Baseball Bugs ^{What's up, Doc?} carrots→ 05:05, 2 January 2015 (UTC)

I believe Masters and Johnson tried to get more truthful answers by building in the assumption that "everyone does it". So, instead of asking "Do you masturbate ?" they would ask "How often do you masturbate ?". The same thing could apply to cheating. StuRat (talk) 06:37, 2 January 2015 (UTC)

I doubt M&J's clients expected to be asked about Coke vs. Pepsi. And the OP's premise leaves out a factor: That interviewees might purposely lie in the other direction for egoistic reasons or whatever. (This was one of the criticisms of the Kinsey Report, as I recall - the suspicion that the interviewees were making stuff up in order to impress the interviewer.) ←Baseball Bugs ^{What's up, Doc?} carrots→ 09:50, 2 January 2015 (UTC)

For this reason, surveys on drug use sometimes include some made-up names. AndrewWTaylor (talk) 13:00, 2 January 2015 (UTC)