Parallel lines have so much in common.
It’s a shame they never get to meet.
sometimes education is the removal of false notions.
pi therapy
Robert E. Buxbaum, January 4, 2017. Aside from the beauty of math itself, I’ve previously noted that, if your child is interested in science, the best route for development is math. I’ve also noted that Einstein did not fail at math, and that calculus is taught wrong, and probably is.
A few months ago, I wrote a rather depressing essay based on Nobel Laureate, Kenneth Arrow’s work, and the paradox of de Condorcet. It is mathematically shown that you can not make a fair election, even if you wanted to, and no one in power wants to. The game is rigged.
To make up for that insight, I’d like to show from the work of John Forbes Nash (A Beautiful Mind) that you, personally, can win, basically all the time, if you can get someone, anyone to coöperate by trade. Let’s begin with an example in Nash’s first major paper, “The Bargaining Problem,” the one Nash is working on in the movie— read the whole paper here. Consider two people, each with a few durable good items. Person A has a bat, a ball, a book, a whip, and a box. Person B has a pen, a toy, a knife, and a hat. Since each item is worth a different amount (has a different utility) to the owner and to the other person, there are almost always sets of trades that benefit both. In our world, where there are many people and everyone has many durable items, it is inconceivable that there are not many trades a person can make to benefit him/her while benefiting the trade partner.
Figure 3, from Nash’s, “The bargaining problem.” U1 and U2 are the utilities of the items to the two people, and O is the current state. You can improve by barter so long as your current state is not on the boundary. The parallel lines are places one could reach if money trades as well.
Good trades are even more likely when money is involved or non-durables. A person may trade his or her time for money, that is work, and any half-normal person will have enough skill to be of some value to someone. If one trades some money for durables, particularly tools, one can become rich (slowly). If one trades this work for items to make them happy (food, entertainment) they can become happier. There are just two key skills: knowing what something is worth to you, and being willing to trade. It’s not that easy for most folks to figure out what their old sofa means to them, but it’s gotten easier with garage sales and eBay.
Let us now move to the problem of elections, e.g. in this year 2016. There are few people who find the person of their dreams running for president this year. The system has fundamental flaws, and has delivered two thoroughly disliked individuals. But you can vote for a generally positive result by splitting your ticket. American society generally elects a mix of Democrats and Republicans. This mix either delivers the outcome we want, or we vote out some of the bums. Americans are generally happy with the result.
A Stamp act stamp,. Used to tax every transaction, the British made it impossible for ordinary people to benefit by small trades.
The mix does not have to involve different people, it can involve different periods of time. One can elect a Democrat president this year, and an Republican four years later. Or take the problem of time management for college students. If a student had to make a one time choice, they’d discover that you can’t have good grades, good friends, and sleep. Instead, most college students figure out you can have everything if you do one or two of these now, and switch when you get bored. And this may be the most important thing they learn.
This is my solution to Israel’s classic identity dilemma. David Ben-Gurion famously noted that Israel had the following three choices: they could be a nation of Jews living in the land of Israel, but not democratic. They could be a democratic nation in the land of Israel, but not Jewish; or they could be Jewish and democratic, but not (for the most part) in Israel. This sounds horrible until you realize that Israel can elect politicians to deliver different pairs of the options, and can have different cities that cater to thee options too. Because Jerusalem does not have to look like Tel Aviv, Israel can achieve a balance that’s better than any pure solution.
Robert E. Buxbaum, July 17-22, 2016. Balance is all, and pure solutions are a doom. I’m running for water commissioner.
I thought I’d try to semi-derive, and explain a remarkable mathematical paper that was published last month in The Proceedings of the Royal Society A (see full paper here). The paper demonstrates that too much agreement about a thing is counter-indicative of the thing being true. Unless an observation is blindingly obvious, near 100% agreement suggests there is a hidden flaw or conspiracy, perhaps unknown to the observers. This paper has broad application, but I thought the presentation was too confusing for most people to make use of, even those with a background in mathematics, science, or engineering. And the popular versions press versions didn’t even try to be useful. So here’s my shot:
Figure 2 from the original paper. For a method that is 80% accurate, you get your maximum reliability at 3-5 witnesses. More agreement suggests a flaw in the people or procedure.
I will discuss only on specific application, the second one mentioned in the paper, crime (read the paper for others). Lets say there’s been a crime with several witnesses. The police line up a half-dozen, equal (?) suspects, and show them to the first witness. Lets say the first witness points to one of the suspects, the police will not arrest on this because they know that people correctly identify suspects only about 40% of the time, and incorrectly identify perhaps 10% (the say they don’t know or can’t remember the remaining 50% of time). The original paper includes the actual factions here; they’re similar. Since the witness pointed to someone, you already know he/she isn’t among the 50% who don’t know. But you don’t know if this witness is among the 40% who identify right or the 10% who identify wrong. Our confidence that this is the criminal is thus .4/(.4 +.1) = .8, or 80%.
Now you bring in the second witness. If this person identifies the same suspect, your confidence increases; to roughly (.4)2/(.42+.12) = .941, or 94.1%. This is enough to make an arrest, but let’s say you have ten more witnesses, and all identify this same person. You might first think that this must be the guy with a confidence of (.4)10/(.410+.110) = 99.99999%, but then you wonder how unlikely it is to find ten people who identify correctly when, as we mentioned, each person has only a 40% chance. The chance of all ten witnesses identifying a suspect right is small: (.4)10 = .000104 or 0.01%. This fraction is smaller than the likelihood of having a crooked cop or a screw up the line-up (only one suspect had the right jacket, say). If crooked cops and systemic errors show up 1% of the time, and point to the correct fellow only 15% of these, we find that the chance of being right if ten out of ten agree is (0.0015 +(.4)10)/( .01+ .410+.110) = .16%. Total agreement on guilt suggests the fellow is innocent!
The graph above, the second in the paper, presents a generalization of the math I just presented: n identical tests of 80% accuracy and three different likelihoods of systemic failure. If this systemic failure rate is 1% and the chance of the error pointing right or wrong is 50/50, the chance of being right is P = (.005+ .4n)/(.01 +.4n+.1n), and is the red curve in the graph above. The authors find you get your maximum reliability when there are two to four agreeing witness.
Confidence of guilt as related to the number of judges that agree and the integrity of the judges.
The Royal Society article went on to a approve of a feature of Jewish capital-punishment law. In Jewish law, capital cases are tried by 23 judges. To convict a super majority (13) must find guilty, but if all 23 judges agree on guilt the court pronounces innocent (see chart, or an anecdote about Justice Antonin Scalia). My suspicion, by the way, is that more than 1% of judges and police are crooked or inept, and that the same applies to scientific analysis of mental diseases like diagnosing ADHD or autism, and predictions about stocks or climate change. (Do 98% of scientists really agree independently?). Perhaps there are so many people in US prisons, because of excessive agreement and inaccurate witnesses, e.g Ruben Carter. I suspect the agreement on climate experts is a similar sham.
Robert Buxbaum, March 11, 2016. Here are some thoughts on how to do science right. Here is some climate data: can you spot a clear pattern of man-made change?
There are two main obstacles students have to overcome to learn statistics: one mathematical one philosophical. The math is somewhat difficult, and will be new to a high schooler. What’s more, philosophically, it is rarely obvious what it means to discover a true pattern, or underlying cause. Nor is it obvious how to separate the general pattern from the random accident, the pattern from the variation. This philosophical confusion (cause and effect, essence and accident) is exists in the back of even in the greatest minds. Accepting and dealing with it is at the heart of the best research: seeing what is and is not captured in the formulas of the day. But it is a lot to ask of the young (or the old) who are trying to understand the statistical technique while at the same time trying to understand the subject of the statistical analysis, For young students, especially the good ones, the issue of general and specific will compound the difficulty of the experiment and of the math. Thus, I’ll try to teach statistics with a problem or two where the distinction between essential cause and random variation is uncommonly clear.
A good case to get around the philosophical issue is gambling with crooked dice. I show the class a pair of normal-looking dice and a caliper and demonstrate that the dice are not square; virtually every store-bought die is not square, so finding an uneven pair is easy. After checking my caliper, students will readily accept that these dice are crooked, and so someone who knows how it is crooked will have an unfair advantage. After enough throws, someone who knows the degree of crookedness will win more often than those who do not. Students will also accept that there is a degree of randomness in the throw, so that any pair of dice will look pretty fair if you don’t gable with them too long. I can then use statistics to see which faces show up most, and justify the whole study of statistics to deal with a world where the dice are loaded by God, and you don’t have a caliper, or any more-direct way of checking them. The underlying uneven-ness of the dice is the underlying pattern, the random part in this case is in the throw, and you want to use statistics to grasp them both.
Two important numbers to understand when trying to use statistics are the average and the standard deviation. For an honest pair of dice, you’d expect an average of 1/6 = 0.1667 for every number on the face. But throw a die a thousand times and you’ll find that hardly any of the faces show up at the average rate of 1/6. The average of all the averages will still be 1/6. We will call that grand average, 1/6 = x°-bar, and we will call the specific face average of the face Xi-bar. where i is one, two three, four, five, or six.
There is also a standard deviation — SD. This relates to how often do you expect one fact to turn up more than the next. SD = √SD2, and SD2 is defined by the following formula
SD2 = 1/n ∑(xi – x°-bar)2
Let’s pick some face of the dice, 3 say. I’ll give a value of 1 if we throw that number and 0 if we do not. For an honest pair of dice, x°-bar = 1/6, that is to say, 1 out of 6 throws will be land on the number 3, going us a value of 1, and the others won’t. In this situation, SD2 = 1/n ∑(xi – x°-bar)2 will equal 1/6 ( (1/6)2 + 5 (5/6)2 )= 1/6 (126/36) = 3.5/6 = .58333. Taking the square root, SD = 0.734. We now calculate the standard error. For honest dice, you expect that for every face, on average
SE = Xi-bar minus x°-bar = ± SD √(1/n).
By the time you’ve thrown 10,000 throws, √(1/n) = 1/100 and you expect an error on the order of 0.0073. This is to say that you expect to see each face show up between about 0.1740 and 0.1594. In point of fact, you will likely find that at least one face of your dice shows up a lot more often than this, or a lot less often. To the extent you see that, this is the extent that your dice is crooked. If you throw someone’s dice enough, you can find out how crooked they are, and you can then use this information to beat the house. That, more or less is the purpose of science, by the way: you want to beat the house — you want to live a life where you do better than you would by random chance.
As a less-mathematical way to look at the same thing — understanding statistics — I suggest we consider a crooked coin throw with only two outcomes, heads and tails. Not that I have a crooked coin, but your job as before is to figure out if the coin is crooked, and if so how crooked. This problem also appears in political polling before a major election: how do you figure out who will win between Mr Head and Ms Tail from a sampling of only a few voters. For an honest coin or an even election, on each throw, there is a 50-50 chance of head, or of Mr Head. If you do it twice, there is a 25% chance of two heads, a 25% chance of throwing two tails and a 50% chance of one of each. That’s because there are four possibilities and two ways of getting a Head and a Tail.
Pascal’s triangle
You can systematize this with a Pascal’s triangle, shown at left. Pascal’s triangle shows the various outcomes for a coin toss, and shows the ways they can be arrived at. Thus, for example, we see that, by the time you’ve thrown the coin 6 times, or polled 6 people, you’ve introduced 26 = 64 distinct outcomes, of which 20 (about 1/3) are the expected, even result: 3 heads and 3 tails. There is only 1 way to get all heads and one way to get all tails. While an honest coin is unlikely to come up all heads or tails after six throws, more often than not an honest coin will not come up with half heads. In the case above, 44 out of 64 possible outcomes describe situations with more heads than tales, or more tales than heads — with an honest coin.
Similarly, in a poll of an even election, the result will not likely come up even. This is something that confuses many political savants. The lack of an even result after relatively few throws (or phone calls) should not be used to convince us that the die is crooked, or the election has a clear winner. On the other hand there is only a 1/32 chance of getting all heads or all tails (2/64). If you call 6 people, and all claim to be for Mr Head, it is likely that Mr Head is the true favorite to a confidence of 3% = 1/32. In sports, it’s not uncommon for one side to win 6 out of 6 times. If that happens, it is a good possibility that there is a real underlying cause, e.g. that one team is really better than the other.
And now we get to how significant is significant. If you threw 4 heads and 2 tails out of 6 throws we can accept that this is not significant because there are 15 ways to get this outcome (or 30 if you also include 2 heads and 4 tail) and only 20 to get the even outcome of 3-3. But what about if you threw 5 heads and one tail? In that case the ratio is 6/20 and the odds of this being significant is better, similarly, if you called potential voters and found 5 Head supporters and 1 for Tail. What do you do? I would like to suggest you take the ratio as 12/20 — the ratio of both ways to get to this outcome to that of the greatest probability. Since 12/20 = 60%, you could say there is a 60% chance that this result is random, and a 40% chance of significance. What statisticians call this is “suggestive” at slightly over 1 standard deviation. A standard deviation, also known as σ (sigma) is a minimal standard of significance, it’s if the one tailed value is 1/2 of the most likely value. In this case, where 6 tosses come in as 5 and 1, we find the ratio to be 6/20. Since 6/20 is less than 1/2, we meet this, very minimal standard for “suggestive.” A more normative standard is when the value is 5%. Clearly 6/20 does not meet that standard, but 1/20 does; for you to conclude that the dice is likely fixed after only 6 throws, all 6 have to come up heads or tails.
From xkcd. It’s typical in science to say that <5% chances, p< .05. If things don’t quite come out that way, you redo.
If you graph the possibilities from a large Poisson Triangle they will resemble a bell curve; in many real cases (not all) your experiential data variation will also resemble this bell curve. From a larger Poisson’s triange, or a large bell curve, you will find that the 5% value occurs at about σ =2, that is at about twice the distance from the average as to where σ = 1. Generally speaking, the number of observations you need is proportional to the square of the difference you are looking for. Thus, if you think there is a one-headed coin in use, it will only take 6 or seven observations; if you think the die is loaded by 10% it will take some 600 throws of that side to show it.
In many (most) experiments, you can not easily use the poisson triangle to get sigma, σ. Thus, for example, if you want to see if 8th graders are taller than 7th graders, you might measure the height of people in both classes and take an average of all the heights but you might wonder what sigma is so you can tell if the difference is significant, or just random variation. The classic mathematical approach is to calculate sigma as the square root of the average of the square of the difference of the data from the average. Thus if the average is <h> = ∑h/N where h is the height of a student and N is the number of students, we can say that σ = √ (∑ (<h> – h)2/N). This formula is found in most books. Significance is either specified as 2 sigma, or some close variation. As convenient as this is, my preference is for this graphical version. It also show if the data is normal — an important consideration.
If you find the data is not normal, you may decide to break the data into sub-groups. E.g. if you look at heights of 7th and 8th graders and you find a lack of normal distribution, you may find you’re better off looking at the heights of the girls and boys separately. You can then compare those two subgroups to see if, perhaps, only the boys are still growing, or only the girls. One should not pick a hypothesis and then test it but collect the data first and let the data determine the analysis. This was the method of Sherlock Homes — a very worthwhile read.
Another good trick for statistics is to use a linear regression, If you are trying to show that music helps to improve concentration, try to see if more music improves it more, You want to find a linear relationship, or at lest a plausible curve relationship. Generally there is a relationship if (y – <y>)/(x-<x>) is 0.9 or so. A discredited study where the author did not use regressions, but should have, and did not report sub-groups, but should have, involved cancer and genetically modified foods. The author found cancer increased with one sub-group, and publicized that finding, but didn’t mention that cancer didn’t increase in nearby sub-groups of different doses, and decreased in a nearby sub-group. By not including the subgroups, and not doing a regression, the author mislead people for 2 years– perhaps out of a misguided attempt to help. Don’t do that.
Dr. Robert E. Buxbaum, June 5-7, 2015. Lack of trust in statistics, or of understanding of statistical formulas should not be taken as a sign of stupidity, or a symptom of ADHD. A fine book on the misuse of statistics and its pitfalls is called “How to Lie with Statistics.” Most of the examples come from advertising.
Imagine a highly infectious, people-borne plague for which there is no immunization or ready cure, e.g. leprosy or small pox in the 1800s, or bubonic plague in the 1500s assuming that the carrier was fleas on people (there is a good argument that people-fleas were the carrier, not rat-fleas). We’ll call these plagues zombie invasions to highlight understanding that there is no way to cure these diseases or protect from them aside from quarantining the infected or killing them. Classical leprosy was treated by quarantine.
I propose to model the progress of these plagues to know how to survive one, if it should arise. I will follow a recent paper out of Cornell that highlighted a fact, perhaps forgotten in the 21 century, that population density makes a tremendous difference in the rate of plague-spread. In medieval Europe plagues spread fastest in the cities because a city dweller interacted with far more people per day. I’ll attempt to simplify the mathematics of that paper without losing any of the key insights. As often happens when I try this, I’ve found a new insight.
Assume that the density of zombies per square mile is Z, and the density of susceptible people is S in the same units, susceptible population per square mile. We define a bite transmission likelihood, ß so that dS/dt = -ßSZ. The total rate of susceptibles becoming zombies is proportional to the product of the density of zombies and of susceptibles. Assume, for now, that the plague moves fast enough that we can ignore natural death, immunity, or the birth rate of new susceptibles. I’ll relax this assumption at the end of the essay.
The rate of zombie increase will be less than the rate of susceptible population decrease because some zombies will be killed or rounded up. Classically, zombies are killed by shot-gun fire to the head, by flame-throwers, or removed to leper colonies. However zombies are removed, the process requires people. We can say that, dR/dt = kSZ where R is the density per square mile of removed zombies, and k is the rate factor for killing or quarantining them. From the above, dZ/dt = (ß-k) SZ.
We now have three, non-linear, indefinite differential equations. As a first step to solving them, we set the derivates to zero and calculate the end result of the plague: what happens at t –> ∞. Using just equation 1 and setting dS/dt= 0 we see that, since ß≠0, the end result is SZ =0. Thus, there are only two possible end-outcomes: either S=0 and we’ve all become zombies or Z=0, and all the zombies are all dead or rounded up. Zombie plagues can never end in mixed live-and-let-live situations. Worse yet, rounded up zombies are dangerous.
If you start with a small fraction of infected people Z0/S0 <<1, the equations above suggest that the outcome depends entirely on k/ß. If zombies are killed/ rounded up faster than they infect/bite, all is well. Otherwise, all is zombies. A situation like this is shown in the diagram below for a population of 200 and k/ß = .6
Fig. 1, Dynamics of a normal plague (light lines) and a zombie apocalypse (dark) for 199 uninfected and 1 infected. The S and R populations are shown in blue and black respectively. Zombie and infected populations, Z and I , are shown in red; k/ß = 0.6 and τ = tNß. With zombies, the S population disappears. With normal infection, the infected die and some S survive.
Sorry to say, things get worse for higher initial ratios, Z0/S0 >> 0. For these cases, you can kill zombies faster than they infect you, and the last susceptible person will still be infected before the last zombie is killed. To analyze this, we create a new parameter P = Z + (1 – k/ß)S and note that dP/dt = 0 for all S and Z; the path of possible outcomes will always be along a path of constant P. We already know that, for any zombies to survive, S∞ = 0. We now use algebra to show that the final concentration of zombies will be Z∞ = Z0 + (1-k/ß)S0. Free zombies survive so long as the following ratio is non zero: Z0/S0 + 1- k/ß. If Z0/S0 = 1, a situation that could arise if a small army of zombies breaks out of quarantine, you’ll need a high kill ratio, k/ß > 2 or the zombies take over. It’s seen to be harder to stop a zombie outbreak than to stop the original plague. This is a strong motivation to kill any infected people you’ve rounded up, a moral dilemma that appears some plague literature.
Figure 1, from the Cornell paper, gives a sense of the time necessary to reach the final state of S=0 or Z=0. For k/ß of .6, we see that it takes is a dimensionless time τ of 25 or to reach this final, steady state of all zombies. Here, τ= t Nß and N is the total population; it takes more real time to reach τ= 25 if N is high than if N is low. We find that the best course in a zombie invasion is to head for the country hoping to find a place where N is vanishingly low, or (better yet) where Z0 is zero. This was the main conclusion of the Cornell paper.
Figure 1 also shows the progress of a more normal disease, one where a significant fraction of the infected die on their own or develop a natural immunity and recover. As before, S is the density of the susceptible, R is the density of the removed + recovered, but here I is the density of those Infected by non-zombie disease. The time-scales are the same, but the outcome is different. As before, τ = 25 but now the infected are entirely killed off or isolated, I =0 though ß > k. Some non-infected, susceptible individuals survive as well.
From this observation, I now add a new conclusion, not from the Cornell paper. It seems clear that more immune people will be in the cities. I’ve also noted that τ = 25 will be reached faster in the cities, where N is large, than in the country where N is small. I conclude that, while you will be worse off in the city at the beginning of a plague, you’re likely better off there at the end. You may need to get through an intermediate zombie zone, and you will want to get the infected to bury their own, but my new insight is that you’ll want to return to the city at the end of the plague and look for the immune remnant. This is a typical zombie story-line; it should be the winning strategy if a plague strikes too. Good luck.
Robert Buxbaum, April 21, 2015. While everything I presented above was done with differential calculus, the original paper showed a more-complete, stochastic solution. I’ve noted before that difference calculus is better. Stochastic calculus shows that, if you start with only one or two zombies, there is still a chance to survive even if ß/k is high and there is no immunity. You’ve just got to kill all the zombies early on (gun ownership can help). Here’s my statistical way to look at this. James Sethna, lead author of the Cornell paper, was one of the brightest of my Princeton PhD chums.
In case it should ever come up in conversation, only the picture at left shows a brass monkey. The other is a bronze statue of some sort of a primate. A brass monkey is a rack used to stack cannon balls into a face centered pyramid. A cannon crew could fire about once per minute, and an engagement could last 5 hours, so you could hope to go through a lot of cannon balls during an engagement (assuming you survived).
Small brass monkey. The classic monkey might have 9 x 9 or 10×10 cannon balls on the lower level.
Bronze sculpture of a primate playing with balls — but look what the balls are sitting on: it’s a dada art joke.
But brass monkeys typically show up in conversation in terms of it being cold enough to freeze the balls off of a brass monkey, and if you imagine an ornamental statue, you’d never guess how cold could that be. Well, for a cannonball holder, the answer has to do with the thermal expansion of metals. Cannon balls were made of iron and the classic brass monkey was made of brass, an alloy with a much-greater thermal expansion than iron. As the temperature drops, the brass monkey contracts more than the iron balls. When the drop is enough the balls will fall off and roll around.
The thermal expansion coefficient of brass is 18.9 x 10-6/°C while the thermal expansion coefficient of iron is 11.7 x10-6/°C. The difference is 7.2×10-6/°C; this will determine the key temperature. Now consider a large brass monkey, one with 400 x 400 holes on the lower level, 399 x 399 at the second, and so on. Though it doesn’t affect the result, we’ll consider a monkey that holds 12 lb cannon balls, a typical size of 1750 -1830. Each 12 lb ball is 4.4″ in diameter at room temperature, 20°C in those days. At 20°C, this monkey is about 1760″ wide. The balls will fall off when the monkey shrinks more than the balls by about 1/3 of a diameter, 1.5″.
We can calculate ∆T, the temperature change, °C, that is required to lower the width-difference by 1.5″ as follows:
-1.5″ = ∆T x 1760″ x 7.2 x10-6
We find that ∆T = -118°C. The temperature where this happens is 118 degrees cooler than 20°C, or -98°C. That’s a temperature that you could, perhaps reach on the South Pole or maybe deepest Russia. It’s not likely to be a problem, especially with a smaller brass monkey.
Robert E. Buxbaum, February 21, 2015 (modified Apr. 28, 2021). Some fun thoughts: Convince yourself that the key temperature is independent of the size of the cannon balls. That is, that I didn’t need to choose 12 pounders. A bit more advanced, what is the equation for the number of balls on any particular base-size monkey. Show that the packing density is no more efficient if the bottom lawyer were an equilateral triangle, and not a square. If you liked this, you might want to know how much wood a woodchuck chucks if a woodchuck could chuck wood, or on the relationship between mustaches and WWII diplomacy.
I don’t know quite why people persist in claiming that Einstein failed high school math. Perhaps it’s to put down teachers –who clearly can’t teach or recognize genius — or perhaps to stake a claim to a higher understanding that’s masked by ADHD — a disease Einstein is supposed to have had. But, sorry to say, it ain’t true. Here’s Einstein’s diploma, 1896. His math and physics scores are perfect. Only his English seems to have been lacking. He would have been 17 at the time.
Albert Einstein’s high school diploma, 1896.
Robert Buxbaum, December 16, 2014. Here’s Einstein relaxing in Princeton. Here’s something on black holes, and on High School calculus for non-continuous functions.
A few weeks back, I decided to do something about the low quality of experiments in modern chemistry and science sets; I posted to this blog some interesting science experiments, and some more-interesting experiments that could be done at home using the toxic (poisonous dangerous) chemicals available under the sink or on the hardware store. Here are some more. As previously, the chemicals are toxic and dangerous but available. As previously, these experiments should be done only with parental (adult) supervision. Some of these next experiments involve some math, as key aspect of science; others involve some new equipment as well as the stuff you used previously. To do them all, you will want a stop watch, a volt-amp meter, and a small transformer, available at RadioShack; you’ll also want some test tubes or similar, clear cigar tubes, wire and baking soda; for the coating experiment you’ll want copper drain clear, or copper containing fertilizer and some washers available at the hardware store; for metal casting experiment you’ll need a tin can, pliers, a gas stove and some pennies, plus a mold, some sand, good shoes, and a floor cover; and for the biology experiment you will need several 9 V batteries, and you will have to get a frog and kill it. You can skip any of these experiments, if you like and do the others. If you have not done the previous experiments, look them over or do them now.
1) The first experiments aim to add some numerical observations to our previous studies of electrolysis. Here is where you will see why we think that molecules like water are made of fixed compositions of atoms. Lets redo the water electrolysis experiment now with an Ammeter in line between the battery and one of the electrodes. With the ammeter connected, put both electrodes deep into a solution of water with a little lye, and then (while watching the ammeter) lift one electrode half out, place it back, and lift the other. You will find, I think, that one of the other electrode is the limiting electrode, and that the amperage goes to 1/2 its previous value when this electrode is half lifted. Lifting the other electrode changes neither the amperage or the amount of bubbles, but lifting this limiting electrode changes both the amount of bubbles and the amperage. If you watch closely, though, you’ll see it changes the amount of bubbles at both electrodes in proportion, and that the amount of bubbles is in promotion to the amperage. If you collect the two gasses simultaneously, you’ll see that the volume of gas collected is always in a ratio of 2 to 1. For other electrolysis (H2 and Cl2) it will be 1 to1; it’s always a ratio of small numbers. See diagram below on how to make and collect oxygen and hydrogen simultaneously by electrolyzing water with lye or baking soda as electrolyte. With lye or baking soda, you’ll find that there is always twice as much hydrogen produced as oxygen — exactly.
You can also do electrolysis with table salt or muriatic acid as an electrolyte, but for this you’ll need carbon or platinum electrodes. If you do it right, you’ll get hydrogen and chlorine, a green gas that smells bad. If you don’t do this right, using a wire instead of a carbon or platinum electrode, you’ll still get hydrogen, but no chlorine. Instead of chlorine, you’ll corrode the wire on that end, making e.g. copper chloride. With a carbon electrode and any chloride compound as the electrolyte, you’ll produce chlorine; without a chloride electrolyte, you will not produce chlorine at any voltage, or with any electrode. And if you make chlorine and check the volumes, you’ll find you always make one volume of chlorine for every volume of hydrogen. We imagine from this that the compounds are made of fixed atoms that transfer electrons in fixed whole numbers per molecule. You always make two volumes of hydrogen for every volume of oxygen because (we think) making oxygen requires twice as many electrons as making hydrogen.
At home electrolysis experiment
We get the same volume of chlorine as hydrogen because making chlorine and hydrogen requires the same amount of electrons to be transferred. These are the sort of experiments that caused people to believe in atoms and molecules as the fundamental unchanging components of matter. Different solutes, voltages, and electrodes will affect how fast you make hydrogen and oxygen, as will the amount of dissolved solute, but the gas produced are always the same, and the ratio of volumes is always proportional to the amperage in a fixed ratio of small whole numbers.
As always, don’t let significant quantities of use hydrogen and oxygen or pure hydrogen and chlorine mix in a closed space. Hydrogen and oxygen is quite explosive brown’s gas; hydrogen and chlorine are reactive as well. When working with chlorine it is best to work outside or near an open window: chlorine is a poison gas.
You may also want to try this with non-electrolytes, pure water or water with sugar or alcohol dissolved. You will find there is hardly any amperage or gas with these, but the small amount of gas produced will retain the same ratio. For college level folks, here is some physics/math relating to the minimum voltage and relating to the quantities you should expect at any amperage.
2) Now let’s try electro-plating metals. Using the right solutes, metals can be made to coat your electrodes the same way that bubbles of gas coated your electrodes in the experiments above. The key is to find the right chemical, and as a start let me suggest the copper sulphate sold in hardware stores to stop root growth. As an alternative copper sulphate is often sold as part of a fertilizer solution like “Miracle grow.” Look for copper on the label, or for a blue color fertilizer. Make a solution of copper using enough copper so that the solution is recognizably green, Use two steel washers as electrodes (that is connect the wires from your battery to the washers) and put them in the solution. You will find that one side turns red, as it is coated with copper. Depending on what else your copper solution contained, bubbles may appear at the other washer, or the other washer will corrode.
You are now ready to take this to a higher level — silver coating. take a piece of silver plated material that you want to coat, and clean it nicely with soap and water. Connect it to the electrode where you previously coated copper. Now clean out the solution carefully. Buy some silver nitrate from a drug store, and dissolve a few grams (1/8 tsp for a start) in pure water; place the silverware and the same electrodes as before, connected to the battery. For a nicer coat use a 1 1/2 volt lantern battery; the 6 V battery will work too, but the silver won’t look as nice. With silver nitrate, you’ll notice that one electrode produces gas (oxygen) and the other turns silvery. Now disconnect the silvery electrode. You can use this method to silver coat a ring, fork, or cup — anything you want to have silver coated. This process is called electroplating. As with hydrogen production, there is a proportional relationship between the time, the amperage and the amount of metal you deposit — until all the silver nitrate in solution is used up.
As a yet-more complex version, you can also electroplate without using a battery. This was my Simple electroplating (presented previously). Consider this only after you understand most everything else I’ve done. When I saw this the first time in high school I was confused.
3) Casting metal objects using melted pennies, heat from a gas stove, and sand or plaster as a cast. This is pretty easy, but sort of dangerous — you need parents help, if only as a watcher. This is a version of an experiment I did as a kid. I did metal casting using lead that some plumbers had left over. I melted it in a tin can on our gas stove and cast “quarters” in a plaster mold. Plumbers no longer use lead, but modern pennies are mostly zinc, and will melt about as well as my lead did. They are also much safer.
As a preparation for this experiment, get a bucket full of sand. This is where you’ll put your metal when you’re done. Now get some pennies (1970 or later), a pair of pliers, and an empty clean tin can, and a gas stove. If you like you can make a plaster mold of some small object: a ring, a 50 piece — anything you might want to cast from your pennies. With parents’ help, light your gas stove, put 5-8 pennies in the empty tin can, and hold the can over the lit gas burner using your pliers. Turn the gas to high. In a few minutes the bottom of the can will burn and become red-hot. About this point, the pennies will soften and melt into a silvery puddle. By tilting the can, you can stir the metal around (don’t get it on you!). When it looks completely melted you can pour the molten pennies into your sand bucket (carefully), or over your plaster mold (carefully). If you use a mold, you’ll get a zinc copy of whatever your mold was: jewelry, coins, etc. If you work at it, you’ll learn to make fancier and fancier casts. Adult help is welcome to avoid accidents. Once the metal solidifies, you can help cool it faster by dripping water on it from a faucet. Don’t touch it while it’s hot!
A plaster mold can be made by putting a 50¢ piece at the bottom of a paper cup, pouring plaster over the coin, and waiting for it to dry. Tear off the cup, turn the plaster over and pull out the coin; you’ve got a one-sided mold, good enough to make a one-sided coin. If you enjoy this, you can learn more about casting on Wikipedia; it’s an endeavor that only costs 4 or 5 cents per try. As a safety note: wear solid leather shoes and cover the floor near the stove with a board. If you drop the metal on the floor you’ll have a permanent burn mark on the floor and your mother will not be happy. If you drop hot metal on your you’ll have a permanent injury, and you won’t be happy. Older pennies are made of copper and will not melt. Here’s a video of someone pouring a lot of metal into an ant-hill (kills lots of ants, makes a mold of the hill).
It’s nice to have assistants, friends and adult help in the laboratory when you do science. Even without the castle, it’s what Dr. Frankenstein did.
4) Bringing a dead frog back to life (sort of). Make a high voltage battery of 45 to 90 V battery by attaching 5-10, 9V batteries in a daisy chain they will snap together. If you touch both exposed contacts you’ll give yourself a wicked shock. If you touch the electrodes to a newly killed frog, the frog legs will kick. This is sort of groovy. It was the inspiration for Dr. Frankenstein (at right), who then decides he could bring a person back from the dead with “more power.” Frankenstein’s monster is brought back to life this way, but ends up killing the good doctor. Shocks are sometimes helpful reanimating people stricken by heat attacks, and many buildings have shockers for this purpose. But don’t try to bring back the long-dead. By all accounts, the results are less-than pleasing. Try dissecting the rest of the frog and guess what each part is (a world book encyclopedia helps). As I recall, the heart keeps going for a while after it’s out of the frog — spooky.
5) Another version of this shocker is made with a small transformer (1″ square, say, radioshack) and a small battery (1.5-6V). Don’t use the 90V battery, you’ll kill someone. As a first version of this shocker, strip 1″ of insulation off of the ends of some wire 12″ long say, and attach one end to two paired wires of the transformer (there will usually be a diagram in the box). If the transformer already has some wires coming out, all you have to do is strip more insulation off the ends so 1″ is un-inuslated. Take two paired ends in your hand, holding onto the uninsulated part and touch both to the battery for a second or two. Then disconnect them while holding the bare wires; you’ll get a shock. As a nastier version, get a friend to hope the opposite pair of wires on the uninsulated parts, while you hold the insulated parts of your two. Touch your two to the battery and disconnect while holding the insulation, you will see a nice spark, and your friend will get a nice shock. Play with it; different arrangements give more sparks or bigger shocks. Another thing you can do: put your experiment near a radio or TV. The transformer sparks will interfere with most nearby electronics; you can really mess up a computer this way, so keep it far from your computer. This is how wireless radio worked long ago, and how modern warfare will probably go. The atom bomb was detonated with a spark like this.
If you want to do more advanced science, it’s a good idea to learn math. This is important for statistics, for engineering, for quantum mechanics, and can even help for music. Get a few good high school or college books and read them cover to cover. An approach to science is to try to make something cool, that sort-of works, and then try to improve it. You then decide what a better version would work like, modify your original semi-randomly and see if you’re going in the right direction. Don’t redesign with only one approach –it may not work. Read whatever you can, but don’t believe all you read. Often books are misleading, or wrong, and blogs are worse (I ought to know). When you find mistakes, note them in the margin, and try to explain them. You may find you were right, or that the book was right, but it’s a learning experience. If you like you can write the author and inform him/her of the errors. I find mailed letters are more respectful than e-mails — it shows you put in more effort.
Robert Buxbaum, February 20, 2014. Here’s the difference between metals and non-metals, and a periodic table cup that I made, and sell. And here’s a difference between science and religion – reproducibility.
There is a general problem when looking for climate trends: you have to look at weather data. That’s a problem because weather data goes back thousands of years, and it’s always changing. As a result it’s never clear what start year to use for the trend. If you start too early or too late the trend disappears. If you start your trend line in a hot year, like in the late roman period, the trend will show global cooling. If you start in a cold year, like the early 1970s, or the small ice age (1500 -1800) you’ll find global warming: perhaps too much. Begin 10-15 years ago, and you’ll find no change in global temperatures.
Ice coverage data shows the same problem: take the Canadian Arctic Ice maximums, shown below. If you start your regression in 1980-83, the record ice year (green) you’ll see ice loss. If you start in 1971, the year of minimum ice (red), you’ll see ice gain. It might also be nice to incorporate physics thought a computer model of the weather, but this method doesn’t seem to help. Perhaps that’s because the physics models generally have to be fed coefficients calculated from the trend line. Using the best computers and a trend line showing ice loss, the US Navy predicted, in January 2006, that the Arctic would be ice-free by 2013. It didn’t happen; a new prediction is 2016 — something I suspect is equally unlikely. Five years ago the National Academy of Sciences predicted global warming would resume in the next year or two — it didn’t either. Garbage in -garbage out, as they say.
Arctic Ice in Northern Canada waters, 1971-2014 from the Canadian ice service 2014 is not totally in yet , but is likely to exceed 2013. If you are looking for trends, in what year do you start?
The same trend problem appears with predicting sea temperatures and el Niño, a Christmastime warming current in the Pacific ocean. This year, 2013-14, was predicted to be a super El Niño, an exceptionally hot, stormy year with exceptionally strong sea currents. Instead, there was no el Niño, and many cities saw record cold — Detroit by 9 degrees. The Antarctic ice hit record levels, stranding a ship of anti warming activists. There were record few hurricanes. As I look at the Pacific sea temperature from 1950 to the present, below, I see change, but no pattern or direction: El Nada (the nothing). If one did a regression analysis, the slope might be slightly positive or negative, but r squared, the significance, would be near zero. There is no real directionality, just noise if 1950 is the start date.
El Niño and La Niña since 1950. There is no sign that they are coming more often, or stronger. Nor is clear evidence that the ocean is warming.
This appears to be as much a fundamental problem in applied math as in climate science: when looking for a trend, where do you start, how do you handle data confidence, and how do you prevent bias? A thought I’ve had is to try to weight a regression in terms of the confidence in the data. The Canadian ice data shows that the Canadian Ice Service is less confident about their older data than the new; this is shown by the grey lines. It would be nice if some form of this confidence could be incorporated into the regression trend analysis, but I’m not sure how to do this right.
It’s not so much that I doubt global warming, but I’d like a better explanation of the calculation. Weather changes: how do you know when you’re looking at climate, not weather? The president of the US claimed that the science is established, and Prince Charles of England claimed climate skeptics were headless chickens, but it’s certainly not predictive, and that’s the normal standard of knowledge. Neither country has any statement of how one would back up their statements. If this is global warming, I’d expect it to be warm.
Robert Buxbaum, Feb 5, 2014. Here’s a post I’ve written on the scientific method, and on dealing with abnormal statistics. I’ve also written about an important recent statistical fraud against genetically modified corn. As far as energy policy, I’m inclined to prefer hydrogen over batteries, and nuclear over wind and solar. The president has promoted the opposite policy — for unexplained, “scientific” reasons.
Here’s a fairly simple model for nuclear reactor decay heat versus time. It’s based on a fractal model I came up with for dealing with the statistics of crime, fires, etc. The start was to notice that radioactive waste is typically a mixture of isotopes with different decay times and different decay heats. I then came to suspect that there would be a general fractal relation, and that the fractal relation would hold through as the elements of the mixed waste decayed to more stable, less radioactive products. After looking a bit, if seems that the fractal time characteristic is time to the 1/4 power, that is
heat output = H° exp (-at1/4).
Here H° is the heat output rate at some time =0 and “a” is a characteristic of the waste. Different waste mixes will have different values of this decay characteristic.
If nuclear waste consisted of one isotope and one decay path, the number of atoms decaying per day would decrease exponentially with time to the power of 1. If there were only one daughter product produced, and it were non-radioactive, the heat output of a sample would also decay with time to the power of 1. Thus, Heat output would equal H° exp (-at) and a plot of the log of the decay heat would be linear against linear time — you could plot it all conveniently on semi-log paper.
But nuclear waste generally consists of many radioactive components with different half lives, and these commpnents decay into other radioactive isotopes, all of whom have half-lives that vary by quite a lot. The result is that a semi-log plot is rarely helpful. Some people therefore plot radioactivity on a log-log plot, typically including a curve for each major isotope and decay mode. I find these plots hardly useful. They are certainly impossible to extrapolate. What I’d like to propose instead is a fractal variation of the original semi-log plot: a plot of the log of the heat rate against a fractal time. As shown below the use of time to the 1/4 power seems to be helpful. The plot is similar to a fractal decay model that I’d developed for crimes and fires a few weeks ago.
After-heat of nuclear fuel rods used at 20 kW/kg U; Top graph 35 MW-days/kg U; bottom graph 20 Mw-day /kg U. Data from US NRC Regulatory Guide 3.54. A typical reactor has 200,000 kg of uranium.
A plausible justification for this fractal semi-log plot is to observe that the half-life of daughter isotopes relates to the parent isotopes. Unless I find that someone else has come up with this sort of plot or analysis before, I’ll call it after myself: a Buxbaum Mandelbrot plot –Why not?
Nuclear power is attractive because it is a lot more energy dense than any normal fuel. Still the graph at right illustrates the problem of radioactive waste. With nuclear, you generate about 35 MW-days of power per kg of uranium. This is enough to power an average US home for 8 years, but it produces 1 kg of radioactive waste. Even after 81 years the waste is generating about 1/2 W of decay heat. It should be easier to handle and store the 1 kg of spent uranium than to deal with the many tons of coal-smoke produced when 35 MW-days of electricity is made from coal, still, there is reason to worry about the decay heat.
I’ve made a similar plot of decay heat of a fusion reactor, see below. Fusion looks better in this regard. A fission-based nuclear reactor to power 1/2 of Detroit, would hold some 200,000 kg of uranium that would be replaced every 5 years. Even 81 years after removal, the after-heat would be about 100 kW, and that’s a lot.
After-heat of a 4000 MWth Fusion Reactor built from niobium-1%zirconium; from UWMAC III Report. The after heat is far less than with normal uranium fission.
The plot of the after-heat of a similar power fusion reactor (right) shows a far greater slope, but the same time to the1/4 power dependence. The heat output drops from 1 MW at 3 weeks to only 100 W after 1 year and far less than 1 W after 81 years. Nuclear fusion is still a few years off, but the plot at left shows the advantages fairly clearly, I. think.
This plot was really designed to look at the statistics of crime, fires, and the need for servers / checkout people.
Dr. R.E. Buxbaum, January 2, 2014, edited Aug 30, 2022. *A final, final thought about theory from Yogi Berra: “In theory, it matches reality.”