Monthly Archives: December 2013

Toxic chemistry you can do at home

I got my start on science working with a 7 chemical, chemistry set that my sister got me when I was 7 years old (thanks Beverly). The chemicals would never be sold by a US company today — too much liability. What if your child poisons himself/herself or someone else, or is allergic, or someone chokes on the caps (anything the size of a nut has to be labeled as a hazard). Many of the experiments were called magic, and they were, in the sense that, if you did them 200 years earlier, you’d be burnt as a witch. There were dramatic color changes (phenolphthalein plus base, Prussian Blue) a time-delay experiment involving cobalt, and even an experiment that (as I recall) burst into fire on its own (glycerine plus granulated potassium permanganate).

Better evil through science. If you get good at this, the military may have use of your services.

“Better the evil you know.” If you get good at this, the military may have use of your services. Yes, the American military does science.

Science kits nowadays don’t do anything magically cool like that, and they don’t really teach chemistry, either, I think. Doing magical things requires chemicals that are reasonably reactive, and that means corrosive and/or toxic. Current kits use only food products like corn-starch or baking soda, and the best you can do with these is to make goo and/ or bubbles. No one would be burnt at the stake for this, even 300 years ago. I suppose one could design a program that used these materials to teach something about flow, or nucleation, but that would require math, and the kit producers fear that any math will turn off kids and stop their parents from spending money. There is also the issue of motivation. Much of historical chemistry was driven by greed and war; these are issues that still motivate kids, but that modern set-makers would like to ignore. Instead, current kits are supposed to be exciting in a cooperative way (whatever that means), because the kit-maker says so. They are not. I went through every experiment in my first kit in the first day, and got things right within the first week — showing off to whoever would watch. Modern kits don’t motivate this sort of use; I doubt most get half-used in a lifetime.

There are some foreign-made chemistry sets still that are pretty good. Here is a link to a decent mid-range one from England. But it’s sort of pricy, and already somewhat dumbed down. Instead, here are some cheaper, more dangerous, American options: 5 experiments you can do (kids and parents together, please) using toxic household chemicals found in our US hardware stores. These are NOT the safest experiments, just cheap ones that are interesting. I’ll also try to give some math and explanations — so you’ll understand what’s happening on a deeper level — and I’ll give some financial motivation — some commercial value.

1) Crystal Drano + aluminum. Crystal Drano is available in the hardware store. It’s mostly lye, sodium hydroxide, one of the strongest bases known to man. It’s a toxic (highly poisonous) chemical used to dissolve hair and fat in a drain. It will also dissolve some metals and it will dissolve you if you get it on yourself (if you do get it on yourself, wash it off fast with lots of water). Drano also contains ammonium nitrate (an explosive) and bits of aluminum. For the most part, the aluminum is there so that the Drano will get hot in the clogged drain (heat helps it dissolve the clog faster). I’ll explain the ammonium nitrate later. For this experiment, you’re going to want to work outside, on a dinner plate on the street. You’ll use additional aluminum (aluminum foil), and you’ll get more heat and fun gases. Fold up a 1 foot square of aluminum foil to 6″ x 4″ say, and put it on the plate (outside). Put an indent in the middle of the foil making a sort of small cup — one that can stand. Into this indent, put a tablespoon or two of water plus a teaspoon of Drano. Wait about 5 minutes, and you will see that the Drano starts smoking and the aluminum foils starts to dissolve. The plate will start to get hot and you will begin to notice a bad smell (ammonia). The aluminum foil will turn black and will continue to dissolve till there is a hole in the middle of the indent. Draino

The main reaction is 2 Al + 3 H2O –> Al2O3 + H2; that is, aluminum plus water gives you aluminum oxide (alumina), and hydrogen. The sodium hydroxide (lye) in the Drano is a catalyst in this reaction, something that is not consumed in this reaction but makes it happen faster than otherwise. The hydrogen you produce here is explosive and valuable (I explain below). But there is another reaction going on too, the one that makes the bad smell. When ammonium nitrate is heated in the presence of sodium hydroxide, it reacts to make ammonia and sodium nitrate. The reaction formula is: NH4-NO3 + NaOH –> NH3 + NaNO3 + H2O. The ammonia produced gives off a smell, something that is important for safety — the smell is a warning — and (I think) helps keep the aluminum gunk from clogging the drain by reacting with the aluminum oxide to form aluminum amine hydroxide Al2O3(NH3)2. It’s a fun experiment to watch, but you can do more if you like. The hydrogen and ammonia are flammable and is useful for other experiments (below). If you collect these gases, you can can make explosions or fill a balloon that will float. Currently the US military, and several manufacturers in Asia are considering using the hydrogen created this way to power motorcycles by way of a fuel cell. There is also the Hindenburg, a zeppelin that went around the world in the 1930s. It was kept aloft by hydrogen. The ammonia you make has value too, though toxic; if bubbled into water, it makes ammonium hydroxide NH3 + H2O –> NH4OH. This is a common cleaning liquid. Just to remind you: you’re supposed to do these experiments outside to dissipate the toxic gases and to avoid an explosion in your house. A parent will come in handy if you get this stuff on your hand or in your eye.

Next experiment: check that iron does not dissolve in Drano, but it does in acid (that’s experiment 5; done with Muriatic acid from the hardware store). Try also copper, and solder (mostly tin, these days). Metals that dissolve well in Drano are near the right of the periodic table, like aluminum. Aluminum is nearly a non-metal, and thus can be expected to have an oxide that reacts with hydroxide. Iron and steel have oxides that are bases themselves, and thus don’t react with lye. This is important as otherwise Drano would destroy your iron drain, not only the hair in it. It’s somewhat hard on copper though, so beware if you’ve a copper drain.

Thought problem: based on the formulas above figure out the right mix of aluminum, NaOH, water and Ammonium nitrate. Answer: note that, for every two atoms of aluminum you dissolve, you’ll need three molecules of water (for the three O atoms), plus at least two molecules of ammonium nitrate (to provide the two NH2 (amine) groups above. You’ll also want at least 2 molecules of NaOH to have enough Na to react with the nitrate groups of the ammonium nitrate. As a first guess, assume that all atoms are the same size. A better way to do this involves molecular weights (formula weights), read a chemistry book, or look on the internet.

Four more experiments can be seen here. This post was getting to be over-long.As with this experiment, wear gloves and eye protection; don’t drink the chemicals, and if you get any chemicals on you, wash them off quick.

Here are a few more experiments in electrochemistry and biology, perhaps I’ll add more. In the meantime, if you or your child are interested in science, I’d suggest you read science books by Mr Wizard, or Isaac Asimov, and that you learn math. Another thought, take out a high school chemistry text-book at the library — preferably an old one with experiments..

Robert Buxbaum, December 29, 2013. If you are interested in weather flow, by the way, here is a bit on why tornadoes and hurricanes lift stuff up, and on how/ why they form. 

Genetically modified food not found to cause cancer.

It’s always nice when a study is retracted, especially so if the study alerts the world to a danger that is found to not exist. Retractions don’t happen often enough, I think, given that false positives should occur in at least 5% of all biological studies. Biological studies typically use 95% confidence limits, a confidence limit that indicates there will be false positives 5% of the time for the best-run versions (or 10% if both 5% tails are taken to be significant). These false positives will appear in 5-10% of all papers as an expected result of statistics, no matter how carefully the study is done, or how many rats used. Still, one hopes that researchers will check for confirmation from other researchers and other groups within the study. Neither check was not done in a well publicized, recent paper claiming genetically modified foods cause cancer. Worse yet, the experiment design was such that false positives were almost guaranteed.

Séralini published this book, “We are all Guinea Pigs,” simultaneously with the paper.

As reported in Nature, the journal Food and Chemical Toxicology retracted a 2012 paper by Gilles-Eric Séralini claiming that eating genetically modified (GM) maize causes cancerous tumors in rats despite “no evidence of fraud or intentional misrepresentation.” I would not exactly say no evidence. For one, the choice of rats and length of the study was such that a 30% of the rats would be expected to get cancer and die even under the best of circumstances. Also, Séralini failed to mention that earlier studies had come to the opposite conclusion about GM foods. Even the same journal had published a review of 12 long-term studies, between 90 days and two years, that showed no harm from GM corn or other GM crops. Those reports didn’t get much press because it is hard to get excited at good news, still you’d have hoped the journal editors would demand their review, at least, would be referenced in a paper stating the contrary.

A wonderful book on understanding the correct and incorrect uses of statistics.

A wonderful book on understanding the correct and incorrect uses of statistics.

The main problem I found is that the study was organized to virtually guarantee false positives. Séralini took 200 rats and divided them into 20 groups of 10. Taking two groups of ten (one male, one female) as a control, he fed the other 18 groups of ten various doses of genetically modified grain, either alone of mixed with roundup, a pesticide often used with GM foods. Based on pure statistics, and 95% confidence, you should expect that, out of the 18 groups fed GM grain there is a 1- .9518 chance (60%) that at least one group will show cancer increase, and a similar 60% chance that at least one group will show cancer decrease at the 95% confidence level. Séralini’s study found both these results: One group, the female rats fed with 10% GM grain and no roundup, showed cancer increase; another group, the female rats fed 33% GM grain and no roundup, showed cancer decrease — both at the 95% confidence level. Séralini then dismissed the observation of cancer decrease, and published the inflammatory article and a companion book (“We are all Guinea Pigs,” pictured above) proclaiming that GM grain causes cancer. Better editors would have forced Séralini to acknowledge the observation of cancer decrease, or demanded he analyze the data by linear regression. If he had, Séralini would have found no net cancer effect. Instead he got to publish his bad statistics, and (since non of the counter studies were mentioned) unleashed a firestorm of GM grain products pulled from store shelves.

Did Séralini knowingly design a research method aimed to produce false positives? In a sense, I’d hope so; the alternative is pure ignorance. Séralini is a long-time, anti GM-activist. He claims he used few rats because he was not expecting to find any cancer — no previous tests on GM foods had suggested a cancer risk!? But this is mis-direction; no matter how many rats in each group, if you use 20 groups this way, there is a 60% chance you’ll find at least one group with cancer at the 95% confidence limit. (This is Poisson-type statistics see here). My suspicion is that Séralini knowingly gamed the experiments in an effort to save the world from something he was sure was bad. That he was a do-gooder twisting science for the greater good.

The most common reason for retraction is that the article has appeared elsewhere, either as a substantial repeat from the authors, or from other authors by plagiarism or coincidence. (BC Comics, by Johnny Hart, 11/25/10).

It’s important to cite previous work and aspects of the current work that may undermine the story you’d like to tell; BC Comics, Johnny Hart.

This was not the only major  retraction of the month, by the way. The Harrisburg Patriot & Union retracted its 1863 review of Lincoln’s Gettysburg Address, a speech the editors originally panned as “silly remarks”, deserving “a veil of oblivion….” In a sense, it’s nice that they reconsidered, and “…have come to a different conclusion…” My guess is that the editors were originally motivated by do-gooder instinct; they hoped to shorten the war by panning the speech.

There is an entire blog devoted to retractions, by the way: A good friend, Richard Fezza alerted me to it. I went to high school with him, then through under-grad at Cooper Union, and to grad school at Princeton, where we both earned PhDs. We’ll probably end up in the same old-age home. Cooper Union tried to foster a skeptical attitude against group-think.

Robert Buxbaum, Dec 23, 2013. Here is a short essay on the correct way to do science, and how to organize experiments (randomly) to make biassed analysis less likely. I’ve also written on nearly normal statistics, and near poisson statistics. Plus on other random stuff in the science and art world: Time travel, anti-matter, the size of the universe, Surrealism, Architecture, Music.

When to enter a neighbors war or family dispute

As I write this, our favored insurgents in Syria have been over-run by our disfavored insurgents, who may be over-run by the government we are trying to topple. We have also committed to help Japan and Vietnam in their disputes with China. I’ve also had the experience of dealing with a couple going through a bitter divorce. So here are five thoughts for myself and president Obama on getting involved in other people’s problems. I’ll hope that at least one person (me) listens.

1. Learn how to wait without committing to either side so you don’t step in something really smelly. Commiserate with both sides; yes you have grievances, yes what they’ve done isn’t nice. Suggest outside review. Just don’t commit until you feel comfortable sticking with this one side in victory, defeat, or (possible) reconciliation.

In a war, even simple gifts of food or transport are support; avoid these gifts, and especially avoid gifts to both sides. Assume any support to a side will be considered treason from the other side. Supporting both sides just causes havoc, and it’s always possible that your gifts will fall in the hands of the wrong side, as in Syria.

Being helpful isn't always helpful. Matthew Deffee, The New Yorker

Being helpful isn’t always helpful, or appreciated. Learn to wait. Matthew Deffee, The New Yorker

Remind yourself that disputes are a normal part of life, that peace always comes eventually, and that disputes are sometimes good in the long run. Offer sympathy only until you really want to support one side or the other — or until they make peace. When peace comes, it’s possible that the resolution will be better than the status quo-anti. As such, perhaps long-term non-intervention is the best cure. Time often answers what wisdom does not.

2.  If you choose to support a side, only support one that openly, and traditionally supports us. No Syrian leaders have openly pledged support to the US and its allies; why ally with someone who won’t support you? The enemy of your enemy might be another enemy, as with the Taliban. In a marriage dispute, lean to support your close relative or friend — it’s less offensive than the opposite, and less likely to cause hurt. As bad as it is when two sides attack each other, it’s worse when both attack you.

Only support someone who could rule reasonably honestly and well. Chaos is worse than a dictator. Kanin from the New Yorker.

Only support someone who could rule reasonably well. Chaos is worse than a dictator. Kanin from the New Yorker.

3. If you feel it’s important to act in a neighbor’s dispute, you don’t always have to ally with either side. You can retaliate for someone blowing up a ship or killing an advisor, or beating their children by intervening at a distance. Perhaps you can use a missile (ideally against a pointless target), or sanctions, or by the UN or a volunteer force (this tends to work for the US). In family disputes, it’s often best to send a councilor or the police or child protective services. There is room to escalate or de-escalate an action like this depending on how things play out. And it’s easier to distance yourself from a 3rd party’s actions than from one’s own. It is not necessary to support either side to achieve a personal goal or protect children in a divorce.

4.  If you decide to choose sides, make sure to keep in mind the end you seek: what good you want to do, what reasonable peace you seek, then act. Do not worry that you can not do everything, but make sure you target a viable end, and that you support a side that could win and rule. Try to pick a side that’s moral and perceived as legitimate from within, but if you can’t, at least pick one that could rule the country or manage the family without your help. Don’t support a loser, or one who can’t stand on his/her own. Chaos is worse than a crooked dictator; see, for example, the French Revolution. In a fight between parents, make sure the one you support could actually raise the kids. And once the goal is achieved, don’t stay too long. If a friend tells you to go, as in Afghanistan, leave quickly. Independence is the goal we hope for — for our children, our friends, and our neighbors.

Being a fair broker of peace is a great role -- in the proper time. From the New Yorker

Being a fair broker of peace is a great role — but only for the right person in the proper time. From the New Yorker

5. Be willing to serve as an honest broker of the peace. An honest broker is very valuable, and it requires that you’re perceived as unbiassed by both sides. Wait till the right moment before offering this service, and offer it like the precious jewel it is. Offer it when asked or when the fighting dies down. If the offer is refused, be willing to go away and return to the first rule. T. Roosevelt won the Nobel peace prize for ending the Russo-Japanese war because he was a good, honest broker: someone who understood the situation and could stand back when not needed.

Robert E. Buxbaum, Dec 18, 2013. Blessed are the peacemakers. 

My failed process for wood to green gasoline

Most researchers publish the results of their successful projects, and ignore the rest. It’s an understandable failing given the cost and work to publish and the general sense that the project that flops indicated a loser – researcher. Still, it’s a shame, and I’d like to break from it here to describe a worthwhile project of mine that failed — turning wood into green gasoline. You may come to believe the project worthwhile too, and figure that you might learn from my story some pathways to avoid if you decide to try it. Besides I figure that it’s an interesting tale. All success stories are similar, I find; failure can come in many ways.

Failure can come from incorrect thinking – assumptions that are wrong. One basis of my thinking was the observation that gasoline, for the most part, was crude-oil that had been fluffed up with hydrogen. The density you buy weighs about 5.5 lb/gallon while crude oil weighs 9 lb/gallon. The difference is hydrogen. Perhaps wood too could be turned into gasoline if hydrogen were added. Another insight was that the structure of wood was the structure of a long chain -alcohol,  —(CHOH)-(CHOH)-(CHOH)—. My company had long experience breaking alcohols to make hydrogen. I figured we could do something similar with wood, fluffing up the wood by breaking the long-chain alcohols to short ones.

A possible first reaction step would be to break a C-O-C bond, a ketone bond, with hydrogen:

—(CHOH)-(CH2O)-(CHOH)— + H2 –>  —(CHOH)-CH2OH + CH2OH—

The next reaction step, I imagined was de-oxygenation:

—(CHOH)-CH2OH  +  H2 –>  —(CHOH)-CH3  + H2O

At this point, we are well on the way to making gasoline, or making a gasoline-relevant alcohol like C6H11-OH. The reactions I wanted were exothermic, meaning they would probably “go” — in actuality -∆G is the determinate of reaction favorability, but usually a -∆H and -∆G go together. Of course there are other reactions that I could have worried about –Ones that produce nasty goop. Among these:

–(CHOH)-(CH2O)-(CHOH)—  –> –(CO)-(C)-(CHOH)— + H2O +H2

I didn’t worry about these reactions because I figured I could outrun them using the right combination of a high hydrogen pressure, the right (low) temperature and the right catalyst. I may have been wrong. Then again, perhaps I picked the wrong catalyst – Fe2O3/ rust, or the wrong set of conditions. I picked Fe2O3 because it was cheap and active.

I convinced myself that Fe2O3 was sufficiently specific to get the product to a good 5-6 carbon compounds for gasoline. Wood celluloses are composed of five and six-carbon ring structure, and the cost of wood is very low per energy. What could go wrong? I figured that starting with these 5-6 carbon ring structures, virtually guaranteed getting high octane products. With the low cost and all the heat energy of the wood, wood + H2 seemed like a winning way to store and transport energy. If i got 6 carbon alcohols and similar compounds they’d have high-octane and the right vapor pressures and the products should be soluble in ordinary gasoline.

And the price was right; gasoline was about $3.50/ gallon, while wood was essentially free.  Hydrogen isn’t that expensive, even using electrolysis, and membrane reactors (a major product of our company) had the potential to make it cheaper. Wood + Hydrogen seemed like the cheaper version of syngas: CO +H2, and rust is similar to normal Fischer Tropsch catalyst. My costs would be low, and I’d expected to get better conversion since I should get fewer low molecular weight products like methane, ethane and methanol. Everything fundamental looked like it was in my favor.

With all the fundamentals in place, I figured my only problem would be to design a reasonably cheap reactor. At first I considered a fluidized bed reactor, but decided on a packed bed reactor instead, 8″ long by 3/4″ OD. This was a tube, filled with wood chips and iron oxide as a catalyst. I introduced high pressure hydrogen via a 150 psi hydrogen + 5% He mix. I hoped to see gasoline and water come out the other end. (I had the hydrogen – helium mix left over from a previous experiment, and was paying rental; otherwise I would have used pure hydrogen). I used heat tape and a controller to keep the temperature near-constant.

Controlling the temperature was key, I thought, to my aim of avoiding dehydration and the formation of new carbon-carbon bonds. At too high a temperature, the cellulose molecules would combine and lose water to form a brown high molecular weight tar called bio-oil, as well as methane and char. Bio-oil is formed the same way you form caramel from sugar, and as with sugar, it’s nothing you’d want to put in a car. If I operated at too low a temperature (or with the wrong catalyst) the reaction would be too slow, and the capital costs would be excessive. I could keep the temperature in the right (Goldilocks) temperature, I thought with the right catalyst and the right (high) hydrogen pressure.

No matter how I did this, I knew that I’d get some carbon-carbon bond formation, and perhaps a little char, but so long as it wasn’t too much it should be manageable. I figured I could hydrogenate the tar and remove the char at the end of the process. Most of the gasoline energy would come from the trees, and not the hydrogen, and there would be little hydrogen wasted forming methane. Trees would always be cheap: they grow quickly, and are great at capturing solar energy. Many cities pay for disposal of their tree waste, so perhaps a city would pay us to take their wood chips. With cheap wood, the economics would be good so long as used all the hydrogen I put in, and got some reasonable fraction of energy from the wood. 

i began my reaction at 150°C with 50 psi hydrogen. At these conditions, I saw no reaction; I then raised the temperature to 200°C, then raised the pressure to 100 psi (still nothing) and then tried 250°C, still at 100psi. By now we were producing water but it was impossible to tell if we were hydrogenating the cellulose to gasoline, or dehydrating the cellulose to bio-oil.

As it turned out we were getting something worse that bio-oil: bio-oil gunk. Instead of the nasty brown liquid that’s made when wood is cooked to dehydration (water removal, caramelization), I got something that was nastier than I’d imagined possible. The wood molecules did not form nice chains but combined to form acidic, aromatic gunk (aromatic in both senses: benzine-type molecules and smelly) that still contained unreacted wood as a sort of press-board. The gunk was corrosive and reactive; it probably contained phenol, and seemed bent on reacting to form a phenolic glue. I found the gunk was insoluble in most everything: water, gasoline, oil, methanol (the only exception was ethanol). As best I can tell, you can not react this gunk with hydrogen to make gasoline as it is non-volatile, and almost impossible to get out of my clogged reactor. Perhaps a fluidized bed would be would be better, but I’m afraid it would form wood clumps even there. 

I plan to try again, perhaps using higher pressure hydrogen and perhaps a liquid hydrogen carrier, to get the hydrogen to the core of the wood and speed the catalysis of the desired products. The key is finding a carrier that is not too expensive or that can be easily recovered.

Robert E. Buxbaum, Dec 13, 2013. Here’s something on a visit to my lab, on adding hydrogen to automobile engines, and on the right way to do science. And here’s my calculation for how much wood a woodchuck chucks if a woodchuck could chuck wood, (100 lbs/ night) plus why woodchucks do not chuck wood like beavers.

Near-Poisson statistics: how many police – firemen for a small city?

In a previous post, I dealt with the nearly-normal statistics of common things, like river crests, and explained why 100 year floods come more often than once every hundred years. As is not uncommon, the data was sort-of like a normal distribution, but deviated at the tail (the fantastic tail of the abnormal distribution). But now I’d like to present my take on a sort of statistics that (I think) should be used for the common problem of uncommon events: car crashes, fires, epidemics, wars…

Normally the mathematics used for these processes is Poisson statistics, and occasionally exponential statistics. I think these approaches lead to incorrect conclusions when applied to real-world cases of interest, e.g. choosing the size of a police force or fire department of a small town that rarely sees any crime or fire. This is relevant to Oak Park Michigan (where I live). I’ll show you how it’s treated by Poisson, and will then suggest a simpler way that’s more relevant.

First, consider an idealized version of Oak Park, Michigan (a semi-true version until the 1980s): the town had a small police department and a small fire department that saw only occasional crimes or fires, all of which required only 2 or 4 people respectively. Lets imagine that the likelihood of having one small fire at a given time is x = 5%, and that of having a violent crime is y =5% (it was 6% in 2011). A police department will need to have to have 2 policemen on call at all times, but will want 4 on the 0.25% chance that there are two simultaneous crimes (.05 x .05 = .0025); the fire department will want 8 souls on call at all times for the same reason. Either department will use the other 95% of their officers dealing with training, paperwork, investigations of less-immediate cases, care of equipment, and visiting schools, but this number on call is needed for immediate response. As there are 8760 hours per year and the police and fire workers only work 2000 hours, you’ll need at least 4.4 times this many officers. We’ll add some more for administration and sick-day relief, and predict a total staff of 20 police and 40 firemen. This is, more or less, what it was in the 1980s.

If each fire or violent crime took 3 hours (1/8 of a day), you’ll find that the entire on-call staff was busy 7.3 times per year (8x365x.0025 = 7.3), or a bit more since there is likely a seasonal effect, and since fires and violent crimes don’t fall into neat time slots. Having 3 fires or violent crimes simultaneously was very rare — and for those rare times, you could call on nearby communities, or do triage.

In response to austerity (towns always overspend in the good times, and come up short later), Oak Park realized it could use fewer employees if they combined the police and fire departments into an entity renamed “Public safety.” With 45-55 employees assigned to combined police / fire duty they’d still be able to handle the few violent crimes and fires. The sum of these events occurs 10% of the time, and we can apply the sort of statistics above to suggest that about 91% of the time there will be neither a fire nor violent crime; about 9% of the time there will be one or more fires or violent crimes (there is a 5% chance for each, but also a chance that 2 happen simultaneously). At least two events will occur 0.9% of the time (2 fires, 2 crimes or one of each), and they will have 3 or more events .09% of the time, or twice per year. The combined force allowed fewer responders since it was only rarely that 4 events happened simultaneously, and some of those were 4 crimes or 3 crimes and a fire — events that needed fewer responders. Your only real worry was when you have 3 fires, something that should happen every 3 years, or so, an acceptable risk at the time.

Before going to what caused this model of police and fire service to break down as Oak Park got bigger, I should explain Poisson statistics, exponential Statistics, and Power Law/ Fractal Statistics. The only type of statistics taught for dealing with crime like this is Poisson statistics, a type that works well when the events happen so suddenly and pass so briefly that we can claim to be interested in only how often we will see multiples of them in a period of time. The Poisson distribution formula is, P = rke/r! where P is the Probability of having some number of events, r is the total number of events divided by the total number of periods, and k is the number of events we are interested in.

Using the data above for a period-time of 3 hours, we can say that r= .1, and the likelihood of zero, one, or two events begin in the 3 hour period is 90.4%, 9.04% and 0.45%. These numbers are reasonable in terms of when events happen, but they are irrelevant to the problem anyone is really interested in: what resources are needed to come to the aid of the victims. That’s the problem with Poisson statistics: it treats something that no one cares about (when the thing start), and under-predicts the important things, like how often you’ll have multiple events in-progress. For 4 events, Poisson statistics predicts it happens only .00037% of the time — true enough, but irrelevant in terms of how often multiple teams are needed out on the job. We need four teams no matter if the 4 events began in a single 3 hour period or in close succession in two adjoining periods. The events take time to deal with, and the time overlaps.

The way I’d dealt with these events, above, suggests a power law approach. In this case, each likelihood was 1/10 the previous, and the probability P = .9 x10-k . This is called power law statistics. I’ve never seen it taught, though it appears very briefly in Wikipedia. Those who like math can re-write the above relation as log10P = log10 .9 -k.

One can generalize the above so that, for example, the decay rate can be 1/8 and not 1/10 (that is the chance of having k+1 events is 1/8 that of having k events). In this case, we could say that P = 7/8 x 8-k , or more generally that log10P = log10 A –kβ. Here k is the number of teams required at any time, β is a free variable, and Α = 1-10 because the sum of all probabilities has to equal 100%.

In college math, when behaviors like this appear, they are incorrectly translated into differential form to create “exponential statistics.” One begins by saying ∂P/∂k = -βP, where β = .9 as before, or remains some free-floating term. Everything looks fine until we integrate and set the total to 100%. We find that P = 1/λ e-kλ for k ≥ 0. This looks the same as before except that the pre-exponential always comes out wrong. In the above, the chance of having 0 events turns out to be 111%. Exponential statistics has the advantage (or disadvantage) that we find a non-zero possibility of having 1/100 of a fire, or 3.14159 crimes at a given time. We assign excessive likelihoods for fractional events and end up predicting artificially low likelihoods for the discrete events we are interested in except going away from a calculus that assumes continuity in a world where there is none. Discrete math is better than calculus here.

I now wish to generalize the power law statistics, to something similar but more robust. I’ll call my development fractal statistics (there’s already a section called fractal statistics on Wikipedia, but it’s really power-law statistics; mine will be different). Fractals were championed by Benoit B. Mandelbrot (who’s middle initial, according to the old joke, stood for Benoit B. Mandelbrot). Many random processes look fractal, e.g. the stock market. Before going here, I’d like to recall that the motivation for all this is figuring out how many people to hire for a police /fire force; we are not interested in any other irrelevant factoid, like how many calls of a certain type come in during a period of time.

To choose the size of the force, lets estimate how many times per year some number of people are needed simultaneously now that the city has bigger buildings and is seeing a few larger fires, and crimes. Lets assume that the larger fires and crimes occur only .05% of the time but might require 15 officers or more. Being prepared for even one event of this size will require expanding the force to about 80 men; 50% more than we have today, but we find that this expansion isn’t enough to cover the 0.0025% of the time when we will have two such major events simultaneously. That would require a 160 man fire-squad, and we still could not deal with two major fires and a simultaneous assault, or with a strike, or a lot of people who take sick at the same time. 

To treat this situation mathematically, we’ll say that the number times per year where a certain number of people are need, relates to the number of people based on a simple modification of the power law statistics. Thus:  log10N = A – βθ  where A and β are constants, N is the number of times per year that some number of officers are needed, and θ is the number of officers needed. To solve for the constants, plot the experimental values on a semi-log scale, and find the best straight line: -β is the slope and A  is the intercept. If the line is really straight, you are now done, and I would say that the fractal order is 1. But from the above discussion, I don’t expect this line to be straight. Rather I expect it to curve upward at high θ: there will be a tail where you require a higher number of officers. One might be tempted to modify the above by adding a term like but this will cause problems at very high θ. Thus, I’d suggest a fractal fix.

My fractal modification of the equation above is the following: log10N = A-βθ-w where A and β are similar to the power law coefficients and w is the fractal order of the decay, a coefficient that I expect to be slightly less than 1. To solve for the coefficients, pick a value of w, and find the best fits for A and β as before. The right value of w is the one that results in the straightest line fit. The equation above does not look like anything I’ve seen quite, or anything like the one shown in Wikipedia under the heading of fractal statistics, but I believe it to be correct — or at least useful.

To treat this politically is more difficult than treating it mathematically. I suspect we will have to combine our police and fire department with those of surrounding towns, and this will likely require our city to revert to a pure police department and a pure fire department. We can’t expect other cities specialists to work with our generalists particularly well. It may also mean payments to other cities, plus (perhaps) standardizing salaries and staffing. This should save money for Oak Park and should provide better service as specialists tend to do their jobs better than generalists (they also tend to be safer). But the change goes against the desire (need) of our local politicians to hand out favors of money and jobs to their friends. Keeping a non-specialized force costs lives as well as money but that doesn’t mean we’re likely to change soon.

Robert E. Buxbaum  December 6, 2013. My two previous posts are on how to climb a ladder safely, and on the relationship between mustaches in WWII: mustache men do things, and those with similar mustache styles get along best.