Friday, December 28, 2018

247: Wombat Geometry

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Recently as I was watching TV with my wife, it suddenly occurred to me that we didn’t have an episode yet on one of the most important mathematical news developments of 2018.    Perhaps you’re thinking I should be featuring new developments in high-energy physics, or something about the recently awarded 2018 Fields Medal winners?    True, those would make good topics, but the one I’m thinking of came up as we were watching “Crikey It’s the Irwins”, that show about cute animals at the Australia Zoo.   I am, of course, talking about new advances in our understanding of cubic wombat dung.

Wombats, in case you haven’t seen one, are cute, furry marsupials related to koalas, and can be found mainly in Australia.    And yes, you heard me right— they are the only animals that generate cube-shaped poop.   I thought this was a joke at first, but numerous sources confirm it; you can find many photos online if you’re skeptical.    For a long time it was a mystery how wombats do this.   In real life, cubes are not an easy shape to generate:  in human-run manufacturing, you basically need to start with some kind of cubic mold, or directly cut materials into cube shapes.   So advances in our understanding of how this happens can have real economic benefits and applications to industry.    Recently a team of scientists led by Patricia Yang at Georgia Tech did some new experiments to discover exactly how this works.

You might guess at first that wombat dung is made up of some kind of crystal, since crystals are one major source of regular shapes with sharp angles in nature.   But that’s not it at all— this dung is similar smelly stuff to what other mammals generate, though it is a bit drier due to wombat metabolism, which helps it hold its shape.    You might then guess that there is effectively some kind of extrusion mold in the wombat’s bodily structure, similar to how our factories would generate that kind of shape.   But that’s not it either.   Somehow the wombats generate this cubic dung purely through soft-tissue activity, and don’t have any explicit square or cube shapes visible anywhere in their body.

Yang’s team ordered some roadkill wombat bodies from Australia, and did some experiments where they inflated balloons in their intestines to measure the elasticity, or stretchiness, at various points along their digestive tract.   They then did the same for pigs, as a control.  They found that the pigs’ intestines had roughly uniform elasticity, leading to the roundish dung generated by pigs and most other animals.   But wombats’ intestines were very irregular, containing some more and less stretchy parts, and in particular two groove-like stretchy areas.   Yang believes these are they key to shaping the dung as it travels down the tract.   Of course, there are a lot of followup questions to answer, such as how two stretchier areas lead to a full cubic shape, but the experiments are continuing.

At this point, you probably are also thinking of another question that Yang didn’t address:   *why* the wombats generate dung in this shape.   There must be some evolutionary advantage, right?    According to most online sources, the key is probably in the fact that they use the dung to mark territory.   Popular Mechanics suggests that the cubes can be stacked to build walls, but other sources note that wombats have never been observed to do this.  However, they do often leave their dung markers in precarious locations like on top of rocks or logs— this probably is able to signal to competitors from farther away than if left on the ground.    And if you want something to stay on top of a log or rock, a shape that is less likely to roll away provides a clear advantage.

I think I know the real secret, however.   Deep down in their burrows, wombats like to play Dungeons and Dragons during their downtime from foraging for food.   They must especially like playing wizards, and need lots of 6-sided dice to roll damage for their fireball spells.   Maybe as D&D geeks grow to dominate the human race, we too will develop this evolutionary manufacturing shortcut.    I prefer the smell of plastic dice personally, but I’m probably just less evolved than the common wombat.

And this has been your math mutation for today.


Sunday, December 2, 2018

246: Election Soutions Revisited

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Way back in podcast 172, I proposed a solution for the issue of contentious election recounts and legal battles over very close vote totals.   With all the anger, bitterness, and waste that handling these close-margin races causes, I thought it would be good to review this proposal again, and really give it some serious thought.

Our nation collectively spends millions of dollars during each election cycle on such issues, and the number of contentious races seems to have exploded in recent years.   One example is the recent Florida senate race, where over 8 million votes were cast, and the victory margin was within 10000, around an eighth of one percent.  Due to the increasing partisan divide, these close margins are now universally accompanied by accusations of small-scale cheating in local precincts or at the margins.   And we can’t deny that due to the all-or-nothing stakes, and the ability of a relatively tiny set of local votes to swing an election with national ramifications, the temptation for local partisans on both sides to cheat in small ways, if they spot an opportunity, must be overwhelming.   

Now a lot of people will shrug their shoulders and say these contentious battles and accompanying waste are an inevitable consequence of democracy.   But as I pointed out in the earlier podcast, if we think carefully, that’s not quite true.   The first key point to recognize, as pointed out by authors such as Charles Seife, is that there is a *margin of error* to the voting process.    Rather than saying that Rick Scott got 4,099,805 votes and Bill Nelson got 4,089,472 votes, it might make more sense just to say that each got approximately 4.1 million votes.    There are errors due to mishandled ballots, natural wear and tear, machine failures, honest mistakes, and even local small-scale cheating.   Once you admit there is a margin of error, then the silliness of recounts becomes apparent:  a recount is just a roll of the dice, introducing a different error into the count, with no real claim to be more precise or correct.   So under the current system, if you lose a close election, it would be foolish not to pour all the resources you have into forcing a recount.

But if we agree that the voting process has a margin of error, this leads to a natural solution, suggested by Seife:   if the votes are close enough, let’s just agree to forsake all the legal battles, recounts, and bitter accusations, and flip a coin.   It would be just as accurate— just as likely to reflect the true will of the people— and a much cheaper, faster, and amicable method of resolution.   But if you think carefully, you will realize that this solution alone doesn’t totally solve the issue.   Now we will repeat the previous battles wherever the vote is just over the margin that would trigger a coin flip.    For example, if we said that the victory margin had to be 51% or greater in a 1000-vote race to avoid a coin flip, and the initial count gave the victor precisely 510 votes, there would be a huge legal battle by the loser to try to shave off just one vote and trigger his random shot.

This leads to my variant of Seife’s proposal:  let’s modify the system so that there is *always* a random element added to the election, with odds that vary according to the initial vote count.   We will use a continuous bell curve, with its peak in the middle, to determine the probability of overturning the initial result.   In the middle, it would be a 50% probability, or a simple coin flip.   In our 1000-vote election, at the 510 vote mark the probability would be very close to 50-50, but a smidgen higher, depending on how we configure the curve, something like 51%.    Now the difference between a 50% chance of winning and a 49%, or 51%, will probably not seem very significant to either candidate:  rather than fighting a legal battle over the margin, they will probably want to go ahead, generate the random number, and be done with it.    Of course we will agree that once the random die is cast, with the agreement of all parties to the election, both winners and losers accept the result without future accusations or legal battles.   Due to the continuity of the curve, there will never be a case where a tiny vote margin will seem to create life-or-death stakes.   The probability of overturning the election will fall smoothly as the margin increases, until it gets down to approximately zero as the vote results approach 100%.   So Kim Jong Un would still be safe.

You can probably see the natural objection to this scheme: as you move further out from the center of the curve, having *any* possibility of overriding the result starts to seem rather undemocratic.   Do we really want to have a small chance of overturning the election of a victor who got 60% of the vote?   If this system is implemented widely, such a low-probability result probably will happen somewhere at some point.   But think about all the other random factors that can affect an election:  a sudden terrorist attack, mass layoff at a local company, random arrest of a peripheral campaign figure, a lurid tabloid story from a prostitute, a sudden revelation of Math Mutation’s endorsement—- the truth is, due to arbitrarily timed world events, there is always a random factor in elections to some degree.   This is just a slightly more explicit case.   Is it really that much less fair?    And again, think of the benefits:  in addition to saving the millions of dollars spent on legal battles and recounts, the reduction in the need for bitter partisan battles in every local precinct on close elections has got to be better for our body politic.

So, what do you think?    Is it time for our politicians to consider truly out-of-the-box solutions to heal our system?   Maybe if all the Math Mutation listeners got together, we could convince a secretary of state somewhere to try this system out.   Of course, I know I’m probably just dreaming, outright nuclear war in Broward County, Florida is a much more likely solution to this issue.   At least I’m located pretty far outside the fallout zone for that one.

And this has been your math mutation for today.