Monday, October 29, 2018

Thesis Psalm

Some time ago while re-reading an old John Barrow volume, “Pi In the Sky” I ran across this delightful bit of parody at the end of a section essentially on the foundational math conflict between formalism and intuitionism:

The Lord is my thesis adviser; I shall not err.
He arranges for me to be published in the
respectable journals; he teaches me how
to use the reductio argument.
He enshores my validity; he leads me by
the classical logic, for the truth's sake.
Yea, though I walk through the valley of
the existence proofs, I will fear no contradiction;
for he edits my work.
The Axiom of Choice and Zorn's Lemma, they comfort me.

He invites a colloquium on classical analysis,
for my participation, in the absence of the constructivists;
He frequently and approvingly abstracts me
in Mathematical Reviews;
my reputation flourishes internationally.
Surely, honors and grants shall follow me
all the days of my career,
and I shall rise in the ranks of the
to Emeritus.


[the verse was not attributed to anyone, so I’m not clear if Barrow penned it himself — a quick internet search didn’t help, but if someone knows the author is other than Barrow please let us know]

Friday, October 26, 2018

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Monday, October 22, 2018

Ponytail Math

(via WikimediaCommons & Quinn Dombrowski)
The most recent issue of Chalkdust Magazine (downloadable as a pdf) includes an article on the mathematics of ponytails (because, of course, math IS everywhere ;)

I’ve always liked ponytails (on females), and imagine most guys do; never cared much for pigtails, braids, bouffants, beehives, or a bunch of other contortions women put their hair through, but simple ponytails are coool! And according to the above piece, ponytails have “Rapunzel numbers” that help achieve a wide, bushy effect under the influence of gravity (and yes, there's a specific equation for it)… so read up! 
This got me to wondering if anyone else had looked at ponytails from a mathematical angle. And googling a bit, wouldn’t you know, Keith Devlin addressed the topic for NPR back in 2012:

Wherein he writes in part:
Unilever makes hair care products. And we're talking about a $40 billion global market in shampoos and conditioners. So they got some mathematicians to come up with a mathematical formula to understand how to make more attractive ponytails so that the ponytail curves around in a beautiful, sexy, attractive arc, rather than just hanging limply. That's a $40 billion question.”
Keith never specifically mentions the “Rapunzel” factor but does conclude that density, elasticity, and curliness, are the 3 hair fiber components to an attractive ponytail.
The abstract for the actual British study Dr. Devlin was reporting on is here:
Worth noting too that this study won the Ig Nobel Prize for physics in 2012!: ;)  

Finally, an updated 2017 version of the topic (from one of the original authors, and including plenty of real math) is here:
It turns out that even the Wikipedia page for "ponytail" includes a section on the ponytail math/equation:

Now, if someone would just kindly point me to some mathematical studies of male pattern baldness (sometimes known as the non-hairy ball theorem)…. 
[also, lots more to enjoy in the current issue of Chalkdust, linked to above].

Friday, October 19, 2018

Chi-i-i-i-i-i-ll Friday *

[*  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Wednesday, October 17, 2018

Seeing Ain't Believing...

             "Somethin's happenin' here; what it is ain't exactly clear..."
                                                        -- Buffalo Springfield

At some point last week John Baez sent out this tweet and optical illusion and I found it quite remarkable:

There are a great many such “motion” illusions. They are fascinating and a constant reminder of how easily manipulated human perception is.
Here’s another favorite of mine:

Some additional ones can be seen here (and there are many more):

Anyway, a day or two after John's post Mike Lawler (and others) tweeted out this “motion” illusion, remarking how amazing it was; some even calling it one of the strongest illusions they had ever seen! 

Yet I, and some other folks got little from this example.  The prior examples are those of motion that is mis-perceived, while this is one where motion that doesn’t even occur is perceived to take place. There are many examples of this genre of illusion, and I’ve probably seen at least 20+ I find more effective than this one — you can google “motion optical illusions” to find many; embiggen them for better effect (...FWIW, I also always had problems seeing the old “magic eye” pictures). For that matter, there are a multitude of static (motionless) optical illusions I find far more impressive than the above heralded illusion.
So I’m very curious what is going on with those of us who see little motion with this particular one? What divides the individuals who do and don’t “fall” for this illusion? Are there divisions along gender or age, left-handed vs. right-handed, Chinese-speaking vs. English, or other significant categories? Some explanations lean toward differences in depth perception... but then do those differences themselves fall along any category lines, or are they just randomly dispersed in the population?
We know from court cases, and psychology experiments for that matter, how weak eyewitness testimony can be even though people have great confidence in their visual conclusions. I wonder too, more generally, what differences may exist between people who seem to handle high-level abstraction (as in math) fairly easily, and those who have great difficulty dealing with concepts they can't "picture" in their head -- is there a difference in how these groups "see" certain optical illusions? So many questions???
I suspect the differences in responses to various illusions has been well-studied before, perhaps even some PhD. theses; it certainly seems fodder for analysis.

Meanwhile, not at all an illusion, but simply a mesmerizing visual delight, I'll close out with this deeeep zoom into the Mandelbrot Set... enjoy!:

Sunday, October 14, 2018

Good Math + Bad Drawings = Great Book!

   At some point awhile back I wrote that Ben Orlin’s mind was either a marvel or a mutation… and I wasn’t sure which. Having finished his marvelous new book I’ll now opt for the former (though a final check with his wife and 23-and-Me might still be in order).
Won’t do a full review of “Math With Bad Drawings” but just make some comments/blurbs about the volume. The first of which is simply, BUY this book, if you enjoy math… or, want to enjoy math. It is suitable for a wide swathe of readers, from young people to college professors — all will gain insights from it, just different ones depending on your age and knowledge. Younger folks can be introduced to a lot of real life math, while adults/teachers/profs can learn new ways/examples with which to approach or think about topics. Its broad possible appeal reminds me a bit of Steve Strogatz's "The Joy of X," another sort of everyman book.
Despite only 5 sections (comprising 24 chapters) the volume touches on an impressive array of topics… with consistent wit, insight, humor, clarity. If you’re not already a Ben Orlin fan you’ll be one by the time you finish these 350 grin-inducing pages. You can figure on 4 chuckles per page (OK, so maybe 7 chuckles on some pages and 1 on others). Ben is simply one of the cleverest writers of math content going… you know this already if you’ve followed his blog for some time.
Amazingly, the book is all NEW material. When he originally announced the volume I assumed it would be a compendium of the best stuff from his blog — something surely to look forward to. Then he said it was all fresh material not from the blog, and I was amazed, even skeptical… and sad to think that so much great prior content would not be included; BUUUUT, it does not disappoint!  I especially like the middle parts on probability and statistics, which represent a large percentage of the book, and which are frequent/vital topics these days, that Ben still manages new ways to present.
Indeed Dr. Orlin seems incapable of writing bad or boring content; you get a feeling of precision, quality, tight control throughout this volume, despite the ever-present quirkiness… and bad (or, deceptively good) drawings. And the publisher, Black Dog & Leventhal, has simply done a fabulous job of presentation here — seriously, one of the best publishing feats in popular math I’ve ever seen, with, shall I say, oddball material. When I interviewed Ben back HERE, he mentioned the publisher did a great job, noting:
…they make gorgeous, colorful books so luscious you want to eat them. Having seen the final product I am 300% sure it was the right call.”
I thought he might be exaggerating… but, no, he wasn’t. The book is luscious (initially I had an urge to slather it with whipped cream ;).
My only beef at all with the volume is that there is no index provided. Dr. Orlin covers a rollicking range of topics, but if you want to look one up to see if it’s included and go directly to it, you’re out of luck (though the table of contents at front gives at least some guidance).
I’d be hardpressed to think of any math volume as original and creative as this one. And the best news may be that Ben is working on a followup volume, introducing calculus… with, of course, deliciously bad drawings. Pass me the derivatives and gravy!

Friday, October 12, 2018

Chi-i-i-i-i-i-ll Friday *

[*  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

[...on Sunday evening, or Monday morning at latest, I'll say a little more about Ben Orlin's new book, "Math With Bad Drawings"]

Wednesday, October 10, 2018

Can I Hear You Now?...

No math today, just a personal anecdote perhaps having relevance to hearing-impaired others. Over the years I’ve had casual conversations with a number of scientists, mathematicians, and personal acquaintances, about hearing loss (afflicting close to 50 million folks as they get older), and what follows is just a personal experience from this sample-size of one:
[hearing loss comes in a multitude of forms, so my experience may not apply to others]

Close to 20 years ago I awoke one morning and quickly realized much of my hearing in one ear was gone! Bingo. (other ear unaffected) After some retrospection I realized my hearing had probably been gradually declining over the prior year and simply passed some threshold that made it readily apparent to me that morning. At any rate, the doctor could figure out no cause for the loss and was quite baffled, even though I logically deduced what I believe with near certainty was the cause (but won't take time to explain here) — every hearing doctor/otolaryngologist I’ve talked to says my explanation isn't valid, which frankly, only makes me distrustful of their understanding of hearing complexities! 

The doctor further said that the kind of hearing loss I had would get gradually worse over time, and that hearing aids would do little good; eventually a cochlear implant would be needed. Also, I should be checked every year to track the progression of the loss. 

Stubborn oaf that I am (when it comes to many medical matters) I’ve never been re-checked in the almost 20 years since, but have watched my hearing very slowly decline further. Hearing aids generally aren’t covered by insurance and are expensive, on the order of $2000 - $3000 or more including all the office visits — there are additional maintenance costs along the way, and they may only be good for 5 years or less, before you need to upgrade. In short, there are a few dozen causes (even more in the Trump era) I’d prefer to see that money go to than to my right ear (and getting by with one good ear is do-able, if limiting).

Over the years I’ve tried a few cheap Personal Sound Amplifier Products (PSAPs) sold over-the-counter, which are quick to emphasize they are NOT “hearing aids” but simply sound amplifiers.  They do what they say they’ll do — amplify sound, all sound (although you can purchase more expensive models that have some filtering and directional capabilities). I’ve been surprised though by how helpful they are — for listening to a TV, lecture, movie, or other point-source of sound; they are less helpful in noisy rooms/gatherings, and the like. So even the cheapest ones, ~$20, when compared to thousands of dollars for hearing aids or tens of thousands of dollars for a cochlear implant, actually have some value. They can however be awkward, inconvenient, and eat through expensive batteries, and be less usable during athletics or other physical activities, so I was never terribly thrilled with them. The quality and variety of these products seems to be significantly improving however, with time and expanding customer-base. 

Recently I had occasion to get one (again for $20) that is small, discreet, usable during many sports, and best-of-all, re-chargeable (no need to buy batteries every month). I’ve been pleasantly surprised at just how well it works — each year I’ve gotten closer to looking into cochlear implants, but at least for the moment, feel I can once again put that off for quite awhile.
I won’t name the cheapie model I got that beat my expectations, but will say there are LOTS of these small behind-the-ear PSAPs available now, ranging up to several hundred dollars (and given how well this $20 model works for me, I’d definitely consider upgrading to a more expensive one later). If you Google “hearing amplifiers” you’ll learn of a slew of them with varying features/styles (and various pros and cons).

One reason I put off for many years buying this particular device is because it not only looked and was cheap (and made in China), but I saw several bad reviews of it on the Web (also saw good reviews, but thought the number of bad reviews was telling). In retrospect, my cynical-self now wonders if any (many?) of those bad reviews are ‘planted’ by the hearing-aid industry who probaby don’t want folks looking at $20-$100 devices that might do some good, when they’re trying to sell you $2000+ devices? (and many of my friends with real hearing aids have significant issues with them, even after spending that amount of money). Of course you have to be somewhat leery of all reviews on the internet anyway! As the population continues to age, hearing aids are definitely a boom product with huge profits to be made, and a lot of competitors vying for a slice of those profits.

If you’ve been putting off dealing with hearing loss (as most do) for the usual sorts of reasons, these more-reasonably priced devices may be worth exploring (while also acknowledging that a do-it-yourself approach is not advisable in many cases). I've had good luck my entire adult life treating myself for typical health ailments, while also sensing when professional medical intervention may be called for. Your mileage may vary. ;) And if one day self-care or treatment costs me dearly, well, que sera sera... (I'm stoic about such matters).
Anyway, as someone who more-often-than-not is disappointed in products I purchase, this has been an instance of low expectations being happily well-exceeded... just a personal anecdote, for any who may get some encouragement from it.

ADDENDUM:  just a bit of update (1/5/19) for whatever it's worth…
after 4 months of frequent use the $20 PSAP became a little more glitchy… requiring a bit of fidgeting and more frequent battery re-charges — still, I was quite pleased with it, especially given that I wore it during sports and heavy sweating at the end of summer, despite cautions that it shouldn’t (because of electronics) be exposed to dampness. I still use it part of every day.  But I was curious what a more expensive model might do, so after much research ordered another model of PSAP for ~$75. It’s slightly sturdier, but also slightly bulkier (with a less snug, comfortable fit, especially for all-day wear); it gives slightly improved hearing features/sound filtering, but also after a few weeks required more frequent re-charging and fidgeting, and combined with the lower comfort-level (for me) I can’t view it as 3-4 times “better” than my prior choice (though it was ~4 times costlier). I won’t likely need another PSAP anytime soon, but when I do I’ll be torn between simply going back for another $20 model or trying out something in the $120 - $150 range. As time goes on I can only imagine these devices will continue to improve, become even more readily-available, and lower in price.  I also suspect the selection of re-chargeable models will continue to grow -- currently models running on batteries remain the bulk of choices.
[One further note I'll make is that both devices I currently have were made in China, and I do now wonder if ones produced in Germany or Japan might have better overall quality control and be worth looking at more closely.  Also, after much wear, the re-chargeable products I've chosen, require more frequent charging than they did initially -- it may be worth considering battery operated versions, of which there are MANY, even though the cost of batteries would be recurring.]

Sunday, October 7, 2018

Insure THIS!

I found Chapter 14 (on “Weird Insurance”) of Ben Orlin’s delightful volume “Math With Bad Drawings” to be one of the most fun (of so many). It falls within an overall section on probability, and describes several non-standard insurances I’d never given much thought to. One of them he calls, “Oh No, My Employees All Won the Lottery” Insurance. We’ve all read stories in recent years of employees at a workplace banding together to purchase many tickets in some multi-million dollar lottery and then if one person wins, splitting the proceeds… and, then turning in their job resignations en masse.  As Ben puts it, it’s “like some kind of natural disaster” for the manager/proprietor of such a business. Oy.

Ben goes on to report that “In the 1990s, over a thousand British businesses invested in the National Lottery Contingency Scheme. If your staff won the jackpot, insurance payouts would help to cover recruitment, retraining, revenue losses, and temporary labor.

Ben also points out that the employer could potentially self-insure by simply joining the employee pool and being one of the winner recipients (of course that assumes the employees even let the boss know what they're up to: "hey Boss, we're all joining together to win the lottery and quit this blankety-blank, friggin' workplace!").

…And if that’s not weird enough for you, shortly thereafter comes a section on “Alien-Abduction Insurance” :)  [...where Ben reports that a British company that sold 37,000 such plans hasn’t made a single payout yet (go figure!), and one of the managers involved says, “I’ve never been afraid of parting the feeble-minded from their cash”... so much for British understatement.]
Yeah, insurance is an interesting, creative business (selling you something you hope to never have to use).

Anyway, Ben’s book is a chuckle-a-minute, so if you want to insure against boredom, get it!
[The sections on probability and statistics constitute close to half the book and are especially fantastic, but the entire volume is a joy ride!]

Friday, October 5, 2018

Chi-i-i-i-i-i-ll Friday *

[*  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]