Computer Poetry? Oh 01101110 01101111 01100101 01110100 01110010 01111001!

February 28, 2014

Computer poetry isn’t bad poetry. In fact, it’s not even un-human in many cases:

There’s some amazing poetry on the linked to site, botpoet.com, and I encourage you to check it out.

Can you write a program to create a human-like poem? Can you write a poem that’s totally computer-like? Put your attempts in the comments!

%   Created by David A. Gross. Copyright 2014.

T = 15;    B = 5;    L = 10;
enjamb = toeplitz(1:(T+2*B),(T+2*B+1):-1:2)';
enjamb = enjamb(randi(10,T+2*B,1),:);
A = [ repmat(' ',L,B) ...
      reshape(char(randi(255,L*T,1)),L,T) ...
      repmat(' ',L,B)];
A(end+1,:) = [repmat(' ',1,T+2*B-13) ' said the cat'];
A(sub2ind( ...
    [L+1,T+2*B], ...
    repmat((1:L+1)',1,T+2*B), ...
    enjamb(1:L+1,:)))

gives us:

          ñ          2s˜]ŸÊUK²]2©
          Pb          \caRRH_Y¡ô
          t          W[L‹Nžç{mڅõK
                …yþ!¸rÚg€ÞÍÐ?
          Ì          D—rŸŽR¹ƒ¬¿-+Å
          ý          ×JA}1¥ŒD©¨ëë
                –ِWhò+uÝßÚCE
                    õû–©˜¤?ç§ö˜jê F
                 ZAB"rÄ\[%4
                  /­M
          ¬#ÁW²ú*áØ
                       said the cat

Enter the Rosser matrix

January 8, 2014

I love matrices. They can encode love affairs, process images – heck, things like representation theory let us use matrices for practically anything.

Rosser Matrix

I also watch Cleve Moler‘s MathWorks blog, Cleve’s Corner, like a hawk. So when he recently posted about the Rosser Matrix I was left disappointed by what he didn’t talk about. The matrix itself is interesting because of its place in eigenvalue history. Eigenvalue: the word is just awesome. If it’s not comfortable for you, just think of the eigenvalues of a matrix like they are your…values. When you go out in the world, you make an impact and push things in the direction of the values you believe in, and certain values are more important to you than others. Matrices do the same thing with the (eigen)values they espouse.

So it’d be great if we could compute the eigenvalues of a matrix: they tell us a lot (or at least something) about who they are. These days, this is straightforward, there are many (even free) computational tools to do it. Back in the day, however, eigenvalues were a difficult thing to find, and some were harder than others. For example, eigenvalues that are really close to one another are hard to pin down precisely, and when an eigenvalue is repeated (that’s a thing) we’d like to find every copy of it.

So, back to Rosser. He makes this test matrix in 1950 that’s got a lot of good stuff in there that they could compute exactly:

  • A double eigenvalue.
  • Three nearly equal eigenvalues.
  • Dominant eigenvalue of opposite sign.
  • A zero eigenvalue.
  • A small, nonzero eigenvalue.

Then they could benchmark proposed eigenvalue-finding-algorithms (which would run for days on behemoth computers) against how close they were to the actual eigenvalues.

I love this steampunk mathematics, but the juiciest parts seemed to be left out of Cleve’s post: what algorithms were they actually using back then and (more importantly) how does one make a test matrix? It appeared that it wasn’t just Cleve leaving out the good stuff either, MATLAB itself doesn’t tell us anything interesting about how to make the Rosser matrix:

%   Copyright 1984-2005 The MathWorks, Inc.
%   $Revision: 5.10.4.2 $  $Date: 2005/11/18 14:15:39 $

R  = [ 611.  196. -192.  407.   -8.  -52.  -49.   29.
       196.  899.  113. -192.  -71.  -43.   -8.  -44.
      -192.  113.  899.  196.   61.   49.    8.   52.
       407. -192.  196.  611.    8.   44.   59.  -23.
        -8.  -71.   61.    8.  411. -599.  208.  208.
       -52.  -43.   49.   44. -599.  411.  208.  208.
       -49.   -8.    8.   59.  208.  208.   99. -911.
        29.  -44.   52.  -23.  208.  208. -911.   99.];

After more slightly digging than expected, I found Rosser’s original paper on the subject (and an incredible bible of math I hadn’t heard of before). The first thing I noticed was that there were many other people involved than just Rosser, none of which were slouches: Lanczos has eponymous algorithms, Hestenes with him crushed some linear systems, and Karush killed it at nonlinear programming. Another name I saw which deserves mention here is in the footnote below:

Miss Fannie M. Gordon, numerical analyis

There’s isn’t much on the internet about Miss Gordon, but it appears she was working at INA along with Lanczos. In his paper on “his” algorithm (not yet named as such) to which the Rosser matrix paper is a direct follow-on, another footnote talks about her in much more grateful detail:

Indebted to Miss Gordon

While she didn’t go down in the record books like Lanczos and friends, it’s great to see that her work behind the scenes was appreciated and talked about, a part of mathematical history we don’t talk about now as much as we should. For another peak into this corner of the mathematical world, check out the list of all of the NBS/NIST staff members mentioned in A Century of Excellence in Measurements, Standards, and Technology: A Chronicle of Selected NBS/NIST Publications, 1901-2000 [Text, Google Books].

With all this information at my fingertips, I could get a much clearer picture of how to get your hands dirty and find an eigenvalue. It’s only in another appendix, however, that Rosser tells us how to make actually make a test matrix, the key ingredient that was used to benchmark algorithms across decades of computational and mathematical advancement. There, on the bottom of page 293, are the 64 entries of the matrix (color coded in the image above), just as they are in rosser.m:

The original Rosser matrix

I had to see how it actually worked, so in the paste below you’ll find a MATLAB Rosser recipe, the way sausage is actually made (you can skip the code for a visual explanation):

%   Created by David A. Gross. Inspired by [1].
%   Construction from [2,3].
%
%REFERENCES
%   [1] http://blogs.mathworks.com/cleve/2014/01/06/ ...
%   the-rosser-matrix/, accessed on 2014/01/07
%
%   [2] Rosser, J.B.; Lanczos, C.; Hestenes, M.R.; Karush, W.
%   Separation of close eigenvalues of a real symmetric matrix
%   (1951), J. Res. Natl. Bur. Stand., Vol. 47, No. 4, p. 291,
%   Appendix 1, https://archive.org/details/jresv47n4p291,
%   accessed on 2014/01/07
%
%   [3] T. Muir, History of Determinants III, 289 (Macmillan
%   and Co., Ltd., London, 1920), http://igm.univ-mlv.fr/ ...
%   ~al/Classiques/Muir/History_3/, accessed on 2014/01/07

% make our eigenvalues in 2x2 matrices
M1 = [102  1 ;  1 -102]; % lambda = ± sqrt(102^2 + 1)
M2 = [101  1 ;  1  101]; % lambda = 101 ± 1
M3 = [  1 10 ; 10  101]; % lambda = 51 ± sqrt(51^2-1)
M4 = [ 98 14 ; 14    2]; % lambda = 100, 0

B = zeros(8);

% explode M[1...4] into an 8x8 matrix
B([1,6],[1,6]) = M1;
B([2,8],[2,8]) = M2;
B([4,5],[4,5]) = M3;
B([3,7],[3,7]) = M4;

sylvester88_A = @(a,b,c,d) [ ...
    a  b  c  d ; ...
    b -a -d  c ; ...
    c  d -a -b ; ...
    d -c  b -a ];

sylvester44 = @(a,b,c,d) [ ...
    a  b  c  d ; ...
    b -a  d -c ; ...
    c -d -a  b ; ...
    d  c -b -a ];

% make Sylvester's "penorthogonant" of determinant 10^8
P = blkdiag(sylvester88_A(2,1,1,2),sylvester44(1,-1,-2,2));

% P'*P = 10I
R = P'*B*P;

It’s quite cool, actually. Four 2×2 symmetric matrices are constructed to have the desired eigenvalues, and those matrices are exploded into an 8×8 sparse matrix (sparse in that it’s all zeros where there aren’t any dots):

How the sausage matrix gets made

Lastly, a special matrix (magenta & yellow, above) is smashed on either side of our sparse matrix and BAM!–you’ve got yourself a full test matrix with the eigenvalues you wanted.

There’s a lot of this that’s wonderfully clear and clever, in hindsight: how Rosser forced and hand-calculated the eigenvalues he wanted, how he kept the matrix symmetric. But there are many things that were left up to Rosser to decide, almost artistically, about how the matrix should be made. The special smashing matrix, for example, actually scales up the eigenvalues of the original four matrices by a constant factor. A guy named Sylvester said that it was easy to make it scale up by powers of 2, but that if you were careful you could make a matrix that scales up by any number you want. Rosser had to cleverly find the integer entries of that special matrix that would give him a scale that meaningfully preserved the original eigenvalues he picked (for usability and clarity) and he chose to scale them up by 10.

Another artistic choice Rosser made was how to explode the original matrices into the sparse 8×8 matrix. What I mean is all of the following matrix explosions:

Matrix explosions

(and 2,511 other possibilities) have the same eigenvalues, but would make completely different full matrices after being smashed. Computational eigenvalue history would have looked very–well, slightly–different had Rosser picked any of these as the base for his test matrix. Maybe there’s something deeper to uncover here about his choices, but I’d like to think that Rosser loved matrices as much as I do, and that’s just one that he liked more than the rest.

If you don’t love matrices now, that’s ok. What originally started as a small coding exercise turned into a much deeper and richer look at the history of computational linear algebra (matrix algorithm stuff). I hope that some of you take the code and have fun making matrices that match up with your own values, and that others learned a little about how math was done almost 65 years ago.

Anyone can say poop and be funny

January 6, 2014

Image from: The Poo Prejudice | The Arid Land Homesteaders League www.plantfreak.wordpress.com

Image from: The Poo Prejudice | The Arid Land Homesteaders League

http://www.plantfreak.wordpress.com

My partner @opsimaths found this gem, and not since last year’s poop transplants have I seen such a strong science/poop crossover story.  But language is so interesting.  When is it better, or more useful, to use “excrement” for “waste”, “poop” for “fecal matter”, “take care of business” for “excretion”?

Click through to see various bloggers’ treatments, and check out a much more extensive list of poop names here.

My thought, though, is that writers have to deal with poop carefully.  A poll from PoopReport suggests that 92% of respondents are not ambivalent toward poop.  Whether they think it’s funny or gross, there exists quite of bit of language that especially invokes the yin or the yang of animalian digestive tract waste products.

A theatrical director and good friend of mine told me once that “Anyone can say poop and be funny”.  This was in the context of trying to stop using it to get quick laughs.  When an audience laughs at poop when you say it, it usually has nothing to do with you being a special snowflake comic genius.  Instead, it usually has to do with the universality of people’s responsiveness to poop: anyone can say poop and it will be “funny”.

This is surely an oversimplification, as is any reduction of “two sides of a thing” to that of an explicit dichotomy, or yinyang.  I’d love to have the time to write some longer creative pieces that try to explore the ideas of waste/poop each in a funny/repulsive light, but I’ll put some rough experiments in the comments.  I encourage you to experiment as well, here or in the safety of a smaller or larger audience.

Poop strong.

unsubscribe or: how i resolve to support things i actually care about

January 1, 2014

I'm a GENDER scout!

So, this is the new year. I’m very fortunate in that I was given (and can take) the opportunity to relax, defocus, and reflect, and something that I come back year after year is the intention to write. Write more, write better, write for me, write for others. It doesn’t seem to happen. In mathematics, when you can’t get at what you want the direct way, you can come at it from the other side. So I started thinking about how I read.

I’ve always thought that I’m a slow reader and it turns out that I subvocalize when I read (and when I do math). That I’ve been punishing myself with subvocalization was quite a realization and I can now put a label on it and take steps to improve my reading, my comprehension, and my overall purpose while reading.

Which gets to the heart of the matter: why am I reading anything at all? What’s in it for me? I feel as if I’ve not consciously asked this question of myself in my waking memory. To date, I know that I am reading a book to finish it, to acquire knowledge, to integrate it into my being so that if someone asks me about it I can give them an honest and relevant and direct and precise support or criticism. This has proven to be an insurmountable goal, for I read everyone’s prose as a mathematical proof, where every word has been isolated and selected to provide necessary meaning.

Sincerely thinking hard about why I am reading something has already helped me read faster–skim paragraphs, pages even–at the cost of being able to recite the book when asked (which I couldn’t even do). That’s okay.

So what does this have to do with my support of the Gender Book? Various things, the first of which is that I support people participating in the current and important dialog about non-binary gender and gender self-identification. Another, though, is that the Gender Book is something real, personal, engaging, and I think necessary. The notification that there’s a new Japanese whiskey on sale from a liquor store in NYC from which I once ordered a small batch of vodka is not necessary to me. Neither is the fact that there’s a class at the Tustin REI that I can’t go to, nor that StumbleUpon have found more unnecessary things for me to read.

Sure, by opting out I am opting out: I won’t learn about certain offers or deals or current events in the same timescale or at all. I may opt-in again at some point. But my choice to unsubscribe from the unnecessary and legitimately support the ones I find necessary is what I’m trying out as a 2014 resolution. Maybe from this absurd reduction to only reading what’s necessary, I’ll get a better handle on the things I find it necessary to write. We’ll see how far it gets me.

Happy New Year and 2014.

HSMF 2013 was gross (the fun kind)

July 12, 2013

July 4th weekend means grilling, fireworks, and drinking.  That’s if you don’t go to High Sierra Music Festival in Quincy, CA.  If you do, then July 4th weekend means hyperbolic, superlative-laden band descriptions, over 100 hours of music and musicianship that actually earns such praise, camping adventures, the best festival food around, and some of the most beautiful California scenery there is.

I’m talking about High Sierra Music Festival, and this is my seventh time attending.  This is not the kind of festival that you wait for the lineup to sign up for:

“Are you going to High Sierra?”  

“Yes.”  

“But who’s playing?”  

“I dunno–it doesn’t matter.  We just go.”

This year was no exception, and the lineup was phenomenal.  Notably, I’ve marked (at the link) which shows I went to and I’d highly recommend that you check out those (and other) bands.

Robert Plant has still got major pipes.  The Revivalists have some serious gusto.  The Hot 8 Brass Band knows how to party.  Primus rocked the house.  Mike Dillon Band made punk trombone make sense.  Thievery Corporation can make anyone move to the beat.  And Lee Fields & The Expressions know the magic of soul.

Not only is there music–there’s food.  Amazing food.  Ghanaian, Southern BBQ, Organic, Raw, Blended, Fried, Iced, Brewed–you name it.  Everyone of them had a tasty dish to sustain us through the weekend.  And then there are vendors.  Sandals, clothes, wraps, skirts, henna, massages, you name it.  And then were these sunglasses who made 20x what they asked for on Kickstarter.

But what are the major things to take away from this year’s High Sierra?  This is the key list of “do’s” that will make for, in the future, a great HSMF 2014:

  • Bring tarps.  Ground cover isn’t important, but shade is.  Camping in the right spot (Hillsides) will let you string them up for a shade complex above your communal area
  • Bring rope.  See above.
  • Plan on eating some festival food.  It’s just too good to pass up.  In other words, if you’re going to bring prepared food, or campsite food, it’s just not realistic that you’ll eat every meal from your personal stores.  Having said that, I really could eat the bean salad we brought like every day of the week.
  • Bring a shovel.  If you’re planning to camp at Hillsides (which you should) you’re going to want to do some terraforming.  Life at 15-20% grade is doable–45% is not.  But also: leave no trace!  You can figure out how to balance those things out for yourself.
  • Camp at Hillsides.  Did I mention this already?  Shady Grove used to have a stage, and the Meadow fills up on Wednesday afternoon with the early arrivals.  Hillsides is appropriately private, but with enough neighbors to ground you and a legitimate view of the Main Stage experience, right from your home away from home.
  • Drink water.  It gets hot up in the valley, there.  Water is free from spigots all over the fairgrounds so bring at least _one_ water bottle and just don’t forget to keep filling it up.
  • Bring clothes / sleeping gear to keep you (very) warm. It gets cold up there at night: much colder than you would expect given the highs that can be achieved during the day.
  • Walkie Talkie’s are a plus.  Cell service is poor up there and I doubt you can keep the battery charged for four days without awkwardly stealing power from the side of the Funk’n Jam House or sitting in your hot car for an hour.
  • Bring a Solar USB Charger.  Do this for genius status.
  • Keep your cell phone off.  Do this to unplug for four days.  Takes some serious commitment, but it’s totally worth it if you trust in the world outside the festival handling their junk without you for a weekend.  Totally acceptable to either (a) stay connected to help friends and family (b) indulge in the delusion that you’re the center of everyone’s universe.  Totally unacceptable to stay connected to read your personalized Big Lots! email ads or Facebook updates from people not at the festival.  Go see some music!
  • Set up your tent at home to check for gotchas.  I broke this sacred rule of camping this year and forgot that my tent poles were packed separately.  MAJOR CHOKE!
  • Get a quick-drying, super-light towel and/or yoga-mat from REI.  These are way smaller than a cotton towel and will help you out for the 6 hours of daily yoga.
  • Bring a table.  Coolers have a top–yes–but they are meant to be opened.  I plan do finally do ourselves a favor next year by bringing a folding table.  Then again, i also said this last time…
  • Make a plane and keep your promises.  We’re procrastinators, me and my friends.  We totally kick ass at packing 3 days or less before a week long camping trip in the woods and dirt and being wildly successful.  If you’re not like us, make sure to take the time, make a spreadsheet, and figure out what you’re going to bring in time to find out if you have it.  
  • Pace yourself.  There’s a lot of excitement at High Sierra.  Make sure you’re not forcing yourself to be on high-alert energy-level for four days without a break.  There are _lots_ of places to take a peaceful break at the festival–that’s kind of a thing it does better than any other, partly to do with it’s small size and partly to do with everyone’s great attitude.  Which leads me to the last, but most important thing to remember:
  • Don’t be un-festival.  It’s as simple as it sounds.  Don’t be the guy hurrying people on with their showers.  Don’t be the guy who tells that guy to chill out in the wrong tone.  Don’t be the one to yell at a little kid for spraying water on you without asking (but do remind them to ask next time after you say thank you).  Don’t be smug.  Don’t judge.  Don’t laugh at someone without laughing at yourself at the same time.  And definitely don’t be the one calling everybody on their un-festival crap.  Be compassionate–Don’t be un-festival.  Be good to the festival, and the festival will be good to you.

math is fun, not gross

April 21, 2013

I’m finishing my evening, reflecting on my math brains.  This morning I took the GRE Subject test in Mathematics [PDF] (on the chance that I want to apply to any post-graduate programs in the next 5 years), and I prepared by taking one Math GRE a day for the last week.  What I found was that everyone must think that math is boring and gross.  I believe that this couldn’t be farther from the truth.  This is probably more a comment on standardized tests, but out of all of the disciplines, I would hope that the mathematicians could make word problems interesting.

I am probably not allowed to comment on what was and wasn’t on today’s exam, but I can say that it was practically the same material as the practice exams I took: there were people in committees, students waiting in line for lunch, dice being cast (although of the two problems, at least one involved a dodecahedron), coins being flipped–there was a differential equation, but none of the group theory questions had any context, and definitely not the set theory or topology.  Linear algebra and complex analysis were context free as well, and even the geometric questions were as bland as “here is an object, apply divergence theorem” or “here is a circle, compute some chords”.

During the week that I was studying / practicing / drilling / training* I accumulated the following (amazing) sample of material in my feedly, which is wholly opposite that material I was working on: These are engaging, interesting, and intellectually challenging accounts instead of numbing drills:

What a roundup!  What if the material generated _just_this_week_ was the kind of stuff that the GRE tested people on?  What if we could ask people how to think _creatively_ about _new_ problems?  Write programs instead of deciphering them?  How many times will students be asked to identify the programmatic output of a Collatz Sequence or Euclid’s Algorithm?  Obviously, it’s hard to standardize good problems, but we’ve had over 150 years of Residue Calculus–can’t someone come up with a complex function with poles that means something?

Anyway, I’m not really ranting.  I thoroughly enjoyed brushing up on my math.  Wronskian?  Adjugate?  L’Hôpital’s Rule (to the max)?  These are things I don’t use in my day job.  Lie groups and matrix invertibility, FFTs and signal processing, and, every once in a while, some Fundamental Theorem (of Algebra, Invertible Matrices, Calculus, &c), but not much, and not in very wide company.  But diving into the tips and tricks was actually a joy for me–and that’s because of where I come from.  My personal (Math Nerd**) and educational (Math Mudder) backgrounds get me excited about what are, truly, “Math Tips and Tricks”.  But wouldn’t it be great if we didn’t test our future mathematicians on those, but instead on exciting, engaging material?  What if people learned something from standardized tests, and what they learned was that they _want_ to answer hard problems with interesting techniques?  I know that should be happening in the classroom (in person), but why can’t we manage to make it happen on paper, too?  Don’t mind the rhetoric too much, and let me know if I’m way off base here.  I hope, in either case, that either the Tips and Trick become interesting on their own to everyone, or we all work very hard to make questions about thinking, and not about Tips and Tricks.

* I was actually doing these tests on my train commute to work, half on the way there and half on the way home.  Happily, everyone was very considerate and didn’t bother me with book, paper, pencils, and countdown timer spread across the tables.  Sadly, no one engaged me about what I was doing so I couldn’t teach any lay-residue theory or integration by parts.

** This is the book I did an independent study with in High School to continued my jump-started career in math (kicked off in earnest by Mr. Sisley’s introduction to Spivak and Mr. Robinson’s introduction to Chaos and Dynamical Systems in 11th and 12th grades, respectively).

ideal vs. real – wikipedia weighs in on pants

March 17, 2013

I’ve been reading up on this month’s announcement that the 3-body problem has 13 more solutions, and came across a wonderful little nugget of a wikipedia disambiguation:

Pair of pants

From Wikipedia, the free encyclopedia


This article is about an object in hyperbolic geometry. For the article of clothing, see trousers.

It’s usually the case that disambiguations favor primary topics by usage but, in same strange twist of fate, the following image is described with the awesomely gross sentence:

Six pairs of pants sewn together to form an open surface of genus two with four boundary components.

So think about that next time you’re at the sewing machine, trying to patch your punctured spheres.

[Edit: "Trousers" is now the front page of the "Pair of pants" redirect, accessed April 2013]

Links

[1] sciencemag.org article: physicists-discover-a-whopping.html
[2] wiki/Pair_of_pants
[3] wiki/Wikipedia:Disambiguation#Is_there_a_primary_topic.3F
[4] wiki/File:Worldsheet.png
[5] wiki/Riemann_surface#Punctured_spheres

pi-kus for pi day

March 14, 2013

I’ve been looking around the twitter-spheroid and blago-blogs and finding that lots of people are writing “pi-ku”s today, a haiku about pi, in honor of pi day:

you go around once
and make an infinity,
of digits that is

But what is a pi-ku, really?  Is a “haiku about pi” the best we can do?  What about my wife’s suggestion (which she came across from Powell’s Bookstore) , where the syllables pay homage to pi’s most well known digits?  Here’s the formula:

— First line: 3 syllables
— Second line: 1 syllable
— Third line: 4 syllables

and used in a sentence poem:

i know, of
pi,
pi squared digits

But we can get grosser than that.  What about longer pi-ku sequences, traversing the decimal-dance of pi’s digits:

(3) from where does
(1) pi
(4) originate?

(1) is
(5) it an integral
(9) half (neg why dee ex plus ex dee why)?

(2) maybe
(6) riemann zeta at 2,
(5) times six, square root is

But I digress.  How can you contort pi into your poetry?  Leave your poems in the comments, and don’t forget to enjoy your favorite kind of pi to celebrate the sweetness and the arbitrary transcendental numbers that permeate  our limited understanding of the universe.  Today I enjoyed smitten kitchen’s apple pie cookies.  Smaller size, same great ratio of circumference to diameter.

Links

[1] Powell’s Bookstore’s Facebook Conversation, Pi Day 2013
[2] my favorite approximations of pi, on github
[3] smitten kitchen’s apple pie cookies

gross contraptions

June 14, 2012

the whole “good things come in small packages” thing is relatively recent [1877], and it’s only in the last 40 years or so that we could really say the same thing about technology. i’ve been sitting on a few great blog posts for the last while now that really show off some of these bigger, and awesome machines that we (as humanity) came up with:

Punch Tape Programmable Metal Mill from 1952

The Machines That Made The Jet Age

Early Russian Hydraulic Computer

Analog Tide Computers and the D-Day Invasion


but then again, some machines haven’t gotten much smaller…

Gorgeous 1940s Lathe Restoration

bamf climbing

May 29, 2012

i’ve realized that i’ve found some RIDICULOUS links about some incredible climbers to share.  despite the frustratingly low frequency with which i utilize my membership at hangar 18, i have a developed a deep love and respect for climbing and those who do it well.  collected here are three generations of climbers that will make your jaw drop.  Hat tips to Eric, Yael, and my Grandpa.

Tiny Hand Over Hand:
Ashima Shiraishi, 11, Conquers Difficult Bouldering Climbs


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