## Jordan Ellenberg’s new book, “Shape”

The full title is “Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else,” and I wasn’t looking forward to it. Yes, I’m a fan of Jordan Ellenberg, a practicing mathematician who’s also written general-interest books, but I have unpleasant memories of the math olympiad program where they were always trying to shove stupid geometry theorems down our throat, all these tricky proofs where this angle equals that angle and we’re supposed to think it’s so cool that these points fall in a circle and . . . jeez, I just wanted to give everything Cartesian coordinates and solve everything by brute force. Also when I was a kid someone gave me an obnoxious book about the beauty of the so-called Golden Ratio, and I’m like, just call it by its true name, (1 + sqrt(5))/2. So, I really enjoy Ellenberg’s writing—he achieves the unusual combination of being informative, surprising, and reasonable—but I wasn’t looking forward to reading a couple hundred pages about the beauty of geometry etc. I guess if it had enough fractals in it, that would be cool—another memory I have, from a few years after those earlier recollections, is snobby mathematicians informing me that fractals aren’t all that, when actually fractals are all that, and those mathematicians were just too boring to accept that something new could come along—but pages and pages of triangles and the spiral of Archimedes and the wonders of the ancients and something about how Archimedes was so damn brilliant and his catapult really worked too—that I wasn’t ready for.

So imagine my joy when I encountered this passage, right on page 2:

Reader, let me be straight with you about geometry: at first I didn’t care for it. Which is weird, because I’m a mathematician now. Doing geometry is literally my job! It was different when I was a kid on the math team circuit. . . . And on that circuit I was well-known among my peers for balking whenever presented with “show angle APQ is congruent to angle CDF,” or the like. Not that I didn’t do those questions! But I did them in the most cumbersome possible way, which meant assigning numerical coordinates to each of the many points in the diagram, then grinding out pages of algebra and numerical computation in order to compute the areas of triangles and lengths of line segments. Anything to avoid actually doing geometry in the approved manner. Sometimes I got the problem right, sometimes I got the problem wrong. But it was ugly every time.If there’s such a thing as being geometric by nature, I’m the opposite.

Jordan! My man! I think I’m gonna like this book.

And I did.

The only thing I wished was that he had a chapter on fun geometry problems, like how it is that movers can carry your couch up the stairs and get it through your doorway when it looks like there’s no way they can possibly get it to fit. In all seriousness, there must be some math to that.

P.S. Also I regret that Jordan followed up the above passage with a statement that he’s no good at “visualizing shapes and what they would look like if rotated or glued together . . . You know the little picture on the credit card machine at the gas station that shows you how to orient the card when you swipe it? That picture is useless to me. It’s beyond my mental capabilities to translate that flat drawing into a three-dimensional action.” I have no doubt that Ellenberg is telling the truth here, but I fear he’s also just giving ammunition to James “cancer cure is coming in minus 21 years” Watson who notoriously said, “The one aspect of the Jewish brain that is not first class is that Jews are said to be bad in thinking in three dimensions . . . it is true.” But I guess it would be a bit single-issue of me to judge an entire book based on whether it provides aid and comfort to the James Watsons of the world. After all, someone might someday invent a time machine and go back to the year 2000 and cure cancer. And that would be good, right?

1. Z says:

>The only thing I wished was that he had a chapter on fun geometry problems, like how it is that movers can carry your couch up the stairs and get it through your doorway when it looks like there’s no way they can possibly get it to fit. In all seriousness, there must be some math to that.

I once used movers to move into a place and they did this, but I didn’t use movers when I moved out and had to leave the couch behind. Somehow didn’t work in rewind…

• Phil says:

When I was in fifth grade, my parents bought a new house so my brother and I could go to school in a good school system instead of a lousy one.(It’s sad that there are these huge imbalances. I sometimes wonder how much my life would have been different if my parents hadn’t had the money and foresight to make that move. Don’t let anybody tell you environment doesn’t matter much and it’s all about genes. I got to go to school with Andrew, and with someone who went on to win one of those MacArthur Genius Grants, instead of going to a high school that had race riots. On the other hand, the first person from my high school class to be mentioned on the New York Times front page was on there for defacing a synagogue in a case that eventually made it to the Supreme Court, so it’s not like the high school environment is completely defining.)

But I digress.

Anyway, my parents bought a new house, and sold our old one, but there was a three- or four-month period when we had to move out of the old one but couldn’t yet move into the new one, so we rented a house for just a few months. My parents and a friend moved everything from the old house into the rental. My dad and the friend had a heckuva time with the couch: you had to carry it up the exterior stairs onto a narrow porch, and then it had to rotate 90 degrees to go through the door into the hall, and then it had to rotate 90 degrees again to head for the living room. It was obvious from the start that they had to use three dimensions rather than two — tilt the couch up so it didn’t extend so far in the x-y plane — but then the back of the couch would hit the top of the door frame, or the arms would hit something else. They eventually figured out a way to do it: roll it and tilt it this way to get it partway into the hall, then keep it at that angle and roll it the other way so the back would clear the door frame, then roll it back the other way.

My parents hired movers to move the furniture from the rental house to the new house a few months later. My dad intended to tell the movers the secret of moving the couch, when the time came to move it. He was in the kitchen or bedroom or something when he heard them say “let’s take the couch”, and he came out just a few seconds later and they had already flipped the couch into the first of the magic positions and were maneuvering it into the doorway, where they did the special roll to get it properly oriented, and then moved it partway, and then did the other roll to get it all the way out. My dad said they didn’t even discuss it, they just both knew what to do and instantly did it. He was really impressed. He told one of them that he had really struggled to get it in and assumed he would need to tell the movers how to do it, and one of them said “oh, we’ve done this a hundred times.”

• Reminds me of Douglas Adams, the Dirk Gently series. In one of them he’s using his brand new Macintosh II to do geometry calculations to figure out how the couch got stuck in his hallway in a position that it seems it could never have gotten into. I think they eventually proved that it couldn’t have ever gotten into that position (this was a clue about some trans-dimensional weirdness). I miss Douglas Adams. He died way too young.

• Phil says:

I somehow never read the Dirk Gently series, in spite of having really liked the HGttG series. Thanks for reminding me of its existence.

From the breezy, casual tone of Hitchhiker’s I had assumed Adams wrote quickly and easily, but a few years ago I read a sort of memoir by…I have no recollection who by. By some other writer. They said Adams would spend an hour writing a paragraph, dithering about every word, and then delete it and start over.

Yeah, he did die way too young.

• Alejandro says:

In the ’90s someone created a screensaver that tried to “solve” the couch conundrum from “Dirk Gently’s Holistic Detective Agency”. I remember running the screensaver using the After Dark program on what must have been a Macintosh 5200 or 5300. The creator of the screensaver shared a bit of the history and some screenshots: http://holisticsofa.com/category/graphics/

• Marshall says:

In my 20s I worked a little while in a job that involved moving furniture periodically. The old guys just knew how to do it, but it didn’t take very long before I did, too (usually). You don’t have to do it hundreds of times. Neuronal networks and cerebella are apparently good at this sort of thing.

2. The “couch problem” is actually a rather interesting unsolved problem in mathematics. You can numerically solve for how wide a hall/corner has to be to get a particular width couch around the corner. But then another interesting take on the problem is, given a particular width hallway and corner, what is the shape of couch giving you the maximum cross-sectional area that you can get around that corner? There are some interesting animations showing couches you wouldn’t believe going around a corner, but they do. Barely.

3. Matt Skaggs says:

“It’s beyond my mental capabilities to translate that flat drawing into a three-dimensional action.”

OMG, there are other people like me! And I’m not even Jewish, despite the shape of my nose. I once bought a set of straps for putting a canoe on top of my car, replete with ratchets and hooks at the end. After hours and hours of struggling, I put them away…came back and struggled for hours, and finally just cut them and knotted them. They take forever to install now, but at least I can carry my canoe on my car.

“The “couch problem” is actually a rather interesting unsolved problem in mathematics.”

This same question, but with a ladder instead of a couch, was offered up by my High School math teacher as the one example he could think of where we might use calculus in real life. I am chagrined to learn that I can’t even solve one real-life problem with calculus. The most difficult math I had to do in 33 years of engineering work was to calculate the volume of a toroid, which is trivial of course.

4. Dmitri says:

The only problem with Jordan’s book is that there is not enough geometry of music in it!

• Jordan S Ellenberg says:

You are absolutely right and my only excuse is that I was already four months past deadline and had a manuscript somewhat longer than what I’d been asked for — but I at least MENTION it in the section where it would have fit nicely so that interested readers know where to go for more! (There are a lot of paths like that in the book that I just ran out of room to go down but I tried to give readers a signpost so they can walk the trail on their own.)

Anyway, everybody intrigued by the idea of modeling chords as points on an orbifold and voice leading as related to paths on that orbifold, check out the article

https://dmitri.mycpanel.princeton.edu/sciencearticle.html

and if you want more, the whole book

https://dmitri.mycpanel.princeton.edu/geometry-of-music.html

• Jordan S Ellenberg says:

Oh yeah and I also pitched Dmitri’s stuff on Daniel Lelchuk’s podcast, which we recorded last week so should go up pretty soon (he’s a cellist who likes science)

5. Nick Firoozye says:

Love the comments. Always hated high school geometry and even more combinatorial geometry (Erdos? Who cares? Except I think I do have an Erdos number of 3? One can’t really apologise enough for one’s coauthors though!).

Funny enough I thought differential geometry was kind of cool (before settling on calculus of variations and pde) but it never had the numerological astrological magical overtones of geometry that were such a huge turnoff.

Irrespective it is great to hear other mathematicians say that there are areas of math that never really turned them on.

• expr says:

Somewhat the revers for me did well in 10th grade geometry and got an A in Engineering Drawing by being able to draw the 3rd view of two given views quickly
When I work on construction projects, my friends give me the complicated 3D stuff to figure out Hit the wall in algebraic topology when the professor started drawing 5D space on the board and acting like he could see it
Hated analysis. all those fiddly estimates.
When I went through (sputnik when I was in 10th grade) we did not have contests or math camp (at least in St. Louis) we did take the MAA SOA test for the first time
when I was in 11th grade but no other contests

6. Anonymous says:

This book just registered on my radar a couple of days ago, and was high on my ‘take a look’ list. It just moved up to the top, I think, because, I _like_ geometry – in fact I need it – a lot of math I understand in a fundamentally geometric way, but I have never had patience or interest in those “show angle APQ is congruent to angle CDF,” questions either. That is not close to what springs to mind when I hear the word “geometry”.

7. Jordan says:

“The one aspect of the Jewish brain that is not first class is that Jews are said to be bad in thinking in three dimensions . . . it is true.”

Maybe aphantasia actually is more common in Jews? I am a Jew with aphantasia, as is everyone else I know who has it.

• Andrew says:

Jordan:

You’ll have to ask James Watson. According to him, cancer was cured 21 years ago so I’m sure he’s had a lot of time to think about this one since then.

8. Matt Skaggs says:

“Maybe aphantasia actually is more common in Jews?”

I score terribly at translating 2D into 3D, and I also do poorly trying to flip a 3D object in my mind. But I most definitely do not have aphantasia. Instead, I have a photographic memory. I used to be a spelling champion because I can “see” any word I have ever seen before (alas, these skills slip with aging).

My ability to “see” scenes and words in my mind helps me not at all in thinking in 3D, so they can’t be the same thing.

• And I have Aphantasia but am really good at comparing two 3D models and rotating their orientations etc. I do it in some kind of “pure space” since there’s no notion of visualness.

9. Mikhail Shubin says:

The question I have about movers is how do they manage to be so physically strong? I was moving recently and mover I hired was able to lift a crate with all my math books! And I will get a back pain if I lift like 5 at once!

10. Darf says:

William Feller and his wife were once trying to move a large circular table from their living room into the dining room. They pushed and pulled and rotated and maneuvered, but try as they might they could not get the table through the door. It seemed to be inextricably stuck. Frustrated and tired, Feller sat down with a pencil and paper and devised a mathematical model of the situation. After several minute he was able to prove that what they were trying to do was impossible. While Willy was engaged in these machinations, his wife had continued struggling with the table, and she managed to get it into the dining room.

– Mathematical Apocrypha

11. rm bloom says:

Andrew: the route from classical geometry to Cartesian geometry is surprisingly subtle, if one wishes to shore up what one does by rote with coordinates. For it is then necessary — one way or another — to prove that the slope of a line segment really is an invariant of any family of similar right triangles. So one needs to revisit the old hierarchy of similarity theorems. That old baggage, taught by rote to Junior high-school students, is where the subtle moves are found. And when I looked into the matter a few years ago, I discovered that the proofs of the fundamental theorems on similarity are stand at bottom on simple lemmas concerning *areas*. It was my impression — unless there are other routes to similarity — that *area* is the more primitive notion; and *slope* is derived. Another surprise finding was an old theorem of Lagrange in which the pythagorean theorem is shown to be logically equivalent to the parallel postulate.

12. Jake says:

Very entertaining post and responses. I enjoy reading your books and writing and this is just another case in point. On the topic of books, every couple weeks or so I check back on the status of Applied Regression and Multilevel Models, and wondering when “around the end of 2020” is coming. Anyway, just wanted to express my appreciation of you sharing and my looking forward to your (et al.) book.

13. Rahul says:

Following on Andrews rant about esoteric geometry theorems I kinda feel the same way about undergraduate math courses these days.

So much of esoteric crap about ODEs and PDEs, analytical solutions etc. that’s barely ever used. Brute force and solvers do the trick rather nicely.

They should rather focus on formulation of problems.

• Andrew says:

Rahul:

I recommend the books Street-Fighting Mathematics and The Art of Insight in Science and Engineering by Sanjoy Mahajan.

• dl says:

Think about what that kind of training would do to the pipeline of high level mathematicians.

• Rahul says:

True!

I think we are teaching the right things if the goal is to become a math / stat / econ / physics PhD.

nd probably the wrong things for everyone else.

• dl says:

yeah but you don’t always know in advance who will be a high level mathematician and who will be “everyone else”

14. Jacob says:

I had a friend in High School who played a game where you were given the net of a polyhedron (so the surface of the shape unfolded into two dimensions) and two points were marked. You had to find the shortest path along the surface of the polyhedron to link the two points. He played the level where you got icosahedra. It was impossible!!

Incidentally, on the whole math team of Jewish, Indian, and Chinese descent, he was the only Hungarian. Make of that what you will.

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