Like many an arrogant kid before me, when I graduate from high school in my podunk hometown (no, it wasn't marshy, and I say podunk with all the warm feelings of a idyllic childhood), I was filled with confidence that I was one of the smartest people I knew. Oh, I'd never say it, and yes I knew I was good mostly at only one small thing, mathematics, but I'm pretty certain looking back that I was a pretty confident ass. As you can well imagine, then, transitioning from my high school to Caltech, an institution filled with near-perfect-SAT-scoring students, Nobel laureate faculty members, and a wide range of just frickin' brilliant people, resulted in a large dislocation in my perspective concerning my own capabilities. But over time, I began to realize that, while I wasn't the sharpest cookie in the cookie jar, every once in a very rare while I could do something worthy of interest to my fellow genii in grooming (mostly jokes, mad rantings, or random acts of bizarreness, if you must know.) Thus I came to the perspective that there was no such thing as a universal genius, that possibly, just possibly, there are people who are good at differing things---little genii of their own domains. It's often disheartening to sit in a room with a large number of brilliant people, until I remind myself of this fact. And Monday, while doing exactly this form of sitting, I began to ponder the different ways in which these people have their own styles of brilliance. Or, in short, I made a list.

Dave's horribly biased and totally incomplete list of the different facets of genius among the theorists he knows (just to cut out the comment before it happens, of course these are not mutually exclusive):

**The Human Calculators**(aka the**Problem Solvers**) When we all think of the stereotypical brilliant person, the picture most of us have in our head is someone from this category. These are the people who, if you give a hard problem, will be able to out problem-solve, out calculate, out prove everyone else around them. The legends about this group tend to involve feats of amazing mathematical prowess: proving an unsolved problem posed as a homework problem by an evil professor, inverting matrices in their head, or summing infinite series without batting an eyelash. The great thing about these people is that they have a great chance of solving any problem you give them. The hard thing must be that no one wants to talk to them about the problem they are working on...do that and the person might just jump the gun on you and solve the problem before you.**The Random Generators**These are the people who generate ideas at a mile a minute. They sometimes have only a cursory understanding of details, and rely heavily on a "feeling" for the direction they are taking. They're the ones you'll see spending more time around the coffee table, spinning a yarn, or debating some off topic idea about a new iPhone application. Sometimes you'll see them say things out of the blue that amaze even themselves. As scientists we often give lip service to being creative, but truly pushing on the boundaries of the box isn't, I think, really recognized as a form of brilliance unless it is repeated more than a few times ("he was just lucky!" if you have only one interesting idea.) The great thing about these people is that they are a constant fount of inspiration. The hard thing about them is that they are a constant fount.**The Field Jumpers Detailed domain knowledge is often of little use in its original field, but can vastly impact seemingly unrelated fields. There are people who have not just mastered a tool, but have the flexibility of mind to apply it outside of where it was original designed to apply. We often say that people coming across the disciplinary divide have a "fresh perspective", but more often than not what they really have is a perspective shaped by their prior field. A strong marker for a field jumper is a history of changing fields: this characteristic is thus often latent until the day after tenure. The great thing about field jumpers is that they stir the pot. The dangerous thing is that when moving into a new field, they don't often do adequate background to understand what has been done and not done before (Physicists are notorious field jumpers in this respect.)****The Connectors (aka the Encyclopedia Connectica) A joke I used to tell: You can tell you're a theorist if someone describes jail to you (room, board, quiet) and you ask "Do they give me a pen and paper?" While there is no doubt that quiet contemplation aids the intellectual digestion of most theorists, sometimes its possible that you just don't have the mind (brains or frame of) to solve the problem you are working on. This is where the connectors come in. These are the renaissance scientists who have a vast encyclopedic knowledge of who did what when and, even better, when you talk to them about your problem can (a) understand your problem and (b) point you to the person or research that might best help you solve your problem. Now, you say, where is the genius in being able to point someone in the right direction? Well if it was so easy, why couldn't you just do it on your own. The encyclopedic keepers of our intellectual history, combined with the exceedingly efficient search engine known as "talking to them" (eat your heart out Google) is, to me, a true form of brilliance. The great thing about connectors is that you can talk to them about anything and they will often lead you into an entirely new direction or connect you to someone you'd never meet otherwise. The dangerous thing is that they might connect someone to you!****The Communicators The most ridiculed of the forms of genius, the communicator is able to take the results of deep science and turn them into something comprehensible. Don't think this is true genius? You obviously haven't read enough seminal papers lately---not that I'm a literature snob or anything, but complete sentence would be nice. The amazing thing about a communicator is that they can bridge the most difficult subject by illustrations that remain true to the spirit of the problem. Having currently been bashing my head up against writing book chapters, I can attest to the genius it takes to synthesize and coherently link together a body of knowledge in order to produce, for example, a textbook. Have you ever picked up a paper which upon reading the paper brushed aside hours you spent frustratingly trying to read written by experts in the field? That's the work of a communicator. Communicators filter and shape the language and form we have for our deepest results. On the other hand, their propaganda can fossilize the view of a particular field.****The Refactoratti So you've worked on your problem and solved it. You're exhausted from the focus you've had on simply getting to the end of the problem, but really happy to start talking with people about what you've done. If you happen, after doing this, to encounter one of the refactoratti, you might even be lucky enough to understand what you've just done. The refactoratti are able to see through your result and understand it for what it truly adds to our knowledge. These are the deep people we all seek out to discuss our results. I suspect that they have very coherent pictures of all of their knowledge in their head, and can thus more easily see how what you've done is innovative. They are usually the opposite of scatterbrained. Refactoratti are awesome for their reflected light. They are also dangerous, of course, because the reflected light might be that someone has already done what you just did!**

One reason I find this interesting to think about has to do with the two talks and many blog posts written by Michael Nielsen on the general theme of open science. A point which Michael discusses is that the lack of openness in science (including mechanism for reward in such systems) is bad for many reasons, including the fact that one can be toiling away at a problem which could be easily solved by some other expert (or this is my interpretation of one of Michael's points.) An important piece to the puzzle of why their is reluctance to push into open science is, in my mind, that we often don't properly reward people for their contributions. We don't view the careful crafting of a paper as a skill worthy of major attribution nor of being the person who connected the two groups who were able to put their heads together to solve the problem. Because we view our work as PROBLEM...SOLUTION, and not as the more complex craft required to push science forward, we tend to think about science as a task of "me, solve problem, reward." But it shouldn't be this way, and we should recognize the manifold nature of genius in pushing science forward. With this, I think, comes a better ability to understand how to bring science out into the open. Or at least that's my optimistic hope sitting in a room surrounded by Human Calculators, Field Jumpers, and Refactoratti.

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Your post reminded me of John Huchra's 7-dimensional classification of researchers in his essay, "On Being an Astronomer". An excerpt:

It looks like there's some overlap with your classification scheme: Human Calculators -> Raw Intelligence, Random Generators -> Creativity, Connectors -> Knowledge, Communicators -> Public Relations.

you're an idiot for even posting such a list.

Zombie: "Mmmm, brains, mmmm."

Thanks for the opinion, though, it's highly valued. Even better is you ability to post like an anonymous coward. Where did you learn that trick?

It was enormously valuable for my academic and personal development to transition from the famous Stuyvesant High School in New York City (which today would be called a Math, Science, Engineering Magnet School) to the even more famous Caltech. Where I discovered that I was...

*** shock! ***

... below average.

You find that you're not on the Fast Track of genius when someone who came from the same high school at the same year to the same Caltech ends up a Provost and Vice President, and an overlapping classmate wins a Nobel prise for work done in his first year of Grad School.

If I may address your phenomenological classification:

(1) The Human Calculators: I've even had one as a student. One exam question involved multiplying a pair of 4x4 matrices (with few if any zero elements). He wrote down the correct answer with no derivation (and this was a no-calculator exam). When I accused him of cheating, and cheating stupidly by not using scratch paper, he insisted that he'd done it in his head. So I gave him another pair of 4x4 matrices. he stared into space for no more than 4 seconds and wrote down the correct product. Ever since then,I've openly accused students of cheating if I've caught them red-handed (crib sheet, copying from adjacent student, or texting/web tools mid-exam).

(2) The Random Generators: as I've recounted before, I asked Linus Pauling what was his secret for this. He nailed it, although it took me decades before I realized that he was imparting wisdom rather than sound bites: (a) learn how to have many many ideas; (b) learn how to be excellent at telling the good ones from the bad one.

(3) The Field Jumpers: Part of Caltech's place in history is the number of distinguished Physicists there who jumped to found modern Molecular Biology. Of course, even Nobel Laureates who fieldjump get accused of being crackpots.

(4) The Connectors: This is a niche that my collaborators praise. Well, at age 57, having been reading and writing copiously and promiscuously in a dozen fields, I'd better be useful this way, or else I've wasted half my life.

(5) The Communicators: Well, you and I both have double B.S. degrees from caltech, with one each being English Literature, so we are hardly objective. They laughed at Carl Sagan, and blackballed him from the National Academy of Sciences. They laughed at Brian Greene and Lisa Randall (to pick two students from literally the same class at Stuyvesant) for their best sellers. They laughed at Isaac Asimov, but gave him a PhD anyway. They laughed at Feynman for his pranks and for playing the bongo drums. Of course, they also laughed at Bozo the Clown.

(6) The Refactoratti: If anyone knew how to train people for this role, that country would take over the world. We have no idea. It's a kind of 2nd order genius.

You don't think that this is an orthonormal basis for the 6-space of Scientists, do you?

Excellent post. I can definitely relate to moment of discovering that, intellectually, your perception of yourself is perhaps somewhat inflated. I still hold onto my perception of myself as genius, if only because it keeps me motivated. I don't think it's productive to continually remind oneself how average one is.

I'm pretty sure there's a place in one of the Feynman books where they talk about people hiding their work from him, in fear that he would just spout out the answer to something they'd been working on for six months. Sure, it seems antithetical to the goals of science, but science is a human endeavour, done by humans, and so it's more than understandable, I think. I'm not sure what I would do if I were in that position...

Nice essay. Pay no attention to the zombie.

I like your closing paragraph. I just wish people would take your sentiments seriously. Perhaps, in an odd way, it points to another type of genius I might call "mass genius." That is, the collective small steps of many, many people over time that, individually, might not be considered genius, but when taken as a whole most certainly is.

Zombie is correct by his manner of thinking, rather than a troll. If you'd like, I can defend him right down to "logic is broken without assumptions," but there's no point in that.

What I'm curious about is this: Is the typo in this essay intentional or not?

Email me for sure if it is, if you want if it's not:

Email deleted by Dave Bacon 5/11/09Dave, You forgot to include my category, slacker:)

Jeff

Sorry for the double-post, all. If I could edit/delete my previous comment, I would.

@Jonathan: Your summation about Refactoratti might be true, but we can't have everyone be a pure Refactoratti. They'd just refactor all over themselves and cause our extinction. Also, I'm pretty sure the answer to your question is "no." David might be an idiot, but he's not that much of an idiot.

This list is a little (actually, very, but it doesn't damage its readability too much for me because I'm currently part of academia) academia-biased.

Also, Refactoratti are only useful in research, or if you want to refine your mode of thinking. In any other case, they're annoying. Really annoying. In fact, people are likely to think they're annoying simply for the sake of being annoying.

Aren't you worried that people will now slot themselves into one of these narrow classifications... pretty much making it unbearable to practice science?

K.B.: from the post "just to cut out the comment before it happens, of course these are not mutually exclusive"

Luke: from the post "Dave's horribly biased and totally incomplete list of the different facets of genius among the theorists he knows"

I strongly disagree that Refactoratti are annoying.

Anyone who's read Darwin should not be too quick to say that primary sources are difficult to read.

On the other hand, Darwin worked with his idea for many years before publishing it. Which leads me to another point:

Ideas, like cross-species pathogens, usually need to evolve before they will be highly contagious. An idea may be correct, but formulated in a way that's not very useful to others. (The medium is the message?) Or it may be partially correct. Or it may be an offshoot of the idea that's actually the most useful.

The refactoratti and the communicators appear to provide major value by evolving ideas to make them more generally useful (contagious).

(The rest of this post owes its inspiration to discussions with Jonathan Vos Post.)

Swinging the pendulum back from the idea-evolvers, through the idea-originators, we come to the crackpots. These are people whose work has poor grounding in science, but who have made an observation that's interesting enough so that an idea starts trying to spread itself through them.

Many crackpots, of course, produce work with no broader value. But some manage to find things that are genuinely interesting - though we can't know it till later, after their ideas have evolved to become more transmissible (and usually more correct).

Mesmerism was a crackpot idea - surely as flaky, in its initial form, as (e.g.) Reich's "orgone energy." But Mesmer managed to transmit his idea, which evolved into hypnotism.

I suspect that homeopathy has enriched our study of placebos.

On JVP's mailing list is a person who believes that electricity has little or nothing to do with electrons. It's worth noting that in the original (Catt) version of the theory, electrons don't even exist. The idea is evolving visibly.

So, if they can find a refactoratti who will listen and be inspired without catching the disease, this line of thought might - I say might - lead to, say, a new formulation of existing electromagnetic equations, which might be useful. Or their mental model of electricity might inspire an experimentalist to develop a new class of device. There's even the faint possibility that, like Mesmer, they've noticed something that's actually interesting in its own right.

Perhaps a crackpot is merely an extreme form of field jumper. But in another sense, the process that they (occasionaly) spark provides an unusually accessible illustration of a part of the scientific process: interesting wrong ideas developing into useful correct ideas. Since crackpots, almost by definition, don't refine their ideas to be palatable to science, we can study the evolution of their ideas in a way that we can't study the evolution of ideas within a single great brain like Darwin's.

How about "The De-constructors"? Is there not a certain genius attached to being able to quickly and correctly identify the flaw in an experiment or a line of theoretical reasoning?

"I have no great quickness of apprehension or wit which is so remarkable in some clever men..."

[from autobiography of Charles Darwin]

I thank Chris Phoenix for increasing the fitness of the ideas in that discussion of the ecological value of people labeled as crackpots.

Intellectual Selection

By DEBBY APPLEGATE

Published: January 29, 2009

âTrying to write intellectual history is like trying to nail jelly to a wall,â the historian William Hesseltine once observed. Standing hammer in hand, there are three obvious ways to grasp hold of this slippery subject. The first is to focus on the thinkers or, to continue the craftsman metaphor, the producers of the ideas. The second method is to concentrate on the genetic development of the ideas themselves, or the product. The third is to focus on the consumers; that is, to trace the transmission and interpretations of these ideas among the wider population."

"In 'Banquet at Delmonicoâs: Great Minds, the Gilded Age, and the Triumph of Evolution in America,' Barry Werth seeks to show how the concept of evolution evolved in the decades after the Civil War, to become a dominant lens through which Americans viewed their rapidly changing world. A slippery subject, indeed. But Werth, the author of 'The Scarlet Professor' and several other books, takes firm hold by keeping a tight, almost cinematic, focus on the intellectual producers â the fossil-hunters, biologists, preachers, economists, historians, industrialists, politicians and editors â 'who accepted the overall truth of gradual development as a principle of biological descent, but disagreed sharply among themselves on Âother essential questions, and on the deeper implications for society, and for God.'"

[truncated]

It wasn't my intention to claim that your classifications were mutually exclusive (or that you believed them to be). It just seems to be the case (at Caltech too) that people really like to think of themselves as belonging to some category or niche in terms of their intelligence.

Seeing a published list of such categories on this blog... I feared too many would be pushed over the edge.

I found myself at the end of the list! The Refactoratti. Cool name.

ANY time I come across a posting on Caltech I can't help but smile, regardless of what is going on in my life.

As a mom of 2 sons, one who attended Caltech(even going back for more 'punishment' as a grad student)and the other MIT, people always asked me which school I preferred-like I was the little genius who got admitted. As if....

Truth be told, I always had a soft spot for Caltech. I admired their quirky stabs at fun, loved the nurturing they gave their students, and looked forward to hearing about their next prank.

But most of all I liked what it gave my son - a sense of perspective, where one can be brilliant in high school, yet be humbled by ALL the genius floating around campus. I liked their collaborated efforts, as they all (except for the true geniuses!)labored into the night to complete a seemingly impossible problem set. Caltech taught team work, and that one should never say never.

MIT, the jewel that it is, is a little less personal, and the geniuses tend to work more on their own. But if it's intimacy that you crave on your , Caltech is the place to be.It is an oasis on the crazy, left coast.

Either way, I ADORE Techers!

Thanks for a delightful read.

benjamin franklin nicely encompasses all catagories.

Gosh, I can add two more classes:

Team-Builders: Von Neumann, Oppenheimer, Ramo, Wooldridge, Poincare, Thorne, Venter (to name just a few). Even Isaac Newton ... whose team at the Royal Mint accomplished the useful purpose of making Isaac Newton exceedingly wealthy! :)

Political Radicals: Priestly, Grothendieck, Leibniz, Galois (and many others)

I really like this list. Thanks.

Jeff: "slackers" probably hits too close to home :)

K.B.: Definitely everyone likes to have their own view of where they fit. I oscillate between whether this is important (understand yourself!) or limiting (by confining yourself to what you view are your own limits.)

I realized you forgot a pair of categories (I guess they might be super-categories that encompass several of the above). David Galenson, an economist at U. of Chicago, studied geniuses and concluded there were generally two broad categories: those who peaked in their relative youth and those who reached their pinnacle later in life - sometimes much later. This was not always simply associated with the type of field a particular genius was in (e.g. historians often complete their seminal work later in their careers). Here's a link to an article about him and his work.

I'm not a scientist by any means - just a retired software geek. At the tender age of 19 I was mentored by a great Communicator who was able to express very complex math to me in terms that made sense to me. What a gift he had and what a gift he gave me. Over my career I've encountered a few in your other categories, too. It's an interesting and insightful list that's inspired me to think back about some of the exceptional folks I've encountered over the years.

Thanks Mary. Everyone should click on Mary Foxworthy's link to see her lifestyle choice: the sailing life :)

Ian: so there are no mid-career geniuses? Doh.

I really like this post, I know people who fit perfectly into each type. I was a human calculator when I was younger, now I am a random generator and yes like you said I am a constant fount, I have trouble keeping my mouth shut.

Ah, the life of an oblivious white male. Of course, my experience was the opposite. Quite convinced that, despite my outstanding abilities in my small high school group, I would be confronted at university with crowds of superior minds. Alas, this was not to pass.

Putting people in certain categories sure is fun. It is also stupid and limits your perception of them.

There are 10 types of people who put other people in categories...

(1) The Human Calculators: they have memorized the Caltech/JPL phonebook and can tell you in 1.3 seconds in which category to place any name you read them from said phonebook.

(2) The Random Generators: they can create a new category at the drop of a hat. If they are about to become parents, they make a first-draft list of 1,001 possible names for the child, and hand that to their spouse/significant other.

(3) The Field Jumpers: if you tell them into which categories you have placed a number of Physicists that you know, they will do the same for a list of molecular biologists.

(4) The Connectors: you give them your list of categories, and they give you a stack of papers on Theory of Mind, Psychometrics, and tell a funny story about why the Navy administers the tests to people on NSF grants about to winter over at Antarctica;

(5) The Communicators: you give them your list of categories, and the next thing you know, they're explaining it on The Tonight Show.

(6) The Refactoratti: you give them your list of categories, and they organize a seminar on Epistemology and Ontology.

(7) The De-constructors: they ask "if someone categorizes all the people who DON'T categorize themselves, does that person categorize themself?"

(8) The Team-Builders: they arrange a lunch with you, a biographer from the History Department, and a gal from HR to draft the university standards for categorization;

(9) The Political Radicals: First they say: "I am not a number. I am a human being," and then they explain why categorizing people is a relic of the white establishment phallocracy which must be destroyed for the good of the ecosystem.

(10) Slackers: you try to give them your list of categories, and they say: "Dude, can't you just Twitter that to me?"

What about humor? I realize you guys are talking about what we might call "scientific brilliance" but the truly funny people in the world are, well, geniuses in their own way. Sort of hyper-refactorrati. Think of George Carlin: a superb communicator, for one, and possessing a remarkable ability to cut through B.S. and put things into context, and rate them in terms of value, usefulness, etc. I doubt he could do quantum mechanics, but I'd bet he could listen to you explain your work and give you a unique and original perspective on its place in the Big Scheme of Things.

There's actually a common theme to all of these categories. Does anyone else see what the theme is? No, not the science as profession part.

These are all symptoms or secondary effects of mild to moderate ADD and ADHD. So ADD is a disease, is it?

ADD/ADHD is to creativity what sickle-cell is to malaria resistance.

The Magicians:

Those who see something, that everyone is doing in a standard way, in a completely new and highly fruitful way; who create theories out of pure imagination that now one else would even have looked for; who are truly, really, amazingly original.

Everyone else is no genius--they merely have talent. Genius is a term used too often---the are very few geniuses.

Exanples: Heisenberg and Matrix Mechanics;

Von Neumann and the C^{\star} approach to QM, Faraday and Flux Tubes, Morse and Morse Theory, Einstein and General Relativity,

Nash and Isometric Embedding, Gromov and

Symplectic Topology, Feynman and Path Integrals/Wiener and Wiener Measure.

The inhumanly clever: Example, Ramnujuan.

A Ramanujuan amusement, what is the sum of

sqrt{1+sqrt{2 +sqrt{3.... ?

Hint: The solution is very short using freshman calculus.

Have fun!!

Eric the Red: From the post "just to cut out the comment before it happens, of course these are not mutually exclusive"

Mark: Very funny.

Penny: Of course we can quibble over the meaning of genius, but limiting it to five people seems to small to me. (Just kidding.) But of course I have a broader definition of genius: geniuses are people who do things that I can't, cause I'm no genius.

Ian: so there are no mid-career geniuses? Doh.

Have no fear Dave. There's still time.

Generalizations of penny's question:

Weisstein, Eric W. "Nested Radical." From MathWorld--A Wolfram Web Resource.

Also, sequence A072449 of the Online Encyclopedia of Integer Sequences:

A072449 Decimal expansion of the limit of the nested radical sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... )))).

1, 7, 5, 7, 9, 3, 2, 7, 5, 6, 6, 1, 8, 0, 0, 4, 5, 3, 2, 7, 0, 8, 8, 1, 9, 6, 3, 8, 2, 1, 8, 1, 3, 8, 5, 2, 7, 6, 5, 3, 1, 9, 9, 9, 2, 2, 1, 4, 6, 8, 3, 7, 7, 0, 4, 3, 1, 0, 1, 3, 5, 5, 0, 0, 3, 8, 5, 1, 1, 0, 2, 3, 2, 6, 7, 4, 4, 4, 6, 7, 5, 7, 5, 7, 2, 3, 4, 4, 5, 5, 4, 0, 0, 0, 2, 5, 9, 4, 5, 2, 9, 7, 0, 9, 3 (

COMMENT

Herschfeld calls this the Kasner number, after Edward Kasner. - Charles R Greathouse IV, Dec 30 2008

No closed-form expression is known for this constant.

"It was discovered by T. Vijayaraghavan that the infinite radical sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ..., where a_n >= 0, will converge to a limit if and only if the limit of (log a_n)/2^n exists" - Clawson, p. 229. Obviously if a_n = n, the limit of (log a_n) / 2^n as n -> infinity is 0.

The continued fraction is 1, 1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...

REFERENCES

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.1.

Aaron Herschfeld, "On Infinite Radicals", American Mathematical Monthly 42:7 (1935), pp. 419-429.

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, London, England, 1997, page 30.

Stephen Wolfram, "A New Kind Of Science," Wolfram Media, 2002, page 915.

LINKS

Eric Weisstein's World of Mathematics, Nested Radical

Eric Weisstein's World of Mathematics, Nested Radical Constant

EXAMPLE

Sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... =~ 1.757932756618004532708819638218138527653...

MATHEMATICA

RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Range[100]]], 10, 111][[1]] A New Kind Of Science

PROGRAM

(PARI) (gives at least 180 correct digits) s=200; for(n=1, 199, t=200-n+sqrt(s); s=t); sqrt(s)

CROSSREFS

Cf. A072450.

AUTHOR

Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 01 2002

See also:

A099874 Decimal expansion of a nested radical: CubeRoot(1 + CubeRoot(2 + CubeRoot(3 + CubeRoot(4 + ...

A099876 Decimal expansion of a nested radical: sqrt(1! + sqrt(2! + sqrt(3! + ...

A099877 Decimal expansion of Nested radical: Sqrt(1^2 + CubeRoot(2^3 + 4thRoot(3^4 + 5thRoot(4^5 + ...

A099878 Decimal expansion of a nested radical: Sqrt(1 + CubeRoot(2 + 4thRoot(3 + 5thRoot(4 + ...

And, finally, with me playing the role of The Connector:

A105546 Decimal expansion of prime nested radical.

2, 1, 0, 3, 5, 9, 7, 4, 9, 6, 3, 3, 9, 8, 9, 7, 2, 6, 2, 6, 1, 9, 9, 3, 9, 6, 4, 9, 6, 8, 5, 3, 2, 5, 4, 4, 4, 0, 4, 2, 1, 6, 2, 2, 8, 8, 2, 4, 0, 0, 1, 3, 8, 7, 2, 9, 8, 6, 8, 7, 2, 8, 4, 5, 6, 3, 8, 8, 5, 1, 7, 0, 8, 4, 8, 3, 7, 3, 6, 2, 3, 2, 1, 8, 4, 6, 6, 9, 7, 4, 7, 6, 3, 3, 5, 5, 2, 1, 9, 4, 4, 9, 4, 0, 9

COMMENT

A105547 is the continued fraction representation of this prime nested radical. A105548 is the similar semiprime nested radical. A105548 is the Fibonacci nested radical. Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(4 + ... = ~ 1.75793275661800... "It was discovered by T. Vijayaraghavan that the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; cf. A072449]. We know the asymptotic limit of primes and hence that the Prime Nested Radical converges.

REFERENCES

Borwein, J. M. and de Barra, G., Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 and 229.

S. R. Finch, Analysis of a Radical Expansion, Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.

LINKS

Eric Weisstein's World of Mathematics, Nested Radical Constant.

FORMULA

Sqrt(2 + Sqrt(3 + Sqrt(5 + Sqrt(7 + Sqrt(11 + ... + Sqrt(Prime(n))))).

EXAMPLE

2.10359749633989726261993964968532544404216228824001387298687284563...

MATHEMATICA

RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Prime[ Range[ 80]]]], 10, 111][[1]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 31 2005)

CROSSREFS

Cf. A000040, A072449.

AUTHOR

Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 12 2005

penny:

Interesting problem! It's closely related to an easier one in which you compute sqrt(x + sqrt(x + sqrt(x + ...))), for any x. To solve this one, write S = sqrt(x + sqrt(x + sqrt(x + ...))), and note that S^2 = x + S. This is a quadratic equation that can be solved for S!

Jonathan,

I always enjoy your interesting posts.

Ramanujuan gave a solution to the problem that I posed. It is a reduction formula for a certain integral. That he found it is--to me--an indication of his almost superhuman intellect.

Zach,

Note that the problem you pose, when x=1 gives rise to the golden mean.

Speaking of "Facets of Genius" -- here is a link to the audio podcast of a recent symposium on this topic:

http://guides.lib.fsu.edu/aecontent.php?pid=99561&sid=747385