]]>An unexpected connection exists between compatibility criteria for quantum states and symmetric informationally complete POVMs. Beginning with Caves, Fuchs and Schack’s “Conditions for compatibility of quantum state assignments” [Phys. Rev. A 66 (2002), 062111], I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. Along the way, I correct two mathematical errors in Caves, Fuchs and Schack’s paper. One error is a minor nit to pick, while the other is a missed opportunity.

Me encouraging scientists to use Twitter sometimes feels like Bill Hicks saying, “I am available for children’s parties, by the way.”

]]>From Wikipedia’s page on Duchamp’s *Nude Descending a Staircase, No. 2* (today’s version):

It has been noted disquisitively [link] that the number 1001 of Duchamp’s entry at the 1912 Indépendants catalogue also happens to represent an integer based number of the Golden ratio base, related to the golden section, something of much interest to the Duchamps and others of the Puteaux Group. Representing integers as golden ratio base numbers, one obtains the final result

1000.1001. This, of course, was by chance—and it is not known whether Duchamp was familiar enough with the mathematics of the golden ratio to have made such a connection—as it was by chance too the relation to Arabic Manuscript of The Thousand and One Nights dating back to the 1300s._{φ}

*Euhhhhh, non.*

As best I can tell, all this is saying is that the catalogue number of Duchamp’s painting contains only 0s and 1s.

The idea behind the “golden ratio base” is that we can write integer numbers in terms of the golden ratio, $\phi$, if we add up different *powers* of $\phi$. For example, anything to the zeroth power is 1, so $1 = \phi^0$. Less obviously, we can say from the definition of $\phi$ that

$\frac{1}{\phi} = \phi – 1.$

Squaring both sides of this equation,

$\frac{1}{\phi^2} = (\phi – 1)^2 = \phi^2 – 2\phi + 1.$

So,

$\frac{1}{\phi^2} + \phi = \phi^2 – \phi + 1 = \phi(\phi – 1) + 1.$

Referring back to our first equation,

$\frac{1}{\phi^2} + \phi = \phi\left(\frac{1}{\phi}\right) + 1,$

which means that

$\frac{1}{\phi^2} + \phi = 2.$

Another way of writing this would be to say

$2 = \phi^{-2} + \phi^1.$

With more cleverness, we can write any positive integer as a sum of powers of $\phi$:

$N = \phi^{k_1} + \phi^{k_2} + \cdots + \phi^{k_n},$

where the numbers $k_1$ through $k_n$ are distinct integers. Notice that we don’t have any coefficients in front of the terms—or, to say it more carefully, the coefficient of any term in the sum is either zero or one. So, “1001″ could be a representation of a number in the golden-ratio base, if we read it as

$1001_\phi = 1\cdot\phi^3 + 0\cdot\phi^2 + 0\cdot\phi^1 + 1\cdot\phi^0.$

In the same way, “1000.1001″ can stand for a number in base $\phi$. It’s the number we normally write as **5.** It is not the “final result” of “representing integers as golden ratio base numbers.”

I tried making sense of the disquisition to which Wikipedia credits this observation. The stuff about writing numbers in the golden-ratio base isn’t even there. What we do get is that the number 1001 is

le nombre figuré pentagonal en relation avec le mythique « nombre d’or » que l’on retrouve dans toute forme pentagonale et dans l’étoile à cinq branches.[the pentagonal number in relation with the mythic "golden number" which one finds in all pentagonal forms and in the five-pointed star.]

It’s true: 1001 is a pentagonal number (so are 1, 2, 5, 7, 12, 15, 22, 26, 35, …). The sense of the argument appears to be, “1001 is a pentagonal number [true], and because pentagon therefore GOLDEN RATIO!” The golden ratio occurs in a *regular* pentagon, as the ratio of the diagonal length to the side length. That doesn’t make the free word association of “pentagon” and “mystical golden number” a valid argument.

But hey, when you feel the need for uninhibited babble slicked over with a superficial veneer of pseudoscholarship, there’s no place like an encyclopaedia article, right?

*“Painters who definitely did make use of GR include Paul Serusier, Juan Gris, and Giro Severini, all in the early 19th century, and Salvador Dali in the 20th, but all four seem to have been experimenting with GR for its own sake rather than for some intrinsic aesthetic reason. Also, the Cubists did organize an exhibition called “Section d’Or” in Paris in 1912, but the name was just that; none of the art shown involved the Golden Ratio.”*

—Keith Devlin

It is not possible that what is common to several classes should have any quality which is excluded from one of them. If, for example, no bankers are poor and no lawyers are honest, it is impossible that lawyers who are bankers should be either poor or honest.

From “On the Algebra of Logic” in *Studies in Logic,* Charles Sanders Peirce, editor, pp. 17–71 (1883).

]]>Chromosomes, as Anne Fausto-Sterling details in

Sexing the Body,can’t be relied on as indicators of the other traits here — sets exist beyond XX and XY, as do humans in whom both are found and outwardly ‘female-bodied’ people with the latter. Anatomy comes in endless combinations, such that estimates of ‘ambiguous’ sets’ commonness vary wildly, with some as high as one in twenty-five (John Money, cited in Fausto-Sterling’s work). Bodies with the ‘wrong’ features — height, hair, breast tissue, Adam’s apples — are common. Everyone preadolescent, postmenopausal or otherwise infertile is sexless judging by sperm and ova. Hormones, like most of these attributes, can be altered at will.When not all these tests are passed, which overrule which? Milinovich describes people with ‘female’ anatomy and XY chromosomes as male, for example — suggesting, confusingly, that she doesn’t think maleness requires physical traits. What reason is there to choose genes rather than body parts when diagnosing sex, and not vice versa? In practice, things tend to go the other way: medics who judge a foetus’s sex via ultrasound, for instance, do so only by identifying outer sex organs, and I know nothing about my chromosomes, interior sex organs, hormones or fertility. The fact (or assumption) I have a penis is seen as enough, most of the time, to classify my sex as male, but why should it outweigh these unknown factors?

It’s common enough for adult cisgender men — deemed male at birth, with bodies read straightforwardly that way — not to grow facial hair. I know two or three who don’t; so probably do you. This isn’t seen to affect their physical sex. Why then, barring blunt intuition, should the absence of a penis? We can argue facial hair is only a secondary sex characteristic, and penises a primary one, but this relies itself on defining sex by reproductive role: the logic is circular. From that standpoint, moreover, why not make testes the sole determinant, so people possessing them and a vulva were ‘males’? Testes have, after all, the more distinct and self-contained function of sperm production. A penis, being a shell for the urethra, is just another pipe among the plumbing — we’ve no grounds except cultural ones to treat it differently from a vas deferens. So why is it more necessary for ‘maleness’?

Milinovich calls sex a static, stubborn fact, then moves inconsistently between ideas (see above) about what it is. If she herself can’t pick a definition, what does this suggest?

- Things That Cost More Than Space Exploration
- Population genetics on islands connected by an arbitrary network: An analytic approach
- “never, ever, trust Ross Douthat when he tells you what “the left” is thinking”
- “Kentucky marriage policy will now be dictated from places like Boston and San Francisco.” Don’t worry; we Bostonians will be gentle.
- “The “exceptional men” doing shots of Maker and grinding on anonymous girls at house parties are not always looking for a wife.”
- Let’s do the gender-relations time warp again
- Termite-inspired robots on the cover of
*Science* - Continuous-time models of group selection, and the dynamical insufficiency of kin selection models
- Higher-order structure and epidemic dynamics in clustered networks
- A Bayesian Characterization of Relative Entropy
- Tea Party’s fringe isolation: How a conspiracist mind-set poses long-term electoral danger
- 9th December 1961: The Beatles play to 18 people in Aldershot
- Self-published ebooks: the surprising data from Amazon
- Reply to comments on “Weak value amplification is suboptimal for estimation and detection” – I love the “Of course, we are not the resource counting police” part.
- In which once again Richard Dawkins makes people go “wha??”
- Mistakes are Interesting
- “the idea of a single effective population that can be used to describe different aspects of the behavior of a population does not work”
- Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold
- “Yeah, but it’s a non-violent game, which is why it seems so weird to us.”

- How
*do*cats land on their feet when falling, anyway? - “In this case of life imitating art — or at least television” OH SNAP
- 3 in 4 TED Talks are by dudes
- Rare are the tweets which could rightly inspire their own erotic fanfiction
- How I made a whiteboard movie
- Optical Society to launch new Gold Open Access journal
- Perils of the Lady Gamer
- When trouble comes, the philosophy department “uses pseudo-philosophical analyses to avoid directly addressing the situation”
- ‘A Panorama of Toxicity’: On Being a Trans Woman Online
- Fear and Loathing of the English Passive [N.B.: this is
*not*about being told to lie back and think of England] - How ‘computer geeks’ replaced ‘computer girls’

hmmmmmmmmm a chance to begin again in a golden land of opportunity and adventure mmmmmmvrrrrrrmmmm

A skeptic who is told, “I saw a UFO!” and responds, “No, you didn’t,” is denying themselves the very data they require in order to have something to be skeptical about. The better response is not to deny the

We don’t have to take at face value the claims of the TV psychic or the faith healer. What we cannot disregard is that the currency of such beliefs in human minds has done its part to make history.

To put it another way: is there a necessary contradiction between the statement “the gauge group of the Standard Model is *SU*(3) × *SU*(2) × *U*(1)” and the statement “a dream can change a person’s life”? Since we teach physics with stories about Einstein dreaming of riding on a beam of light, we had better be able to make this reconciliation!

- Oh, look, the “Jane Austen Was a Game Theorist” guy. Anyone checked him for fabricated Bob Dylan quotes yet?
- “New Math” as a 1980s thing? Isn’t that about 20 years off?
- And here is the proof that Google autocomplete can’t always be right.
- R. Crumb’s rendition of Philip K. Dick’s gnostic experience
- A Simple Model of Group Selection that cannot be analyzed with Inclusive Fitness
- We Millennials will be tying onions to our belts soon enough. “Last Exit to Springfield”, origin of the onion/belt reference, will be 21 years old this March. #TheMoreYouKnow
- Some Negative Remarks on Operational Approaches to Quantum Theory
- Reinventing Explanation
- Atlas Shrugged: Strawman Has a Point
- Did ancient aliens build giant stone boxes in Egypt?
- Surreal Analysis: An Analogue of Real Analysis for Surreal Numbers
- 5 (more) words that are older than you think: unlike (verb), flash mob, weapons of mass destruction, innit, and hipster
- “Christ, I hate Blackboard”
- “disconnect between perception and reality, among such a powerful segment of the population, is in itself dangerous”

- Contextuality supplies the magic for quantum computation
- Stochastic patterns in a 1D Rock-Paper-Scissor model with mutation
- Wanted – suggestions for alternatives to Mendeley (socially aware reference collections)
- Venture Capitalist Godwins Self in Public
- Boston, San Francisco, and The Bonfire of The Vanities
- Fossils and DNA tell different stories about ant evolution. Or do they?
- Memoirs by scientists that successfully combine the professional and the personal. I can’t second the Feynman recommendation, however—there’s really very little of his professional accomplishments in
*Surely You’re Joking.* - This is What it’s Like to Be a Woman at a Bitcoin Meetup — it’s depressing that unwanted touching to the point of making one “squirm uncomfortably” doesn’t mentally count as mistreatment.
- Coverage of Nonexistent Hookup Culture Makes Students Feel Left Out of Nonexistent Hookup Culture
- How to Rip Off Photographers for Fun and Profit
- Relative Entropy in Evolutionary Dynamics
- “To John Archibald Wheeler, the race to explain time was personal.”
- Syria’s Polio Epidemic: The Suppressed Truth
- Turing patterns from cells, rather than molecules

`#scio14`

.
What can one do? I revile air travel more every year. I don’t have time/energy to prepare for the conference beforehand, or to follow up on anything discussed there after. My proposal for the session I was to moderate was, to summarize only slightly, “hey let’s build this website”. Must I travel for that??

I should mention another factor. As indicated, I was to be a session moderator. I had expected to be sharing this load with a colleague also attending the conference. During the registration process, a surprise got sprung on me: one and only one moderator per session. To be honest, this change in policy left a bad taste in my mouth. It seemed like an attempt to reduce the number of “insiders” at the expense of running a good conference—and I doubted it would do much to change the perception that the meeting was full of “insiders”. Indeed, I thought it would be counterproductive: fewer moderators meant fewer opportunities for new joiners to take on responsibilities and become visibly active players. Moreover, when moderator slots were eliminated, who would fail to suspect that the people who had been Of The Body for longer would be the ones to *keep* their positions?

Other problems come to mind. First, moderating the kinds of sessions which typify ScienceOnline is *hard work,* and it helps to have a partner to share the neural load. Yes, wisdom is distributed around the room, but coordinating that collection of varied experiences can itself be a two-person job. Second, some topics *need* a double perspective. As David Dobbs pointed out, a session like “Can’t Writers and Researchers Get Along?” pretty much requires co-moderation. Third, more generally, a second moderator can intervene when the first begins to dominate the conversation. Fourth, sometimes people have to back out. (I feel like I was just recently discussing reasons why this could happen.) When that happens, you don’t want to be stuck without a spare. The remaining moderator can also help with recruiting a replacement.

- is seven feet (2.13 m) tall,
- carries three hundred pounds (136 kg) without effort,
- can easily tear straps meant to withstand 5000 pounds of force (the weight of over 2 tonnes),
- can sing beautifully whilst waving musicians like batons, and
- neutralizes gravity itself through the power of sheer awesomeballs.

But to a certain mindset, which fancies itself much put-upon and misunderstood, Harrison Bergeron could burst through the door waving a BFG9000 and riding a T-rex, all to nary a chuckle.

]]>*Why* is this piece of gee-whizzery being waved about, when the popularized “explanation” of it is so warped as to be misleading? Is the goal of “popularizing mathematics” just to inflate the reader’s ego—the intended result being, “Look what *I* understand!,” or, worse, “Look at what those [snort] *professional mathematicians* are saying, and how obviously wrong it is.”

Today’s instalment (noticed by my friend Dr. SkySkull): the glib assertion going around that

$$ 1 + 2 + 3 + 4 + 5 + \cdots = -\frac{1}{12}. $$

Sigh.

It’s like an Upbuzzdomeworthy headline: *These scientists added together all the counting numbers. You’ll never guess what happened next!*

“This crazy calculation is actually used in physics,” we are solemnly assured.

Sigh.

The physics side of the story is, roughly, “Sometimes you’re doing a calculation and it looks like you’ll have to add up $$1+2+3+4+\cdots$$ and so on forever. Then you look more carefully and realize that you shouldn’t—something you neglected matters. It turns out that you can swap in $$-1/12$$ for the *corrected* calculation and get a good first stab at the answer. More specifically, swapping in $$-1/12$$ tells you the part of the answer which doesn’t depend on the particular details of the extra effect you originally neglected.”

For an example of this being done, see David Tong’s notes on quantum field theory, chapter 2, page 27. For the story as explained by a mathematician, see Terry Tao’s “The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation.” As that title might hint, these do presume a certain level of background knowledge, but that’s kind of the point. This is an instance where the *result itself* requires at least moderate expertise to understand, unlike, say, the four-colour theorem, where the premise and the result are pretty easy to set out, and it’s the stuff in between which is much harder to follow.

**ADDENDUM (19 January 2014):** I’ve heard the argument in favour of this gee-whizzery that it “gets people excited about mathematics.” So what? A large number of people are misinformed; a tiny fraction of that population goes on to learn more and realize that they were, essentially, lied to. Getting people interested in mathematics is a laudable goal, but you need to pick your teaser-trailer examples more carefully.

And I see Terry Tao has weighed in himself with a clear note and some charming terminology.

]]>Responsive Ed has a secular veneer and is funded by public money, but it has been connected from its inception to the creationist movement and to far-right fundamentalists who seek to undermine the separation of church and state.

Infiltrating and subverting the charter-school movement has allowed Responsive Ed to carry out its religious agenda—and it is succeeding. Operating more than 65 campuses in Texas, Arkansas, and Indiana, Responsive Ed receives more than $82 million in taxpayer money annually, and it is expanding, with 20 more Texas campuses opening in 2014.

Along with the usual evolution-denialist drivel, those taxpayer funds are buying a threat to public health:

The only study linking vaccines to autism was exposed as a fraud and has been retracted, and the relationship has been studied exhaustively and found to be nonexistent. But a Responsive Ed workbook teaches, “We do not know for sure whether vaccines increase a child’s chance of getting autism, but we can conclude that more research needs to be done.”

Anti-vax lunacy from the religious right? Who would have thunk it?

Well, other than people who have looked at the data, that is.

]]>Betcha people at *Science* and *PLOS* are giggling fit to burst right about now.

Yours,

Blake