deodands

The nonneutron teaspoon

Recent theoretical studies and a literature controversy have shown the need for a determination of the volume of the teaspoon. We have carried out a series of volume determinations and report the surprising results here.

Literature review

The teaspoon has long been asserted by authorities in the field to have a volume of 5ml (author's spouse, pers comm). In the context of recent developments in the neutron teaspoon debate, our laboratory conducted a literature search (Google Inc, Stanford) for neutron star teaspoon, followed by heuristic refinement, and discovered a NASA estimate of 1.5ml for the volume of a teaspoon. As both of these sources have been known to be error in the past (for example the Hubble space telescope for one and a still classified example for the other) we decided to carry out new determinations.

Materials

All identifiable teaspoons from two locations (author's cutlery drawer, including the rarely seen bit right at the back with the pastry crimper; dishwasher contents) containing teaspoons believed not to have been previously used to unscrew things were gathered. These comprised both shiny metallic ('steel') and plastic spoons. In steel, 5 teaspoons (John Lewis, acquired at time of author's marriage to teaspoon expert, with subsequent resupply at time of critical inability to find any teaspoons at all in this bloody house); 2 IKEA teaspoons, two unbranded spoons (China), two Disney spoons (Pooh and ballons; Pooh and tree, both in the superior EH Shepard version); in plastic, 3 spoons (Teletubbies, Mothercare, and Heinz Baby Basics), and the breadmaker measuring spoon. A 5ml medicine spoon (provenance large national healthcare purchasing operation via author's spouse) was used as positive control , and volumetric studies were carried out with 5ml syringe of the same provenance. Water supplies (infrastructure by Cambridge municipal pride, billing by brand-consultancy-named French multinational rather regretting overacquisition) were un-ionised, not unlike the multinational.

Methods

Preliminary studies revealed surprising difference between the volume of water contained in a level teaspoon and the maximum loadable using surface tension to create a liquid 'heaped teaspoon'. Accordingly, each spoon was evaluated at its level load and its maximum load. Assessment of the point of level loads wa subject to parallax errors, especially when evaluated from the height of the six year old technical assistant, although these were minimised by the beginning of Gladiators on Challenge TV and the subsequent author-only experiments. Assessment of maximum loads were performed by watching for the first drip, and required a steadier hand than likely available. Volumes greater than 5ml were not experimentally accessible given this constraint on singlehandedly refilling the syringe. Replications were not performed so as not to complicate the ANOVA.

Results

Plastic-based teaspoons were found to be consistently smaller level loaders than steel ones and were discarded from the subsequent analysis on the grounds that no one visualizes a Teletubbies spoon full of neutron star. The plastic spoons did have comparable heaped loads to the metal ones, possibly because of increased surface tension. The 11 steel teaspoons had a mean level volume of 2.66 ml (sd 0.37ml) and heaped volume of 4.2ml (sd 0.46 ml). Intriguingly, the positive control medicine spoon had a level load of only 3.2ml (5ml heaped) leading to a reclassification of it as not a positive control, and a redefinition of laboratory medical procedures.

Conclusions

Teaspoons do contain about 5ml, but only when you heap them. Variation amongst our sample was about 20%. NASA is wrong. Water in surface tension looks rather beautiful close up.

NASA, 100 million tons (this time)

NASA again. Note the difference of 900 million tons from here:

One teaspoon of neutron star would weigh 100 million tons on Earth.

The Wow of physics: the amazing variability of the neutron star teaspoon

Neutron stars are small and very dense stars. So astonishingly, mindbogglingly, dense that they're a rather useful rhetorical device for the people who do public understanding of science. They want to persuade a grudging audience that there is a point to theoretical physics and astronomy. If neutron stars can be dense enough to make you say 'Wow', then they have made some sort of progress. That's why you'll probably have heard of the amazing mass of a teaspoon of neutron star. What this entry is about is how amazing the variability is in the subsequent estimates, and what this teaches us about physics. It uses sarcasm and SI units.

Neutron stars do really exist. Long after the protons and electrons have long given up the struggle to maintain their identity against the force of gravity, all that is left is neutrons, pressed together into one big atomic nucleus a few kilometres across. They're the pulsars that populate the radio sky with signals so regular and apparently the product of a conscious agent that their discoverers worried about how they would talk to their scientific colleagues about the existence of Little Green Men.

There are problems with the Wow approach to physics. Some people, for example, might object to representing theoretical physics as a collection of Wow things, rather than a high minded pursuit of truth. Since I don't think it's very high minded, and there are other more interesting kinds of scientific explanations of the world anyway, I don't mind that. But there is a more practical problem, common to every attempt to explain the mindboggling, which is that the mind gets boggled before it gets to the explanation.

The reasonable solution to the boggle block for the gigantic quantities of cosmology is to shift at least one of the quantities involved to a human scale. This is where the teaspoon appears: as the standard Public Understanding of Science (they really do call it PUS) unit of neutron star volume. A teaspoon is reckoned by cookbooks at 5mls=5x10-6 m-3. The maximum theoretical density of a neutron star is roughly 2x1018 kg m-3 . So a teaspoon of neutron star would have a mass of up to around 1013 kg.

1013 kg is still too boggling to fit in the mind, so the PUSsians still have some work to do. We can lose three orders of magnitude by converting to the still apparently intuitive unit of the tonne. (Apparently intuitive, because we think we know what a tonne is, but there is no conceptually graspable difference between a billlion kilograms and a billion tonnes.) A teaspoon of neutron star might have a mass of 1010 tonnes. Finally we can fit that past the boggle filter by converting it into words. A neutron teaspoon has a mass of 10 billion tonnes. (Actually, some go further and try and convert that figure into the mass of a mountain or - a prevalent recent meme - the mass of all the cars, lorries and trucks on Earth. There's a whole other investigation there, someday).

Wow. 10 billion tonnes. Give these people university departments, superconducting supercolliders, and a cameo on the Simpsons.

And yet. Try a google for 'neutron star teaspoon', a triple which yields almost no false positives. According to NASA, who after all have the most acute PUS mission of any organisation on the planet, a teaspoon of neutron star 'weighs about a billion tonnes'. Well, only a factor of 10 different to my estimate. Perhaps they use those tiny little souvenir teaspoons with an enamelled picture of the Belgian flag on the handle. Another hit, from the University of Tasmania: 500 million tonnes. I suppose it would be quite fiddly to fill the whole teaspoon. Another hit, NASA again, but they seem to have spilt some: 100 million tonnes. From the astrophysics course at Cornell (so not really PUS), a 'teaspoon of neutron star material would weigh about 10 million tonnes'. Ooh. That's a really small teaspoon. In the other direction, the Canadians say 50 billion tonnes , and my world google record so far comes (of course) in California, where the neutron teaspoon has a mass of 1 trillion tonnes . To meta-teaspoon this, you could balance one Californian's neutron teaspoon against one hundred thousand New York teaspoons. When we got married we were given six teaspoons: I would have to get married over 16 thousand times to acquire enough NY ones to balance the Californian, and given UK divorce laws would have to live more than 30000 years.
Wow. Fund that SSC after all.

(It would probably be unfair to mention the student project page at Penn State that assigns the neutron teaspoon a massive 5 tonnes, so I will. Similar sightings welcome).

So where does this teaspoon variability come from? Even the fiddliest Belgian souvenir spoon can't differ by more than a factor of ten in volume from my wedding present ones. Is there difference in the science?
Time for the science bit then, which relies on Shapiro and Tekukolsky, a graduate text from 1983 which starts with the heartening words

An understanding of polytropic stellar models or the equation of an ideal Fermi gas is likely to be useful to the student forever.

Stars are big balls of gases. Their size is determined by the balance between two opposing forces: gravity pulling the gas inwards, and pressure pushing it outwards. Just like the pressure of air in a balloon, pressure reflects the fact that it's hard to push things together. The pressure depends on how many things you're trying to push together (density), but it also depends on how hard they are to push together. At higher temperatures, the air molecules have more energy, so it takes more effort to keep them from bouncing off each other. There's a relation, then, between the amount of matter and the pressure it exerts in a given setting, which is called the equation of state. We have a fairly good idea of this relationship for the interior of stars like our own sun. Eventually, as our sun radiates energy away, the internal pressure will fall and the gravitational force will increase the density until the point at which electrons are forced together (or, more precisely into degenerate states) forming a white dwarf, and for these conditions we also have a fairly good idea of the equation of state. But for more massive stars, the collapse keeps going past this point and is only halted when the remaining neutrons are forced too 'close' together. And at this point we reach some uncertainty (at least according to 1980 era graduate texts) in what the equation of state is. So there is some real scientific uncertainty in the mass of the neutron teaspoon.

Another source of variability is whereabouts in the neutron star we are going to scoop our teaspoon out from. Since it's really, mindbogglingly, dense, remember , there's a lot of gravity about, which accounts for a pretty steep gradient in density from the outside to the inside. Shapiro and Teukolsky give two examples for a neutron star of typical mass, on p251, in which the outer crust can have a density of between 1014 kg m-3 and 1017 kg m-3 depending on the choice of the equation of state, but the variation in the core density is only between 1.28 and 1.93 x 1018 kg m-3 . Oops, soory, too mindboggling...that 's a two-yearly remarriage to keep up for a period between 36 days (outer crust) and 300 years (core). It seems to me the core density is the natural unit to teaspoonize, but I suppose one might also want to use the mean density. Another textbook (Michel, Theory of Neutron Star Magnetospheres, University of Chicago Press, 1991) gives a mean density of 2.67x1017 kg m-3 with an astonishing degree of precision and lack of references.

The next question is how massive is the neutron star we are interested in. The larger the mass, the greater the core density. S&T (Fig 9.6) give a range of observed masses for seven observed neutron stars, of between about one and two times solar mass. But actually Figure 9.2 suggests that within that range, different choices for equations of state account for more change in core density prediction than differences in mass choice, and whatever the mass choice there is a range of core densities of between 3x1017 to 3x1018 kg m-3 . (It's not clear to me how much weight should be given to those different state equation choices, and whether some can be ruled out: this Figure is drawn from a review paper from 1979 so with luck more is known now. ) The upper limit seems to be about the place where the theory of how neutrons pack together gets a bit foggy, and S&T leave open the possibility that exotic, denser phenomena might still be possible.
But it seems clear that if its the core density you are interested in, and neutron (rather than pion) stars is what you are talking about, then theory is confident to within an order of magnitude. The teaspoon of core contains a mass of between 1012 and 1013 kg: between a billion and and ten billion tonnes. If you want to take a mean density, which would take some fancy teaspoon work, perhaps an order of magnitude or so less. If you scoop from the crust, you can push the teaspoon down to maybe 109 kg: a million tonnes, but you probably weren't meaning to do that because at that density you aren't getting the exciting neutron-neutron packing that makes neutron stars do pulsar things and all the other worthwhile theoretical targets you were trying to PUS in the first place. So NASA (1 billion tonnes) gets full marks, Tasmania (500 million tonnes) and Canada (50 billion tonnes) pass with allowance for teaspoon variation, and there are fails for Cornell (10 million tonnes, and they imply they mean the core not the crust), and UC Riverside (10 trillion tonnes).

Why all these errors? None of these pages show their working, which is fine for the two PUS ones (NASA and Canada) who actually get the answer right, but more embarrassing for the academic pages. My guess is that the ten trillion tonnes was meant to be ten trillion kg, but I can't quite imagine where the 10 million came from.

How can these errors persist? Because they don't matter. You still say Wow even if the neutron teaspoon comes from Penn State (at a measly 5 tonnes) because that is already off the scale of how heavy teaspoons can reasonably be. So the effect is achieved in any case. No one does anything with these estimates, which is why they don't get thought about once made. Which slightly makes one wonder about the point of them. Neutron stars are real objects, with a physical presence in our lives from the time a graduate student sat in a field, not far from where I am typing this, and heard the first pulsar. There's an extraordinary human achievement in understanding as much as we do about them: that's the real Wow.

I finish with my favourite estimate for the mass of the neutron teaspoon. It comes via a Yahoo news story, sadly no longer online, but preserved here:

Neutron stars are almost unimaginably dense: a teaspoon of neutron star material weighs a billion tons (1.016 billion tonnes)

(NB I have probably made a mistake or two in the numbers. For the reasons I've suggested it doesn't matter, but let me know anyway. The science is likely to be out of date. Thanks to Andrew Brown for relaying the question that sparked this. I am happily married and have no plans for further teaspoons.)

Famous for a day

Today's fleeting fame brought an email alerting me to Bob Johnson's Fibonacci resources page for his Durham mathematics undergraduates, which includes a couple of interesting articles such as one by George Markowsky neatly demolishing some of the sillier myths about the Golden Ratio and a 1995 Scientific American article by Ian Stewart. Stewart, as ever, explains things beautifully; it builds up on the Douady and Couder experiments, and a version of the same material, without the hokey dialogue device, made its way into Stewart's book Life's Other Secret. It's a shame Stewart was apparently unaware of Graeme Mitchison's much earlier Science paper which captured most of the key ideas: I think he could have made characteristically good popular use of it.

As featured in the Guardian...really

Whoops. I now seem to be in the paper paper as well as the, um, nonpaper paper. It wasn't really meant to happen. I had sent an email to the journalist involved, mainly out of sheer gratitude at being linked to, but it got routed to the letters section instead and they told me they were going to publish it at quarter to five (when I do real work) and I didn't get a chance to think about whether it was wise or not.

The letter 1) it doesn't make any sense if you don't know about the way people use jargon like 'emergent', 2) it doesn't make much sense if you haven't spent hours thinking about Turing and phyllotaxis (and I know no one else in the world who gives a toss)
3) on reflection I think it's maybe also wrong, so its good that it's incomprehensible. One function of the original letter was self-advertisement, so there was a result.

The spark was that someone said that the study of 'emergent' phenomena goes back to Turing's work in the 50s on the development of plant structure. It's true that he developed a model of how things like stripes could apparently 'emerge' from homogenous tissue. But the way people use words like 'emergent', particularly the Santa Fe complexity theory types is more to do with the notion of a self-organising complexity that arises as a feedback between the state of the system and the response of individual agents to that state (traffic jams, termite mounds, tipping points, blah), and that you can't analyse in terms of that individual response. The point of the letter was that Turing was doing 'good old fashioned applied mathematics': ie exactly doing that kind of analysis you're not meant to be able to do for emergent systems. And if that counts as analysing emergent phenomena then like that man in Moliere, applied mathematicians have been doing complexity theory all along.

Why I think I might be wrong is that there might, after all be a qualitative difference between, say, using Maxwell's equations to predict radio waves, which is a very domain specific, experimentally-bound kind of applied mathematics, and the Turing approach which is much more an approach asking how in principle certain kinds of structure can come from certain kinds of inputs. Turing's is perhaps much closer to the complexity theory viewpoint. I'm just in the middle of reading an interesting book by Evelyn Fox Keller on why biologists loathe this latter view.

But I am now the sort of person that writes letters to the Guardian.

Self-organisation is not reductionist

Why Turing stripes are not a self-organising system....

"Can things self-organize ? Yes, any system that takes a form that is not imposed from outside (by walls, machines or forces) can be said to self-organize. The term is usually employed however in a more restricted sense by excluding physical laws (reductionist explanations), and suggesting that the properties that emerge are not explicable from a purely reductionist viewpoint" Self-Organizing Systems FAQ for Usenet newsgroup comp.theory.self-org-sys

In other words, if you have a proper theory, it's not a self-organised, emergent system

Manchester details updated

I have updated the section of my Turing archive bibliography dealing with the Manchester NAHC archive.

As featured in the Guardian...

Sort of. Linked to in the online edition but despite excitedly digging through the recycling box to find last weeks paper (see, Lomborg, there is a point to recycling) not in the paper copy.

It's not quite clear from the article, but it sounds to me like an Eric Bonabeau talk at ETCon claimed that the concept of emergence goes back to Turing's morphogenesis work. (Hence a link to my site). Lovely to have my site referenced, but whoever said it, it's a weak claim. If 'emergent' means anything, it means properties that you can only see in a large enough interacting system, whereas Turing's reaction diffusion theory is good old fashioned applied mathematics. It only predicts emergent pattern in the sense that radio waves are an emergent property of Maxwell's equations. You might claim that his theory of Fibonacci phyllotaxis comes a bit closer: it has at least an idea of complexity increasing through time via a simple iterated dynamics, but the complexity is that the parastichy numbers are 89 and 55, say, not 3 and 5. These are differences of degree, not of kind: there's not anything to 'emerge'. Andrew Hodges does say somewhere that Turing was in a position where he could have gone towards chaos theory, but that remains a might have been.

I think the silliness most likely resides in the mushiness inherent whenever people start using words like 'emergent'. As the great and very cute John Maynard Smith says of complexity theory: 'Absolute fucking crap. Crap with good PR but crap' (via Andrew Brown's Darwin Wars).

Microtubules

I got all excited when I came across a reference to Fibonacci numbers in microtubules in Roger Penrose's Shadows of the Mind. But now I'm not excited, or rather only excited about microtubules.

Microtubules are hollow tubes, around 25nm in diameter, which help form the cytoskeleton. Microtubules are a polymer of tubulin dimers, which are connected head to tail to form protofilaments which, when assembled side by side, form the cylinder, although that's probably not how they are actually made in vivo. Now the exciting thing was that one common arrangement is to make the cylinder from 13 parallel lines of protofilaments rolled together, and 13 is a Fibonacci number, right? Exciting because we understand enough about Fibonacci structure in plants to see that under the right circumstances you should be able to see it elsewhere (like Douady and Couder's oil drops), but in truth there are no other biological examples. There are some chemical ones I think.

So I read some papers: Li et al, Structure 10: 1317 (2002) and Inclan et al J Cell Science 114:413 (2000). These are why modern molcular biology is so exciting - sequence analysis, structure data, molecular modelling, functional studies, astonishing imaging coming together to offer the beginnings of a real explanation of cellular processes. It turns out that the 13 protofilament tubule has a rise of one and a half tubulin dimers per turn, so that there is a seam running along the tube where the two different monomers become neighbours. First problem for a phyllotaxis like explanation - with aperiodic boundary conditions you're lost. It also turns out that, while most microtubules have 13 protofilaments, they can have between 9 and 16. Second problem. Final problem: it looks as if the assembly mechanism is template driven in some way, which makes it hard to envisage a continuous change along a parastichy tree as an explanation.

So, lose enthusiasm for that idea. Let's go back to Penrose and have another look. He does to his credit say

However, one should not get carried away with such considerations; for example,the "9" that occurs in the bundles of microtubules in cilia and centrioles is not a Fibonacci number. Penrose, Shadows of the Mind

You might wonder what all the fuss was about 13, then. Also, Penrose gives a picture, Fig 7.8, imagining a microtubule split open and displaying a 5+8 parastichy (but not the 1 1/2 unit rise). Where did this picture come from? The only reference in the text is that 'it is apparently found (at least normally) that this pattern is made up of 5 right-handed and 8 left-handed helical arrangements' but there's no reference given for this claim.

Why I am going on about this? Well I already had a caricature view of Penroses's thesis, something like "we perform Turing uncomputable operations when we think, and we couldn't do that with deterministic, digital computer-like brains, and the way it can happen is because quantum mechanics, and quantum coherence in particular, are not averaged out up at the biophysical level of the brain". Always thought it seemed like nonsense, and now I have the book on my desk it's a chance to find out what this manifestly bright man knows that I don't that makes him say such apparently silly things.

Unfortunately I flunked my chance. Philosophy makes my eyes glaze, no matter how I try, and a benefit of my long mathematical education was to learn skip all formulae, so that's most of the book unread. It's towards the end of the book where Penrose tries to figure out where the QM is getting to act at this scale and points the finger at those microtubules. But that's an unconvincing discussion when it talks about things I know about, which is always pretty discouraging. So maybe Penrose will remain unread, and the mystery of the apparently silly will remain unsolved, at least until I'm in my philopause.

Another original view

From Golden Section in Enigmatic Martian Monuments and Terrestrial Flora and Fauna, by Dr Demetrio G Lahoz in Tetet-95, whatever that is. Nice pictures, though:

As if ETI [extra-terrestrial intelligence], knowing that phi was going to be so good for sunflowers alleviated to them the billion-year long process of naturally learning to encode it in their genes AND the creator ETI Himself artificially implanted phi in their quasi-soul just a few megayears ago.

Turing after the instability

Most of my site is about Turing's approach to the specific morphogenetic problem of Fibonacci phyllotaxis, so it's good to see a paper (Allaerts W: Fifty years after Alan M. Turing. An extraordinary theory of morphogenesis. Belgian Journal of Zoology 133(1): 3-14 2003) on Turing's more general reaction-diffusion theory based not on the well known 1952 paper but on his later works finally published in 1992.

On the other hand, the Belgian Journal of Zoology don't seem to have mastered online publishing ("free full text of special issues"). Personally I have pretty much stopped reading any new academic papers that I can't get online, so I hope Dr Allaerts' paper doesn't have the same fate as a chemistry paper published in German about the time all the chemists started publishing in English instead (1914?) or a Domesday book published for a special edition of the BBC Micro.

UC Riverside: 1 trillion tons

From the University of California at Riverside, course notes I think:

...their density is enormous, a teaspoon of neutron-star material would weigh about 1012 (one trillion) tons on the Earth's surface.

Best estimate of all

This is my favourite. It's already turned up in people's .sig files. It came via Yahoo to a mailing list

Neutron stars are almost unimaginably dense: a teaspoon of neutron star material weighs a billion tons (1.016 billion tonnes), or as much as all cars, trucks and buses on Earth.

Canada: 50 billion plus bang

I think Triumf is some Canadian high energy physics consortium.

How dense? A teaspoon of this material would weigh 50 billion tons on Earth!

Tasmania: 500 million plus bang

Research opportunities at the University of Tasmania on
pulsars

The density of neutron stars is around 1017kg m3 (sic). A teaspoon of this matter would have a mass of 500 million tonnes!

Cornell: 10 million

From an astronomy course at Cornell

A teaspoon of neutron star material would weigh about 10 million tons.

NASA: 1 billion

From NASA's Pulsar Tutorial:

A teaspoon of neutron star material weighs about a billion (1,000,000,000) tons.

Penn: 5 tons

A student page, I think, at Penn State

A neutron star has a diameter of only a dozen or so miles (NASA), and densities of about 5 tons per teaspoon (Bennett, 297).

The Bennett reference is: Bennett, Jeffrey, et al. The Cosmic Perspective (Brief Edition). New York: Addison Wesley Longman, Inc., 2000. But I haven't looked it up.