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Category theory and chemistry

Yes, they're connected! Next week, +Blake Pollard and I will talk about our work on categories where the morphisms are open reaction networks, like the thing shown here. We’ll do this at Dynamics, Thermodynamics and Information Processing in Chemical Networks, a workshop at the University of Luxembourg organized by Massimiliano Esposito and Matteo Polettini. We’ll do it on Tuesday, 13 June 2017, from 11:00 to 13:00, in room BSC 3.03 of the Bâtiment des Sciences. So if you’re around, please stop by and say hi!

If you can't slip off to Luxembourg, never fear: you can already see my slides here:

• The mathematics of open reaction networks,

Abstract. To describe systems composed of interacting parts, scientists and engineers draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Feynman diagrams and the like. In principle all these different diagrams fit into a common framework: the mathematics of monoidal categories. This has been known for some time. However, the details are more challenging, and ultimately more rewarding, than this basic insight. Here we explain how various applications of reaction networks and Petri nets fit into this framework.

(I just finished making these slides while watching a crew of hawks soaring in front of the Alps. I'm at a farm in Rubigen, a town near Bern in Switzerland. When workers harvest hay, mice run out. Hawks, who know this, are circling. My wife is busy listening to talks at a workshop on the role of emotions in classical China and Greece. I've been busy writing my talk for next week. But this afternoon we'll be rewarded for our labors: a bunch of us will hike up into the mountains.)

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This compound explodes if you touch it with a feather. In fact, it's the only compound so explosive that it can be set off by exposing it to small amounts of alpha radiation - that is, high-speed helium nuclei.

Puzzle 1: What is it?

Puzzle 2: When it explodes, what color smoke does it make?

Puzzle 3: If it's so unstable, how do you ever get enough of it in one place to explode?

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I have a new favorite molecule: adamantane. Someone is ‘adamant’ if they are unshakeable, immovable, inflexible, unwavering, uncompromising, resolute, resolved, determined, firm, rigid, or steadfast. But ‘adamant’ is also the name of a legendary mineral - and the word comes from the same root as ‘diamond’.

The molecule adamantane is shown here. It's the simplest of the diamondoids: molecules made of carbon and hydrogen where the carbons are arranged just like a small portion of a diamond crystal!

For more pretty pictures of diamondoids, and some fun math puzzles about adamantane, read my blog article:

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Metal-Organic Framework 5

I like the look of this thing!   It's a metal-organic framework - a compound made of metal ions connected by organic stuff.   The picture here is just part of a structure that keeps repeating in all directions. 

The blue tetrahedra are made of an oxygen atom surrounded by 4 atoms of zinc.  They're connected by a kind of latticework made of an organic molecule called 1,4-benzodicarboxylic acid. 

The whole thing is called Metal-Organic Framework 5 or MOF5 for short.  There are lots of other kinds.

But what about the huge yellow ball?

That's not a real thing.  It's empty space where you can put something - like a molecule of hydrogen! 

And indeed, metal-organic frameworks are used for storing hydrogen - you can actually pack more hydrogen into a MOF than you can easily squeeze into an empty tank!  So they're not only beautiful, they're practical.

For a bigger view of MOF5, go here:

For more about metal-organic frameworks, go here:

Also check out my new chemistry collection:

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The structure of a diamond crystal is fascinating.  But there’s an equally fascinating form of carbon, called the triamond, that’s theoretically possible but never yet seen in nature.

In the triamond, each carbon atom is bonded to three others at 120° angles, with one double bond and two single bonds. Its bonds lie in a plane, so we get a plane for each atom.

But here’s the tricky part: for any two neighboring atoms, these planes are different. In fact, if we draw the bond planes for all the atoms in the triamond, they come in four kinds, parallel to the faces of a regular tetrahedron!

If we discount the difference between single and double bonds, the triamond is highly symmetrical. There’s a symmetry carrying any atom and any of its bonds to any other atom and any of its bonds. However, the triamond has an inherent handedness, or chirality. It comes in two mirror-image forms.

Some chemists have argued that the triamond should be metastable at room temperature and pressure: that is, it should last for a while but eventually turn to graphite. Diamonds are also considered metastable, though I’ve never seen anyone pull an old diamond ring from their jewelry cabinet and discover to their shock that it’s turned to graphite. The big difference is that diamonds are formed naturally under high pressure — while triamonds, it seems, are not.

It's the mathematics behind the triamond that really interests me.  It's a topological crystal that can be constructed starting from the complete graph on four vertices.  For the details of how this works, see my Visual Insight blog article:

The picture here was made by Greg Egan.
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Silly sounding elements

Forget Trump. We have until November to prevent scientists from naming an element oganesson

I don't have anything against Yuri Oganessian, a pioneer in the study of  highly radioactive, short-lived elements.   I just think the word "oganesson" stumbles off the tongue like a dazed, jet-lagged passenger staggering off a plane and falling down the stairway. 

And it's a noble gas!   A noble gas should sound noble.  

Neon.  Argon.  Krypton.  Xenon.  Oganesson.  

Which one does not belong?  Which name was created by somebody without a shred of poetry in their soul?

The International Union of Pure and Applied Chemistry, or IUPAC, has begun a five-month public review, ending 8 November 2016, before it names these elements:

Element 113: Nihonium (Nh)
Element 115, Moscovium (Mc)
Element 117, Tennessine (Ts)
Element 118, Oganesson (Og)

I have no opinions except about how the names sound and look.  Nihonium and Moscovium sound okay to me.   The word "Tennessine" is awkward.   I like the state of Tennessee.  I have nothing against it having its own highly radioactive element.   I just don't like this word.  This element is a halogen, and again it's the least pretty of the bunch:

Fluorine.  Chlorine.   Bromine.   Iodine.   Astatine.   Tennessine.

But "oganesson" is worse.  IUPAC could have done better hiring that unemployed guy who used to make up names for elements on Star Trek.  The only good thing about "oganesson" is that it has such a short half-life that we'll hardly ever need to say that word. 

I would gladly accept tennessine if it would stop "oganesson" from lurching onto the periodic table.  In fact, I'd volunteer to eat the world's entire supply of tennessine.

If you agree with me, or have other opinions, write a polite letter to Dr. Lynn M. Soby, the Executive Director of IUPAC, at

PS - yes, I know helium is also a noble gas.  People gave it a name suitable for a metal, not a noble gas, before they knew better.  It's too late to call it helion.
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Superionic ice

There are over 15 kinds of ice.  Different kinds are stable at different pressures and temperatures.  Some of the weirdest may exist inside ice giants: planets like Uranus and Neptune, which have also been found orbiting other stars.  Most of what we know about these kinds of ice comes from computer simulations, since they only exist at very high pressures.

They're called superionic ices, because while the oxygen atoms get locked in a crystal structure, the hydrogen atoms become ionized, breaking apart into protons and electrons.  The protons can then move around like a liquid between the oxygen atoms!  

The first phase of superionic ice was predicted in 1999 by a group of Italian scientists.   They predicted that this ice exists at pressures 500,000 times the atmospheric pressure here on Earth, and temperatures of a few thousand Kelvin.  In this kind of ice, the oxygen atoms form a crystal called a body centered cubic.

In 2012, Hugh F. Wilson, Michael L. Wong, and Burkhard Militzer predicted the new phase shown here.  This may show up above 1,000,000 times atmospheric pressure.  The oxygen atoms, shown as blue spheres, form a pattern called a face centered cubic.  The protons are likely to be found in the orange regions.

Hugh Wilson said:

Superionic water is a fairly exotic sort of substance.  The phases of water we're familiar with all consist of water molecules in various arrangements, but superionic water is a non-molecular form of ice, where hydrogen atoms are shared between oxygens. It's somewhere between a solid and a liquid—the hydrogen atoms move around freely like in a liquid, while the oxygens stay rigidly fixed in place. It would probably flow more like a liquid, though, since the planes of oxygen atoms can slide quite freely against one another, lubricated by the hydrogens.

These simulations are hard, and newer papers are reporting different results.  You can also try to make superionic ice in the lab, but that's even harder!   In 2005 Laurence Fried tried to make it at the Lawrence Livermore National Laboratory in California.  He smashed water molecules between diamond anvils while simultaneously zapping it with lasers.  He seemed to find evidence for superionic ice.

Eventually theory and experiment will converge on the truth.  Only then will we understand the hearts of the ice giants.

You can read more here:

and for some even newer results, try this:

Here's the paper on the first kind of superionic ice:

• C. Cavazzoni, G. L. Chiarotti, S. Scandolo, E. Tosatti, M. Bernasconi, and M. Parrinello, Superionic and metallic states of water and ammonia at giant planet conditions, Science 283 (1999), 44-46.  Available free with registration at

and here's the second kind:

• Hugh F. Wilson, Michael L. Wong, Burkhard and Militzer, Superionic to superionic phase change in water: consequences for the interiors of Uranus and Neptune.  Available free at

#spnetwork arXiv:1211.6482 #ice
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Chemical reactions in Copenhagen

This is a famous harbor called Nyhavn.  I haven't been there yet!  I'm in Copenhagen at a workshop on Trends in Reaction Network Theory, and I've been sweating away in hot classrooms listening to talks. 

But don't feel sorry for me!  (You probably weren't.)  I've been loving these talks, loving the conversations with experts and the new ideas — and after the workshop is over, I'm going to spend a few days walking around this town.

A reaction network is something like this:

2 H₂ + O₂ → 2 H₂O
C + O₂ → CO₂

just a list of chemical reactions, which can be much more complicated than this example.   If we know the rate constants saying how fast these reactions happen, we can write equations saying how the amounts of all the chemicals changes with time! 

Reaction network theory lets you understand some things about these equations just by looking at the reaction network.  It's really cool.

The biggest open question about reaction network theory is the Global Attractor Conjecture, which says roughly that for a certain large class of reaction networks, the amount of chemicals always approaches an equilibrium. 

It's a hard conjecture: people have been trying to prove it since 1974.  In fact, two founders of reaction network theory believed they'd proved it in 1972.  But then they realized they had made a basic mistake... and the search for a proof started. 

The most exciting talk so far in this workshop — at least for me — was the one by Georghe Craciun.  He claims to have proved the Global Attractor Conjecture!  He's a real expert on reaction networks, so I'm optimistic that he's really done it.  But I haven't read his proof, and I don't know anyone who says they follow all the details. 

So, there's work left for us to do.  His paper is here:

• Georghe Craciun, Toric differential inclusions and a proof of the global attractor conjecture,

There's a branch of math called 'toric geometry', which his title alludes to... but I asked him how much fancy toric geometry his proof uses, and he laughed and said "none!"   Which is a pity, in a way, because it's a cool subject.  But it's good, in a way, because it means mathematical chemists don't need to learn this subject to follow Craciun's proof.

There have been a lot of other good talks here.  You can read about some on my blog:

including the comments, where I'm live-blogging. 

I gave a talk called 'Probabilities and amplitudes', about a mathematical analogy between reaction network theory and particle physics, and you can see my slides.  Alas, the talks haven't been videotaped, and most of the other speaker's slides aren't available.  I have, however, collected links to some papers.

I've gotten at least two ideas that seem really promising, both from a guy named Matteo Polettini, who is interested in lots of stuff I'm interested in.  I won't tell you about them until I work out more details and see if they hold up.  But I'm excited!  This is what conferences are supposed to do.   They don't always do it, but when they do, it's really worthwhile.

The picture here was taken by a duo called angel&marta.  You can see more of their fun photos of Europe here:

Finally, here is the abstract of Craciun's paper:

Abstract. The global attractor conjecture says that toric dynamical systems (i.e., a class of polynomial dynamical systems on the positive orthant) have a globally attracting point within each positive linear invariant subspace -- or, equivalently, complex balanced mass-action systems have a globally attracting point within each positive stoichiometric compatibility class. We introduce toric differential inclusions, and we show that each positive solution of a toric differential inclusion is contained in an invariant region that prevents it from approaching the origin. In particular, we show that similar invariant regions prevent positive solutions of weakly reversible k-variable polynomial dynamical systems from approaching the origin. We use this result to prove the global attractor conjecture.

#spnetwork arXiv:1501.02860 #chemistry #reactionNetworks #globalAttractorConjecture   #mustread  
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Boron - not boring

This is boron carbide, an extremely hard ceramic material used in macho gear like tank armor, bulletproof vests, and engine sabotage powder. 

(Engine sabotage powder?  Yes, you can pour this into the oil supply, and it will make a car engine grind itself to death.)

If diamond has a hardness of 10, this comes in at 9.497.  But its crystal structure is even cooler than diamond!

A group of 12 boron atoms likes to form an icosahedron.   You can see 8 of these icosahedra here - the green things.  These form the corners of a rhombohedron - a kind of squashed cube.  These repeat over and over, forming a rhombohedral lattice.  

But that's not all!   The icosahedra are connected by struts!  These struts have carbon atoms at their ends and a boron in the middle.  Only one strut is shown in detail here.  The carbon atoms are the black balls and the boron is the little green ball.

Overall there are 4 boron atoms per carbon atom, so people call boron carbide B₄C. 

Puzzle 1: why are there 4 borons per carbon?  I haven't done the counting, so I don't understand this.

Puzzle 2: what's the difference between a rhombus and a parallelogram?

Puzzle 3: what's the difference between a rhombohedron and a paralleliped?

Puzzle 4: what's the difference between a rhombohedral crystal and an 'orthorhombic' crystal? 

Another macho application of boron carbide is to shielding and control rods for nuclear reactors!  The reason is that boron can absorb neutrons without forming long-lived radioactive isotopes.

The structure of boron carbide has even more subtle features, which I don't understand.  Maybe I'm not looking at the pictures carefully enough!

Puzzle 5: where are the octahedra made of boron atoms?

For clues, read this:

The picture here was made by 'Materialscientist' and placed on Wikicommons:

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Beautiful poison

Cinnabar is a mineral made of mercury - the silver balls - and sulfur - the yellow ones.  It's fascinated people for thousands of years.  When you grind it up, you get vermilion: a brilliant red pigment.

Vermillion was used in murals in Çatalhöyük, one of the world's oldest cities, in Turkey, back around 7000 BC.   It's been used in the art and lacquerware of China since the Han Dynasty!  You'll also find it in the Tomb of the Red Queen built by the Mayans around 650 AD.

It was precious in Rome, used for art and decoration.  Since mercury is poisonous, a term in working in the cinnabar mines was a virtual death sentence.  Pliny the Elder wrote:

Nothing is more carefully guarded. It is forbidden to break up or refine the cinnabar on the spot. They send it to Rome in its natural condition, under seal, to the extent of some ten thousand pounds a year. The sales price is fixed by law to keep it from becoming impossibly expensive, and the price fixed is seventy sesterces a pound.

The Chinese were probably the first to make a synthetic vermilion, back in the 4th century BC. A Greek alchemist named Zosimus of Panopolis mentioned the process around the 3rd century AD. In the early ninth century the alchemist Jabir ibn Hayyan described it in a book - and it then spread to Europe.

The process is pretty simple.   You mix mercury and sulfur together, forming a black compound called Aethiopes mineralis. You heat it in a flask. The compound vaporizes, and recondenses on the top of the flask. Then you break the flask, take out the vermilion, and grind it.  At first the stuff is almost black, but the more you grind it, the redder it gets.

Puzzle 1: Where did they get the mercury in the first place, if not from cinnabar? 

Puzzle 2: If they had cinnabar, why not just grind that to make vermillion?

Puzzle 3: Why does the stuff start out black?

Here's one possible answer to Puzzle 3.  Cinnabar contains one crystal form of mercury sulfide, the so-called alpha form, shown here.  It's a hexagonal crystal, and it's red.  But there's also another form, the beta form, which is black.  This is sometimes called metacinnabar - a cool word if I ever saw one.

In Taoist alchemy in China, cinnabar and gold were used in various potions that were supposed to give long life.  Cinnabar was considered to have a lot of yang and gold a lot of yin.  According to their theories, gold naturally transmutes into cinnabar over time, much as yin becomes yang (and vice versa). The evidence?   Deposits of cinnabar are sometimes found beneath veins of gold. 

Unfortunately, some people got mercury poisoning thanks to these potions! 

Isaac Newton also spent a lot of his later life doing alchemy.  This is not as dumb as it sounds, because at that time alchemy included what we now call 'chemistry', along with more mystical things.  Some hairs from Newton's body have been found to contain 4 times as much lead, arsenic and antimony as normal - and 15 times as much mercury!  This might explain Newton's tremors, severe insomnia, and paranoia.

I love the look of this crystal!  The picture, made by Ben Mills, is on Wikipedia:

Some of my text is quoted or paraphrased from these articles:

For the use of cinnabar in Taoist alchemy, see:

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