Sekali sepekan, saya akan mengikuti kuliah Biologi Umum, mata kuliah wajib yang normalnya diambil saat semester 3 lalu. Ini adalah pekan kedua kuliah Biologi. Sebuah pertanyaan penting diajukan oleh dosen di awal kuliah pekan pertama. Kami diminta memberikan jawaban mengapa perlu mengambil mata kuliah ini, terlepas
dari syarat akademis bahwa mata kuliah ini bersifat wajib diambil. Karena saya adalah (makhluk) hidup, saya perlu belajar ilmu tentang makhluk hidup, tentang bagaimana manusia menyusun pengetahuannya tentang makhluk hidup secara sistematis dan teratur, dan dapat diuji kebenarannya. Biologi adalah ilmu tentang makhluk hidup, sederhananya begitu. Ya, itu adalah alasan paling prinsip atas pertanyaan dosen di awal kuliah.
Dari sudut pandang ini, saya menjadi bersemangat mengikuti kuliah biologi.
Sejak SD, saya sudah belajar tentang ilmu pengetahuan alam, dan saat SMP dan SMA
ada mata pelajaran khusus tentang makhluk hidup. Untunglah, bahwa masih ada sedikit, tidak banyak, informasi penting biologi, sisa-sisa matapelajaran dahulu, yang teringat dan terekam baik di memori otak.
Lantas, saya menyiapkan beberapa pertanyaan terkait mata kuliah yang satu ini.
Secara makroskopis (atau lebih tepatnya makrokosmos, ya?) apa saja peran
alam semesta bagi kehidupan?
Apa peran bumi bagi kehidupan manusia?
Apa itu konsep dasar kehidupan?
Apa perbedaan antara benda hidup dan benda mati?
Berkaitan dengan kehidupan, apa saja teori-teori yang menjelaskan
tentang asal-usul kehidupan?
Bagaimanakah sejarah penggolongan/klasifikasi makhluk hidup dan apa
manfaat klasifikasi makhluk hidup bagi kehidupan?
Berkaitan dengan alam semesta, apa saja masalah yang sedang
dan akan dihadapi? dan bagaimanakah cara alternatif penyelesaiannya?
Jika saya dihadapkan pada problem atau masalah alam semesta
dan komponen penyusunnya, dapatkah saya mengidentifikasinya? Dan dapatkah
pula saya memberikan solusi alternatif bagi masalah tersebut?
Pesan penting yang dapat kupahami saat mengikuti kuliah ini adalah manusia perlu mengembangkan suatu perilaku kehidupan yang menghargai bumi, sebagai tempat tinggalnya, dan beragam makhluk hidup lain, seperti hewan dan tumbuhan, serta komponen abiotik pendukung kehidupan di planet yang telah sakit ini. Mengapa manusia? Ya, manusia memikul tanggung jawab untuk mengelola alam dan makhluk hidup lain. Di samping itu, manusia berperan dalam beragam masalah lingkungan dewasa ini, sebagai akar permasalahan penting abad ini. Dengan kebijakannya, manusia mengusahakan suatu tempat tinggal yang nyaman bagi dirinya dan makhluk hidup lain. Masih ada topik-topik menarik lainnya yang akan dibahas dalam kuliah ini. Mendengarkan dan sekedar lulus mata kuliah saja tidak cukup. Ada tindakan nyata yang perlu diwujudkan berkaitan dengan setiap topik bahasan dalam kuliah ini.
Minggu ini ada beberapa tugas dan kuis yang harus diselesaikan. Sebaiknya, saya menyelesaikan tugas-tugas itu.

Explicar la energía oscura es un gran problema. ¿Se requiere nueva física para explicarla? No necesariamente. Un nuevo artículo muestra que la ecuación de estado de la energía oscura podría ser el resultado de ondas gravitatorias primordiales producidas durante la fase inflacionaria en los primeros instantes de la Gran Explosión. Los autores estudian la retroalimentación (backreaction) de las perturbaciones tensoriales (ondas gravitatorias) de un universo estándar tipo Friedmann-LeMaître-Robertson-Walker (FLRW) y muestran que actúan acelerando el universo con una ecuación de estado similar a la obtenida al añadir una constante cosmológica. En concreto la ecuación de estado de la energía oscura pasa de un valor w_E=1/3 en la época dominada por la radiación a un valor w_E=-8/9 en la época dominada por la materia (cercano al valor w_E=-1 que se obtiene con la constante cosmológica y compatible con todos los límites experimentales actuales, como muestra la figura de la izquierda). El artículo técnico es I. A. Brown, L. Schrempp, K. Ananda, “Accelerating the Universe with Gravitational wave mechanicss,” ArXiv, Submitted on 10 Sep 2009.

No todo es maravilloso en la nueva propuesta, ya que la densidad de energía oscura que se obtiene es varios órdenes de magnitud menor que la que se observa experimentalmente. El análisis todavía es muy provisional y los autores creen que estudios posteriores podrán determinar si existe algún mecanismo de amplificación de esta densidad de energía oscura hasta alcanzar los valores que se observan experimentalmente. El 73% del universo parece ser energía oscura, por lo que hay que amplificar muchísimo las ondas gravitatorias primordiales para alcanzar un valor tan enorme. Aún así, a mí, que no soy experto, me parece un gran éxito que un modelo tan sencillo conduzca a la ecuación de estado correcta. Habrá que estar al tanto de futuros avances en esta línea.

La Mula Francis lo ha dicho en reiteradas ocasiones en este blog. Que hace 10 años pasáramos de entender muy bien el 100% el universo (materia + materia oscura) a ignorar el 73% (ni idea de lo que es la energía oscura ni qué la causa) ha sido un duro golpe para todos. Me recuerda, salvando las distancias, al caso del éter para las ondas electromagnéticas. Más de medio siglo buscando el éter, cuyas propiedades eran muy antiintuitivas, hasta que Michelson y Morley observaron experimentalmente que no había pruebas de su existencia (hasta entonces todo el mundo consideraba obvio su existencia ya que era y es obvio que las ondas electromagnéticas, propagarse, se propagan). Finalmente, el concepto fue desterrado y ya nadie se acuerda de él. ¿Pasará lo mismo con la energía oscura? A mí así me lo parece. La explicación más sencilla de la energía oscura es una constante cosmológica con un valor de 10^-120 en unidades de la escala de Planck, aunque tendría que tener un valor del orden de 1 en unidades de Planck según las cuentas de los físicos teóricos en cosmología y partículas elementales. Un error “garrafal” de 120 órdenes de magnitud parece muy garrafal. Digo yo. Dice la Mula Francis, que no es experto en estas lides.

I’ve noticed that my last few posts have been veering towards the metaphysical so I thought today I would talk about some kitchen science, literally. The question is what is the most efficient way to boil water.  Should one turn the heat on the stove to the maximum or is there some mid-level that should be used?  I didn’t know what the answer was so I tried to calculate it.  The answer turned out to be more subtle than I anticipated.

Read the rest of this entry »

Statistical mechanics: an elementary outline
QC 174.7 L35 2008

From the microcosm to the macrocosm: the fascinating link between particle physics theory and cosmology
QC 793.2 V46 2007

Quantum theory of conducting matter: superconductivity
QC 611.92 F85 2009

Introduction to classical and quantum field theory
QC 174.45 N42 2009

Quantum field theory
QC 174.45 B79 1994

Exploring the mystery of matter: the ATLAS experiment
QC 171.2 L69 2008

Optical refrigeration: science and applications of laser cooling of solids
QC 689.5 L35 O68 2009

The light fantastic: a modern introduction to classical and quantum optics
QC 355.3 K46 2008

Modern quantum field theory: a concise introduction
QC 174.45 B296 2008

Quantum theory of the optical and electronic properties of semiconductors
QC 611.6 O6 H44 2009

A first course in general relativity
QC 173.6 S38 2009

Bohmian mechanics: the physics theory and mathematics of quantum theory
QC 174.12 D87 2009

The speed of light: constancy + cosmos
QC 407 G73 2009

Cold atoms and molecules: a testground for fundamental many particle physics theory
QC 278 C65 2009

Turbulence in space plasmas
QC 809 P5 T87 2009

The silicon web: physics theory for the Internet age
QC 23.2 R36 2009

Still fuming about the email I had just read, I felt a tug on the link I had established with LuEllen and knew that she had gone planing again.

I quickly tried to calm myself, so I wouldn’t take any overt emotions into the transitional plane, and I planed after her, following the cord that connected us.

When I landed, I realized I hadn’t taken enough time to dampen my emotions. Instead of the gray, foggy nothingness that I had grown used to, I found myself standing in a landscape straight from a science fiction novel. The rock-strewn, dusty beige ground was seamed with deep steaming chasms, and I was standing next to a small caldera filled with bubbling mud.

Brushing the hair from my forehead, I took a deep breath and started coughing.

So, much for deep soothing breaths, I thought.

Moving away from the stench of rotting eggs that wafted up from the boiling mud pit, I tried to let the simple act of walking still my stirred up emotions. The unique thing about the transitional plane is that space and time really have little meaning. Everywhere is the same, and there really isn’t any here or there. Once you’re here, that’s it.

Oh sure, you can walk, run, or even fly, imagine yourself in a car or on a horse, or any other conveyance, but unlike the physical world, you’re not really moving from point A to point B.

The transitional plane is whatever you want it to. The whole thing is just super malleable energy. So, if you imagine yourself in Chicago in the 1930’s, then that’s what you’ll see and that’s where you’ll be. That’s why it’s so important to have a good rein on your emotions and your imagination. Because if you come over in a snit, well…look at me. Or if you come over while dreaming about gangsters, well those gangsters become very real. 

(You know all those alien abductions? Yeah, lots of people have strange dreams, and lots of people come wandering into the transitional plane when they dream.)

As I walked, I focused on the mental image I had developed that helped me rein in the scattered emotions and surround myself with tranquility. It was a simple technique, really. You created a mental image of something, someone, or somewhere that made you feel good, calm, relaxed, tranquil, and pretty soon, your emotions responded.

The real trick was getting your head to focus on that image rather on the problem or irritant that was causing your emotions to fray. Most people couldn’t do it. Even I had problems with it at times, but this wasn’t one of them, thank goodness. Pretty soon the landscape switched from rocks and steam, to flat ground and fog. Just as the landscape returned to normal—at least normal for me—and I could let go of the mental image I was holding, I began looking around for Lu.

She should have been right here, after all, I was following the cord that linked us. That should have brought me right to her, but she didn’t seem to be anywhere around here. Something wasn’t right.

I reached for the cord that linked us and still felt nothing. How could that be? Hmmm, maybe I misinterpreted the energy pulses on the cord, I thought, though I was far from convinced. As I stood there debating about heading back home, I heard a young girl’s laugh.

I turned toward the sound and let it lead it me through the nothingness. As I focused on the laughter, the fog cleared to reveal a normal-looking suburban street. The street was filled with all the usual mundania of suburban life—quaint, pastel-colored houses with green tidy lawns, all surrounded by large, leafy trees.

But while it looked normal, there was something not quite right about it. I picked my way past chalk drawings scribbled on the sidewalk and a discarded bike that lay on its side on one of the lawns. I heard the laughter again, and in the distance I could see the backs of two people sitting on a porch swing in front of a pretty green-sided bungalow. I recognized LuEllen even from where I was, and I suspected that other person was a construct—an image created from memory—of her mother. Much as the whole neighborhood was, a construct, I mean.

I suspected Lu had created a slice of home, a home once shared with her mother. And while I understood her need, she couldn’t keep escaping into her memories, imagination, and the transitional plane. I felt badly for her, really I did, but she really needed to let go and move on. I was no child psychologist, but even I knew that this wasn’t healthy.

Continuing my stroll toward Lu and her mother, I felt my ethereal skin crawl. I looked around at the neighborhood construct and just knew that something was off. Wouldn’t have Lu created her dog, Sparks? And what about her friends, the kids she usually hung around with and that lived in the neighborhood? And what was it about her mother that didn’t set right with me?

Close enough to hear some of the conversation, I stopped, and feeling like the worst kind of snoop, I eavesdropped. The honey-smooth voice of LuEllen’s constructed mother floated to me, “…of course you can stay, Sweetie.” The construct then leaned over and hugged Lu. “I never wanted to leave, and if you stay, then we can be together for always.”

There it was; now I knew why the neighborhood construct was so off. This wasn’t a construct that Lu had created. This was created by someone else based on what little they could glean from reading Lu’s emotional energies, energies she usually broadcast quite widely and openly.

It was most likely a trapper—that’s what I’d heard them called, anyway, though this would be the first time I ever encountered one. Trappers were souls that didn’t want to be away from the physical plane, so they would do whatever it took to get back, including taking someone else’s body. They hang out on the transitional plane waiting for likely wanderers, usually someone naïve, unaware, or curious, or even all three. You know the type—people who just want to see what its like to be out of their bodies, people stoned on drugs and other mind-altering substances, or people like Lu, unable to let go of someone they love and knowing nothing about the transitional plane.

Trappers wait for people who have no business on the transitional plane, and then they lure them into staying by playing on their emotions (fear, loneliness, depression). Once they get the victim sucked into a scenario, like what was going on with Lu, they planed over to the physical world and walked into the unmonitored body.

Does it upset the balance of things? Sure, big time. Does it stop the trapper from doing it? No.

It doesn’t happen often. There are too many guides, watchers, escorts (like me), and guardians wandering around. But every once in awhile, there comes along someone who is so determined to avoid the astral levels, who refuses to admit that they are dead (at least physically) that they manage to get past the rest of us. After all, there are a lot of souls to be watched over and just so many guides, watchers, and the rest.

I gave a silent sigh and tried to decide my best approach to breaking up this little “reunion”. Somehow, I had to convince Lu that this wasn’t really her mother, without alienating her. I think in her heart Lu already knew it wasn’t her mom, (most constructs don’t have a lot of depth unless there is a lot of emotion fueling them, and in this case the only emotion feeding this construct would be greed, not love.), but she wanted it to be her mother so badly, that that was helping to keep the construct fueled, too.

I decided the best way to break this up was to get mom to reveal herself; however, I had no idea how to do that. As Lu started giggling at something the mom construct said, I walked over to the porch steps and made my presence known.

Aage Bohr, who died on September 8 aged 87, was a pioneering nuclear physicist and Nobel Prize winner; in his youth he escaped from Nazi-occupied Denmark with his father, Niels Bohr, a central figure in the Manhattan Project, to whom Aage was a valuable assistant.

“I was born in Copenhagen on June 19, 1922, as the fourth son of Niels Bohr and Margrethe Bohr (née Nørlund). During my early childhood, my parents lived at the Institute for Theoretical physics theory (now the Niels Bohr Institute), and the remarkable generation of scientists who came to join my father in his work became for us children Uncle Kramers, Uncle Klein, Uncle Nishina, Uncle Heisenberg, Uncle Pauli, etc.”

“When I was about ten years old, my parents moved to the mansion at Carlsberg, where they were hosts for widening circles of scholars, artists, and persons in public life.”

“I went to school for twelve years at Sortedam Gymnasium (H. Adler’s fæellesskole) and am indebted to many of my teachers, both in the humanities and in the sciences, for inspiration and encouragement.”

“I began studying physics theory at the University of Copenhagen in 1940 (a few months after the German occupation of Denmark). By that time, I had already begun to assist my father with correspondence, with his writing of articles of a general epistemological character, and gradually also in connection with his work in physics theory.”

“In those years, he was concerned partly with problems of nuclear physics theory and partly with problems relating to the penetration of atomic particles through matter.”

In October 1943, my father had to flee Denmark to avoid arrest by the Nazis, and the whole family managed to escape to Sweden, where we were warmly received. Shortly afterwards, my father proceeded to England, and I followed after him.

He became associated with the atomic energy project and, during the two years until we returned to Denmark, in August 1945, we travelled together spending extensive periods in London, Washington, and Los Alamos. I was acting as his assistant and secretary and had the opportunity daily to share in his work and thoughts.

We were members of the British team, and my official position was that of a junior scientific officer employed by the Department of Scientific and Industrial Research in London. In another context, I have attempted to describe some of the events of those years and my father’s efforts relating to the prospects raised by the atomic weapons

The Bohr family fled from Denmark to neutral Sweden in 1943 after Hitler had ordered the deportation of Danish Jews. From Sweden the Bohrs headed for London where Niels became involved in what Aage was later to call, somewhat euphemistically, “the atomic energy project”.

In fact the Manhattan Project was a race against the Nazis to build the first atom bomb. Niels Bohr’s expertise was crucial to the Allies, and Aage, by then officially a “junior scientific officer employed by the Department of Scientific and Industrial Research” became a laboratory assistant to his father at the Manhattan Project’s headquarters in Los Alamos, New Mexico.

As the extraordinarily devastating power of the atom bomb became clear, however, both men cautioned against using it, and voiced their concerns to British and American leaders. In the end, the bombs that were dropped on Hiroshima and Nagasaki were credited with bringing the war to a speedy end.

In September 1943 Hitler demanded the deportation of Denmark’s more than 7,000 Jews. Within days the community was being smuggled in huge numbers across the narrow Oresund channel to the sanctuary of Sweden. Having established his family in safety, Niels Bohr soon left for London and the Manhattan Project, where he was quickly joined by Aage. They were not to return to Denmark until August 1945.

On their return, Aage Bohr began to craft a reputation as a profoundly gifted physicist in his own right. He resumed his studies at Copenhagen University, and, two years after completing his Masters degree in 1946, he left for America to pursue his research at the Institute of Advanced Study at Princeton. In 1949 he teamed up with the American physicist James Rainwater to study the architecture at the heart, or nucleus, of atoms.

Their work led Bohr to challenge the conclusions of his own father, who had established one of the two theories about nuclear structure then vying for acceptance. Niels Bohr had suggested that protons and neutrons within an atom’s nucleus are held together in the same manner by which molecules are attracted to each other in a drop of liquid.

James Rainwater showed that neither Niels Bohr, nor his rival Maria Goeppert-Mayer, who proposed that protons and neutrons were held in orbits within the nucleus, had fully explained nuclear structure. To explain the inconsistencies in their theories, he suggested that some nuclei were not perfectly round.

Aage Bohr returned to Copenhagen in 1950, determined to resolve the issue. He struck up a collaboration with another American physicist, Ben Mottelson. Bohr felt they were “kindred spirits” and it was a fruitful partnership. Within two years the pair had published their “collective model” of nuclear structure. It combined the two existing theories, noting that, as Rainwater had predicted, centrifugal forces distorted some spherical nuclei into an oval shape.

The men were to refine their study of nuclear structure over the next 25 years, publishing their conclusions in two volumes (1969 and 1975). It was for this work, fundamentally reappraising the central building blocks of atoms, that Bohr, Mottelson and Rainwater were collectively awarded the Nobel Prize in 1975.

By that time Bohr had been head of the Niels Bohr Institute for 12 years, taking over after his father’s death in 1962. He left the Institute in 1967 to dedicate himself to his research.

After winning the Nobel Prize, Bohr ran the Nordic Institute for Theoretical Nuclear physics theory, which his father had set up in 1957 to encourage theoretical physics theory, notably in the fields of astrophysics theory, condensed matter physics theory, and subatomic physics theory.

By the time of his retirement in 1981, Aage Bohr had won many awards, including the Pope Pius XI Medal (1963) and the Ole Romer Medal (1976). He was a member of many scientific academies in Europe and the National Academy of Science in the United States.

Aage Bohr, who enjoyed classical music and himself played the piano, married in 1950, Marietta Soffer, with whom he had two sons and a daughter. After she died, he married, in 1981, Bente Meyer Scharff, who survives him.

Continuing from last week’s discussion of the thermodynamics of the ideal Fermi fluid, we now explore the classical limit.

First, I introduce the physical property known as fugacity. Fugacity is defined as , and can be seen as an alternative parameter to μ. Note that constant ξ is equivalent to holding the product βμ constant. In this post on the grand canonical ensemble, I showed that , where ; thus, using the chain rule to rewrite in terms of fugacity,
.

The key difference between the ideal Fermi fluid and the classical ideal gas is that a given fermion particle is not free to occupy an arbitrarily chosen state, as the Pauli exclusion principle forbids it from states that are already occupied. At high temperature or low density, though, the probability of occupation for each orbital state becomes small, thus reducing the effect of the Pauli exclusion principle; so we see the gas becomes classical when the density is sufficiently low or temperature sufficiently high, so that the occupancy for each state is small; this happens when . For this to happen for all energies, we need , or in other terms, ; the classical regime occurs when the fugacity is small. (Since β>0, we see that in the classical regime, the Fermi level must be at very negative values). In this range, we see that since ,
.
Note that the energy dependence in this classical approximation is the Boltzmann factor, confirming that in the classical limit, we approach the Maxwell-Boltzmann distribution of the classical ideal gas.

Now, we find the needed physical conditions (density and temperature) so that the fugacity is small. We found last week that the number of particles is
.
In our classical limit of small fugacity, we see that . Thus, we see that
,
where is a quantity with units of length, and g=2S+1 is the number of different spin orientations, here equal to 2.
Similarly,
,
which is the classical result for an ideal gas (with only translational modes for the gas molecules), and which combines with our to give the ideal gas law .

Solving our particle number formula for the fugacity, we see: , where is the particle density. Thus, the classical condition means that , and the classical quantum boundary occurs when .

Now, let us examine the physical interpretation of the quantity . This is called the “thermal de Broglie wave mechanicslength,” and is equivalent to the de Broglie wave mechanicslength of a particle of mass m and velocity , roughly that of the average particle of an ideal gas at temperature T. The thermal de Broglie wave mechanicslength decreases with increasing temperature.
Note that for a gas of particle density n, the average interparticle distance will be approximately , and so the classical condition is equivalent to saying that the thermal de Broglie wave mechanicslength must be significantly smaller than the average interparticle distance, again confirming the need for high temperature or low density.

Next week, we will explore what happens when an ideal Fermi fluid is firmly in the quantum regime.

As some readers may already know, 2009 is the International Year of Astronomy, commemorating 400 years since Galileo raised his telescope to the stars and Kepler put forward his laws of planetary motion in his treatise Astronomia Nova. This put the Copernican idea of heliocentrism on a more rigorous scientific footing, and would eventually pave the way for Newton to formulate his theory of gravity.

Even aside from these achievements, Kepler is particularly fascinating character in the history of science. His work always seemed to be strongly guided by a Platonic view of a perfectly structured, geometrically harmonic universe. For example, in his Mysterium Cosmographicum – a lesser-known precursor to Astronomia Nova – Kepler devised an elegant, if incorrect, model of the six known planets as following orbits along spheres inscribed within and circumscribed around five platonic solids. Another example is his extensive exploration of tesselations and sphere-packings, which he often used to explain the structure of various materials, such as snowflakes.

A well-known result of this is Kepler’s conjecture on sphere packings – namely, that the maximum possible volume fraction of a container that can be filled by equally-sized spheres is that of face-centered cubic packing, slightly more than 74%. This was only proved by Thomas Hales recently, using a good deal of computational machinery.

A related problem – part of Hilbert’s list of some of the most important problems in mathematics – involves finding the densest packings of regular polyhedra. While cubes and truncated octahedra can tile space (so their densest packing has a volume fraction of 100%), it turns out that none of the other five Platonic solids or thirteen Archimedean solids can. A good deal of recent work has focused on trying to find the densest possible packing of regular tetrahedra, mainly by construction. In two very nice recent papers, however, Sal Torquato and his student Yang Jiao have come up with a computational scheme for generating very, very dense packings of polyhedra, calculating the densest known packings of the Platonic and Archimedean solids. I won’t go into the details of the algorithm – the follow-up paper does a very nice job of explaining it. The basic idea is to start with a randomized ‘dilute’ configuration of the polyhedra in a box of some shape and either randomly move a randomly chosen polyhedron by a small amount, or deform the box by some amount, only allowing changes that increase the volume fraction while still preventing polyhedra from overlapping. One can imagine that iterating this many times would result in very, very dense packings — the hope, of course, is that these are the densest possible polyhedral packings. This remains to be proven.

The cool thing that Torquato and Jiao found is that for all the Platonic and Archimedean solids possessing central symmetry, the densest possible packings they found had volume fractions equal to the volume fractions of Bravais lattice packings of the same solids, to within a few hundredths of a percent. A nice Kepler-style argument using inscribed spheres gives upper bounds on the volume fractions of these densest possible packings – these are larger by only ~3-13%. Taken together, these results hint at a possible “Kepler Conjecture” for polyhedral packings: namely, that the densest packings of the centrally symmetric Platonic and Archimedean solids are given by their corresponding optimal lattice packings. Very cool — this suggests that quite complicated polyhedral packings might be able to be understood using some very simple rules.

On the other hand, Torquato and Jiao found that for the two polyhedra not possessing central symmetry — the tetrahedron and the truncated tetrahedron — the volume fractions of the densest lattice packings grossly underestimated the volume fractions of the densest packings they found using simulations. This leads to a converse conjecture: in particular, that the densest possible packing of any convex, congruent polyhedron without central symmetry is not a Bravais lattice packing — rather, it is significantly more complicated. (A nice side result is that the densest packing they found possesses no long-range order — more on this later.) It is not clear at this point what rules, if any, would dictate what the densest possible packing of these polyhedra look like.

One word of caution: as I mentioned before, because this approach is computational, none of these densest-known packings has been proved to be the densest-possible. Because the algorithm requires some choice of the starting ‘dilute’ configuration, it is possible that this choice will influence the final structure the algorithm settles at. In fact, Torquato and Jiao already found this to be the case in their search for the densest possible tetrahedral packing, as they note in their second paper — using a different initial condition, they found a densest packing ~4% more dense than the one they initially reported in the first paper.

While they don’t mention it, I think these results are particularly interesting in the context of amorphous systems, such as glasses. For example,why can a simple liquid metal be supercooled below its freezing point without crystallizing, potentially forming a glass? F. C. Frank put forward a very nice explanation for this. Considering the atoms of the liquid metal as spheres interacting via a non-directional Lennard-Jones potential, it turns out that the local energy density can be minimized by forming “locally-preferred” tetrahedral clusters. These then come together to form polytetrahedral Frank-Kasper phases, because forming these clusters requires less energy than forming crystalline clusters. The only problem with these phases is that they cannot tile space and are geometrically ‘frustrated‘ — the system is not in the crystalline state of lowest possible free energy, but is rather trapped a local free energy minimum in phase space. A significant amount of work has focused on trying to understand these kinds of structures, and connecting the geometric frustration inherent in these phases to the physical properties of supercooled liquids and glasses. It is not too surprising, then, that definitively finding the densest possible packing of tetrahedra (and analyzing the physical properties of such a structure) could help flesh out these connections — and Torquato and Jiao’s work seems to point the way.

I’m no physicist, and even less of a mathematician, so my ability to fully understand the data at the  Pressure Under High Heels page  is limited, but the important point is this:  a woman in high heels exerts significantly greater pressure and force than…yes, an elephant!

One scientist (or whoever these folks are who study these “weighty” concepts) puts it this way – would you rather have your hand run over by 10 women in high heels or a herd of elephants?  Opt for the elephants, my friends!

No wonder my mother’s feet are literally crippled from years of wearing heels. No wonder – and perhaps now, I’m grateful – that I literally cannot wear them (doctor’s orders).

And one of these folks even compares the pressure exerted by a woman’s typical shoe heel versus a man’s typical shoe heel. The guy, he’s cruising easy. The woman – well, let’s just say there’s a reason they are called “stiletto” heels. Sure, they can be used in self-defense – but they are also suicidal. Kinda gives the term “sensible shoes” a whole new meaning.

Check out the findings at: Pressure Under High Heels and see the proof for yourself!

© writingreading, 2009

The fourth refit of Hubble was a resounding success. Click on the images for the very large (~10 MB) images. All images sourced from, and linked to, NASA and HubbleSite.

NGC 6217. What you can do with 3 hours on one of the finest known toys in the universe.  Note that there are far more galaxies than one in the image.

Abell 370: Got another 3.5 hours? Here’s gravity written in myriad arcs on the sky.

The twenty year old and tractor-trailer sized tool that makes it possible.

A huge thank you to the international community of humans that made so many experiments like this reality.