Why Only Five Platonic Solids in Our Geo-Bio-Chemical World?

The DNA-surrounding capsid of the cold virus and the atomic arrangement of  a certain allotrope of  boron consists of  20 triangles arranged in a three dimensional shape known as a icosahedron.

If you form a three-dimensional structure with 8 triangles, you get an octahedron. An example is the molecular structure of the electrical insulator and potent greenhouse gas, sulfur hexafluoride, SF₆. It can also appear in the mineral pyrite.

If you limit the number of triangles to four, you get a tetrahedron. The tetrahedral silicate unit,  SiO₄^2- , is the basic component of most silicates in the Earth’s crust. On our planet’s surface, any time carbon makes four single bonds with other carbons or other elements in a wide variety of life’s hydrocarbons, we also get a tetrahedral arrangement. This allows the four bonding pairs of electrons to get as far from each other as possible.

Changing polygon, we can use 6 squares to make a cube. The similarly-sized ions of cesium and chloride can form an arrangement where each whole ion centers 8 vertices occupied by an ion of the opposite charge. But viewed within the lattice, the boundaries of the cube are such that only 1/8 of each ion is at a vertex. Given that there are 8 vertices, this maintains the ratio of one cesium for every one chloride ion. In the mineral halite, which is composed of NaCl, the chloride ion is considerably bigger than the sodium. They don’t pack into a cube in the same manner, but overall they still form the same shape.

Then there is the dodecahedron, consisting of a dozen pentagons. Cubes of pyritohedron form the macro-illusion of a dodecahedron, but otherwise this “perfect” solid is rare in nature.

Are there more than these 5 possible perfect or Platonic solids? No. But why?

There is a relationship that holds for all solids of this type. If you let V represent the number of vertices, E = number of edges and F = the number polygonal faces, 

you will observe that in a tetrahedron V= 4, E = 6 and F = 4.

For a cube,

Notice that for both solids, the following simple formula(called Euler’s Formula)holds true:

V + F – E = 2.                                 Equation (1)

That doesn’t constitute a proof for why it should apply to all cases, but you can find one here. We can use this formula to prove that there are only a limited number of Platonic solids.

First let’s introduce two other variables, N = the number of sides in a polygonal face and R = number of edges that meet at a vertex.

Since each edge is shared by two vertices, if we multiply R by the number of vertices,V , and divide by two, we will get the total  number of  edges:

RV / 2  = E;   Solving for V we get

V = 2E / R                                        Equation (2)

The number of edges can also be obtained by the number of faces. Each face has one edge for each of the number of sides, N. But each edge is shared with another face, so again not to count things twice:

NF / 2 = E.   Solving for F we get

F = 2E / N                                        Equation (3)

Substituting equations (2) and (3) into equation 1:

2E / R  + 2E / N   – E = 2.     

Now divide each term by 2E:

1/R + 1/N – 1/2 = 1/E. 

We need at least three edges to get a 3D shape so R ≥ 3. Similarly to get a polygon, N ≥ 3. Interestingly N and R cannot simultaneously be greater than 3,  because as they create progressively smaller fractions, 1/R  and 1/N will add up to a maximum sum of 1/2 ( if R=N=4), which in the formula will yield 0 = 1/E.  

Letting N = 3,  if R = 3  then E = 6. Using this result and equation(3),  F = 2E/N = 2(6)/3= 4: the tetrahedron.

Having no choice due to the restriction we mentioned, we have to keep either N or R at 3 while increasing the other, so

letting N = 3,  if R = 4, E = 12 and F = 8, the octahedron. 

Reversing the values and letting N = 4 and R = 3, E = 12 and F = 2(12)/4= 6,  the cube.

We can then try the combinations of N= 3,  R= 5 and N= 5, R= 3, which will solve for the icosahedron ( E = 30; F = 20) and dodecahedron ( E = 30; F = 12), respectively.

But 5 is the limit because if we try values of 3 and 6:

1/3 + 1/6 – 1/2 = 0, which means the impossibility of no edges. A value larger than 6 yields a negative value for E.

Given that there are only these 5 solutions to Euler’s formula , then only five Platonic solids can exist in three dimensional space.

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The Aesthetics and Chemistry of Petrified Forests

Someone can observe a fair number of sunrises without becoming jaded. There is often an unprecedented observable variation in the pattern and thickness of clouds, which changes the corresponding array of colours. Add to it the sense of scale and the feeling of rejuvenation with every dawn, and each sunrise has the potential of inspiring a line like Cat Stevens’ “Morning has broken like the first morning”.

Petrified Forest National Park (pic by the author)

So far I have only walked through Arizona’s Petrified Forest once, but I cannot imagine being jaded from repeated hikes. Almost every log of stone appears unique, and its perception, as every good photographer realises, is influenced by lighting conditions. Touching them is not monotonous either. In some petrified wood, the ends of the logs are rugged; in others the mineral has been cleaved, as if by a blade. In others, the ridges and the grain of the original tree are preserved.

And what intensifies the sense of wonderment is the mechanism that created these masterpieces of natural history. Remarkably the first rigorous laboratory study was only published recently in May of 2016. As the authors Marisa Acosta and George Mustoe pointed out, prior to their research explanations of petrified wood’s colour were derived from an unpublished analysis of limited rock samples and from speculation.

The Earth at the time of the late Triassic. Image from the Australian Museum

The story of Arizona’s petrified wood started about 225 million years ago during the Late Triassic period, when all land was part of a splitting Pangaea, when dinosaurs and large reptiles dominated. Flowering plants had not yet evolved, so there were no poplars, maples or palo verdes, Arizona’s national tree. Cacti, also being angiosperms, had not arisen either, and they would not have lived there anyway because the land within present-day Arizona was then a lush subtropical forest filled with the ancestors of present day conifers. In one area, these clung to eroding riverbanks and the trees fell into streams. Buried in flood plains, they were out of contact with oxygen. Anaerobic conditions slowed their decomposition. That in itself would not have been sufficient for petrification (the process of turning into stone without seeing Medusa’s head 🙂 . There was a lot of volcanic ash nearby, which introduced silica into the ground water,probably in the form of Si(OH)_4. With the cell walls still intact, their cellulose and lignin (components of wood) had an affinity for silica. How the silica(quartz) formed chemically from Si(OH)₄  is a mystery so far because the process has not been replicated in the lab.

This is the process believed to occur in diatoms that deposit SiO2 particles. Whether something similar occurs during petrification is not clear.

Then the intracellular spaces also filled with the same quartz mineral. But there wasn’t just one single episode of permineralization. As we shall see, a given sample’s varied oxidation states of iron in the silica suggest that the solidifying wood was exposed to penetrating solutions exposed to different environmental conditions over the course of time.

It was previously believed that ions of copper, chromium, manganese and aluminate were responsible for the assortment of colours in petrified wood. But the authors collected over 30 samples from 3 sites in Arizona, including the Petrified Natural Park and also from Nevada, Oregon and Zimbabwe. Using a petrographic machine they prepared 200 um slides and then used Laser Ablation Inductively Coupled Plasma Mass Spectrometry(LA-ICP-MS) to identify the key impurity colouring the silica. The technique is highly sensitive in identifying elements directly on solid samples. Using LA-ICP-MS they began by focusing  a laser beam on each thin rock sample to generate fine particles. The ablated particles were then transported to the secondary excitation source of the ICP-MS instrument for digestion and ionization of the sampled mass. The excited ions in the plasma torch were finally sent to a mass spectrometer detector for elemental analysis.

Rainbow patterns in petrified wood especially were believed to be caused by a variety of metal ions. But iron alone was the key to the hues of red, purple and yellow. Strangely iron and not chromium was also responsible for the green colour in 1/3 bright green samples and in ¾ of dark green rocks. It’s already known that both abiotic and biotic pathways can reduce the +3 state of iron in ferric oxyhydroxide minerals to a +2 state to produce green rust or fougerite( Fe²⁺ ₄ , Fe ³⁺ ₂ (OH)₁₂ CO₃.3 H₂O). Those of course are the actual minerals involved in the green parts of petrified wood. But the fact that the different oxidation states of iron in red and green minerals coexist in the same rock sample suggests that permineralization is not a one-shot deal. The following graph reveals how combinations of pH and exposure to the atmosphere influence the electron-ripping (oxidation)or electron-donating abilities (reduction) of solutions. The latter is measured by E_h, a solution’s oxidation potential. As the E_h changes over time, it could use different events to precipitate a different colour in the same log, even if iron is involved in both cases.

Petrified wood from the Long Logs Loop Trail (photographed by the author)

One will also notice abstract art–like patterns in the petrified wood. The authors describe a number of physical processes at work. These include diffusion, dilution, and a form of natural chromatography. The black parts, contrary to the previous speculation, are not due to the presence of manganese, as revealed by the analysis. Total absorption of light by the silica structure could be one cause, or in some cases small amounts of persistent organic matter were responsible.

The Petrified Forest is in essence a nature-made museum of art, history and geochemistry.

The History and Chemistry of Rock–not the Rolling Stones, the Dolomites

The Brentan Dolomites, the only dolomitic group west of the Adige River, photographed by the author at Madonna di Campiglio

If you take a sample of rock from one of the main peaks of Northern Italy’s Dolomite Mountains and add hydrochloric acid, the effervescence will be quite weak. The streaming of carbon dioxide bubbles will be far more vigorous if you add the same acid to rocks from most Alpine peaks. Both samples contain carbonate, the source of CO₂ ,  but the rates differ because the Dolomites consist of mostly CaMg(CO₃)₂ *, a mineral known as dolomite, whereas the Alps are mostly limestone, which contains calcite mineral CaCO₃.

How does dolomite form?

Dinosaur tracks, south of Trento in the Dolomites. They date back to 200 million years ago, around the time that the dolomite material formed. Source: https://www.visittrentino.it/en/dinosaur-tracks_md_2373

First, there has to be some calcium carbonate in the environment. In warm areas, especially in reservoirs of magnesium rich- water in shallow lagoons, magnesium ions will penetrate the calcite and get incorporated into the crystal to yield dolomite:

Mg²⁺_(aq) + 2 CaCO_{3 (s)} –> CaMg(CO₃)_{2 (s)} + 2 Ca²⁺_(aq)

The material of the Dolomite peaks  was made in the Upper Triassic Period, about 200 million years ago when Pangea was splitting up and 70% of species were becoming extinct due to massive volcanism and ensuing global warming. It helped dinosaurs extend a long-lasting advantage over mammals.

How does calcium carbonate form?

Artificial micelles embedded in calcite simulate a similar process in nature. From Nature.

It’s well known that algae, bacteria and especially molluscs are capable of precipitating calcium carbonate. But not to be to reductionist, it should be pointed out that the approximately 5% of a shell that does not consist of CaCO3 makes the shell much stronger. The minor component consists of a protein micelle embedded within the calcium carbonate crystal. Furthermore, the crystal growth itself depends on the protein matrix binding to Ca²⁺.

But where does the carbonate ion come from?

Calcium carbonate is not very water-soluble, so carbonate cannot be obtained directly from limestone in the ocean. Molluscs and bacteria rely on urea hydrolysis. With the help of an enzyme, urea reacts with water to produce both carbamic acid and the alkaline molecule ammonia. Carbamic acid, in turn , can be hydrolyzed to produce carbonic acid (H2CO₃) and more ammonia. Since carbonic acid is in equilibrium with hydrogen carbonate and H+ ions, the hydroxide from the aqueous ammonia-equilibrium can then make carbonate according to the following:

H⁺_(aq) + HCO₃ – _(aq)  +2 OH^–_(aq) –>  CO₃ ^2-_(aq) + 2 H ₂O

What are other sources of hydrogen carbonate?

Also found in baking soda and in every living cell, hydrogen carbonate ion abounds in the sea. And most of it does not come from the hydrolysis of urea. That biochemical cycle serves mostly to provide the ammonia which raises the pH in order to precipitate carbonate. So where does most of the hydrogen carbonate come from? When carbon dioxide dissolves in rainwater, it forms carbonic acid which weathers a variety of rocks, ranging from feldspar to mica.

 

Scanning electron micrograph of a coccosphere of Emiliania huxleyi, composed of plate-shaped calcium carbonate coccoliths. From http://aem.asm.org/content/71/5.cover-expansion

Feldspars are characterized by Si₃O₈^4-, while micas contain Si₃O₁₀^8- . When they react with carbonic acid, they liberate positive ions, transform the silicate polyatomic to a different silicate in the form of clay and also release quartz and hydrogen carbonate ion. Rivers then deliver the ions to the sea where some serve as a reactant for calcium carbonate production in shell material and coccoliths, which originated at the time that the Dolomites’ material was formed.

How did dolomite from the sea become mountains that now reach the clouds?

The Earth’s most common intrinsic igneous rock is granite, which contains quartz along with two forms of feldspar and mica. Granite, has large crystals, suggesting that they cooled slowly underground. The solidified rock only became exposed by uplifting through plate tectonics. The same mechanism is also what lifted the dolomite material out of the sea as the African plate collided with the European one, a process that’s been ongoing for 40 million years and will continue to do so, perhaps closing off the Mediterranean.  The collision not only created the Dolomites but the Alps and the Pyrenees.

Other Sources

Physical Geology, fourth edition, 1988. Plummer and McGeary.

Britannica Macropedia. Minerals. Last Printed Edition, 2010

http://onlinelibrary.wiley.com/doi/10.1111/sed.12001/abstract;jsessionid=EF3999A595CC9E41472559F36FA1CC15.f02t04

https://en.wikipedia.org/wiki/Brenta_group

• * postscript:

 

*In Lake Huron’s Bruce peninsula there are a combination of minerals present. The more erosion-resistant dolomite is at times underneath layers of other sedimentary rock. The less resistant layers have been carved out of the cliffs, leading to the attractive formation known as the Grotto.