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,  SiO42- , 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.pyrite_really_cubes_molecular

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, slide1-l

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

For a cube, download

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/N1/2 = 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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Comments On Views of the Late Isaac Asimov

asimovLike society in general, science needs a variety of both skills and viewpoints to survive. Some of its devotees have been ingenious experimentalists; others have devised elegant theories solely from their desks. Some combine both skills and manage research teams.  Many scientists, like natural philosophers of yesteryear, remain focused on providing humanity with insight into the universe. Others want to assist engineers and investors to derive more practical benefits. Since the pursuit of convenience is accompanied by costs to culture, health and the environment, we are also lucky to have scientists who focus on providing checks and balances.

We all tend to gravitate towards people with similar viewpoints, which is probably why I have always looked up to the late Isaac Asimov. He was a biochemistry specialist but abandoned a career path that would have blocked off the beautiful branches of what he called the “orchard of science.” Instead he devoted the rest of his life to science fiction and science popularization, reminding us that science’s survival also depends on generalists and educators.

Through a series of developments of absorbing lack of interest (as far as these pages are concerned), I found myself doing research on a biochemical topic. In that area of study I obtained my Ph.D., and in no time at all I was teaching biochemistry at a medical school.

But even that was too wide a subject. From books to nonfiction, to science, to chemistry, to biochemistry—and not yet enough. The orchard had to be narrowed down further. To do research, I had to find myself a niche within biochemistry, so I began work on nucleic acids

And at about that point, I rebelled! I could not stand the claustrophobia that clamped down upon me. I looked with horror, backward and forward across the years, at a horizon that was narrowing down and narrowing down to so petty a portion of the orchard. What I wanted was all the orchard, or as much of it as I could cover in a lifetime of running…

I have never been sorry for my stubborn advance toward generalization. To be sure, I can’t wander in detail through all the orchard, any more than anyone else can, no matter how stupidly determined I may be to do so. Life is far too short and the mind is far too limited. But I can float over the orchard as in a balloon.

As a voting citizen, a scientist can never isolate himself from politics. Leaders serve us well if they do their best to address a variety of society’s selfless pursuits. But it’s all too easy to pretend and deceive while in a position of power. Asimov often complained that while a good scientist will be ruined professionally by being dishonest, many politicians and businessmen thrive on purposely distorting the truth.

Having said that, young scientists should not fall into the trap of thinking that the world would be better off if power was all in the hands of scientists. Being human they can easily be biased by self-interest and shallow ideologies. The world has had leaders who had a science background, but from a utilitarian point of view, they fall on various spots on a spectrum of quality.  What we need is for leaders to tap the best qualities of a variety of people. And honesty is not only the mark of a good scientist, but it’s essential in all occupations, from plumber to president. When that quality was lacking in a President, Asimov was never shy to speak out:

Asimov vehemently opposed Richard Nixon, considering him “a crook and a liar”. He closely followed Watergate, and was pleased when the president was forced to resign. But Asimov was dismayed over the pardon extended to Nixon by his successor: “I was not impressed by the argument that it has spared the nation an ordeal. To my way of thinking, the ordeal was necessary to make certain it would never happen again.”[205]

high-angle-view-of-crowd-waiting-at-crosswalk-to-cross-road-556706617-58d0377f5f9b581d72f70c90The more we crowd ourselves on the planet, the more likely we are to communicate diseases amongst ourselves, take ourselves for granted, and stress our planet for the resources we need to survive. Asimov, who was usually very optimistic, felt strongly about this serious problem which we have mostly ignored after a surge of interest in the 1960s and 1970s.

Overpopulation is going to destroy it all… if you have 20 people in the apartment and two bathrooms, no matter how much every person believes in freedom of the bathroom, there is no such thing. You have to set up, you have to set up times for each person, you have to bang at the door, aren’t you through yet, and so on. And in the same way, democracy cannot survive overpopulation. Human dignity cannot survive it. Convenience and decency cannot survive it. As you put more and more people onto the world, the value of life not only declines, but it disappears.[214]

 

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