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.


Zebra Finches At a STEM Conference

In the spring of 2016, I attended a STEM conference. Expanding the acronym as science, technology, engineering and math doesn’t shed too much light on the intentions and philosophy of STEM.  The premise is that math, science and technology subjects should not be taught in isolation; there should be more integration and emphasis on applications. All of this is largely inspired by the job market’s need for a larger number of better-trained people in these specialised fields. It all seems reasonable as long as the approach is not taken to an extreme.

A society, regardless of its bent, functions best when a wide range of talents are cultivated, even if they seem to serve no practical purpose. Similarly, we have a healthier situation in schools and colleges when educators don’t sail on the same ship. There was at least one organiser at the conference who shared my views because a particular lecture went against the grain of STEM.  95% of the auditorium featuring the lecture was empty and attended mostly by the speaker’s university students, a couple of bird-lovers and a little cluster of Canada Wide Science Fair attendees. Rudely, the latter group even walked out before it ended. But if you stick to the premise that attendance at public events is very often inversely proportional to its quality, you don’t worry about numbers.

Parentese or “baby talk“ is far from being just indulgence on the part of parents. In humans it helps draw attention from babies and promotes the learning of speech. Regardless of language, there are universal characteristics of parentese. Voice pitch is modulated; speech is slower, more repetitive and more attention-grabbing than adult-talk.

Image of baby zebra finch from http://www.singing-wings-aviary.com/zebrafinches.htm

A similar situation arises in a small Australian bird known as the zebra finch. For those of you unfamiliar with the small bird, one of its distinguishing characteristics is its song, which is reminiscent of the squeaky sound of a rubber duckie. With their form of baby talk, adult zebra finches change their vocalisations when singing to young birds. They slow them down and use more repetition. The juvenile finches in return pay more attention to such songs than to those used between adults.In their young lives, they the simpler versions. With time, in the physical presence of adults, the chick’s song converges with that of their tutor. When zebra finches were isolated and exposed to mere recordings, they developed different songs.

The social interaction between tutor and chick stimulates communication between the midbrain’s ventra legmental area (VLA) and regions of the cerebrum. The VLA is partly a reward centre and uses dopamine.  When humans acquire language, neural bridges of that type are also made. In case of the finch, the evidence for such a pathway comes from the fact that in the absence of tutor’s physical presence, a marker for gene expression of catecholamines (which include dopamine) remained inactive.

Another revelation which made my morning was that the zebra finch researcher had originally majored in economics, reinforcing my notion that to get to an island you don’t have to board any specific boat.



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