Just another teaser

Below is a picture that I took at the University Gardens in University of Florida. It's an insect with a semicircular saw-like organ on the back and named as wheel-bug. It's not a complete wheel and certainly not a locomotive wheel! Every now and then, we come across creatures and plants which mimic other objects in nature, often useful for their survival and explained by the law of natural selection. The wheel was the second-most important thing in the civilization after fire and it's only a few thousand years old. How come is there no macroscopic creature in nature with a locomotive wheel-like or sphere-like body part after millions of years of evolution? Wouldn't such an organ beg for strong natural selection? It's a biological question, but probably has a bio-physical answer! Would you like to think a bit again?

More drink for thoughts

Euclidean tiling (with convex regular polygons):

My research lives on lattices or tiles if you want to look at it from a slightly different perspective. It's well-known that the flat two dimensional plane can be covered with regular triangles, squares, regular hexagons but not regular pentagons. It's not very difficult to prove! But would you be amazed if I say that this limited possibility can be captured in a very simple equation 2 (N + E) - NE = 0, symmetric in E and N where E is the number of sides of the covering tiles and N is the number of neighbors that a point in the tiling has? The condition is obviously that {N, E} has to have integer solutions and there are only three such pairs! {N=6, E=3} corresponds to a triangular lattice, {N=4, E=4} corresponds to a square lattice and {N=3, E=6} corresponds to a hexagonal lattice.

Platonic solids:

Now, you might have heard that there are also five possible paltonic solids: namely the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. Again, you can also prove that there can't be more than five platonic solids using an inequality rather than an equation! Now the inequality is 2 (N + E) - NE > 0

There are five possible solutions for these: {N=3, E=5: dodecahedron}, {N=3, E=4: cube}, {N=3, E=3: tetrahedron}, {N=4, E=3: octahedron}, {N=5, E=3: icosahedron}

Just think that the starting from a vertex, a platonic polyhedron is just regular tiling of a curved two dimensional plane rather than flat two dimensional plane!

(Image source: https://www.arxiv-vanity.com/papers/2001.00188/)

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