# SNIPPETS....

The power of 11111

This is called a repunit i.e. a repeated sequence of ones. There are repunits which are also primes like 11. After 11, a string of nineteen ones i.e. 111...(repeated 19 times and written as R19) is another prime, other examples of repunit primes are R23, R317 and R1031.

No larger repunit prime than R1031 is perhaps known! The number one is itself unique in many ways; i.e. apart from it being used to denote the top or number one position in any situation or subject. One and zero are of course unique binary digits, the universal basis for all digital systems and computations.

Anything multiplied by one, remains the same, but anything multiplied by zero also vanishes. Anything (i.e. any number), raised to the power of zero, becomes one, but anything (however large or small) raised to the power of one does not change and is the same number! This has led to the amusing adage “Show me a useless exponent and I will show you one!”. In the adventures of Alice in Wonderland (or rather in its sequel Through the looking glass) one of the riddles posed to Alice (by the Red Queen) is “What is one and one and one...?” Alice replies, “I have lost count!” Of course you can build up all the numbers using only zero and one which is actually the basis for the binary system.

The number eleven again has interesting properties. The powers of eleven give the binomial co-efficients in the binomial expansion, as is well known in school algebra, i.e. 11, 121, 1331 etc. If you want to expand (a+b) to the power of n, ex: n=2, coefficients are 1, 2, and 1 for a squared, product ab and b squared. A convenient way to remember this is the so-called Pascal’s triangle, which successively gives the power of eleven. The first few powers of eleven are also palindromic (i.e. read the same both ways), ex: 11, 121, 1331 etc.

Eleven dimensions also play a unique role in unified theories of basic interactions in physics. The universe is supposed to have started with eleven dimensions, four expanding and the remaining shrinking and becoming compact to minuscule dimensions which could be probed in the Large Hadron Collider! The eleventh of November this year will also have the sequence 111111.

C Sivaram

Imagining a new skin that can repel ice

De-icing aircraft with chemicals is expensive, environmentally unfriendly and time-consuming, as stranded passengers during winter blizzards may be well aware. For the last decade researchers have been exploring the possibility of building planes with hydrophobic, or water-repellent, materials that would not require de-icing. But now, researchers from MIT report that this approach is flawed. Although a surface might be water-repellent, it may not be ice- or frost-repellent. Their findings appear in the journal
Applied Physics Letters. “Water can go directly from a vapour stage to a solid state,” said Kripa Varanasi the study’s lead author and a professor of mechanical engineering at MIT.

“When ice forms this way – typically called frost – on a super hydrophobic surface it can pretty much coat up the entire surface.” The result is a surface covered in frost that is no longer hydrophobic, but incredibly hydrophilic, or water-attracting, he said. Varanasi believes that frost build-up can be better controlled by creating a surface with nanoscale texturing. The technology would have applications outside aviation as well. Wind turbines tend to collect large amounts of ice, and aside from introducing significant drag and reducing performance, a spinning turbine can hurl out large chunks of ice that can cause serious damage.

The real challenge lies in creating an ideal texture that deflects unwanted frost formation and is also durable and scalable, Varanasi said.

Sindya N Bhanoo

New York Times News Service