Gabriel’s Horn and the Painter’s Paradox

By Anupum Pant

If you take the plot of y=1/x and plot it from 1 to infinity, you’ll see that the plot seems to never meet the x axis. Now, take this plot and spin it fast with the x axis as the axis of rotation. You’ll then have a horn shaped solid object which is endlessly long. A mathematical object also known as the Gabriel;s horn or Torricelli’s trumpet.

GabrielHorn

Mathematically, this object is interesting because it can contain a finite amount of volume, but it’s surface area is infinite. That is to say, you can fill the horn/trumpet with a finite amount of paint, yet the whole paint it contains would not be enough to paint the inside surface of the object – known as the painter’s paradox. However, there’s a catch about painting the inner part of this horn. As WIkipedia puts it.

In fact, in a theoretical mathematical sense, a finite amount of paint can coat an infinite area, provided the thickness of the coat becomes vanishingly small “quickly enough” to compensate for the ever-expanding area, which in this case is forced to happen to an inner-surface coat as the horn narrows. However, to coat the outer surface of the horn with a constant thickness of paint, no matter how thin, would require an infinite amount of paint.

Calculating Lego Combinations

By Anupum Pant

When Godtfred Kirk Christiansen was the third son of Ole Kirk Christiansen, the founder of LEGO went to the patent office to get a patent for lego blocks, he was asked a question by the patent officer to which he had no good answer. He only had an estimate. The question was – “How many combinations can 6 of the lego blocks be used to make different compound objects.”

He said he had calculated a number which was close to 102,981,000 combinations. He wasn’t quite right.
The question was so hard, that it took years to get to the exact answer. Soren Eilers, a professor of Mathematics at the University of Copenhagen put his mind to work and figured out the answer. In fact, his mind did not do him much service here. It was a computer which spit out the answer to him after a week long calculation.
He found that the actual answer to this mathematical conundrum is – 915,103,765

His computer can now calculate this number in a matter of minutes. But as you increase the number of blocks, the time needed to arrive at the answer increases hundred folds. So, the number of combinations from 7 blocks takes 2 hours. 8 blocks – around 20 days. But calculating the same for 9 blocks might take hundreds of years.

The Man Who Counted to a MIllion

By Anupum Pant

In the summer of 2007, Jeremy Harper, a software professional took some time off his work to be at home and count to a million for a good cause. Once this started, for 3 long months, counting for 16 hours a day, he kept doing this. While he was doing this, he streamed his performance live over the internet on a website where anyone could see him and donate to the charity for which he was doing this.

By the end of it he was able to raise $10,000 for a charity Push America. He ended up getting his name etched in the guiness book of world records for the largest number ever counted. His record still remains.