Tag: klein

These Intelligent Crows Will Make You Smile

By Anupum Pant

Background

One interesting thing to note about crows is the size of their brains. Among all the bird species, crows have the largest brain to body ratio.
Besides that, the anatomy of their brain is a lot similar to our own brains. The forebrain which is responsible for the intelligence of an organism, is highly developed in crows. This makes them very adaptable. They are able to easily adjust and learn things as per the changing situations.

Although crows aren’t adored, probably because of the annoying sounds they make, their amazing feats of dexterity are bound to elicit an awe. May be valporate can help them with the voice.

Notably, crows are found living with human beings and are known to exploit us to get their work done. Only in rare cases are crows found to be living very away from human settlements. Joshua Klein, in a TED talk, shares a couple of anecdotes about how good crows can be at problem solving. One of them was as follows:

Crows and Hooks

Crows are normally seen holding sticks in the beaks to take food out of narrow holes in the wild. When one crow in a laboratory was given a piece of hard wire to draw a piece of meat from inside of a narrow tube, it wasn’t working very well for it. The crow, despite never having learned any related trick beforehand, found a solution for the problem completely on its own.

Solution: It used the surroundings to bend the wire to make a hook out of it; then simply put it in the tube to use the hook to retrieve meat!

But that is just a warm up of kinds of problem solving crows can handle. Here is another one.

Amazing Problem Solving Crows

In the suburbs of Tokyo, where crows often came across a kind of nut they were not able to dig into, because of its hard shell, they found an extremely intelligent way to deal with it. They dropped it on the roads for the vehicles to break the nuts for them! That was not all…

While doing this, they were faced with another problem – As cars, passed by at high speeds, it was risky for them to get back the nuts. Crows found a solution for that too. I’m not that smart. I bet I could have never thought of that.

Solution: They dropped it on the zebra crossing. As the cars stopped for the humans to cross, crows walked behind humans to get back their broken nuts safely. Interestingly, some crows who had devised this technique taught it to all the crows in the surrounding areas. Now all of the suburban crows around Tokyo know the trick.

If you liked this, you’ll definitely love ants and their amazing counting skills.

Klein Bottle – A Bottle That Contains Itself

By Anupum Pant

To appreciate the beauty of mathematics and nature there is no escaping without learning about a Klein Bottle. A three-dimensional representation of a Klein bottle looks like this – [image]

There are number of phrases you can use to describe (not exhaustively) it. A few of them are as follows:

  • An object with no boundaries.
  • An object with no inside or outside.
  • One sided surface.
  • Non-orientable surface

Wikipedia describes it as:

The Klein bottle is a non-orientable surface; informally, it is a surface in which notions of left and right cannot be consistently defined.

Simplifying things: A Möbius strip is a simpler example of a non-orientable object. That means it has no inside or outside. Add another aspect – having no boundaries – to it, it gets more complex and you end up with a Klein bottle.
If you haven’t heard of Möbius strips, to understand such surfaces, you can make one for yourself now.

  1. Tear off a strip of paper.
  2. Hold it horizontally, straight with both of the short edges in your hands.
  3. Now, twist one of the edges by 180 degrees and join the two short edges. You’ll have something like this in your hands – [image]

Test the surface and edges: On this object you just created, move your finger along the surface. You’ll find that your finger comes  back to the same place eventually. There is no inside or outside for this object, there is just one surface.
The same thing happens with its edge (try moving your finger along the edge). Here is a Music box playing a Harry Potter theme continuous – forward, inverted, forward and so on – manner; Relevant video: [video]

Now spin it (the Möbius Strip) fast. You can NOT practically do it. I mean, spinning it like you spin a circle and get a sphere. There! You have a Klein bottle. It is better than a Möbius strip in a way that it (Klein Bottle) has no boundaries.

Klein bottles cannot actually exist in our three-dimensional worlds, the ones that look like them (Klein Bottles) are just 3D representations of a 4D object. Like a two-dimensional drawing of a 3D cube. These models are available for you to buy. Interestingly, in spite of having no inside or outside, they can be filled with a liquid. But, given the opposing force of air, they are pretty tough to fill. It is important to note that the 3D representation of a 4D Klein bottle has an intersection of material, this doesn’t happen in 4D. It is like the intersecting edges of a 3D cube in the 2D representation.

You’re thinking 3D? At MIT (and other places) 4D printing is already happening.

If you are having a tough time imagining this 4D object, the following 4D animation might help (or leave you perplexed) – [video]
[Extra reading for math geeks] as if they already didn’t know about Klein bottles.