Tag: shape

The Hexagon Storm

By Anupum Pant

Saturn is probably the most beautiful planet we have in our solar system. But did you know, Saturn is also home to a very peculiar phenomenon which has never been seen anywhere else before – a hexagonal hurricane.

A hurricane in the shape of a hexagon (six-sided), not circle. If that doesn’t blow your mind, try this – the storm is an incredibly huge – 30,000 km across! And it is about 100 km deep, with winds of ammonia and hydrogen moving at  more than 320 km per hour. It is large enough to swallow four planets of the size of Earth. This is what the Earth would look like if it were kept beside the storm.

saturns hexagonal storm and earth comparision

It’s only natural for hurricanes to be circular. And yet, researchers at Ana Aguiar of Lisbon University have been able to show that the hexagonal storm raging in the north pole of Saturn is also very natural too. In the year 2010, they proved  to by reproducing a similar effect in the laboratory by using rotating liquids.

According to them, a very narrow jet stream that goes about the hurricane’s edge creates a couple of other tiny hurricanes. These little storms are the ones that push the larger hurricane’s borders and give it a hexagonal shape.

In the 80s, the storm was first spotted by the twin voyager spacecraft.

Evolution of Eggs

By Anupum Pant

Eggs come in a variety of shapes, sizes and colours. Birds, a major group of creatures that descended from reptiles have, for several years, continued to evolve the design of their eggs for millions of years now (not consciously, through natural selection).

Eggs could have been cube shaped. In that case they would have been very difficult to lay. Also, they would have been weakest at the centre points of a face of the cube. Hence, eggs didn’t end up being squarish.

While most eggs have evolved to, well, an egg-shape, some eggs like those of some owls are nearly spherical in shape. But oval and pointy eggs do have an advantage of sort.

Spherical eggs tend to roll easily, and if laid somewhere near a cliff, they’d roll away, never to be seen ever again. Oval eggs normally tend to roll in circles. Usually, they roll in big circles. Still dangerous for birds who perch on cliffs most of the time.

Of all the eggs, the egg of a common guillemot bird is probably the most incredible – in the sense that it has a design that doesn’t let it roll down cliffs very easily.

Common guillemots are sea birds and they normally like to perch on cliffs. To add to the danger of their precarious perching places, they usually perch on such cliffs with a huge group. Also, they don’t even make nests.

Had their eggs been shaped like those of owls, they would have easily gotten knocked by someone from that huge group of perching birds, perching on precarious cliffs. So, their eggs have evolved to survive these conditions.

This is how their eggs look like. They are very awkwardly shaped. But when it rolls, thanks to natural selection, it rolls in very small circles! They don’t fall off cliffs easily. Wonderful!

common guillemot egg

First seen at [io9]

Cutting a Round Cake on Scientific Principles

By Anupum Pant

Background

For years the phrase “cake cutting” has conjured up just one image in my brain – A triangular section of the cake. This way of cutting a cake is so normal that even the tools (especially the spatula) that are made for cake cutting are made in a way that’d work with best when you are making that traditional triangular cut. Turns out, this method of cutting a cake which we’ve all know for years is totally wrong.

Why is it wrong?

It’s wrong mostly for mathematical loners. People who, on their birthday, have no one around to share the cake with, and cannot finish off the whole cake. For them and the ones who have to store the cake after cutting it, are extremely careful about how moist the edges remain when they next eat it, this right way to cut a cake might be of great importance.

The way we’ve always know is “wrong” because when you cut off, say a single section of the cake and decide to store the larger piece in the fridge, some internal part of the cake remains exposed and it dries off. So, the next time you cut off a piece near the area where you started, you’d get a freshly cut moist wall of cake on one side, and a repulsively hard dried up wall on the other. That, some think, is extremely unpleasant.

What’s the Right way?

About 100 years back, a brilliant Polymmath (and a mathematician), Sir Francis Galton, faced a similar annoyance. So, instead of cursing others for having invented an absurdly inefficient way to cut a cake, he decided to develop his own. He ended up developing a very simple and efficient cut which helped him keep the cake wall relatively moist. Here’s how the cut works. (Cut along the dotted line)

the right way to cut a cake

Describing his new way of cutting cakes, he got an article published in the Nature magazine (dated December 20th, 1906). “Cutting a Round Cake on Scientific Principles”

Alex Bellos from the Numberphiles describes it in a video below:

Constant Width Objects – Not Spheres!

By Anupum Pant

Until today, I had no idea that non-spherical objects could have the same diameter at every point! Don’t believe me? Have a look at these wonderful little metal objects that aren’t anything even close to a sphere and are still able to roll a flat surface on them, as spheres would. [Video]

Like Gombocs, these shapes are a Mathematician’s fantasy. They have a generic name – constant width objects. In fact, these carefully machined metal objects could be perfect gifts for your mathematician friend. I know, I would some day, if you think you need these too, you can buy them here. (I’m in no way related to Grad-Illusions, nor am I an affiliate marketer)

Note: Though in some manner they look like Gombocs, they are not Gombocs. To know more about what Gombocs are, read this – [Gomboc – An Object That Never Falls]

How are they made?

Theoretically, to understand how they are made, you need to understand that the 3D objects of constant width are usually* spun out of a 2D object – Just like a sphere can be made by spinning  a circle. Though there is a kind of constant width 3D object that is not a spun version of any 2D curve.

50 Pence coin 1994 - 50th anniversary of D-Day...
50 Pence coin 1994 – 50th anniversary of D-Day – Normandy Landings (Photo credit: ell brown)

The 2D form is called the Reuleaux triangle and it looks like this [image]. The one shown in the link is a constant width curve based on an equilateral triangle (triangle with equal sides). It turns out, you can construct a constant width curve out of any triangle, and a polygon too (like the 50 pence coin shown above made out of a regular heptagon). To make a constant width curve using an equilateral triangle, all you need is a compass, a paper and a pencil:

  • Draw an equilateral triangle.
  • Put the point of your compass on one vertex.
  • Trace out an arc that starts from one of the other vertex and ends at the third one.
  • Repeat the same for rest of the two vertices. There! You have your constant width curve. Cut it out of paper.
  • Now around an axis dividing the shape into half, spin it. You have a theoretical 3D shape that resembles the one shown in the video above.

Vehicle tyres and square hole drills

So, since these shapes can roll things around like circles can, wheels could be made of these shapes too! Then, why aren’t wheels made that way? That is because when these shapes roll, they don’t have their centers at one place. If vehicles had tyres like these, engineers would have had a hard time designing axle systems.

They are in fact used in a Wankel engine. And since the center traces a square when Reuleaux triangle rolls, they have been used in drills that can drill out square holes.

Random constant width facts:

    • There are a few pencils which are manufactured in an extruded-reuleaux triangle shape. These pencils can roll around smoothly like circular pencils.
    • For some reason, even guitar picks are often manufactured in these shapes.
    • Instead of spinning around a constant width 2D object, a 3D constant width object can be made by modifying flat tetrahedron faces using intersections of sphere faces. As it can’t be done on a lathe, these are particularly hard to machine. It is called the Meissner’s tetrahedron or the Reuleaux Tetrahedron.
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Gomboc – An Object That Never Falls

By Anupum Pant

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There’d be hardly anyone among us who hasn’t played with a roly-poly toy during their childhoods. If you know it by some other name, you could think of it as a toy that never falls, no matter how hard you hit it, and sells in variants which look like this. That isn’t exactly what a Gomboc is, but you get an idea about what it does – It does not fall. For more, read on.

What is a Gomboc?

A Gomboc (Gömböc) is a mathematical 3-D shape which has only one position in which it can stand and is made up of a single material  of uniform density. If you try to make it stand in some other way, or try to knock it down, it moves back to that single stable position, gradually. When placed on its side, it starts rocking magically, gains momentum, straightens itself and gradually comes to rest in that single position. Here is a video of a Gomboc doing its thing.

A Gomboc is an object surrounded by a number of complex curves, it takes an immense amount of accuracy to get the surfaces right. An accuracy of  the orders of around 1/10th of a human hair’s thickness is required for it to work properly. For better, people have started 3D printing these complex shapes.

The world’s largest Gomboc was displayed in China in the year 2010 which measured around 3 meters in all directions.

Terrestrial tortoises, who use a similarly shaped shell to get on their feet when turned upside down, were using it long before humans had found a way to construct it. The first time we made it, was in the year 2006. Evolution got there first!

How is it different than a Roly-Poly toy?
A roly-poly toy usually has an internal counter weight made up of a heavier material. But a Gomboc is made up of a single material.

Uses: Use it as a paper weight or to gift it to your friend who is a math geek. Tortoises use it to save their own lives.

Where can I buy one?
You can get one for yourself from an official website of the inventors – Here.

The Standard World Map is Misleading

by Anupum Pant

Mercator’s projection

Most of us have this image of the world in our minds. This kind of a map, today printed in almost every textbook, known as the Mercator’s projection was first created to make work easy for navigators. Even Google Maps uses a Mercator-derived technique to project the world on a flat surface. But, Mercator’s projection has only deceived our idea of geographical area for all these years. For instance, it has led us into believing that Greenland covers an area which is almost equal to Africa (Also, have a look at the size of Antarctica there. Gosh!). The comparison of these two land masses actually looks like this.

According to this infographic, the actual size of Africa is larger than US, China, India, Mexico, Peru, France, Spain, Papua New Guinea, Sweden, Japan, Germany, Norway, Italy, New Zealand, the UK, Nepal, Bangladesh and Greece, all of them put together. In short, Africa is around 14 times larger than Greenland. Do not underestimate its area.

You can try playing with various combinations on this web app – map fight. Try these: Australia vs. Antarctica; US (contiguous) vs. Russia; and of course Greenland vs. Africa; they’ll leave you spellbound.

Why does this happen?

Since our planet is a sphere (an oblate ellipsoid really), to project it on a flat surface like paper, the actual shapes and sizes of landmasses have to be distorted to some extent. There is no way around it. Today, hundreds of different projection methods meant for various purposes are available, but none of them can exactly show the actual shapes & sizes of the landmasses. Some preserve the shape, some preserves the size, and others preserve direction…so on…

Mercator’s projection, the devious one discussed above, for example, uses a cylindrical projection. That means, it stretches the areas on a globe, which are nearer to the poles. Hence, the imprecise size of Greenland and Antarctica.

What is a perfect map, then?

Even after developing hundreds of projection method, we haven’t been able to spot the perfect method, nor will it happen in the future. But, to get the right sense of area, a projection method known as the Peters (also known as Gall-Peters projection) projection, is said to be the most accurate (in terms of area). It is also one of the most controversial maps.

Peter’s projection also has a huge fan following in spite of its terrible appearance.

Bonus Map Facts:

  1. National Geographic started using the, good looking, Robinson projection from the year 1988, and used it for ten years, then, it moved to the Winkel-Tripel in 1998.
  2. An ideal Mercator’s projection would have infinite height if it doesn’t truncate some area near the extreme poles.
  3. Peters pointed that the Mercator’s projection made developing countries seem much smaller than they actually are. He said that these errors made the struggles of developing nations near the equator looks much smaller to the developed world.
  4. XKCD published a comic on projections – “What your favorite map projection says about you.” 977. [see the explanation here]