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]