Article By Nature: Paradox At The Heart Of Mathematics Makes Physics Problem Unanswerable

When Trilobites Ruled the World

infographics by The New York Times

brought to you by Graphic Services for Science and Graphic Biology

Rest in peace, Marie Severin. A visionary artist for Marvel and co-creator of #Spiderwoman, her work shaped comics and inspired millions.

“Spiderwoman” by ayhotte: http://bit.ly/2LG2IGT  

douggsbase This was the 4th big way I have organised at Kjerag, Norway and the local record of 26 jumpers. The last record held here was a 24 way back in 1999. It went incredibly well. Text book in fact! And for most people this was their first big way. 20 jumpers were from @learntobasejump#LTBJA huge thanks to everyone for doing their jobs. I am very proud of all of you!! Big thanks to @sophiimeredithfor the great video angle 🎥

Start the day off right… with a little code! 

In 2015, a 777-200 made the Newyork-London route in 5 hours,16 minutes where the usual journey time is ~7 hours.

The flight reached ground speeds of up to 1200 km/h (745 mph),riding a powerful jet stream of up to 322 km/h (200 mph) tailwinds and breaking the sonic barrier ( 1224 km/h (761 mph)).

Tail and headwinds

The principle is analogous to those high school problems in relative velocity:

“A man rows a boat in a river. The velocity of the boat is … Find the stream velocity”

If you are headed downstream i.e in the same direction as the river stream you will reach your destination faster than if you were rowing upstream.

Similarly a tailwind is one that blows along the same direction of the aircraft increasing the net speed of the aircraft ,and headwind is one that blows in the opposite direction and slows the craft down.

So, does this mean that if you are moving at v kmph and there is a headwind of -v kmph, you would just hover? Hell yeah!

Take a look at this video:

Wind shear

A phenomenon known as ‘wind shear’ occurs when the wind speed changes abruptly, which can cause turbulence and rapid increase/decrease in velocity of flight.

This can be really challenging during landing since if the headwind turns tailwind, there is a possibility of the aircraft overshooting the runway due to the increased velocity.

What causes this ?

The aviation industry takes advantage of trade winds and jet streams in order to cut time off the flight and save fuel.

Tradewinds are caused by the unequal heating of the atmosphere at different latitudes and altitudes and by the effects of the Earth’s rotation (Coriolis effect).

                           Trade wind pattern. Credit: Earth Wind Map

Jet streams on the other hand are this narrow current of fast moving winds in the upper troposphere flowing west to east. And riding one can definitely make your travel time shorter.

                               Jet streams in the northern hemisphere

As a result of jet streams, within North America  the time needed to fly east across the continent can be decreased by about 30 minutes if an airplane can fly with the jet stream, or increased by more than that amount if it must fly west against it.

How do pilots know about this ?

Pilots receive a weather briefing actively during flight. Included in the briefing is the best combination of jetstreams and other wind patterns that the pilot can take advantage of saving time and fuel.

Many airports have runways facing in different directions in order to allow the pilots to use the runway that faces the wind during take off/landing.

Have a great day!

Can you flatten a sphere?

The answer is NO, you can not. This is why all map projections are innacurate and distorted, requiring some form of compromise between how accurate the angles, distances and areas in a globe are represented.

This is all due to Gauss’s Theorema Egregium, which dictates that you can only bend surfaces without distortion/stretching if you don’t change their Gaussian curvature.

The Gaussian curvature is an intrinsic and important property of a surface. Planes, cylinders and cones all have zero Gaussian curvature, and this is why you can make a tube or a party hat out of a flat piece of paper. A sphere has a positive Gaussian curvature, and a saddle shape has a negative one, so you cannot make those starting out with something flat.

If you like pizza then you are probably intimately familiar with this theorem. That universal trick of bending a pizza slice so it stiffens up is a direct result of the theorem, as the bend forces the other direction to stay flat as to maintain zero Gaussian curvature on the slice. Here’s a Numberphile video explaining it in more detail.

However, there are several ways to approximate a sphere as a collection of shapes you can flatten. For instance, you can project the surface of the sphere onto an icosahedron, a solid with 20 equal triangular faces, giving you what it is called the Dymaxion projection.

The Dymaxion map projection.

The problem with this technique is that you still have a sphere approximated by flat shapes, and not curved ones.

One of the earliest proofs of the surface area of the sphere (4πr2) came from the great Greek mathematician Archimedes. He realized that he could approximate the surface of the sphere arbitrarily close by stacks of truncated cones. The animation below shows this construction.

The great thing about cones is that not only they are curved surfaces, they also have zero curvature! This means we can flatten each of those conical strips onto a flat sheet of paper, which will then be a good approximation of a sphere.

So what does this flattened sphere approximated by conical strips look like? Check the image below.

But this is not the only way to distribute the strips. We could also align them by a corner, like this:

All of this is not exactly new, of course, but I never saw anyone assembling one of these. I wanted to try it out with paper, and that photo above is the result.

It’s really hard to put together and it doesn’t hold itself up too well, but it’s a nice little reminder that math works after all!

Here’s the PDF to print it out, if you want to try it yourself. Send me a picture if you do!

Funny pets

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