Science and Math
Science and Math World

IMPOSSIBLE! Right?You may have heard “the interior angles of a triangle always add up to 180 degrees”. This is not always true. Check out the second image, it shows a triangle with 3 right angles for a total of 270 degrees!It is true in flat

Euclideangeometry (the geometry you probably learned in school) however. But there are so many other geometries out there! You may be thinking, are other geometries real though? A mathematician would argue they are just as real as the typical flat geometry you know and love (or hate). Thesealternative geometriescan be practically useful too!The images above show triangles in spherical geometry.

Those aren’t triangles though!Oh but they are! A triangle is just a polygon enclosed by three lines. Looks like it fits the criteria.Wait but those aren’t lines, they are curved!Ah yes. I argue that these are, for all intents and purposes, just as good as lines. We need to ask: What is a line? A line is so basic to us we may not know how to describe it. I offer this definition:A line is the shortest path between 2 points.The 3 curves that make the triangle above are in fact the shortest paths from one vertex to the other on the surface of the sphere (they just so happen to be on circumferences of the sphere, which are often referred to asgreat circles). So it may be more useful to think of lines, in general, aslength minimizing curves.In conclusion, we would consider the shape above to be a triangle as it is enclosed by 3 length minimizing curves on a surface.Spherical geometry can be very useful; think about the Earth. To reduce travel time, airplanes would want to travel along great circles as they are the shortest paths from one place to another. Additionally, this type of thinking (rethinking straight lines as length minimizing curves) is central to Albert Einstein’s general theory of relativity.

read more at http://staffrm.io/@missnorledge/35H6cS1T52

“You want to do calculus with numbers? Accounting is over there, here in math we do calculus with letters.”

— Calculus professor

If one remembers this particular episode from the popular sitcom ‘Friends’ where Ross is trying to carry a sofa to his apartment, it seems that moving a sofa up the stairs is ridiculously hard.

But life shouldn’t be that hard now should it?

The mathematician Leo Moser posed in 1966 the following curious mathematical problem:

whatThis question became known as the

is the shape of largest area in the plane that can be moved around a

right-angled corner in a two-dimensional hallway of width 1?moving sofa problem, and is still unsolved fifty years after it was first asked.The most common shape to move around a tight right angled corner is a square.

And another common shape that would satisfy this criterion is a semi-circle.

But

what is the largest area that can be moved around?Well, it has been

conjectured that the shape with the largest area that one can move around a corner is known as “Gerver’s

sofa”. And it looks like so:## Wait.. Hang on a second

This

sofa would only be effective for right handed turns. One can clearly

see that if we have to turn left somewhere we would be kind of in a tough

spot.Prof.Romik from the University of California, Davis has

proposed this shape popularly know as Romik’s ambidextrous sofa that

solves this problem.Although Prof.Romik’s sofa may/may not be the not the optimal solution, it is definitely is a breakthrough since this can pave the way for more complex ideas in mathematical analysis and more importantly sofa design.

Have a good one!

If one remembers this particular episode from the popular sitcom ‘Friends’ where Ross is trying to carry a sofa to his apartment, it seems that moving a sofa up the stairs is ridiculously hard.

But life shouldn’t be that hard now should it?

The mathematician Leo Moser posed in 1966 the following curious mathematical problem:

whatThis question became known as the

is the shape of largest area in the plane that can be moved around a

right-angled corner in a two-dimensional hallway of width 1?moving sofa problem, and is still unsolved fifty years after it was first asked.The most common shape to move around a tight right angled corner is a square.

And another common shape that would satisfy this criterion is a semi-circle.

But

what is the largest area that can be moved around?Well, it has been

conjectured that the shape with the largest area that one can move around a corner is known as “Gerver’s

sofa”. And it looks like so:## Wait.. Hang on a second

This

sofa would only be effective for right handed turns. One can clearly

see that if we have to turn left somewhere we would be kind of in a tough

spot.Prof.Romik from the University of California, Davis has

proposed this shape popularly know as Romik’s ambidextrous sofa that

solves this problem.Although Prof.Romik’s sofa may/may not be the not the optimal solution, it is definitely is a breakthrough since this can pave the way for more complex ideas in mathematical analysis and more importantly sofa design.

Have a good one!

Mathematicians finally cracked László Fejes Tóth’s zone conjecture, a 44-year-old geometry problem! Learn about it here: http://bit.ly/2Cxqp0C

[#Einsteinsmama]

Koch Snowflake: Finite area, Infinite perimeter.The Koch Snowflake has finite area but infinite perimeter… yeah that happens with fractals. This abstract curve requires an infinite process (depicted in the gif) to construct and is an example of a

fractal–a mathematical set (usually a curve or geometric figure) which exhibits a repeating pattern that displays at every scale. (More about fractals here https://en.wikipedia.org/wiki/Fractal)

But How?It seems clear that the area would be finite since the figure encloses a finite amount of space. To grasp why the Koch Snowflake has infinite perimeter, notice how as the iterations progress, the edges become more and more intricate. Now imagine trying to draw the edges with a pen. Since the construction of the snowflake continues indefinitely, the edges become infinitely intricate and you could never finish detailing these intricacies with your pen (that is the intuitive argument at least. I’ll leave the precise calculations up to you).Fractals may seem so abstract and impractical but they actually have many useful real-world applications. For example, Benoit Mandelbrot (considered the “father of fractals”) found that stock market prices could be modeled with a factual curve. Check the wiki page for a long and diverse list of applications.

Fractal geometry may seem more abstract than traditional geometry but Mandelbrot argues that fractals are

“the geometry of nature”. Objects in nature have random irregularities and are seemingly infinite in their intricacies. Attempting to incorporate this in drawings or animations is extremely difficult. Movie special effects and CGI often use fractals to make objects appear more natural looking. Since fractals can be made with mathematical formulae they are easy to generate with a computer. The firstStar Warsmovies were renowned for their special effects and were some of the first to use fractals to generate life-like explosions and landscapes of other worlds.