Finally I had some time to make a new animation. Hopefully more will follow 🙂

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

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!

Triangulating a circle.

How to do a drawing like this? Start with a circle and divide it into n equal parts. I choose n=100 for this drawing. There is an easy formula for this, the i-th point is (radius*cos(i*2*PI/n),

radius*sin(i*2*PI/n)). Then for each i connect the i-th point to the point (radius*cos(PI-2*i*2*PI/n), radius*sin(PI-2*i*2*PI/n)). This might not be one of the original points, but it is on the circle. And that is it 🙂

Why is this mathematically interesting? This way we get a set of nonparallel lines such that there are a lot of triple intersections between them.

Complex analysis – the perfect subject for combining visual and analytic thinking. I just read “Visual Complex Analysis” by Tristan Needham, a basic book on complex analysis: lots of historical references, uncompromising explanations, lots of problem solving and plenty of beautiful illustrations!

I have this book. It is very good.

Pattern

Based on Joan Taylor’s wanderer tiling. http://tilings.math.uni-bielefeld.de/substitution/wanderer-refl/