fouriestseries:

Rotational Stability

Time for an experiment! Find a book and secure it shut using tape or a rubber band. Now experiment with spinning the book while tossing it into the air. You’ll notice that when the book is spun about its longest or shortest axis it rotates stably, but when spun about its intermediate-length axis it quickly wobbles out of control.

Every rigid body has three special, or principal axes about which it can rotate. For a rectangular prism — like the book in our experiment — the principal axes run parallel to the shortest, intermediate-length, and longest edges, each going through the prism’s center of mass. These axes have the highest, intermediate, and lowest moments of inertia, respectively.

When the book is tossed into the air and spun, either about its shortest or longest principal axis, it continues to rotate about that axis forever (or until it hits the floor). For these axes, this indefinite, stable rotation occurs even when the axis of rotation is slightly perturbed.

When spun about its intermediate principal axis, though, the book also continues to rotate about that axis indefinitely, but only if the axis of rotation is exactly in the same direction as the intermediate principal axis. In this case, even the slightest perturbation causes the book to wobble out of control.

The first simulation above shows a rotation about the unstable intermediate axis, where a slight perturbation causes the book to wobble out of control. The second and third simulations show rotations about the two stable axes.

Unfortunately, as far as my understanding goes, there’s no intuitive, non-mathematical explanation as to why rotations about the intermediate principal axis are unstable. If you’re interested, you can find the stability analysis here.

Mathematica code posted here.

Additional sources not linked above: [1[2] [3] [4]

mathmajik:

Snowy Mathematical Art from Simon Beck

Simon is an artist and is most well-known for making incredibly delicate and detailed art in the snow, just by walking over a fresh snowfall. He literally walks miles in the snow to create these pieces. He could spend hours upon hours creating one design, just to have it be covered by snowfall or blown away by the next day. But he still makes them.

These delicate patterns were created in the beautiful Savoie Valley in France, overlooking Mont Blanc.

He creates large, mathematical patterns that have different effects when viewed from different angles.

Source: viralnova.com Images: Simon Beck

"Using the chain rule is like peeling an onion. You have to deal with every layer at a time and if it’s too big you’ll start crying."
- Calculus professor (via mathprofessorquotes)

Fractals

catscardiganscalculus:

The name “fractals” is derived from the fact that fractals don’t have a whole number dimension—they have a fractional dimension. Initially this may seem impossible—what does a dimension of 2.5 mean, after all—but it becomes clear when we compare fractals with other shapes.

A line has…

chaotic-renan:

“The universe is a pretty big place. If it’s just us, seems like an awful waste of space.” - Carl Sagan

“Two possibilities exist: either we are alone in the Universe or we are not. Both are equally terrifying.” - Arthur C. Clarke

I already quoted those here. But never talked about them.

I personally like those quotes very much, they explain a lot of what I think.

When I was a Christian, the question of extra-terrestrial life wasnt important, for me at least. But as the years passed and I became an Atheist this is became of the things that haunts my mind.

If we are alone in the universe, well, it an awful waste o space, as Sagan said. All those galaxies all for us and we barely went(if ever) to another place that isnt earth. All this while we are here, growing old, paying taxes, slaves to a society that is made of the very persons that want to get out.

On the other hand… If we are not alone. Who are they? Do they know about us? Will we ever meet them? Just imagine the things that they may have seen. What they can teach us. If they arent some kind of algae  or something.

While sometimes I wonder if the human race will survive more than 100 years or will it perish by its own wars? Or some creation or experience that went wrong? We may yet be able to conquer the universe, now not. Later. In the book  “Além da cuvatura da luz”, by Mário Sanchez, the humanity flees from earth as the sun is collapsing. Is this what is waiting for us? Fly away to conquer the space? Answer all those questions about god and purpose or whatever? Or is the future as said in “The world without us”, by Alan Weisman, where we are extinct and the radiation from our technology is echoing through out the space hoping to find someone to listen to it while the earth slowly heals from us?

I usually dont write this much. I hope it does makes sense. But if it doesn’t… Well, who knows?