The Edge of Forever

Cool science related stuff! - Gabriel, 17, Brazil
1ucasvb:

In a previous post, I showed how to geometrically construct a sine-like function for a regular polygon.
I also pointed out how the shape of the function’s graph depends on the orientation of the polygon, since it isn’t perfectly symmetric like the circle.
This animation illustrates how the polygonal sine (dark curve) and polygonal cosines (clear curve) change as the generating polygon rotates.
Derivation
First of all, it is important to point out these functions are not based on the perimeter of the shape, like it is for the unit circle. We’re still sticking to the interior angle here. If we used the perimeter as a substitute for the angle we would just get a deformed linear spline of the sine function, which is rather useless and boring.
In order to find these functions for an arbitrary polygon, we first need to write the polygon in polar form. That is, we want the radius for a given angle. In a circle, this is a constant value.
A general “Polar Polygon” function is:
PPn(x) = sec((2/n)·arcsin(sin((n/2)·x)))
Where n is the number of sides of the polygon. If n is not an integer, the curve is not closed.
Armed with this function, we can quickly find the polygonal sine and polygonal cosine:
Psinn(x) = PPn(x)·sin(x)Pcosn(x) = PPn(x)·cos(x)
As n grows, the functions approximate the circular ones, as expected. To rotate the polygon, just add an angle offset to the x in PPn.
This technique is general for any polar curve. Here’s a heart’s sine function, for instance
So, what is it good for?
I’ve used this several times when I wanted some smooth interpolation between a circle and a polygon, in such a way that the endpoints of the interpolation are a perfect circle and a perfect, pointy polygon. It’s useful in parametric surfaces, such as in this old avatar of mine:

Now you can also listen to what these waves sound like

1ucasvb:

In a previous post, I showed how to geometrically construct a sine-like function for a regular polygon.

I also pointed out how the shape of the function’s graph depends on the orientation of the polygon, since it isn’t perfectly symmetric like the circle.

This animation illustrates how the polygonal sine (dark curve) and polygonal cosines (clear curve) change as the generating polygon rotates.

Derivation

First of all, it is important to point out these functions are not based on the perimeter of the shape, like it is for the unit circle. We’re still sticking to the interior angle here. If we used the perimeter as a substitute for the angle we would just get a deformed linear spline of the sine function, which is rather useless and boring.

In order to find these functions for an arbitrary polygon, we first need to write the polygon in polar form. That is, we want the radius for a given angle. In a circle, this is a constant value.

A general “Polar Polygon” function is:

PPn(x) = sec((2/n)·arcsin(sin((n/2)·x)))

Where n is the number of sides of the polygon. If n is not an integer, the curve is not closed.

Armed with this function, we can quickly find the polygonal sine and polygonal cosine:

Psinn(x) = PPn(x)·sin(x)
Pcosn(x) = PPn(x)·cos(x)

As n grows, the functions approximate the circular ones, as expected. To rotate the polygon, just add an angle offset to the x in PPn.

This technique is general for any polar curve. Here’s a heart’s sine function, for instance

So, what is it good for?

I’ve used this several times when I wanted some smooth interpolation between a circle and a polygon, in such a way that the endpoints of the interpolation are a perfect circle and a perfect, pointy polygon. It’s useful in parametric surfaces, such as in this old avatar of mine:

Now you can also listen to what these waves sound like

(via engrprof)

complexityblog:

Drawing a curve, using only straight lines.
1. draw 2 intersecting lines.
2. mark off an equal number equidistant of tick marks on both axes.
3. connect opposite tick marks with a straight line (lowest on the vertical axis with the furthest on the horizontal axis).
optional: using your daughter’s school supplies while “helping” with her homework.

complexityblog:

Drawing a curve, using only straight lines.

1. draw 2 intersecting lines.

2. mark off an equal number equidistant of tick marks on both axes.

3. connect opposite tick marks with a straight line (lowest on the vertical axis with the furthest on the horizontal axis).

optional: using your daughter’s school supplies while “helping” with her homework.

(via visualizingmath)

mucholderthen:

TOP RESEARCHERS COMMENT AS THE “GRAVITATIONAL WAVE REVOLUTION”  RIPPLES THROUGH THE SCIENTIFIC COMMUNITY
Nature | News by Ron Cowen  || 18 March 2014

The evidence of gravitational waves from the early Universe found by researchers working at the South Pole has been hailed as a landmark discovery in cosmology, astronomy and physics. The announcement was made by astronomer John Kovac on 17 March at the Harvard-Smithsonian Center for Astrophysics in Cambridge, Massachusetts. … Here Nature has collected reactions from leading researchers.

Please go to the open access article in Nature to read the comments (and see who made them) …

‒ “… many great intellectual discoveries are never confirmed at the time when the authors are still alive. I’m not dead yet and they are already seeing this gravitational-wave signal.” [Comment from one of the discoverers/creators of the theory of cosmic inflation]

‒ “Nobel prize material, no question. It’s not everyday that you wake up and learn something fundamentally new about the Universe, a telegram from the very earliest moments of the Universe.  … just in time for the one-hundredth birthday of Einstein’s general [theory of] relativity next year.”

‒ “If the BICEP2 result holds up, this is really big — as important as the discovery of dark energy, cosmic microwave background anisotropy or the Higgs boson. …”
______________________________

Top  Amundsen-Scott South Pole Station, with the BICEP2 telescope on the right. [Robert Schwarz/University of Minnesota]
Middle/Bottom: credit: BICEP2 Collaboration / Nature Magazine, 17 March 2014

(via s-c-i-guy)