Posts tagged math
Posts tagged math
Artist Nike Savvas transforms mathematic formulas into beautiful sculptures.
Solids and string art.
Circle packing on a sphere.
A nanocentury is about Pi seconds long.
While it would be more accurate to say “the first eight digits of π” rather than “the value of π”, it’s hard to be too upset with a post that merges math and coffee.
Sol LeWitt, Wall Drawing #139 (Grid and arcs from the midpoints of four sides)
(This picture is from the book Math Made Visual by Claudi Alsina and Roger B. Nelsen)
Given any two circles, there are exactly two lines that are tangent to both those circles at the same time. If the circles are different sizes, then these two tangents cross at some point.
If we have three circles, we can build three such points by looking at the common tangents to every pair of circles.
Monge’s theorem states that these three intersections always lie on a line.
The proof in the book is cute:
Let each circle be the “equator” of a sphere. Given a pair of spheres consider the cone generated by the two corresponding tangent lines. Half of the cone will lie above the plane of the circles and half will lie below. Now consider a plane tangent to the three half-spheres. This plane will also be tangent to each of the three cones, and it will intersect the original plane in a line L. Since this plane contains one line from each half-cone, the vertices of the three cones must be located on the intersection line L.
cute? downright adorable.
Very nice. I wonder what the projective version of this theorem is? (I.e., if you replace the circles with conic sections.)
I came across these curves while writing a test: they are the intersections of the unit sphere with helicoids of decreasing “pitch”. I was surprised to find all the curves are smooth (although, in retrospect, I shouldn’t have been). They are parametrized by
f(t) = (sin(t)*cos((n*cos(t)), sin(t)*sin(n*cos(t)), cos(t)).
The ballistic ellipse
This is something I found when I was playing around with ballistic trajectories. I wondered what shape you would get if you connected all the apex points of all trajectories, if you only changed the angle and kept the same initial speed.
Surprisingly, you get an ellipse!
The equation for the ellipse is:
x2 / a2 + (y - b)2 / b2 = 1
Where a = v02 / (2g) and b = v02 / (4g). Naturally, v0 is the initial speed and g is the acceleration due to gravity.
In another curiosity, the eccentricity of this ellipse is constant for all values of v0 and g, and this value is e = √3 / 2.
Obviously, I wasn’t the first to find this. A quick search revealed a paper on arXiv from 2004 describing this. Still, it’s a nice little-known curiosity of a classical physics problem.
Bonus points: for which angles does the trajectory contain the foci of the ellipse?
Everyone should be following this blog.
These visualizations are amazing.
Better late than never. Happy Valentine’s Day!
Here’s a heart’s sine function.
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.
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.
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:
This is how I will always explain inverse functions from now on.
I drew this for one of my calculus classes yesterday to get them thinking in >3 dimensions.
Nothing beats a snow pentagonal dodecahedron
Okay who let the nerds out in the snow.
Found this while looking through my calculus textbook. Maybe if Angela and Brian could bond over their shared love (or hate) of math problems with completely artificial “context”…