Posts tagged math
Posts tagged math
A properly, conformally embedded once-punctured torus in R^3, parametrized by
with 1≤u≤2π, 1≤v≤2π, obvs.
(A surface in R^3 is properly embedded if its intersection with any compact set is compact. Thus, the puncture in this surface is placed “at infinity”. A surface is conformally embedded if angles in the intrinsic metric match angles in the ambient metric. Here, you can see that latitude and longitude lines on the torus meet orthogonally.)
The Julia set of 2*cos(z) on the complex plane. I was inspired to make this by this post.
chaotic-renan replied to your post: If you’re still looking for stuff to do, I think a 4D Sierpinski sponge would be pretty neat to render.How the heck does that work?
Well, first you think of one of these, but one dimensions up. Then, you conceive of a way to spin it around. Then, you intersect it with an appropriate hyperplane. Finally, you express this thought to a machine capable of generating an image from it. With your help, it should be able to produce something like this.
I would pay money to see slices of a 4d Menger sponge.
Five great mathematicians and their contributions, in minimalist posters – the best thing since those minimalist posters celebrating pioneering women in science and philosophy’s major movements distilled in minimalist graphics.
Pair with 17 equations that changed the world.
Mathematics and Science Wall Art by CutnPaste.
That’s some good art.
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)).