Recent content by lavinia

@WWGD I thought you'd like see see why evolutes are critical values of the normal map. For a curve in the plane, the normal map is c(s) + tN(s) where N(s) is a unit normal to the curve c(s). If c(s) is parameterized by arc length and N(s) points in the direction of c'' , a classical equation...

I think for hypersurfaces of Euclidean n-space one gets n-1 evolute like hypersurfaces as the critical values of the normal mapping. This just generalizes the case of a curve in the plane. In the hypersurface case there are specific curves called principal curvatures along which the normal...

For the parabola (x,x^2) plugging into the formula for the evolute gives the parametric equation (-4x^3,1/2+3x^2). This is not a parabola because of the cubic term. I thought perhaps an inductive proof would show that the parametric equation for the n'th evolute would be a pair of polynomials...

i tried to indicate a method of proof in reply #4

The evolute of a parabola contains a cubic term so is not a parabola. I wonder if the degree of iterated evolutes of the parabola is increasing.

One intuitive idea of defining a differentiable structure on a triangulated manifold is to embed the manifold in Euclidean space and then smooth out the vertices and edges to eliminate all of the creases and sharp points. For instance a tetrahedron can be made into a smoothly embedded sphere by...

What I am used to and maybe this is wrong, a trivial bundle is one for which there exists a global trivialization. But there is no need for a specific trivializaton to say that the bundle is trivial. In the case of connected compact smooth manifolds trivializations of the tangent bundle are...

A choice of a product decomposition of a topological space that is homeomorphic to a Cartesian product AxF naturally supplies a projection mapping. This mapping together with the projection onto the first factor defines a trivial fiber bundle. The definition of a fiber bundle includes a choice...

I think it might be worth thinking about how to visulalize the other non-trivial orientable circle bundles over the 2 sphere. All of these are the quotients of the 3 sphere by the action of a finite cylic subgroup of SO(2) on S^3 . It seems that this action is directly reflected in the...

It took me a while to understand what you are saying. Yes. You can't rotate one circle in the link so that it is flipped without banging into the forbidden circle. What I tried to do here was to show various ways to detect and define linkage. The definition which seems most intuitive, to me at...

Technically if sin(a) and sin(b) are both not zero. Without invoking a probability space, the set of pairs of angles with one or both zero has Lebesque measure zero. I think.

Can you make that probaility argument more precsise?

I think not all great circles. Some of them intersect with each other. The great circles in the Hopf fibration do not intersect with each other, this because they are orbits of the action of SO(2) on S^3. If two orbits intersected then where would the intersection point go under the action of...

Here is an image of a twisted cylinder that shows the linked boundary circles. This was taken from Wikipedia.

Clifford parallels and flat tori in the three sphere Start with a square in the xy-plane whose corners are (0,0),(2π,0),(0,2π) and (2π,2π). Consider the mapping of this square into R^4 given by the rule T:(x,y) -> (cos(x), sin(x),cos(y),sin(y)) The image of this mapping is a torus in R^4...