Recent content by lavinia

@mathwonk Thank you for this wonderful reply. It inspires me to enter into this incrediblle world.

@cianfa72 This thought came up while thinking about your thread. A topological 7 spere is two 7 disks that are glued together along their 6 sphere boundaries. Any such gluing creates a topological 7 sphere. But the gluing map still matters since the topological 7 sphere can have more than one...

In the category of topological spaces and homeomorphisms, one can define those manifolds which are made from gluing two disks along their boundaries. It turns out that they are all homeomorphic to the standard sphere. Once this is proved, the topological sphere can be defined as two disks pasted...

@cianfa72 Your thoughts brought to mind a few considerations that I thought would be of interest to you. You start out by saying that the definition of the 2-sphere is the quotient space of two disks with their boundary circles glued together. To me this definition is incomplete because it...

I thought it might be relevant to describe the topology of the real line without any use of the Euclidean metric or any other metric for that matter and without any algebraic structure. A little web research revealed several ways to characterize the topology of the real line or equivalently any...

@cianfa72 This may be obvious but I thought I'd mention it anyway. If one identifies SU(2) with S^3 then the covering map SU(2)→SO(3) maps the upper hemisphere of S^3 onto SO(3) with antipodal points remaining only on the equatorial 2 sphere. The upper hemisphere is a closed 3 ball in...

Without checking the parameterization, assuming it induces a bijection from the closed ball modulo antipodal points on its boundary onto the SO(3) matrices, if it is is continuous(proof?) then you get a continuous bijection from a compact space (proof?) onto SO(3) as a subset of R^9. AS @WWGD...

Yes but it is also used in the theory of manifolds and in many other mathematical contexts. Generally, when one has a topological space, measurement is an added structure.

Not sure what you mean by intrinsic. There are many possible metrics on the continuum. Are they all intrinsic?

The real line without addition and multiplication has structure as a continuum. The continuum generalizes to Euclidean space and then further to manifolds. My sense is that extending the rational points on the line using equivalence classes of Cauchy sequences was motivated by the quest for...

@Klystron There is also this Medieval view of geometry and ruler and compass constructions. https://old.maa.org/press/periodicals/convergence/mathematical-treasure-god-the-supreme-geometer

From the Wikipedia passage on the logarithm quoted in post #3 it seems that the ancient Greek problem of Appolonius was to square the hyperbola. That is: to find a region between 1 and some point on the hyperbola whose area was the same as a unit square. That point is the number, e. So it seems...

When one defines e(x) as the inverse function of the natural logarithm, the Chain Rule gives e'(x)/e(x) = 1 and e is then defined as e(1).

epi and and pi + e cannot both be algebraic since e and pi are both roots of the polynomial x^2-(pi + e)x+epi I think the search for pi goes back to ancient times. One problem was to use a ruler and compass to construct a square whose area is the area of a given circle. This was called squaring...

IMO and based on personal experience, I think that getting good at math means getting good at thinking mathematically. It is a mental discipline. Once that is achieved one can learn math on one's own and how good you get at a particular subject directly reflects how much work and passion you put...