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Day #20: Riding on a Wild Torus

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torus_from_rectangle

Q1: Say, you are actually riding on the surface of a torus, and say, you´re in a hurry! Could you find the shortest path to follow??…

Q2: And, if the surface of a – flexible – square that generates a Torus is just 1. Could you tell radius of its two generating circles and of its cartesian equation??…

A1: The shortest path will always be shortest geodesic joining them! Also see  Method of Lagrange – Wolfram.com.

A2: If the area of the square is L^2 = 1, then the perimeter of the both circles is equal to L = 2Pi R = 2Pi r = 1, thus R = r = 1/2Pi must be set in, for example, the implicit equation in Cartesian coordinates radially symmetric about the zaxis:

\left(R-{\sqrt {x^{2}+y^{2}}}\right)^{2}+z^{2}=r^{2},
R = r = 1/2 Pi

Geometry and Topology

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution.

Real-world examples of toroidal objects include inner tubes, swim rings, and the surface of a doughnut, a bagel or an apple.

A torus should not be confused with a solid torus, which is formed by rotating a disc, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, vadai or vada, many lifebuoys, and O-rings.

In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S1 × S1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into three-dimensional Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane. This produces a geometric object called the Clifford torus, a surface in 4-space.

In the field of topology, a torus is any topological space that is topologically equivalent to a torus.

 A torus is the product of two circles, only one of which is shown in this diagram. The red circle is swept around an axis, which is not shown. R is the radius of the magenta circle; r is the radius of the red one.
ringR > r: Ring torus
hornR = r: Horn torus
spindleR < r: Spindle torus
Bottom-halves and cross-sections of the three classes

 A diagram depicting the poloidal (θ) direction, represented by the red arrow, and the toroidal (ζ or φ) direction, represented by the blue arrow.

A torus can be defined parametrically by:[1]

  {\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta \end{aligned}}

where

θ, φ are angles which make a full circle, so that their values start and end at the same point,
R is the distance from the center of the tube to the center of the torus,
r is the radius of the tube.

R is known as the “major radius” and r is known as the “minor radius”.[2] The ratio R divided by r is known as the aspect ratio. A doughnut[clarification needed] has an aspect ratio of about 2 to 3.

An implicit equation in Cartesian coordinates for a torus radially symmetric about the zaxis is

\left(R-{\sqrt {x^{2}+y^{2}}}\right)^{2}+z^{2}=r^{2},

or the solution of f(x, y, z) = 0, where

  f(x,y,z)=\left(R-{\sqrt {x^{2}+y^{2}}}\right)^{2}+z^{2}-r^{2}.

Algebraically eliminating the square root gives a quartic equation,

{\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).}

The three different classes of standard tori correspond to the three possible aspect ratios between R and r:

  • When R > r, the surface will be the familiar ring torus.
  • R = r corresponds to the horn torus, which in effect is a torus with no “hole”.
  • R < r describes the self-intersecting spindle torus.
  • When R = 0, the torus degenerates to the sphere.

Topologically, the torus can also be described as a quotient of the Cartesian plane under the identifications

  {\displaystyle (x,y)\sim (x+1,y)\sim (x,y+1),\,}

or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA−1B−1.

 Turning a punctured torus inside-out

The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself:

  \pi _{1}(\mathbf {T} ^{2})=\pi _{1}(S^{1})\times \pi _{1}(S^{1})\cong \mathbf {Z} \times \mathbf {Z} .

Intuitively speaking, this means that a closed path that circles the torus’ “hole” (say, a circle that traces out a particular latitude) and then circles the torus’ “body” (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly ‘latitudinal’ and strictly ‘longitudinal’ paths commute. This might be imagined as two shoelaces passing through each other, then unwinding, then rewinding.

If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two different ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of a rectangle together, choosing the other two sides instead will cause the same reversal of orientation.
The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian).

Source: WikipediA

Of what use is a torus??… Apart from the pure and blissful joy of knowledge…. 

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Author: Math - Update

Updating Math In Our Mind & Heart!!...

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