It is arguable that mathematicians possess the most intricate minds on Earth.
Concepts that cause the “average Joe-on-the-street” to stagger at their importance are topics for the water cooler among mathematicians.
The Poincare’ Conjecture is just such a topic. The Poincare’ Conjecture is effectively mathematics in hyper-drive.
Let us hasten to the point, lest this appears to be the stalling technique of a ninth-grade term paper rather than the “Brace Yourself” Preamble in which it is being offered.
The Poincare’ Conjecture is stated as follows. “Every simply connected, closed 3-fold manifold is homeomorphic to the 3-sphere.”
“Originally conjectured by Henri Poincare’, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere.” http://en.wikipedia.org/wiki/Poincaré_conjecture
Now, to the typical reader, trying to decipher this foreign language, coined as “Math-Latin,” the previous definition could pass for a dead-on impression of Charlie Brown’s teacher.
If the best way to eat an elephant is “one bite at a time,” then what say we take a run at it?
The simplest unit of geometry is allegedly the “point.” A point is a hypothetical location without length, without width, and without depth/height.
(For the sake of simplification, we will be using the theoretical definitions, even though we all know that it is logically impossible for the smallest dot, point, or specific location to have some modicum of infinitesimal length, width or height, without also equalling to a concept similar to zero in geometric space.)
Slightly more complex is the “line,” a geometric feature created entirely of length.
Two-dimensional manifolds are comprised of length and width. Examples of these are the square, the rectangle, the circle, the hexagon, the octagon, the trapezoid, the rhombus, the parallelogram, the triangle to list but a few.
Three-dimensional manifolds are comprised of length, width, and height. Examples of these are the sphere of varying lengths, (such as the equi-radial sphere (i.e. ball-shaped,) the elliptical sphere (i.e. hotdog-shaped,) the half-sphere (i.e. the derby hat-shape,) and the spiral sphere (i.e. shaped like the corkscrew of desirable roller-coasters,) the cone, the cylinder, and the cube to name a few.
Okay, so what is all the hub-bub about the Poincare’ Conjecture?
(In order to avoid being misunderstood as to the tenor of the following comments, let the reader consider that the subject designation, includes the word, “explaining,” which seems to give the impression that the concept, under consideration is complex, needing to be simplified for the uninitiated. Hence, the tone being used to explain. Respect for subject is non-negotiable. Dialect must be.)
First, to take the fizz out of the spazz, the “Poincare'” part is named after the man, Henri Poincare’, who is the first man to be credited with having this Million-Dollar Thought, dated 1904.
Secondly, a “conjecture” is the high-brow way to specify that he “took a guess.”
Third, Messieur Poincare’ was apparently not satisfied to stop with the names of geometric shapes (or manifolds) in their simplest forms. He wanted to know the name of a shape (or manifold) that was more complex than the sphere.
For a moment, let’s take a step backward to consider a two-dimensional shape/concept with four-sides. Squares, rectangles, parallelograms, rhombuses, and trapezoids all have four-sides, even though they don’t all fit into a sweet little “cookie-cutter” package of one-size-fits-all four-sided shapes.
Quadrilateral is the term that covers four-sided shapes (or manifolds,) comprised of length and width.
Apparently, Messieur Poincare’ was contemplating the concept of quadrilaterals, when he went in search of the union of three-dimensional shapes that may be termed the quadrispherical.
Simply put, if the commonly-understood ball-shaped sphere is elongated into the shape of a hotdog, is it still a sphere? Well,…Yes.
If the hotdog-shape is squeezed in the middle to make the footlong hotdog look like two regular-sized hotdogs (that have yet to be split into meat for two sandwiches,) is it still a sphere? Well, according to this theory, then the answer still seems to be…Yes.
If the two regular hotdogs are further squeezed in the middle into subsequently smaller and smaller dogs, like the little finger foods that often show-up at wedding receptions, is the whole link of “dogs” to be considered a sphere, even yet? Maintaining the pattern,…Yes.
To summarize, Messieur Henri Poincare’ theorized back in 1904 that any connected three-dimensional shape may be called a “sphere.”
This explanation is admittedly simplistic, and not much more than a primer on the concept of the Poincare’ Conjecture, since myriad aspects of the theory have yet to be expressed, including the Fourth Dimension of Time, which may be expressed through an extension of this theory as the looping of linear time into multidimensional, and highly theoretical concepts,…at least at this point.
Remember, knowledge in the realms of every discipline of science, including mathematics, is still learning and gaining understanding every day.
The Solution for the Poincare’ Conjecture was formulated in 2010. http://en.wikipedia.org/wiki/Solution_of_the_Poincaré_conjecture
Hmmmmm, seems rather recent to me. You?
