Full Flat Sheets

In mathematics, the Brouwer fixed point theorem is an important fixed point theorem that applies to finite-dimensional spaces and forms the basis for several more general fixed point theorems. It is named after Dutch mathematician L. E. J. Brouwer. more...

Home
Bedding
Bed Pillows
Bed Skirts
Bed-in-a-Bag
Bedspreads, Coverlets
Blankets
Canopies, Netting
Children's Bedding
Comforters
Decorative Bed Pillows
Duvet Covers
Feather Beds
Foam Mattresses
Inflatable Mattress, Air...
Mattress Pads, Covers
Other Bedding
Pillow Shams
Quilts
Sheets
Fitted Sheets
California King Fitted...
Full Fitted Sheets
King Fitted Sheets
Queen Fitted Sheets
Twin Fitted Sheets
Flat Sheets
Full Flat Sheets
King Flat Sheets
Queen Flat Sheets
Twin Flat Sheets
Pillowcases
King Pillowcases
Other Pillowcases
Standard/Queen Pillowcases
Sheet Sets
California King Sheet Sets
Full Sheet Sets
King Sheet Sets
Queen Sheet Sets
Twin Extra Long Sheet Sets
Twin Sheet Sets
Waterbed Accessories
Furniture
Home Decor
Miscellaneous
Patio & Grilling

Statement

The theorem states that every continuous function from the closed unit ball D ⁿ to itself has at least one fixed point. In this theorem, n is any positive integer, and the closed unit ball is the set of all points in Euclidean n-space Rⁿ which are at distance at most 1 from the origin. A fixed point of a function f : D ⁿ → D ⁿ is a point x in D ⁿ with f(x) = x.

Notes

The function f in this theorem is not required to be bijective or even surjective.

Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, the theorem equally applies if the domain is not the closed unit ball itself but some set homeomorphic to it (and therefore also closed, bounded, connected, without holes, etc.).

The statement of the theorem is false if formulated for the open unit disk, the set of points with distance less than 1 from the origin. Consider for example

which maps every point of the open unit disk in R² to another point of the open unit disk slightly to the right of the given one.

Illustrations

The theorem has several "real world" illustrations. One works as follows: take two equal size sheets of graph paper with coordinate systems on them, lay one flat on the table and crumple up (but don't rip) the other one and place it any way you like on top of the first. Then there will be at least one point of the crumpled sheet that lies exactly on top of the corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet right beneath it.

An example of the case n=3 is given by an informational display of a map in, for example, an airport terminal. The function that sends points of the terminal to their image on the map is continuous and therefore has a fixed point, usually indicated by a cross or arrow with the text You are here. A similar display outside the terminal would violate the condition that the function is "to itself" and fails to have a fixed point.

Read more at Wikipedia.org