Pizza theorem: Difference between revisions

Content deleted Content added

Inline

Revision as of 23:09, 19 December 2009

In elementary geometry, the pizza theorem states the equality of two areas that arise when one partitions a disc in a certain way.

Let p be an inner point of the disc. Draw an arbitrary line through p and rotate it $4+2n-1$ times by an angle of ${\frac {360^{\circ }}{2\cdot (4+2n)}}$ degrees $(n\in \mathbb {N} _{0})$ to form $4+2n-1$ lines dividing the circle into $2\cdot (4+2n)$ regions ("sectors"). Number those regions consecutively in a clockwise or anti-clockwise fashion. Then the pizza theorem states that:

The sum of the areas of the odd numbered regions equals then sum of the areas of the even numbered regions.

The name of theorem is derived from the fact that the partition of the disc is mimics a traditional pizza slicing technique.

References

Carter, Larry; Wagon, Stan (1994), "Proof without Words: Fair Allocation of a Pizza", Mathematics Magazine, 67 (4): 267.
Mabry, Rick; Deiermann, Paul (2009), "Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results" (PDF), American Mathematical Monthly, 116 (5): 423–438.

External links

Weisstein, Eric W. "Pizza Theorem". MathWorld.
Sillke, Torsten, Pizza Theorem, retrieved 2009-11-24

@@ Line 3: / Line 3: @@
 In elementary [[geometry]], the '''pizza theorem''' states the equality of two areas that arise when one partitions a disc in a certain way.
-Let ''p'' be an inner point of the disc. Now we draw an arbitrary line through ''p'' and rotate it <math>4+2n-1</math> times by an angle of <math> \frac{360^\circ}{2\cdot(4+2n)} </math> degrees (<math> n \in \mathbb{N}_0 </math>), so that we've got  <math>4+2n-1</math> lines dividing the circle into<math> 2\cdot(4+2n) </math> regions ("sectors"). If we number those regions in a clockwise or anti-clockwise fashion, then the following statement holds:
+Let ''p'' be an inner point of the disc. Draw an arbitrary line through ''p'' and rotate it <math>4+2n-1</math> times by an angle of <math>\frac{360^\circ}{2\cdot(4+2n)}</math> degrees <math>(n \in \mathbb{N}_0)</math> to form <math>4+2n-1</math> lines dividing the circle into <math>2\cdot(4+2n)</math> regions ("sectors"). Number those regions consecutively in a clockwise or anti-clockwise fashion. Then the pizza theorem states that:
 :''The sum of the areas of the odd numbered regions equals then sum of the areas of the even numbered regions''.
-The name of theorem is derived from the fact that the partition of the disc is mimicks a traditional pizza slicing technique.  There you pick a point around the center of the pizza and start cutting slices from there with an equal angle.
+The name of theorem is derived from the fact that the partition of the disc is mimics a traditional pizza slicing technique.
 ==References==
+*{{citation|first1=Larry|last1=Carter|first2=Stan|last2=Wagon|title=Proof without Words: Fair Allocation of a Pizza|journal=Mathematics Magazine|volume=67|issue=4|year=1994|page=267|url=http://www.jstor.org/stable/2690845}}.
-*{{MathWorld|urlname=PizzaTheorem|title=pizza theorem}}
+*{{citation|first1=Rick|last1=Mabry|first2=Paul|last2=Deiermann|url=http://www.lsus.org/sc/math/rmabry/pizza/Pizza_Conjecture.pdf|title=Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results|journal=[[American Mathematical Monthly]]|volume=116|issue=5|pages=423–438|year=2009}}.
-*Larry Carter, Stan Wagon: Proof without Words: Fair Allocation of a Pizza. Mathematics Magazine, Vol. 67, No. 4 (Oct., 1994), p.&nbsp;267
-*[http://www.math.uni-bielefeld.de/~sillke/PUZZLES/pizza-theorem University website with collection of proofs of the pizza theorem] (accessed 2009-11-24)
+==External links==
-* Rick Mabry, Paul Deiermann: ''[http://www.lsus.org/sc/math/rmabry/pizza/Pizza_Conjecture.pdf Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results]'' (PDF)
+*{{MathWorld|urlname=PizzaTheorem|title=Pizza Theorem}}
+*{{citation|url=http://www.math.uni-bielefeld.de/~sillke/PUZZLES/pizza-theorem|title=Pizza Theorem|last=Sillke|first=Torsten|accessdate=2009-11-24}}
 [[Category:Geometry]]