Pizza theorem
Encyclopedia
In elementary geometry
, the pizza theorem states the equality of two areas that arise when one partitions a disk
in a certain way.
Let p be an interior point of the disk, and let n be a number that is evenly divisible by four and greater than or equal to eight. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line n/2 − 1 times by an angle of π/n radian
s, and slicing the disk on each of the resulting n/2 lines. Number the sectors consecutively in a clockwise or anti-clockwise fashion. Then the pizza theorem states that:
The pizza theorem is so called because it mimics a traditional pizza
slicing technique. It shows that, if two people share a pizza sliced in this way by taking alternating slices, then they each get an equal amount of pizza.
provide an alternative proof by dissection
: they show how to partition the sectors into smaller pieces so that each piece in an odd-numbered sector has a congruent
piece in an even-numbered sector, and vice versa.
showed, dividing a disk into four sectors, or a number of sectors that is not divisible by four, does not in general produce equal areas. answered a problem of by providing a more precise version of the theorem that determines which of the two sets of sectors has greater area in the cases that the areas are unequal. Specifically, if the number of sectors is 2 (mod 8) and no slice passes through the center of the disk, then the subset of slices containing the center has smaller area than the other subset, while if the number of sectors is 6 (mod 8) and no slice passes through the center, then the subset of slices containing the center has larger area. An odd number of sectors is not possible with straight-line cuts, and a slice through the center causes the two subsets to be equal regardless of the number of sectors.
also observe that, when the pizza is divided evenly, then so is its crust: the crust may be represented as the area between two concentric circles, and since the disks bounded by both circles are partitioned evenly so is their difference. Mathematically, the perimeter of a disk as well as its area is split into equal subsets by the lines. However, when the pizza is divided unevenly, the diner who gets the most pizza area also gets the most crust. As note, an equal division of the pizza also leads to an equal division of its toppings, as long as each topping is distributed in a disk (not necessarily concentric with the whole pizza) that contains the central point p of the division into sectors.
and study the game theory
of choosing free slices of pizza in order to guarantee a large share, a problem posed by Dan Brown and Peter Winkler
. In the version of the problem they study, a pizza is sliced radially (without the guarantee of equal-angled sectors) and two diners alternately choose pieces of pizza that are adjacent to an already-eaten sector. If the two diners both try to maximize the amount of pizza they eat, the diner who takes the first slice can guarantee a 4/9 share of the total pizza, and there exists a slicing of the pizza such that he cannot take more. The fair division
or cake cutting problem considers similar games in which different players have different criteria for how they measure the size of their share; for instance, one diner may prefer to get the most pepperoni while another diner may prefer to get the most cheese.
, a sequence of integers that counts the number of pieces one gets when slicing pizza by lines that do not all go through a single point, and the ham sandwich theorem
, a result about slicing three-dimensional objects whose two-dimensional version implies that any pizza, no matter how misshapen, can have its area and its crust length simultaneously bisected by a single carefully chosen straight-line cut.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the pizza theorem states the equality of two areas that arise when one partitions a disk
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...
in a certain way.
Let p be an interior point of the disk, and let n be a number that is evenly divisible by four and greater than or equal to eight. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line n/2 − 1 times by an angle of π/n radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s, and slicing the disk on each of the resulting n/2 lines. Number the sectors consecutively in a clockwise or anti-clockwise fashion. Then the pizza theorem states that:
- The sum of the areas of the odd numbered sectors equals the sum of the areas of the even numbered sectors .
The pizza theorem is so called because it mimics a traditional pizza
Pizza
Pizza is an oven-baked, flat, disc-shaped bread typically topped with a tomato sauce, cheese and various toppings.Originating in Italy, from the Neapolitan cuisine, the dish has become popular in many parts of the world. An establishment that makes and sells pizzas is called a "pizzeria"...
slicing technique. It shows that, if two people share a pizza sliced in this way by taking alternating slices, then they each get an equal amount of pizza.
History
The pizza theorem was originally proposed as a challenge problem by ; the published solution to this problem, by Michael Goldberg, involved direct manipulation of the algebraic expressions for the areas of the sectors.provide an alternative proof by dissection
Dissection puzzle
A dissection puzzle, also called a transformation puzzle or Richter Puzzle, is a tiling puzzle where a solver is given a set of pieces that can be assembled in different ways to produce two or more distinct geometric shapes. The creation of new dissection puzzles is also considered to be a type of...
: they show how to partition the sectors into smaller pieces so that each piece in an odd-numbered sector has a congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
piece in an even-numbered sector, and vice versa.
Generalizations
The requirement that the number of sectors be a multiple of four is necessary: as Don CoppersmithDon Coppersmith
Don Coppersmith is a cryptographer and mathematician. He was involved in the design of the Data Encryption Standard block cipher at IBM, particularly the design of the S-boxes, strengthening them against differential cryptanalysis...
showed, dividing a disk into four sectors, or a number of sectors that is not divisible by four, does not in general produce equal areas. answered a problem of by providing a more precise version of the theorem that determines which of the two sets of sectors has greater area in the cases that the areas are unequal. Specifically, if the number of sectors is 2 (mod 8) and no slice passes through the center of the disk, then the subset of slices containing the center has smaller area than the other subset, while if the number of sectors is 6 (mod 8) and no slice passes through the center, then the subset of slices containing the center has larger area. An odd number of sectors is not possible with straight-line cuts, and a slice through the center causes the two subsets to be equal regardless of the number of sectors.
also observe that, when the pizza is divided evenly, then so is its crust: the crust may be represented as the area between two concentric circles, and since the disks bounded by both circles are partitioned evenly so is their difference. Mathematically, the perimeter of a disk as well as its area is split into equal subsets by the lines. However, when the pizza is divided unevenly, the diner who gets the most pizza area also gets the most crust. As note, an equal division of the pizza also leads to an equal division of its toppings, as long as each topping is distributed in a disk (not necessarily concentric with the whole pizza) that contains the central point p of the division into sectors.
Related results
show that a pizza sliced in the same way as the pizza theorem, into a number n of sectors with equal angles where n is divisible by four, can also be shared equally among n/4 people. For instance, a pizza divided into 12 sectors can be shared equally by three people as well as by two; however, to accommodate all five of the Hirschhorns, a pizza would need to be divided into 20 sectors.and study the game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
of choosing free slices of pizza in order to guarantee a large share, a problem posed by Dan Brown and Peter Winkler
Peter Winkler
Peter Mann Winkler is a noted research mathematician, author of more than 125 research papers in mathematics and patent holder in a broad range of applications, ranging from cryptography to marine navigation...
. In the version of the problem they study, a pizza is sliced radially (without the guarantee of equal-angled sectors) and two diners alternately choose pieces of pizza that are adjacent to an already-eaten sector. If the two diners both try to maximize the amount of pizza they eat, the diner who takes the first slice can guarantee a 4/9 share of the total pizza, and there exists a slicing of the pizza such that he cannot take more. The fair division
Fair division
Fair division, also known as the cake-cutting problem, is the problem of dividing a resource in such a way that all recipients believe that they have received a fair amount...
or cake cutting problem considers similar games in which different players have different criteria for how they measure the size of their share; for instance, one diner may prefer to get the most pepperoni while another diner may prefer to get the most cheese.
See also
Other mathematical results related to pizza slicing involve the lazy caterer's sequenceLazy caterer's sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a circle that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point, but...
, a sequence of integers that counts the number of pieces one gets when slicing pizza by lines that do not all go through a single point, and the ham sandwich theorem
Ham sandwich theorem
In measure theory, a branch of mathematics, the ham sandwich theorem, also called the Stone–Tukey theorem after Arthur H. Stone and John Tukey, states that given measurable "objects" in -dimensional space, it is possible to divide all of them in half with a single -dimensional hyperplane...
, a result about slicing three-dimensional objects whose two-dimensional version implies that any pizza, no matter how misshapen, can have its area and its crust length simultaneously bisected by a single carefully chosen straight-line cut.