In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the cake number, denoted by ''C
n'', is the maximum of the number of regions into which a 3-dimensional
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
can be partitioned by exactly ''n''
planes. The cake number is so called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped
cake
Cake is a flour confection usually made from flour, sugar, and other ingredients and is usually baked. In their oldest forms, cakes were modifications of bread, but cakes now cover a wide range of preparations that can be simple or elabor ...
. It is the 3D analogue of the
lazy caterer's sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a Disk (mathematics), disk (a pancake or pizza is usually used to describe the situation) that can be made with a given nu ...
.
The values of ''C
n'' for are given by .
General formula
If ''n''! denotes the
factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
, and we denote the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s by
:
and we assume that ''n'' planes are available to partition the cube, then the ''n''-th cake number is:
:
Properties
The cake numbers are the 3-dimensional analogue of the 2-dimensional
lazy caterer's sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a Disk (mathematics), disk (a pancake or pizza is usually used to describe the situation) that can be made with a given nu ...
. The difference between successive cake numbers also gives the lazy caterer's sequence.
The fourth column of
Bernoulli's triangle (''k'' = 3) gives the cake numbers for ''n'' cuts, where ''n'' ≥ 3.
The sequence can be alternatively derived from the sum of up to the first 4 terms of each row of
Pascal's triangle
In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
:
:
Other applications
In ''n'' spatial (not spacetime) dimensions,
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
represent
different independent real-valued equations.
See also
*
Dividing a circle into areas (Moser's circle problem)
*
Lazy caterer's sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a Disk (mathematics), disk (a pancake or pizza is usually used to describe the situation) that can be made with a given nu ...
*
Pizza theorem
In elementary geometry, the pizza theorem states the equality of two areas that arise when one partitions a Disk (mathematics), disk in a certain way.
The theorem is so called because it mimics a traditional pizza slicing technique. It shows tha ...
References
External links
*
*
Mathematical optimization
Integer sequences
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