Illustrated number

In Arithmetic, a number illustrated is a integer which can be represented by a whole of points laid out in a more or less regular way and forming a geometrical figure.

The illustrated numbers are of very old origin. One generally allots to Pythagore the first studies of illustrated numbers (square numbers). Diophante solved several problems with regard to them. Pascal wrote a treaty on the subject.

Example of illustrated numbers

The number illustrated simplest are

the square numbers

the triangular numbers

who draw figure plane

the cubic numbers :

1 8 27 64 who draw a cube in space

the numbers being reproduced on the faces of a cubic die:

Tracks of exploration

The first contact which any thinking being maintains with the number passes by the illustrated number. It is a universal language nonrelated on the writing and the numbering systems. The illustrated number makes it possible to prove that certain animals (the Poulpe for example) are a aware of the number. In pedagogy, the passage by the illustrated number makes it possible to visualize properties like the Commutativité or the Associativité of the laws of Addition and Multiplication - laws which are dictated to them by building lines or tables of points. Relation 2 + 3 = 3 + 2 = 5, for example, which translates the fact that 2 and 3 are permutable for the addition, can be represented by

+ = + =

And the relation 2×3 = 3×2 = 6 (which translates the fact that 2 and 3 are permutable for the multiplication) can be represented by

The study of the illustrated numbers in general consists in even finding a relation between the number him and its row in the series. For example, the triangular Number of row N est $\ frac {N (n+1)}{2}$ . The cubic number of row N is $n^3 \,$ . The concepts of the illustrated numbers utilize implicitly the modern concept of Récurrence.

A second way of research is to determine the properties of the numbers appearing in the same series. For example, it is easy to show that there is no Prime number among the triangular numbers (except 3), squares or rectangles.

Another way of research is to use the numbers illustrated in solutions of equation in $\ mathbb N$ like the extraction of root square and cubic Racine.

Classification

One arranges the numbers illustrated according to dimension of the figure represented. This one is very frequently a Polytope and the number is then called a number polytopic.

In dimension one

the linear numbers

they are the traditional entireties

In dimension two

the polygonal numbers (triangular, square, pentagonal, hexagonal, heptagonal, octagonal or gnomonic)
the centered polygonal numbers (square, pentagonal, hexagonal, heptagonal, octagonal)

In dimension three

tetrahedral numbers
cubic numbers or cubic centered

In higher dimension

The illustrated numbers are not representable any more by figures corresponding to the tangible world but are considered commes sights of the spirit

Nombre hypersolide (in dimension four)
number DNN (in dimension N)

A transversal

There exists a list of illustrated number evolving/moving at the same time on the size of the figure and the dimension which it determines: they are the numbers R - linear topics

numbers in dimension a
triangular numbers in dimension two
tetrahedral numbers in dimension three
numbers pentatoptic in dimension 4

The number $r \,$ - topic of row N is given by the formula

$\ forall N \ in \ mathbb {NR} ^* \ quad P_r (N) = C_ {n+r-1} ^r = {n^ {(R)} \ over {R!}}$

$r! \,$ is the Factorielle of $r$ , $C_n^r = {N \ choose p}$ is a binomial Coefficient, and $n^ {(R)}\,$ is the increasing factorial .

The numbers polytopic for R = 2,3, and 4 are

$P_2 (N) = \ frac {N (n+1)}{2}$ (triangular numbers)
$P_3 (N) = \ frac {N (n+1) (n+2)}{6}$ (tetrahedral numbers)
$P_4 (N) = \ frac {N (n+1) (n+2) (n+3)}{24}$ (numbers pentatopic)

See too

Theorem of Pick

References

Gnomon, From Pharaohs to Fractals. Midhat J. Gazalé, Princeton University Near, Princeton, 1999.
a classification of the illustrated numbers

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