# Dimension

In the common direction, the concept of dimension returns to the size; dimensions of a part are its length, its width and its depth/its thickness, or its diameter if it is a part of revolution.

In Physical and mathematical, the concept of dimension means initially the number of independent directions, then was wide.

## Technique

In the field of mechanics, the term “dimension” renvoit with the size of a part.

In the absolute, dimensions of a part can be selected in a completely arbitrary way, the important one being that they are compatible with the final use of the part.

With an aim of standardization, it is however preferable to use like nominal linear Dimensions values of the “series of Fox”.

In practice of the every day, the term “dimension” renvoit with the size of a objet.
Example and definition:

• Object of: 350 X 250 X 255 mm.
• Description: (L) ongor X (L) argor X (H) auteur.
• Form: D = (L X L X H)

## Physics

In physics, the dimension term gathers two completely different concepts.

### Dimension of a vector space

Physics uses much the mathematical concept of vector Space. One can popularize this concept by saying that

the dimension of a space is the number of variables which are used to define a state, an event.
Thus for example, it is said classically that our universe is with four dimensions, since an event is defined by the position in space ( X , there , Z ) and the moment T to which this event occurs.
• a constant voluminal object (i.e. the properties are independent of time, at least during the study) is known as with three dimensions, because one needs three numbers ( X , there , Z ) to indicate one of its points within the object;

• object plane (as a sheet of paper) which one neglects the thickness is known as with two dimensions, because one needs two numbers ( X , there ) to indicate one of its points within the object;
• object linear (as a wire) which one neglects the thickness is known as with a dimension, because it is enough to only one number X to indicate one of its points within the object (curvilinear X-coordinate);
• object specific (as a point) which one neglects the size is known as of dimension zero, because once one indicated the point, one needs no parameter to find the point…
These concepts are taken again in modeling Informatique (object 2D, 3D).

This concept is the translation of the mathematical concept of dimension (see low).

### Dimension of a size

The dimension of a physical size is its unit expressed compared to the seven basic units of the international system. The units in sizes are retranslated.

For example, speed with dimension a length divided by a time (i.e. the unit speed is the meter a second).

## Mathematics

### Dimension of a vector space

In Mathematical, the concept of dimension corresponds to the dimension of the vector Space of physics:

if a vector space is provided with a bases cardinal finished D , then all the bases of this space have as a cardinal D , and the dimension of this space is D .
(Let us recall that a base is a free and generating family vector space, i.e. any vector can break up into a single linear combination of the vectors of this family.) This is a consequence of the Lemme of Steinitz:
if a vector space has a finished generating family of N vectors, any family of more than N vectors is dependant.

When a vector space does not admit any base of finished cardinal, it is said that it is of “infinite dimension”. Example: the whole of the real continuations is a vector space of infinite size. In such a space there exist finished free families arbitrarily large, but no finished generating family.

### Dimension fractale

But the definition of the dimension given above is insufficient, in particular in the case of the Fractale S.

In a simplified way and at first approximation (cf the article specialized for a better definition), a fractal object is an object having an internal homothety, i.e. a portion of the object is identical to the complete object. Let us consider a simple example, the curve of von Koch: this curve is built in a recursive way, one starts from a segment of right-hand side, and one replaces each segment by a segment with a rafter in the medium.

One repeats this operation ad infinitum. This curve is a line (thus of dimension 1, with the ordinary direction). Its length is infinite, since with each stage one multiplies his length by 4/3, and that there is an infinite number of stages. However, and contrary on an infinite line, one can always find a curve finite length as near as one wants curve of von Koch. One can thus say in fact that if that the length of the curve of von Koch is infinite, it is found is that one evaluates it in “a bad” dimension, and that while measuring “better”, one would have a “useful” measurement, finished.

We need to reconsider the concept of standard in physics:

• the lenght rod is a fixed rule length (dimension 1): to measure a length, one looks at how much rules hold end to end on the curve;
• the standard of surface is a square (square) on fixed side (dimension 2): to measure surface, one looks at how much squares or can pose side-by-side on surface;
• the standard of volume is a paving stone (cubic) of stops fixes (dimension 3): to measure volume, one looks at how much paving stones one can pile up in the object.

One can evaluate the length only of one object of dimension 1: even by taking a tiny rule, a point will be able to never contain it, and contrary on a surface, one can put an infinite number of rules side-by-side (those have a null thickness).

In the same way, one can evaluate the surface only of one object of dimension 2: a point or a curve could never be paved by squares (even very small), and in a volume, one can pile up an infinite number of squares (those have a null thickness).

One can evaluate the volume only of one object with three dimension, since one cannot put of paving stone in a point, a curve nor a surface.

Thus, if one calls do the dimension of the object and de that of the standard, one a:

• if de > do , measurement gives 0: one cannot put only one standard in the object; it is the case for the curve of von Koch when one uses a measurement with a surface, which thus states that its dimension fractale is strictly lower than 2.
• if de < do , measurement gives ∞: one can put standard as much than one wants in the object; it is the case for the curve of von Koch when one uses a measurement with a length, which thus states that its dimension fractale is strictly higher than 1.
• if de = do , measurement can give (if the measured object is not infinite) a finished number, the number of standard which it is necessary to cover the object; our problem is thus to find (if it exists) “good” dimension, that which will give us a finished measurement of course (if the object is finished,).

To make this measurement, the “size” of the standard is not without effect. If the standard is too large, it does not return in the object (measurement is null), but by taking increasingly small standards, one obtains (usually) measurements which approach. If, to measure a line, one uses a rule length ℓ, more ℓ is small, more one will be able to put standards in the object to be measured. The measurement is the product of the number standards by the size of the standard : If one makes hold NR rules length ℓ, measurement will be

M (ℓ) = NR ×ℓ
For a square on side ℓ, the surface of the square will be ℓ ², and if one covers the surface of NR squares, measurement will be
M (ℓ) = NR ×ℓ ²
For a paving stone of ℓ stops, the volume of the paving stone will be ℓ, and if one filled the object with NR paved, measurement will be
M (ℓ) = NR ×ℓ
It is seen that dimension is also the exhibitor intervening in the calculation of measurement.

In the case of a usual line, when one uses a rule length ℓ divided by two (or three, four,… NR), one can put about two (respectively three, four,… NR) time more once the standard in the object: measurement almost does not change, and finally, as the size of the standard is reduced, one obtains a continuation measurements which converges: the exact length of the curve is the limit of M (ℓ) when ℓ tends towards 0, it is a real number.

Let us take the example of a surface; when it is paved squares, only one approximation of his surface is obtained (one approaches surface by a polygon). If one makes hold NR rules length ℓ, measurement will be

M (ℓ) = NR ×ℓ 2
The surface exact length of the curve is the limit of M (ℓ) when ℓ tends towards 0. On the other hand
M (ℓ) = NR ×ℓ
tends towards +∞, and
M (ℓ) = NR ×ℓ 3
tends towards 0. One finds by calculation what one established by geometry.

In the case of the curve of von Koch, one sees well that when one divides the lenght rod by 3 , one can put 4 times more standard. Blow, the continuation of measures of length

M (ℓ/3) = NR ℓ/3 × (ℓ/3) 1 = 4 NR ×ℓ 1/3 = 4/3 × M (ℓ) > M (ℓ)
do not converge and the series of measures of surface
M (ℓ/3) = NR ℓ/3 × (ℓ/3) 2 = 4 NR ℓ/3 ×ℓ 2/9 = 4/9 × M (ℓ) < M (ℓ)
tends towards 0.

But it is possible to imagine a fractional dimension, to vary continuously “dimension”. And in fact, for dimension

do = log 4/log 3 ≈ 1,261  9.
one can make converge measurement for the curve of von Koch while taking:
M (ℓ) = NR ×ℓ C

This can be represented in a more rigorous way by dimension of Hausdorff-Besicovitch.

### Topological dimension

The topological dimension, defined by recurrence, associates with each part P of R N an entirety, equal to algebraic dimension if P is a subspace closely connected, with N if P is of nonempty interior, to 1 if P is a regular curve, to 2 if P is a regular surface, etc In a general way it allots to a usual unit its intuitive dimension which is the number of independent variables necessary to describe it.

### Dimension of Hausdorff - Besicovitch

The dimension of Hausdorff-Besicovitch Dh takes its definition by the quotient logarithmic curve between a number of Homothétie S interns of an object, on the reverse of the reason of this homothety. Therefore,

$D_h = \ frac \left\{\ ln \left(NR\right)\right\}\left\{\ ln \left(\ frac \left\{1\right\} \left\{R\right\}\right)\right\}$.
One will thus have, for a point:
$D_h = \ frac \left\{\ ln \left(1\right)\right\}\left\{\ ln \left(N\right)\right\}= 0$ with natural N > 1,
considering one can draw up a point by a Homothétie intern of reason N . One can say “a point is the product of internal N homotheties of this same point, of reason N ”.

With a line (segment), it can be established with two internal homotheties of reason 1/2, therefore

$D_h = \ frac \left\{\ ln \left(2\right)\right\}\left\{\ ln \ left \left(\ frac \left\{1\right\} \left\{\ frac \left\{1\right\} \left\{2\right\}\right\} \ right\right)\right\}= \ frac \left\{\ ln \left(2\right)\right\}\left\{\ ln \left(2\right)\right\} = 1$.
In this way, one finds for the forms and Euclidean objects, a Isomorphisme between these two established dimensions. However, of the great differences present themselves with the Fractale S.

### Dimension of Minkowski - Bouligand

The dimension of Minkowski-Bouligand Dm is the quotient logarithmic curve between the volume of balls which one needs to cover any Euclidean object, or not Euclidean (of possible ray smallest), which can be contained in a ball of ray R , with the quotient of the rays. One obtains then,

$D_m = \ frac \left\{\ ln \left(NR \left(R, \ rho\right)\right)\right\}\left\{\ ln \ left \left(\ frac \left\{R\right\} \left\{\ rho\right\} \ right\right)\right\}$,
where R is the ray of the external ball, which is covered by smaller balls with ray $\ rho$. NR ( R , ρ) indicates the surface or volume of the ball or disc which recover this figure.

### combinative Dimension of a topological space

This other concept of dimension is especially useful in the case of the Schémas, where topology, in a certain direction, is of a nature more Combinatoire that geometrical.