Physical space
Previous chapter ( Self-similarity ) | Table of contents | Next chapter ( Physical vector ) |
Corresponding Wikipedia article: Space
«The world is chaotic, nothing in it is immutable, and that includes dimensionality. Space dimensionality in the Universe fluctuates and changes smoothly within very wide ranges. The best condition for the emergence of organic life is space dimensionality equal to π (3,14159...). Wide deviations from that value are detrimental to living nature. Space dimensionality in the vicinity of the Solar system is +3,00017... at present, and the proximity of this number to the integer 3 has misled you»
(Unknown author)
Dimensionality
The space is the infinite set of points, for which determined a distance between them (the metric). The infinity of this set does not mean that it’s unbounded. The objects are moving in a space from the points of theirs locations to the nearest neighboring points.
The line (straight line) is the only shortest path between two points, which passes through all the intermediate nearest points belonging to that line.
The angle (plane angle) is an expression of the relative position of two lines, which are intersected at the angle vertex. The angle value has different definitions:
- Traditional radians, degrees, turns etc., are the lengths of a circle arc.
- Cosine and sine, is a pair of the dependent values, which can be merged in one complex number with a module 1 by the Euler's formula \(e^{i\alpha}=\cos\alpha+i\sin\alpha\).
Radians | Complex number | |
---|---|---|
Length of a circle arc | \[\alpha\] | \(\arctan\frac{\mathrm{Im}\;\alpha}{\mathrm{Re}\;\alpha}\)(argument of a complex number) |
Zero angle | \[0\] | \[1\] |
Right angle | \[\pi/2\] | \[i\] |
Straight angle | \[\pi\] | \[–1\] |
Sum of angles | \[\alpha+\beta\] | \[\alpha\cdot\beta\] |
Difference between angles | \[\alpha-\beta\] | \[\alpha\cdot\overline{\beta}\] |
The traditional angular measure is derived from the ancient Babylonian 60-base arithmetic and from the ancient calendar, where were 360 major days. The round astronomical (astrological) map of the celestial sphere, in which the sun makes a complete turn in a year, was divided into 360 degrees, which are then divided into 60 minutes and seconds. Then the scientific radian measure was invented, however, it does not get rid of the complicated trigonometry calculations. And recently, in the complex and matrix methods, the angular measures were directly substituted by the sine and cosine values, which simplify the trigonometric calculations.
The triangle is a triplet of points-vertices, which do not lie on the same line, and lie on an imaginary plane. The expression of sides \(a, b, c\) and their respective opposite angles \(\alpha,\beta,\gamma\), as a complex numbers in this plane, gives the following representation of the known triangle properties:
Convexity | \[\mathrm{Im}\;\alpha>0,\;\;\;\mathrm{Im}\;\beta>0,\;\;\;\mathrm{Im}\;\gamma>0\] |
Sum of angles | \[\alpha\cdot\beta\cdot\gamma+1=0\] |
Law of sines | \[\frac{|a|}{\mathrm{Im}\;\alpha}=\frac{|b|}{\mathrm{Im}\;\beta}=\frac{|c|}{\mathrm{Im}\;\gamma}\] |
Law of cosines | \[|a|^2=|b|^2+|c|^2-2|b||c|\mathrm{Re}\;\alpha\] \(|a|^2=b\cdot\overline{b}+c\cdot\overline{c}-b\cdot\overline{c}-c\cdot\overline{b}\) |
The perpendicular lines are two lines, which are intersected at the point A, and have a property: Any pair of points on the different lines always makes a right triangle with the right angle at the point A.
The integer dimensionality of a space is a maximum number of the orthogonal (perpendicular to each other) lines, which are intersected at any point of this space. The strict properties of the perpendicular and parallel lines are valid only for the ideal space of an integer dimensionality (Euclidian, Hilbert). A point is 0D-space, a line is 1D-space, and a plane is 2D-space.
The object is a subspace, which is specified by the conditions and by the relative distances between certain points. The object dimensionality, by definition, does not exceed the space dimensionality. The characteristic size (or simply size) is a value, which specifies the absolute distances between points.
The similarity of objects is their equality regardless of their size. The factor of similarity is a ratio of their sizes.
The objects intersection is an object, which is produced by an intersection of their sets of points. The intersection dimensionality does not exceed the lowest dimensionality of the intersected objects.
The surface of an object is the object of lower dimensionality, which bounds the volume of the given object.
The hypersphere is a simple basic object of an arbitrary dimensionality, a subset of points, which distance from a center does not exceed the radius. The hypersphere is not attached to the orthogonal axes, and has an infinite number of the symmetry axes. An intersection of the non-concentric hyperspherical surfaces with the lowest dimensionality \(N\), is the hyperspherical surface of dimensionality \((N – 1)\) (negative dimensionality is equal to zero).
Dimensionality N | Object | Surface |
---|---|---|
0 | Point | Point |
1 | Line segment | Pair of points |
2 | Disk | Circle |
3 | Ball | Sphere |
N | N-dimensional hypersphere | N- dimensional hyperspherical surface |
The hypercube is a complex object of an integer dimensionality, which is constructed from the equal orthogonal line segments and is circumscribed into a hypersphere. A cube is 3D-hypercube, a square is 2D-hypercube, and a line segment is a degenerate 1D-hypercube.
The volume of an object is a number of the non-overlapping base objects of a unit volume of dimensionality \(N\), which completely fill this object of dimensionality \(N\). A length is 1D-volume, and the area is 2D-volume. Traditionally, the base object is a hypercube. The choice of a base object reflects on the values of object volumes, but not on their ratios. The volume is a function of the dimensionality \(N\) and the size \(R\) with a geometrical factor \(C(N)\), which depends on the choice of a base object: \[V_N(R)=C(N)R^N\] The geometry is "surveying" in the ancient Greek language. The land pieces are usually rectangular, and their area is convenient to measure in the square units, which were entered into the foundations of mathematics and physics. In fact, the rectangular objects are rare in nature, as opposed to the circular objects. When replacing the square and cube units to the "circular" and "spherical", a hypersphere volume of diameter \(D\) becomes \(V_N=D^N\).
The real dimensionality is a value \(N\) in the ratio of volumes \(V_N\) of two similar objects with sizes \(R\) and \(R_0\): \[\frac{V_N(R)}{V_N(R_0)}=\left(\frac{R}{R_0}\right)^N\] \[N=\frac{\ln V_N(R)-\ln V_N(R_0)}{\ln R-\ln R_0}\] The fractional exponents were not easily accepted, because an irrational exponent has no the arithmetical definition. Also, the fractional spatial dimensionality is waiting to be accepted. Every actual physical value is continuous. The discretization is a product of human mind, who name the positive integers natural (from nature).
The self-similarity of space is the law of the spatial geometry of arbitrary dimensionality. The space is scalable and the set of all similar objects (subspaces) is infinite.
The fractal is a geometric object with some self-similarity. It is constructed by dividing the original object into the proportional parts and by its repetition within these parts recursively. For example, a construction of the orthogonal straight line segments by a fractal principle creates a grid of 2D-space with an infinite number of parallel-perpendicular lines:
The fractional (fractal) spatial dimensionality is a violation of the perpendicular and parallel lines of the integer-dimensional space. According to the law of self-similarity, this deformation is fractal, i.e. it is applied to an infinite number of the subspaces. Increasing of the integer dimensionality \(N\) by less than 1 causes a splitting of the space into an infinite number of subspaces of dimensionality \(N\), which is not a full dimensionality \((N+1)\).
The fractals of a fractional dimensionality are the visual images of essence of a fractional spatial dimensionality, which are constructed from the polygonal chains and from the surfaces with non-right angles. For example, the Koch curve is constructed by dividing the original object into 4 parts and reduction by 3 times, thus producing the dimensionality \(N=\ln 4/\ln 3=1,26\).
The infinitely divisible and folding curve does not approach to a solid 2D-surface, but it increases in its length infinitely, which implies the emergence of a new dimension. More close points are surrounding the object in this space, and the object has more alternatives to move than in the straight 1D-space.
The Dragon curve is another example of a fractal polygonal chain. Each segment is folded in half at the right angles, reducing by a half. It seems that the curve dimensionality should be 1, but the right angles and twisting cause the connection of segments to a grid of the orthogonal lines. This grid approaches to a sold 2D-surface although irregular.
The homogeneous space is a space, which have the same dimensionality for each point. The actual physical space is heterogeneous.
The universe has an average dimensionality of 3 or more, but not less, because the matter becomes unstable. A significant deviation from this value is incompatible with an organic life.
Position
The actual physical coordinates are the distances from a given point A to the \(M\) reference points within a space of dimensionality \(N < M\). This positioning method is known as the trilateration. It is used in the surveying and in the radio navigation (GPS, GLONASS). Any other coordinate system is not absolute and is derived. The point A is an intersection of M hypersperes. Here’s an example for 2D-space:
The actual coordinates of an integer-dimensional space are dependent on each other, that is, in this example, two circles have two points of intersection, and the third has to intersect one of them, having only two permissible radii. If the spatial dimensionality \(2 < N < 3\), than an intersection of two circles contains more than two points, and the third circle has more permissible radii, defining the coordinates exactly. Every coordinate-vector operation can be done by the solution of triangles system. The linear algebra is useless in case of the fractional spatial dimensionality.
Another example of 1D-space:
The right figure shows a known method to build a perpendicular, which exists only in the 2D-space and more. At a dimensionality \(1\leq N<2\), two hyperspheres can have only one point of intersection A.
The physical space has the absolute reference frame, with respect to which the absolute linear velocities of all objects are defined.
Units of area and volume
For the traditional (hypercubic) units of area and volume, the equations for volume \(V_N(R)\) and for the surface area \(S_N(R)\) of a hypersphere are known: \[V_N(R)=C(N)R^N\] \[S_N(R)=\frac{\mathrm{d}V_N (R)}{\mathrm{d}R}=NC(N)R^{N-1}\] \[C(2N)=\frac{\pi^N}{N!},\;\;\;\;\;C(2N+1)=\frac{2^{N+1}\pi^N}{(2N+1)!!}\]
Dimensionality N | Object | Volume | Surface | Angles |
---|---|---|---|---|
1 | Line segment | \[2R\] | two directions | |
2 | Disk | \[\pi R^2\] | \[2\pi R\] | \(2\pi\) radians |
3 | Ball | \[\frac{4}{3}\pi R^3\] | \[4\pi R^2\] | \(4\pi\) steradians |
4 | 4D-hypersphere | \[\frac{1}{2}\pi^2R^4\] | \[2\pi^2R^3\] |
The factor \(C(N)\) is also defined by Euler's gamma function with a real argument: \[C(N)=\frac{\pi^{N/2}}{\Gamma(N/2+1)},\;\;\;\;\;\Gamma(x)=\int_0^{+\infty}{t^{x-1}e^{-t}\mathrm{d}t}\] Application of the hyperspheric units alters the integral calculus and the equations for volume and surface: \[V_N(R)=(2R)^N\] \[S_N(R)=X(N)(2R)^{N-1}\neq\frac{\mathrm{d}V_N(R)}{\mathrm{d}R}\] The \(X(N)\) is found from the following conditions:
- The circle’s circumference or the "disk surface area" in this case is also \(2\pi R\).
- The ratio of the sphere area to the disk area of the same radius is \(4\).
- The ratio of the 4D-hypersphere surface area to the sphere area of the same radius is \(3\pi⁄2\).
These conditions are satisfied by this solution: \[S_N(R)=N\frac{C(N)}{C(N-1)}(2R)^{N-1}\]
Dimensionality | Object | Volume | Surface |
---|---|---|---|
1 | Line segment | \[D\] | |
2 | Disk | \[D^2\] | \[\pi D\] |
3 | Ball | \[D^3\] | \[4D^2\] |
4 | 4D-hypersphere | \[D^4\] | \[\frac{3\pi}{2}D^3\] |
\(D\) – diameter. |
The ellipsoid volume in the hyperspheric units is the product of its diameters.
The following corrections are necessary for a multiple integral in the hyperspheric units:
- Double integral. The area ratio of a square to a disk (the disk is inscribed into the square) \(4/\pi\).
- Triple integral. The volume ratio of a cube to a ball (the ball is inscribed into the cube) \(6/\pi\).
Previous chapter ( Self-similarity ) | Table of contents | Next chapter ( Physical vector ) |