The 4-ball example

The method of the multidimensional ball is a method to be introduced in this research that can be used for the visualization of both 4-dimensional and higher dimensional geometrical figures. It uses an algorithm of stepwise concept induction, beginning with the 1-dimensional projection of the observed figure. In this section for the purpose of defining the main concepts concerning the method, a ball is to be chosen as an example to be researched.

So far although mathematicians have successfully calculated the 4-dimensional volume of a ball and other geometrical figures, they were not successful in representing how their results should be interpreted. Although there was the most considerable success in the research of a hypercube and related polytopes, noticeable in the hypercube graphs, elements construction and rotation, its physical interpretation still lagged behind due to the extra spacelike dimensions (Bowen).

As it can correctly be inferred, the ball is truly defined as a ball only in a 3-dimensional space. In the 2-dimensional space (plane), it is a disc or a 2-ball with radius R and in the 1-dimensional space (line) - a segment line (1-ball) with length 2R. Nonetheless, in the following method the term “ball” is going to be used beyond its ordinary meaning and be applied as a common definition for the n-dimensional geometric figure in order to unify its concept during the different stages of construction. In the example here, the method of the multidimensional ball will be used to visualize and interpret the physical meaning of a 4-dimensional ball (also 3+1-dimensional ball or the set of points defined as the interior of a 3-sphere).

dimensions - figure 1.1

The 1-dimensional ball as already stated above is a segment line with a length equal to 2 times its radius and two points as its boundaries. It is the first stage of this method’s induction and application.

The second stage includes the observation of the 2-dimensional ball, i.e. the disc. It consists of infinitely many mutually neighboring and parallel segment lines (for which here the term “1-dimensional frames” is to be inserted). They form a continuum along a second dimension with a specific interdependence, determined by the ball geometry: each segment line is minimally longer than the former and minimally shorter than the next before the middle segment line; each segment line is minimally shorter than the former and minimally longer than the next after the middle segment line. Furthermore, differentiating this change, we get a derivative that is not a constant, hence, the former is not linear. Thus, the first and the last segment lines are points and the middle one after which the change reverses is the diameter and therefore the longest, whose radius is also the radius of the 2-dimensional ball. The 1-dimensional frames (the segment lines) are just 1-dimensional balls forming a 2-dimensional continuum in a specific non-discrete order with different 1-dimensional magnitudes (lengths).

The third stage is to observe the 3-dimensional ball and to slice it into infinitely many small planes. Its 2-dimensional frames on these planes are infinitely many neighboring parallel discs with the interdependence defined by the ball geometry. Each disc has an area that is minimally larger than the former and minimally smaller than the next before the middle plane and vice versa after the middle plane. Thus, the first and the last disc are points with zero area (2-dimensional magnitude) and the middle one is the largest. The diameter of the 3-dimensional ball lies in the plane of the middle disc. The 2-dimensional frames (the discs) are only 2-dimensional balls with 2-dimensional magnitudes that vary according to the ball geometry.

dimensions - figure 1.2

Following the stages of the method of the multidimensional ball yields a correlated growing dimensionality sequence in which the ball geometry can be used as a tool for the construction of the figure in n dimensions.

The fourth stage of this construction results in a visualization and implication of the 4-dimensional ball. Defining the 3-dimensional balls as a 3-dimensional frames which are mutually neighboring and parallel (embedded outside of the space of each) so that each has minimally larger volume than the former and minimally smaller volume than the next before the middle ball and vice versa after the middle ball is the first conclusion to be made. A more precise definition is given by the ball geometry, which implies that the first and the last one are points as well as that the derivative of the circle function is the exact definition of the change between each pair of parallel neighboring balls. The result is a 3-dimensional ball that expands and shrinks in time with a specific rate[1]. In a continuum of 3-dimensional frames (moments) that can be defined as time, what can be observed is the appearance of a point, then its decreasing growing in size (in every moment its volume is minimally larger than in the former moment and minimally smaller than in the next moment) and, after it reaches the middle moment of its existence, it increasingly shrinks to a single point and disappears.

dimensions - figure 1.3

The model of the 4-dimensional ball can be used in a further fifth stage for visualizing and conceptualizing the 5-dimensional ball but this will be done in the next chapter.

[1] the velocity of growing decreases from infinity to 0 and the velocity of shrinking increases from 0 to infinity, both in the period of the time-radius; for more information see the next chapters