18. BUILDING UP OF SPACE FROM CONGRUENT POLYHEDRA.
If we enquire for those groups of motions in the plane for which a fundamental region exists, we obtain various answers, according as the plane considered is Riemann's (elliptic), Euclid's, or Lobachevsky's (hyperbolic). In the case of the elliptic plane there is a finite number of essentially different kinds of fundamental regions, and a finite number of congruent regions suffices for a complete covering of the whole plane; the group consists indeed of a finite number of motions only. In the case of the hyperbolic plane there is an infinite number of essentially different kinds of fundamental regions, namely, the well-known Poincaré polygons. For the complete covering of the plane an infinite number of congruent regions is necessary. The case of Euclid's plane stands between these; for in this case there is only a finite number of essentially different kinds of groups of motions with fundamental regions, but for a complete covering of the whole plane an infinite number of congruent regions is necessary.
Exactly the corresponding facts are found in space of three dimensions. The fact of the finiteness of the groups of motions in elliptic space is an immediate consequence of a fundamental theorem of C. Jordan,[41] whereby the number of essentially different kinds of finite groups of linear substitutions in
variables does not surpass a certain finite limit dependent upon
. The groups of motions with fundamental regions in hyperbolic space have been investigated by Fricke and Klein in the lectures on the theory of automorphic functions,[42] and finally Fedorov,[43] Schoenflies[44] and lately Rohn[45] have given the proof that there are, in euclidean space, only a finite number of essentially different kinds of groups of motions with a fundamental region. Now, while the results and methods of proof applicable to elliptic and hyperbolic space hold directly for
-dimensional space also, the generalization of the theorem for euclidean space seems to offer decided difficulties. The investigation of the following question is therefore desirable: Is there in
-dimensional euclidean space also only a finite number of essentially different kinds of groups of motions with a fundamental region?
A fundamental region of each group of motions, together with the congruent regions arising from the group, evidently fills up space completely. The question arises: Whether polyhedra also exist which do not appear as fundamental regions of groups of motions, by means of which nevertheless by a suitable juxtaposition of congruent copies a complete filling up of all space is possible. I point out the following question, related to the preceding one, and important to number theory and perhaps sometimes useful to physics and chemistry: How can one arrange most densely in space an infinite number of equal solids of given form, e. g., spheres with given radii or regular tetrahedra with given edges (or in prescribed position), that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as great as possible?
If we look over the development of the theory of functions in the last century, we notice above all the fundamental importance of that class of functions which we now designate as analytic functions—a class of functions which will probably stand permanently in the center of mathematical interest.
There are many different standpoints from which we might choose, out of the totality of all conceivable functions, extensive classes worthy of a particularly thorough investigation. Consider, for example, the class of functions characterized by ordinary or partial algebraic differential equations. It should be observed that this class does not contain the functions that arise in number theory and whose investigation is of the greatest importance. For example, the before-mentioned function
satisfies no algebraic differential equation, as is easily seen with the help of the well-known relation between
and
, if one refers to the theorem proved by Holder,[46] that the function
satisfies no algebraic differential equation. Again, the function of the two variables
and
defined by the infinite series
which stands in close relation with the function
, probably satisfies no algebraic partial differential equation. In the investigation of this question the functional equation
will have to be used.
If, on the other hand, we are lead by arithmetical or geometrical reasons to consider the class of all those functions which are continuous and indefinitely differentiable, we should be obliged in its investigation to dispense with that pliant instrument, the power series, and with the circumstance that the function is fully determined by the assignment of values in any region, however small. While, therefore, the former limitation of the field of functions was too narrow, the latter seems to me too wide.
The idea of the analytic function on the other hand includes the whole wealth of functions most important to science, whether they have their origin in number theory, in the theory of differential equations or of algebraic functional equations, whether they arise in geometry or in mathematical physics; and, therefore, in the entire realm of functions, the analytic function justly holds undisputed supremacy.
[41] Crelle's Journal, vol. 84 (1878), and Atti d. Reale Acad. di Napoli, 1880.
[42] Leipzig, 1897. Cf. especially Abschnitt I, Chaplets 2 and 3.
[43] Symmetrie der regelmässigen Systeme von Figuren, 1890.
[44] Krystallsysteme und Krystallstruktur, Leipzig, 1891.
[45] Math. Annalen, vol. 53.
[46] Math. Annalen, vol. 28.