While both theories are doing an excellent job when used on their own, and their achievements are innumerous — in their combination some fundamental contradictions emerged, deep cracks in the building of physics. The inconsistencies, as became gradually perceptible, are concentrating at two spots: The theories’ notions of  “spacetime” and of  “vacuum” seem to be hardly compatible.

    Newton’s physics had imagined the vacuum as a part of space, which is simply empty, containing absolutely nothing. Such vacuum however is absolutely impossible according to quantum theory. The vacuum of quantum theory is a part of space, which is as empty as compliant with the laws of nature. The vacuum of quantum theory is filled with fluctuations of all types of quantum fields, particularly with the zero-point oscillations of all oscillation modes of those fields. In total, the energy of the zero-point oscillations is infinite! This is no difficulty for the quantum field theories, because to them only energy differences matter. A constant, infinite energy pedestal is somewhat disturbing, but no real problem.

    Quite different in the theory of general relativity. Any type of energy adds to the space’s curvature, and an infinite amount of energy will cause infinite space curvature. Nothing like that is perceived in astronomical observations. There are perceptible space curvatures nearby heavy objects. But in the vast, almost empty intergalactical regions space seems to be practically flat. Therefore quantum theory seems to be wrong, at least in this regard. But still it's just the theory of the vacuum, which enabled quantum electrodynamics to compute atomic spectra with a precision, which was never before achieved in any branch of physics.

    The theories’ conflicting – and irreconcilable – notions of space and time crystallize around the mysterious correlations within extended quantum systems. What seems a “spooky action at a distance”, when viewed through the glasses of GRT, must be acknowledged as the „indivisibility of quantum phenomena” according to quantum theory.

   A puzzling situation. The hopes of many physicists, that sooner or later GRT and QT could be merged into a quantum theory of gravitation – thus eventually completing the consistant, integrative “theory of everything” – is being frustrated since tens of years. Quite the contrary: The incompatibilities inbetween both theories meanwhile have accumulated to massive contradictions, which particularly in cosmology can not anymore be overlooked nor superseded. GRT and QT not only differ fundamentally in their mathematical structure, but also apply quite different, sometimes almost diverging concepts of notion. This probably is a significant source of the inconsistencies.

    The Astrophysical Institute Neunhof is working exactly at the boundary separating GRT and QT. It is concerned with the mathematical-technical aspect of the contradictions inbetween both theories, but most notably with the episthemological investigation of their perpetual parallel existence. While the struggle for the „theory of everything“ is ongoing, and certainly there still is a chance for success, – we must face the possible alternative: The use of incompatible theories – in terms of a generalisation of Bohr’s principle of complementarity – might not only be acceptable, but even an indispensable requisite for a complete description of the physical world. If this should prove true, then of course a much better understanding of the ranges of validity, and the mutual complementary completion of both theories would be required. In any case this field of questions deserves intensive studies. The Astrophysical Institute Neunhof is working to make contributions to the clarification of this issue.

   The APIN’s logo is worth a closer look. The logo is suggesting the construction of a regular pentagon from the radius of its circumscribed circle. First we present a short explanation of the method:

   According to the rules of geometry, as set by the ancient Greeks, any construction must be executed solely with compasses and ruler (without scale). Neither goniometer nor algebraic aids are allowed. Observing these rules, the construction of a regular pentagon is anything but trivial, even if the length of the sides may be arbitrarily chosen. This construction is based on the fact that the ratio of the diagonal BF to the side length BD equals the golden ratio:

It’s the irrational root of 5, which is making the geometry of pentagons at the same time difficult and delightful. How the construction can be performed, was already explained by Euklid in the „Elements”.
The ratio of the pentagon’s sidelength BD to the circumference’s radius MX is irrational too, comprising even the double root of 5:

The construction is accomplished as follows: First the circumscribed circle is drawn. Next the length CX is needed, with CX/MX being the golden ratio. For this purpose, MX is halved, yielding point A. Now an arc is drawn around A through B (B is the intersection point of the vertical diagonal and the circle) onto the horizontal diagonal, thus finding point C.
The ratio of length CM + MX to length MX is just the golden ratio, i.e. the following relation holds:

(To proof this, note that CA = BA, and apply Pythagoras’ theorem to the triangle MBA.) Thus at least the needed factor root of 5 has appeared in the construction. But the real clou comes with the last step:
An arc is drawn around B through C. This arc’s radius is the pentagon’s side length. The arc’s intersections with the circle immediately give vertices D and E. Vertices F and G are found by drawing arcs with same radius around D and E.

   The division of a line in the golden ratio was and is regarded as aesthetically very appealing by many artists. The fact that the golden ratio could be exploited for the depicted construction is but one of innumerous examples demonstrating that beauty is often (but not always) a useful guideline, when searching for the solution of a difficult scientific problem. At the same time, its a typical example by showing that the way may not be straightforward. A perplexing idea was required for the construction’s second step.

   The irrational numbers 1.61803... and 1.175570... are playing a prominent role in the pentagon’s construction. The discovery of irrational numbers is ascribed to the Pythagoreans. The Pythagoreans believed that rational proportions were the very fundament of harmony. E.g. they saw the base for musical harmony in the rational lengths proportions of strings and tubes of musical instruments, and the base for the harmony of the world in rational proportions of the planets’ orbits. Thus it must have been a shock to them when they found out that in geometry, which was considered the realm of perfect harmony, the ratio of a square’s diagonal length to side length cannot be expressed by a rational number.

   Nowadays we have an unrestrained positive attitude towards the irrational numbers: As being one of the most impressive and astonishing creations of human intellect, being evidence that our insight is not limited to what may have been useful in course of our evolution. Why can human beings understand much more than needed in respect of biological needs? We don’t know the answer. But obviously it would be a serious failure to neglect this amazing giftedness and persist in stupidity. We clearly see the obligation, to make best use of our intellectual capability to unravel the secrets of nature, even if we can not tell what might be the „deaper meaning” of our efforts.