Numerical construction and critical behavior of Kaluza Klein black holes The idea of extra dimensions provides a promising approach to overcome various problems in modern physics. This includes theoretical as well as phenomenological aspects such as the unification of the fundamental interactions or the hierarchy problem. Based on the seminal works by Kaluza and Klein that were published nearly 100 years ago we denote theories with at least one compact periodic dimension as Kaluza Klein theories. From a gravitational point of view the question arises what are the fundamental solutions to Einsteins field equations of general relativity under these assumptions. In particular in this work we are concerned with black hole solutions in Kaluza Klein theory. Considering only the static case without electric charge it turns out that there is a much richer phase space than in the usual four dimensional theory where only the Schwarzschild solution exists. There are at least two types of solutions with a completely different horizon topology: localized black holes with an ordinary spherical horizon and black strings with a horizon that wraps the compact dimension. Several arguments favor the conjecture that the solution branches of both types are connected via a singular topology changing solution that is controlled by the so called double cone metric. We study the regime close to this singular transit solution in five and six spacetime dimensions with the help of a highly accurate numerical scheme that we describe in detail. Consequently for the first time we are able to show that in this regime the black objects exhibit a critical behavior indicating that physical quantities are governed by universal critical exponents. Interestingly such exponents were already derived from the double cone metric. We show that our data confirms these values extremely well. This provides compelling evidence in favor of the double cone metric as the local model of the transit solution.