Mike WinklerRuhr-Universität Bochum | RUB · Faculty of Mathematics (1)
Mike Winkler
Ph.D.
About
21
Publications
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Introduction
I am a mathematician and live on a horse farm on the German North Sea coast. My work is mostly in the areas of number theory, combinatorics and discrete geometry, including graph theory. My favorite topics are the 3x + 1 problem, regular matchstick graphs, nonperiodic tilings and Heesch’s problem. Besides math, I love horses and cycling, play the piano and guitar, swim and paint occasionally. More stuff you will find on my homepage: https://www.mikematics.de/
Publications
Publications (21)
The 3x + 1 problem concerns iteration of the map T : Z → Z given by T(x) = x/2 if x ≡ 0 (mod 2) and T(x) = (3x + 1)/2 if x ≡ 1 (mod 2). The 3x + 1 Conjecture states that every x > 1 has some iterate T^s(x) = 1. The least s∈N such that T^s(x) < x is called the stopping time of x. It is shown that the residue classes of the integers x > 1 with a fini...
A matchstick graph is a planar unit-distance graph. That is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph 4-regular if every vertex has only degree 4. Examples of 4-regular matchstick graphs with less than 63 vertices are only known for 52, 54...
A matchstick graph is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph (m; n)-regular if every vertex has only degree m or n. In this article the authors present the latest known (4; n)-regular matchstick graphs for 4 ≤ n ≤ 11 with a minimum numb...
The first part (pp. 1 - 7) of this article presents the currently known examples of 4-regular matchstick graphs with 63 - 70 vertices. The second part (pp. 8 - 15) presents the currently known examples of (2; 4)-regular matchstick graphs with less than 42 vertices which contain only two vertices of degree 2.
It is shown that every Collatz sequence $C(s)$ consists only of same
structured finite subsequences $C^h(s)$ for $s\equiv9\ (mod\ 12)$ or $C^t(s)$
for $s\equiv3,7\ (mod\ 12)$. For starting numbers of specific residue classes
($mod\ 12\cdot2^h$) or ($mod\ 12\cdot2^{t+1}$) the finite subsequences have the
same length $h,t$. It is conjectured that for...
We present aperiodic sets of prototiles whose shapes are based on the well-known Penrose rhomb tiling. Some decorated prototiles lead to an exact Penrose rhomb tiling without any matching rules. We also give an approximate solution to an aperiodic monotile that tessellates the plane (including five types of gaps) only in a nonperiodic way.
We present aperiodic sets of prototiles whose shapes are based
on the well-known Penrose rhomb tiling. Some decorated prototiles
lead to an exact Penrose rhomb tiling without any matching
rules. We also give an approximate solution to an aperiodic
monotile that tessellates the plane (including five types of gaps)
only in a nonperiodic way.
A 4-regular matchstick graph is a planar unit-distance graph whose vertices have all degree 4. Examples of 4-regular matchstick graphs are currently known for all number of vertices ≥ 52 except for 53, 55, 56, 58, 59, 61, and 62. In this article we present 38 different examples with 50 - 62 vertices which contain two, three, or four distances which...
In 2010 it was proved that a 3-regular matchstick graph of girth 5 must consist at least of 30 vertices. The smallest known example consisted of 180 vertices. In this article we construct an example consisting of 54 vertices and prove its geometrical correctness.
In this article we proof the existence of 4-regular planar unit-distance graphs consisting only of unit triangles without additional triangles. It is shown that the smallest number of unit triangles is ≤ 6422.
It is conjectured that a non-periodic tiling of the plane with a set of four prototiles can be realised by a simple decorated monotile and a simple replication-algorithm in a spiral movement. The algorithm uses overlaps and gaps, but there is no need of adding or removing edges. We also show some mutually locally derivable tilings including a set o...
In 2010 it was proved that a 3-regular matchstick graph of girth 5 must consist at least of 30 vertices. The smallest known example consisted of 180 vertices. In this article we construct an example consisting of 54 vertices and prove its geometrical correctness.
This article exhibits the currently smallest known examples of 3-regular matchstick graphs of girth 5 consisting of less than 70 vertices.
In this article we proof the existence of 4-regular planar unit-distance graphs consisting only of unit triangles without additional triangles. It is shown that the smallest number of unit triangles is ≤ 6422.
Fermat's Last Theorem states that the Diophantine equation X^n + Y^n = Z^n has no non-trivial solution for any n greater than 2. In this paper we give an approach to a brief and simple proof of the theorem using only elementary methods.
Chapter 1 to 6 were peer-reviewed. Chapter 7 and 8 are not finished yet.
Keywords and phrases: 3x + 1 problem, Collatz conjecture, Syracuse problem, finite stopping time, directed rooted tree, recursive algorithm, Diophantine equation, A020914, A020915, A022921, A056576, A076227, A100982, A177789, A293308.
A matchstick graph is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph (m; n)-regular if every vertex has only degree m or n. In this article we present the latest known (4; n)-regular matchstick graphs for 4 ≤ n ≤ 11 with a minimum number of ver...
Es wird die Konstruktion eines neuen 4-regulären Streichholzgraphen mit 114 Kanten vorgestellt, welcher vom Autor am 15. April 2016 entdeckt wurde.
Let $\sigma_n=\lfloor1+n\cdot\log_23\rfloor$. For the Collatz 3x + 1 function exists for each $n\in\mathbb{N}$ a set of different residue classes $(\text{mod}\ 2^{\sigma_n})$ of starting numbers $s$ with finite stopping time $\sigma(s)=\sigma_n$.
Let $z_n$ be the number of these residue classes for each $n\geq0$ as listed in the OEIS as A100982.
It...
Conjecture: The quadratic Diophantine equation system q^2 = a^2*alpha-b^2*beta-c^2*gamma, pq = (ad)^2*alpha-(be)^2*beta-(cf)^2*gamma, p^2 = (ad^2)^2*alpha-(be^2)^2*beta-(cf^2)^2*gamma, has no nontrivial integer solution for d ≠ e ≠ f ≠ 0, if there is alpha=beta=gamma=1, or |alpha| ≠ |beta| ≠ |gamma| ≠ 0 with alpha|a, beta|b, gamma|c and |alpha| ≠ a...
The behaviour of a Collatz sequence is clearly related to the way in which the powers of 2 are distributed among the powers of 3 in a stopping time term formula. It is shown, that the seemingly chaotically distribution of the powers of 2 can be generated by an iterative algorithm (conjecture), which also allows an insight into the properties of the...
Questions
Question (1)
Dear colleagues and members,
the researchgate member Budee U Zaman (researchgate.net) copied one of my preprints one-to-one and passed it off as his own work. He only slightly changed the title. His other uploads, many of which he uploads in a short period of time, also give this impression. Is anyone else affected by this scammer besides me? ResearchGate itself does not want to do anything about it.
Kind regards
Mike Winkler, Ph.D.
