Die Mittelsenkrechte – stoffdidaktische Analyse und Bezüge zum Unterricht

The concept of the perpendicular bisector is analysed with a didactical view and different opportunities of definitions are discussed. After that we focus on the implementation in the classroom. Therefore, we look at the definition used in schoolbook but also the previous knowledge of children around the perpendicular bisector. Finally, written explanations of seventh-graders show how they argue with the concept of the perpendicular bisector. Classification: D70, E50, U20. Keywords: geometry, perpendicular bisector, mathematics education.

Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

Circumcenter, Circumcircle and Centroid of a Triangle

Summary We introduce, using the Mizar system [1], some basic concepts of Euclidean geometry: the half length and the midpoint of a segment, the perpendicular bisector of a segment, the medians (the cevians that join the vertices of a triangle to the midpoints of the opposite sides) of a triangle. We prove the existence and uniqueness of the circumcenter of a triangle (the intersection of the three perpendicular bisectors of the sides of the triangle). The extended law of sines and the formula of the radius of the Morley’s trisector triangle are formalized [3]. Using the generalized Ceva’s Theorem, we prove the existence and uniqueness of the centroid (the common point of the medians [4]) of a triangle.

A Range Free Acoustic Localization Algorithm in Wireless Sensor Networks

For accomplishing acoustic location in wireless sensor networks (WSNs), a range free acoustic localization algorithm based on perpendicular bisector partition is proposed, taking into account of reducing computation complexity and reduce the interference of the background noise. Adopting a range free perpendicular bisector partition, the proposed method can find the sub-region of the source, and the time complexity is much lower than that of existing methods. According to extensive analysis on noise, the concept of noise sensitive region is derived. Experimental results show that the proposed method has a high localization precision and low complexity.

Teaching the Perpendicular Bisector: A Kinesthetic Approach

Through movement-a welcome change of pace-students explore the properties of the perpendicular bisector.