A heptagonal triangle is a non-isosceles triangle formed by three vertices of a regular heptagon. Its angles are π 7 , 2π 7 and 4π 7 . As such, there is a unique choice of a companion heptagonal triangle formed by three of the remaining four vertices. Given a heptagonal triangle, we display a number of interesting companion pairs of heptagonal triangles on its nine-point circle and Brocard circ...

Naturally occurring phenomena in space often (if not always) cannot be precisely defined because of the intrinsic uncertainty of their features. The location of animal refuges might not be precisely known, and the path of rivers might be uncertain due to water volume fluctuations and changing land characteristics. The extension of lakes can also change and thus have uncertain areas. All these a...

Let Q be the field of rationals numbers. We prove that: (1) if x, y ∈ R (n > 1) and |x− y| is constructible by means of ruler and compass then there exists a finite set Sxy ⊆ R containing x and y such that each map from Sxy to R n preserving unit distance preserves the distance between x and y, (2) if x, y ∈ Q then there exists a finite set Sxy ⊆ Q containing x and y such that each map from Sxy...

Figure 1. A diagram depicting th e hybrid biological/organic photochemical half-cell. This report describes our progress towards the design, assembly, characterization, and optimization of a biological/organic hybrid photochemical halfcell. This device will couple PS I, which efficiently captures and stores energy derived from sunlight, with either an [FeFe]-H2ase or a [NiFe]-H2ase, which can g...

L awsuits aimed at compelling legislatures to increase school funding have been filed in some 42 states. Courts have found for the plaintiffs in more than half of the cases on the grounds that schools are not “adequately” funded (see Figure 1). These decisions have, in effect, changed the way education appropriations are made, moving decisionmaking from legislatures to the courts. Instead of fl...

Knowing where people look is a useful tool in many various image and video applications. However, traditional gaze tracking hardware is expensive and requires local study participants, so acquiring gaze location data from a large number of participants is very problematic. In this work we propose a crowdsourced method for acquisition of gaze direction data from a virtually unlimited number of p...

cu lty R es ea rc h N ew sl et te r 4 that lie on the circle and take only the ones inside it. Given points P and Q in our space, the "line through P and Q" is interpreted as the arc of the circle through P and Q that intersects C at right angles (see Figure 1). With these interpretations of the words "point" and "line", it can be shown that the fifth postulate fails because given a line L and ...

Consider the following special case of planar point location: preprocess k sets of lines, where each set consists of parallel lines, to support queries of the form “given a point p, what is the line immediately above or below p?” What is the fastest possible query time as a function of k and the total number n of lines? In other words, the n given lines have k distinct orientations. See Figure ...

Fractals can model many classes of time-seri" data. The fractal dimension is an important actertsfic of ftaqo& that coniai, c information . out their geometrical structure at multiple Ntod re a b g ms i l d h ti BC m h d t c s, : s a ass of CJJ4 ent proac ox-coun n ( } et over 1 g c es, e.g., o , o estimate the fractal-dimension. Jr. this paper, the d rential box-counting (DBC) approach, origin...

Paddy is one of major crops grown in China. Fractal dimension is an important parameter used to describe geometrical characteristics of many natural objects. The objective of this study is to investigate mathematical relationships between the fractal dimension and the yield of the paddy spike based on the fractal theory and the image processing technologies. The color images of paddy spikes wer...