Aristotle’s Political Justice and the Golden Ratio between the Three Opposing Criteria for the Distribution of Public Goods among Citizens: Freedom, Wealth and Virtue

In this article, I interpret Book V of the Nicomachean Ethics in which Aristotle presents a geometrical problem to explain which is the Best Criterion for the Distribution of Political and Economic Rights and Duties among Citizens, starting from the empirical evidence that there are three opposing opinions on which is the fairest distribution criterion: for some it is Freedom (Democrats), for others Wealth (Oligarchs), and for others Virtue (Aristocrats). Against the almost unique and most quoted interpretation of the geometrical problem, I present my mathematical solution, which I arrived at thanks to the Doctrine of the Four Causes and the Theory of the Mean. My thesis is that the Mean Term of Distributive Justice is the Golden Ratio between the opposite criteria of distribution, and the unjust distribution is the one that violates this ratio. This solution allows us to understand what is the Rational Principle at the basis of just distribution: that is, Geometrical Equality as opposed to Arithmetical Equality. Indeed, by applying the geometric figure of the Golden Triangle to the different political constitutions, I show, in line with Politics, that the Best Form of Government is the Aristocratic Politeia, i.e., a mixture of Democracy, Oligarchy and Aristocracy.

“GEOMETRIC FIGURES” CLUSTER IN THE LANGUAGE PICTURE OF THE WORLD (RUSSIAN-CHUVASH PARALLELS)

Работа посвящена анализу русских и чувашских лексических единиц, репрезентующих кластер «геометрические фигуры», с точки зрения его отражения в сознании двух языковых коллективов. В центре нашего лингвокультурологического анализа паремии, традиционные народные символы, используемые в фольклоре. Лексика данной тематической группы помогает выявить своеобразие менталитета носителей языков, помогает выделить особенности мировосприятия чувашского и русского этносов. Кластер «геометрические фигуры» не исследовался ни в отношении русской, ни в отношении чувашской языковых картин мира. Методы исследования: анализ словарных дефиниций, сопоставительный, концептуальный, лингвокультурологический. Лексемы, обозначающие геометрические фигуры, своеобразно трактуются в их вторичной номинации разными народами. В центре нашего внимания элементарные фигуры, такие как круг, квадрат (прямоугольник), треугольник (угол), шар. Геометрическая фигура «круг» в чувашском языке имеет положительную коннотацию, а в русском языке как положительную, так и отрицательную; шар в обоих анализируемых языках воспринимается отрицательно; угол и в чувашском, и в русском языках имеет и положительную, и отрицательную коннотацию; эмоциональные и оценочные оттенки в высказываниях, содержащих лексему «четырехугольник», имеются только в чувашском языке. Ассоциативно-образное восприятие геометрических фигур связано с особенностями менталитета этноса и раскрывается в национально-культурной коннотации данных лексических единиц. The paper is devoted to the analysis of the Russian and Chuvash lexical units that represent the cluster “geometric figures” from the point of view of its reflection in the minds of two language communities. At the center of our linguistic and cultural analysis are paremias, traditional folk symbols. The vocabulary of this thematic group helps to identify the specificity of the mentality of native speakers, allows us to highlight the peculiarities of the worldview of the Chuvash and Russian ethnic groups. The cluster “geometric shapes” was not studied either in relation to the Russian or Chuvash language pictures of the world. The research methods employed are the analysis of dictionary definitions, comparative, conceptual, linguoculturological analysis. The lexemes denoting geometric figures are uniquely interpreted in their secondary nomination by different peoples. In the study we focus at such elementary figures as circle, square (rectangle), triangle (angle/corner), and sphere. The geometric figure of a circle in the Chuvash language has a positive connotation, and in Russian both positive and negative connotations; the sphere is perceived negatively in both analyzed languages; the angle/corner in both Chuvash and Russian languages has both positive and negative connotations; emotional and evaluative shades in statements containing the lexeme of rectangular are available only in the Chuvash language. The associative-figurative perception of geometric figures is associated with the peculiarities of the mentality of an ethnic group and is revealed in the national-cultural connotation of these lexical units.

The Method of Computing Diameter of Nano Wood Powder Based on Geometric Figure Fitting

Computing the diameter of nanometer wood powder is the key step of intelligently acquiring wood powder mesh during processing and production in the wood powder manufacturing industry. To obtain the micro image of nano wood powder, the method of hole filling is adopted to fill the binary image of wood powder particles. The contours of wood powder particles are extracted with the use of the edge detection operator, and the control experiment is carried out accordingly. The shape line method is adopted while fitting the geometric shape of wood powder particles, and the longest side or diameter of the figure is solved so as to obtain the diameter. In addition, based on the conversion standard, the mesh number of particles is calculated. The method presented in this study is expected to facilitate the automation of wood powder pellet processing industry, whereas the method is also found to have optimal applicability and reference significance for the measurement of other sorts of particles.

Ideas for improving mathematics teaching in fourth grade of primary school by using the mathematical software GeoGebra

In the paper studying some spatial geometric figures in 4th grade mathematics education is explored. To do this the possibilities of GeoGebra are used. For each of the figures – rectangular parallelepiped, cube, pyramid, cylinder, cone, sphere, drawings are given in order to illustrate the geometric figure as well as to ensure following the basic rule in learning and understanding mathematical concepts – varying the insignificant properties of the concept to make its significant ones stand out.

Hole dissections for planar figures

A geometric dissection is a cutting of a geometric figure (or a finite set of figures) into pieces that we can rearrange to form another geometric figure (or finite set of figures). If our figures are required to be polygons, then there is always a dissection that has just a finite number of pieces. This was established by John Lowry [1], William Wallace [2], Farkas Bolyai [3], and Karl Gerwien [4]. The American Sam Loyd [5] and the Englishman Henry Ernest Dudeney [6, 7] emphasised the goal of minimising the number of pieces that resulted from such a standard dissection.

Final report from a registry study of patients with active cancer undergoing ecological staging.

e22501 Background: The host-cancer relationship can be viewed as a set of tumor supportive and inhibitory interactions, that is as an ecosystem. An advantage of such a perspective might include a more dynamic understanding of the selection pressures affecting tumor evolution. If so then a means of detecting and measuring those forces would be of help in understanding that drive and point to potential therapeutic clinical interventions. Towards this goal a registry study of patients with invasive cancer was conducted to examine readily available clinical measures to inform an ecological staging system. Methods: Eligibility for registry participation included age of 18 years or older, active cancer requiring treatment, life expectancy estimated to be at least six months and willingness to provide informed consent for this IRB-approved study. Ecological staging was then conducted at entry assessing 22 separate clinical parameters from history, physical exam and laboratory known to effect tumor biology, grouped into eight profiles and scored using a trichotomization of data, increasing score corresponding to an increase in positive tumor selection. To aggregate data from the profiles, a geometric figure, an octamer, or ecogram, was constructed, vertices representing the net score of each of the eight profiles. We hypothesize that the area bounded by the polygon, ecogram area (EA) would reflect an index of adverse, tumor-supporting ecological influence. Overall survival was determined from time of diagnosis of cancer until time of registry closure. Results: 15 subjects, eight female, seven male, ages 53-84 (median 64) participated in the registry. Complete ecological staging was obtained from all participating subjects, an indication of feasibility for collection of such data in the clinic. Resulting EA ranged 12-fold (0-12.1) across subjects suggesting sensitivity of EA to differences in selection pressures among the participants; ecogram morphology from individual subjects demonstrated unique shapes, an indication of specificity for character of individual tumor ecological landscape. Mean overall survival for participants from time of cancer diagnosis was 265 weeks. For 12 subjects with advanced stage disease, overall survival was inversely related to ecogram area (r = -.45). Conclusions: Ecological staging of cancer from clinical parameters proved feasible in this registry study of patients with active cancer. Ecological information from those parameters when aggregated using a geometric figure, an ecogram, demonstrated both sensitivity and specificity for differences among participants. For those subjects with advanced stage disease, ecogram area was inversely related to overall cancer survival. Ecological staging may be a useful tool in gauging selection pressures affecting cancer evolutionary tendency and of potential benefit to oncologists in forecasting such development.

On Unique and Nonunique Fixed Points in Metric Spaces and Application to Chemical Sciences

We introduce the notions of a generalized Θ -contraction, a generalized Θ E -weak contraction, a Ψ E -weak JS-contraction, an integral-type Θ E -weak contraction, and an integral-type Ψ E -weak JS-contraction to establish the fixed point, fixed ellipse, and fixed elliptic disc theorems. Further, we verify these by illustrative examples with geometric interpretations to demonstrate the authenticity of the postulates. The motivation of this work is the fact that the set of nonunique fixed points may include a geometric figure like a circle, an ellipse, a disc, or an elliptic disc. Towards the end, we provide an application of Θ -contraction to chemical sciences.

Seed Morphology in Key Spanish Grapevine Cultivars

Ampelography, the botanical discipline dedicated to the identification and classification of grapevine cultivars, was grounded on the description of morphological characters and more recently is based on the application of DNA polymorphisms. New methods of image analysis may help to optimize morphological approaches in ampelography. The objective of this study was the classification of representative cultivars of Vitis vinifera conserved in the Spanish collection of IMIDRA according to seed shape. Thirty eight cultivars representing the diversity of this collection were analyzed. A consensus seed silhouette was defined for each cultivar representing the geometric figure that better adjusted to their seed shape. All the cultivars tested were classified in ten morphological groups, each corresponding to a new model. The models are geometric figures defined by equations and similarity to each model is evaluated by quantification of percent of the area shared by the two figures, the seed and the model (J index). The comparison of seed images with geometric models is a rapid and convenient method to classify cultivars. A large proportion of the collection may be classified according to the new models described and the method permits to find new models according to seed shape in other cultivars.

Seed Morphology in Key Spanish Grapevine Cultivars

Ampelography, the botanical discipline dedicated to the identification and classification of grapevine cultivars, was grounded on the description of morphological characters and more recently is based on the application of DNA polymorphisms. New methods of image analysis may help to optimize morphological approaches in ampelography. The objective of this study was the classification of representative cultivars of Vitis vinifera conserved in the Spanish collection of IMIDRA according to seed shape. Thirty eight cultivars representing the diversity of this collection were analyzed. A consensus seed silhouette was defined for each cultivar representing the geometric figure that better adjusted to their seed shape. All the cultivars tested were classified in ten morphological groups, each corresponding to a new model. The models are geometric figures defined by equations and similarity to each model is evaluated by quantification of percent of the area shared by the two figures, the seed and the model (J index). The comparison of seed images with geometric models is a rapid and convenient method to classify cultivars. A large proportion of the collection may be classified according to the new models described and the method permits to find new models according to seed shape in other cultivars.

Discovering geometry via the Discover command in GeoGebra Discovery

We present a new way to discover statements in a planar geometric figure by using GeoGebra Discovery, an experimental version of GeoGebra, the free dynamic mathematics software package. A new command "Discover" (which is also available as a tool) requires an input point of the figure---as output several properties of the figure are communicated by the program. That is, "Discover" reports a list of the observed geometric properties, including point equality, equal long segments, collinearity, concyclicity, parallelism and perpendicularity. All of the obtained statements are checked symbolically: this means that the verification is done with computer algebra means. The obtained properties are also highlighted with colors or dashed lines in the original figure. The discovery process can always be continued by creating new objects and selecting a new target point to discover. We focus on possible uses in a classroom: two basic examples are shown from an Austrian textbook first. Then some more difficult topics are introduced that are usually covered by the secondary school curriculum. As a final example, we consider the discovery of a more advanced theorem, namely, a proposition according to Napoleon. We learn that discovery can lead to unexpected results, but this is an important characteristic of mathematics. In the paper we give some references to related software systems and the applied mathematical background as well.