Properties Of Euclidean Geometry Research Articles

Developmental differences in children's and adults' use of geometric information in map-reading tasks.

Using maps effectively requires the ability to scale distances while preserving angle and orientation, the three properties of Euclidean geometry. The aim of the current study was twofold: first, to examine how the ability to represent and use these Euclidean properties changes with development when scaling maps in object-to-object relationships and, second, to explore the effects on the scaling performance of two variables of the array of objects, type of angular configuration and relative vector length. To this end, we tested seventy-five 4-, 6-, and 8-year-old children, as well as twenty-five adults, in a simple completion task with different linear and triangular configurations of objects. This study revealed important developmental changes between 4 and 6 years of age and between 8 years of age and adulthood for both distance and angle representation, while it also showed that the configuration variables affected younger and older children's performances in different ways when scaling distances and preserving angles and orientation. This study was instrumental in showing that, from an early age, children are able to exploit an intrinsic system of reference to scale geometrical configurations of objects.

A Mathematical Model of Winter Route Counting: Checking for Falsification

In the course of our model experiment, an attempt was made to falsify (refute) a consequence from the mathematical justification of the winter route counting of hunting animals (WRC) – the consequence of the mathematical expectation of the proportionality coefficient of this counting being equal to π/2. A simulation was carried out in the MapInfo program. The essence of our digital experiment was as follows: in several places of the model territory, which was a circle 5 km in diameter, daily tracks of animals were placed. Next, a grid of parallel routes oriented along 12 equally spaced directions relative to each other was applied to the model territory. The frequency of the route grid (the distance between parallel routes) for different tracks was chosen so that the total number of all intersections of a single track in all directions was not less than a thousand. In total, 19 electronic daily tracks of various animal species were used. 95 different actual values of the proportionality coefficient were obtained for various locations of the tracks in the model territory, no violations of the above mentioned consequence were found. The results of our experiment made it possible to formulate a corollary arising from the fundamental properties of Euclidean geometry on the directly proportional dependence between the number of intersections between the lines of daily animal tracks and route lines and the product of the total lengths of these lines. The mathematical justification of WRC has once again passed a simulation test, already using the means of geographical information systems and data from satellite navigators. At present, there are no grounds for abandoning the practical use of winter route counting for game animals. All possible discrepancies in the estimates of the numbers of game animals determined by the WRC method with their actual values should be attributed to shortcomings in the direct organization of this counting rather than to its mathematical justification.