Volume X , issue 1 ( 2007 ) | back |

Hexagonal systems. A chemistry-motivated excursion to combinatorial geometry | 1$-$10 |

**Abstract**

Hexagonal systems are geometric objects obtained by arranging mutually congruent regular hexagons in the plane. Such geometric features are usually not considered in secondary-school courses of mathematics, although these are easy-to-grasp and probably interesting to students. In this paper the elements of the theory of hexagonal systems are outlined. We indicate some properties of hexagonal systems that secondary-school students could prove themselves, as well as a few less easy results and open problems.

**Keywords:** Hexagonal Systems, Tilings, Combinatorial Geometry, Benzenoid Hydrocarbons, Chemistry.

Didactical analysis of primary geometric concepts II | 11$-$36 |

**Abstract**

Surveyed in historical perspective, the contents of school geometry can be sorted into the following three stages: intuitive, pre-Euclidean and Euclidean. Geometric topics usually found in primary school programs constitute the first stage of intuitive geometry. Although this geometry is related to the perception of the surrounding real world, both the coherence and order in its exposition are equally important. This paper is devoted to that didactical task. At this first stage, pairings of the real world appearances (objects and their collections) with geometric models (iconic signs and their configurations) make the main learning procedures. These pairings are interactive, meaning that the appearances ``re-animate'' the models and the latter serve to the intelligible conceiving of the former. In order to discriminate between the two sides of this process of pairing, we consider two levels of intuitive geometry. When conveyors of geometric meaning are real world appearances, we speak of inherent geometry and when such conveyors are configurations of iconic signs, we speak of visual geometry. These latter signs (called here ideographs) express the meaning of geometric concepts and their deliberate use leads to the assimilation of this meaning. Following Poincaré's views on genesis of geometric ideas, we formulate a cognitive principle stating that perception of solids in the outer space, in the way when all their physical properties are abstracted (ignored), leads to the creation of these ideas. In addition, an intelligent ignoring of extension leads, then to the creation of concepts: point, line and surface. In the final didactical analysis, a series of basic geometrical concepts will be discussed (and this paper includes: point, line, segment, ray and straight line).

**Keywords:** Intuitive geometry, levels of inherent and visual geometry, principle of creation of
geometric ideas, point, line segment, ray, straight line.

Nekotorye didakticheskie aspekty primeneniya professional'nyh programmnyh paketov v prepodavanii matematiki v srednei xkole i universitetah | 37$-$50 |

**Abstract**

Possibilities of applying professional software packages in teaching of mathematics to upper high school students, as well as first grade universities students of technical and economy faculties are presented. A new paradigm of teaching of mathematics is discussed. Some examples are given.

**Keywords:** Professional software packages, MATHEMATICA.

Projections of the twisted cubic | 51$-$62 |

**Abstract**

One gets every cubic curve with rational parametrization by projecting the curve with general point $(t\mid t^2\mid t^3)$ in different ways onto planes. This result shows the power of elementary methods.

**Keywords:** Parabola; space curves; twisted cubic; orthographic projection; divergent parabolas;
trident curve.