Icosahedron Inscribed in an Affine Regular Octahedron in a GS-Quasigroup

A golden section quasigroup or shortly a GS-quasigroup is an idempotent quasigroup which satisfies the identities
a(ab\(\cdot\)c)\(\cdot\)c=a\(\cdot\)(a\(\cdot\)bc)c=b. The concept of a GS-quasigroup was introduced by VOLENEC. A number of geometric concepts can be introduced in a general GS-quasigroup by means of the binary quasigroup operation. In this paper, it is proved that for any affine regular octahedron there is an affine regular icosahedron which is inscribed in the given affine regular octahedron. This is proved by means of the identities and relations which are valid in a general GS-quasigroup. The geometrical presentation in the GS-quasigroup \(\mathbb{C}(\frac{1}{ 2} (1+\sqrt{5}))\) suggests how a geometrical consequence may be derived from the statements proven in a purely algebraic manner.

Key words: GS-quasigroup, GS-trapezoid, affine regular icosahedron, affine regular octahedron

Trigonometric Functions in the m-plane

In this paper, we define the trigonometric functions in the plane with the m-metric. And then we give two properties about these trigonometric functions, one of which states the area formula for a triangle in the m-plane in terms of the m-metric.

Key words: Taxicab metric, Chinese checker metric, alpha metric, m-metric, m-trigonometry
circles

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On Brocard Points of Harmonic Quadrangle in Isotropic Plane

In this paper we present some new results on Brocard points of a harmonic quadrangle in isotropic plane. We construct new harmonic quadrangles associated to the given one and study their properties dealing with Brocard points.

Key words: isotropic plane, harmonic quadrangle, Brocard points

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The Loci of Vertices of Nedian Triangles

In this article we observe nedians and nedian triangles of ratio \(\eta\) for a given triangle. The locus of vertices of the nedian triangles for \(\eta\in \mathbb{R}\) is found and its correlation with isotomic conjugates of the given triangle is shown. Furthermore, the curve on which lie vertices of a nedian triangle for  fixed  \(\eta\), when we iterate nedian triangles, is found.

Key words: triangle, cevian, nedian, nedian triangle, isotomic conjugate

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Non-standard Aspects of Fibonacci Type Sequences

Fibonacci sequence and the limit of the quotient of adjacent Fibonacci numbers, namely the Golden Mean, belong to general knowledge of almost anybody, not only of mathematicians and geometers. There were several attempts to generalize these fundamental concepts which also found applications in art and architecture, as e.g. number series and quadratic equations leading to the so-called "Metallic means" by V. DE SPINADEL [8] or the cubic "plastic number" by VAN DER LAAN [5] resp. the "cubi ratio" by L. ROSENBUSCH [7]. The mentioned generalizations consider series of integers or real numbers. "Non-standard aspects" now mean generalizations with respect to a given number field or ring as well as visualizations of the resulting geometric objects. Another aspect concerns Fibonacci type resp. Padovan type combinations of given start objects. Here it turns out that the concept "Golden Mean" or "van der Laan Mean" also makes sense for vectors, matrices and mappings.

Key words: generalized Fibonacci sequence, Golden Mean, non-Euclidean geometry, number field, ring

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Generalized Conchoids

We adapt the classical definition of conchoids as known from the Euclidean plane to geometries that can be modeled within quadrics. Based on a construction by means of cross ratios, a generalized conchoid transformation is obtained. Basic properties of the generalized conchoid transformation are worked out. At hand of some prominent examples - line geometry and sphere geometry - the actions of these conchoid transformations are studied. Linear and also non-linear transformations are presented and relations to well-known transformations are disclosed.

Key words: conchoid transformation, line geometry, sphere geometry, cross ratio, regulus, Dupin cyclide, Laguerre transformation, equiform transformation, inversion

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Rational Trigonometry in Higher Dimensions and a Diagonal Rule for 2-planes in Four-dimensional Space

We extend rational trigonometry to higher dimensions by introducing rational invariants between k-subspaces of n-dimensional space to give an alternative to the canonical or principal angles studied by Jordan and many others, and their angular variants. We study in particular the cross, spread and det-cross of 2-subspaces of four-dimensional space, and show that Pythagoras theorem, or the Diagonal Rule, has a natural generalization for such 2-subspaces.

Key words: rational trigonometry, subspaces, canonical angles, Diagonal rule, spread, cross

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Survey Analysis of the Great Pyramid

The topic of this paper is an analysis of the survey of Cheops pyramid (also known as the Great pyramid), the most significant of the three pyramids of the Giza complex, the archeological site on the plateau of Giza, situated on the periphery of Cairo. It is assumed that Cheops as well as Khafre and Menkaure pyramids were built around 2686 - 2181 BC, known in the history as the Old Kingdom of Egypt. Our goal was to collect data about geodetic survey of Cheops pyramid and analyze it. Along with that, several hypotheses related to the construction method of the pyramid and possible purposes of the construction itself are described. When analyzing the survey, two numbers, also called "two treasures of geometry", are constantly appearing, these are the number Pi (\(\pi\)) and the Golden ratio or golden number Fi (\(\varphi\)). One of the chapters is dedicated to these numbers.

Key words: geodetic survey, analysis of collected data, Ludolph's constant, Golden ratio

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