Stellae Octangulae in Motion Revisited

It is well-known that two congruent regular tetrahedra T₁ and T₂ forming a Stella Octangula allow a continuous motion of T₂ relative to T₁ such that each edge of T₂ slides along an edge of T₁. Recently the same property has been confirmed for pairs (T₁, T₂) of indirect congruent tetrahedra of general form. It turns out that this overconstrained kinematical systems admits besides some special one-parameter motions also two-parameter motions. We provide a synthetic analysis of the problem. Based on involved quadrics, we study in depth the two-parameter motions and their boundaries. Moreover, we present some generalizations of Stellae Octangulae.

Key words: tetrahedron, Stella Octangula, Euclidean motion, two-parameter movements

Article in PDF.

A Generalization of Archimedean Circles on an Arbelos

In this paper, we extend the classical notion of Archimedean circles, originally discovered by Archimedes in the arbelos, to the broader framework of the arbelos with overhang. By means of new constructions, we establish conditions under which circles in this generalized setting retain the characteristic radius property of Archimedean circles. Our results unify and extend previous findings, revealing deeper symmetries and structural invariants within these geometric figures.

Key words: Archimedean circles, Arbelos, Arbelos with Overhang

Article in PDF.

On Fuss’ Relations for Bicentric Polygons with an Odd Number of Vertices

This paper presents novel analytical forms of Fuss’ relations for bicentric polygons with an odd number of sides and higher rotation numbers. The method is based on Poncelet’s theorem and Radi´c’s theorem and conjecture concerning the connection between Fuss’ relations for different rotation numbers. Explicit analytical expressions are obtained for the bicentric triskaidecagon with k = 2,4,6 and for the bicentric pentadecagon with k = 2, while complete sets of relations are established for the bicentric heptadecagon (k = 1,2,3,4,5,6,7,8) and enneadecagon (k =1,2,3,4,5,6, 7,8,9). The proposed approach simplifies the derivation and enables a systematic extension of known Fuss’ relations to higher-order bicentric polygons and new rotation

Key words: bicentric polygon, Fuss’ relation, rotation number, triskaidecagon, pentadecagon, heptadecagon, enneadecagon

Article in PDF.

A Generalization of the Twin Circles of Archimedes

We consider the arbelos and generalize Archimedean circles and the twin circles of Archimedes.

Key words: arbelos, Archimedean circle, k-Archimedean circle, twin circles of Archimedes, k-Archimedean twins

Article in PDF.

Limits of Triangle Centers

The construction of a triangle center always produces central triangles which again allow for the construction of the respective center. Doing this infinitely many times may in some cases lead to a known triangle center, but in the vast majority, a new center will show up. The symbolic computational approach is limited in many cases due to the complexity of the computations. In order to overcome these difficulties, we shall start with numerical approaches towards several centers’ limits. This gives rise to some conjectures which later allow for an exact determination of the limit of a triangle center.

Key words: triangle center, iterated construction, numerical simulation, limit

Article in PDF.

A Geometric Construction of a Family of Keplerian Ellipses

We investigate a one-parameter family of Keplerian ellipses in a plane sharing a fixed focus F₁ and passing through a prescribed point P with identical instantaneous speed. By means of a purely geometric construction – the reflection of the ray F₁P in the tangent at P – the second focus F₂ is located on a circle f₂, yielding simple loci for the centers M and the secondary vertices C,D (both circles) and for the principal vertices A,B (conchoids of a circle). The family admits an envelope, itself an ellipse whose semiaxes are obtained in closed form. The configuration provides a direct geometric interpretation of the vis-viva relation: All members share the same semimajor axis a, and thus, the same orbital period. When rotated about the axis F₁P, the envelope ellipses form a family of confocal ellipsoids of revolution, thus connecting the planar Kepler construction with the classical geometry of quadrics.

Key words: Keplerian ellipses, envelope ellipse, conchoid, limacon, vis-viva relation, energy equation

Article in PDF.

Projective Parallelians and Related Porisms

We give a projective generalization of the construction of parallelians and the thus defined conics. To any properly chosen point P and line g in the plane of a triangle Δ = ABC, we construct six points that always lie on a conic P, the parallelian conic P of the pivot P with respect to Δ. Further, we find the parallelian tangent conic T , the parallelian inconic I , and two further conics D and J that are related in a natural way with Δ and P. Any pair out of these conics gives rise to a certain porism and even a chain of porisms by means of polarization. We study the regularity and singularity as well as the relative position of these conics with respect to the line g depending on the choice of P and g. We also give a detailed study of the sets of possible pivot points changing the triangle or hexagon porisms of any pair of conics into such with one-parameter families of quadrangles and pentagons.

Key words: parallelian, parallelian conic, porism, triangle cubic, triangle center, algebraic transformation

Article in PDF.

On Remarkable Properties of Number 2025

In this paper we investigate the number 2025 and visualize its regularity. This is a perfect square, but a deeper look reveals much more structure related to counting lattice points in polygons and polyhedra. We will also discuss the frequency of square and regular years and the uniqueness of such a regular year number.

Key words: figure numbers, trigonal numbers, tetrahedral numbers

Article in PDF.