4,493 research outputs found
On the relationship between plane and solid geometry
Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned area
Geometry in the Transition from Primary to Post-Primary
This article is intended as a kind of precursor to the document Geometry for
Post-primary School Mathematics, part of the Mathematics Syllabus for Junior
Certicate issued by the Irish National Council for Curriculum and Assessment in
the context of Project Maths.
Our purpose is to place that document in the context of an overview of plane
geometry, touching on several important pedagogical and historical aspects, in
the hope that this will prove useful for teachers.Comment: 19 page
Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry
A new method to obtain trigonometry for the real spaces of constant curvature
and metric of any (even degenerate) signature is presented. The method
encapsulates trigonometry for all these spaces into a single basic
trigonometric group equation. This brings to its logical end the idea of an
absolute trigonometry, and provides equations which hold true for the nine
two-dimensional spaces of constant curvature and any signature. This family of
spaces includes both relativistic and non-relativistic homogeneous spacetimes;
therefore a complete discussion of trigonometry in the six de Sitter,
minkowskian, Newton--Hooke and galilean spacetimes follow as particular
instances of the general approach. Any equation previously known for the three
classical riemannian spaces also has a version for the remaining six
spacetimes; in most cases these equations are new. Distinctive traits of the
method are universality and self-duality: every equation is meaningful for the
nine spaces at once, and displays explicitly invariance under a duality
transformation relating the nine spaces. The derivation of the single basic
trigonometric equation at group level, its translation to a set of equations
(cosine, sine and dual cosine laws) and the natural apparition of angular and
lateral excesses, area and coarea are explicitly discussed in detail. The
exposition also aims to introduce the main ideas of this direct group
theoretical way to trigonometry, and may well provide a path to systematically
study trigonometry for any homogeneous symmetric space.Comment: 51 pages, LaTe
Hyperbolic geometry in the work of Johann Heinrich Lambert
The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is
one of the founding texts of hyperbolic geometry, even though its author's aim
was, like many of his pre-decessors', to prove that such a geometry does not
exist. In fact, Lambert developed his theory with the hope of finding a
contradiction in a geometry where all the Euclidean axioms are kept except the
parallel axiom and that the latter is replaced by its negation. In doing so, he
obtained several fundamental results of hyperbolic geometry. This was sixty
years before the first writings of Lobachevsky and Bolyai appeared in print. In
the present paper, we present Lambert's main results and we comment on them. A
French translation of the Theorie der Parallellinien, together with an
extensive commentary, has just appeared in print (A. Papadopoulos and G.
Th{\'e}ret, La th{\'e}orie des lignes parall{\`e}les de Johann Heinrich
Lambert. Collection Sciences dans l'Histoire, Librairie Scientifique et
Technique Albert Blanchard, Paris, 2014)
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