Related papers: A different perspective on teaching Geometry at hi…
Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial…
This paper looks at how ancient mathematicians (and especially the Pythagorean school) were faced by problems/paradoxes associated with the infinite which led them to juggle two systems of numbers: the discrete whole/rationals which were…
Euclidean geometry is among the earliest forms of mathematical thinking. While the geometric primitives underlying its constructions, such as perfect lines and circles, do not often occur in the natural world, humans rarely struggle to…
Wigner found unreasonable the "effectiveness of mathematics in the natural sciences". But if the mathematics we use to describe nature is simply a coded expression of our experience then its effectiveness is quite reasonable. Its…
Toposes can be pictured as mathematical universes. Besides the standard topos, in which most of mathematics unfolds, there is a colorful host of alternate toposes in which mathematics plays out slightly differently. For instance, there are…
We distinguish the axiomatic study of proofs in geometry from study about geometry from general axioms for mathematics. We briefly report on an abuse of that distinction and its unfortunate effect on US high school education. We review a…
A physical theory of the world is presented under the unifying principle that all of nature is laid out before us and experienced through the passage of time. The one-dimensional progression in time is opened out into a multi-dimensional…
How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T-geometric one are considered. T-geometry is built only on the basis of information included in the metric (distance between two…
This book is a textbook for the course of foundations of geometry. It is addressed to mathematics students in Universities and to High School students for deeper learning the elementary geometry. It can also be used in mathematics coteries…
In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…
In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…
We present two new constructions in the usual euclidean plane. We only deal with 'Grecian Geometry', with this phrase we mean elementary geometry in the two-dimensional space R 2 . We describe and prove two propositions about 'projections'.…
Technology has helped to innovate in the teaching-learning process. Today's students are more demanding actors when it comes to the environment, they have at their disposal to learn, experiment and develop critical thinking. The area of…
The purpose of this thesis is to use the language of orbifold groupoids to describe the geometry and topology of orbifolds, highlighting advantages and disadvantages of this language as they arise.
Graphs are interesting structures: extremely useful to depict real-life problems, extremely easy to understand given a sketch, extremely complicated to represent formally, extremely complicated to compare. Phylogeny is the study of the…
There is a contemporary trend toward geometrizing all mathematical theories, as proposed by the Langlands program, and, by extension, physical theories as well. Within this paradigm, it becomes possible to represent physical objects as…
Cities are living organisms. They are out of equilibrium, open systems that never stop developing and sometimes die. The local geography can be compared to a shell constraining its development. In brief, a city's current layout is a step in…
The general concept of symmetry is realized in manifold ways in different realms of reality, such as plants, animals, minerals, mathematical objects or human artefacts in literature, fine arts and society. In order to arrive at a common…