Related papers: The ancient Greeks present: Rational Trigonometry
We show that main results of rational trigonometry (as developed by NJ Wildberger, "Divine Proportions", 2005) can be succinctly expressed using projective geometric algebra (PGA). In fact, the PGA representation exhibits distinct…
The main result of this paper, is the complete parametric description of the family of triangles which have integer sidelengths and with one angle being sixty degrees.
As one type of incidence theory, the geometry of pentagram map seems quite classical at first. However, this is an excellent example of such a classical idea developed into a marvellous insight by some modern approach. We introduce an…
By Fermat's method, we show that there are infinitely many Heron triangle and $\theta$-integral rhombus pairs with a common area and a common perimeter. Moreover, we prove that there does not exist any integral isosceles triangle and…
One of the important ways development takes place in mathematics is via a process of generalization. On the basis of a recent characterization of this process we propose a principle that generalizations of mathematical structures that are…
In a previous paper we have introduced the ortho-homological triangles, which are triangles that are orthological and homological simultaneously. In this article we call attention to two remarkable ortho-homological triangles (the given…
Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we study a two-parameter family of roughly hexagonal regions in the hexagonal grid and…
When considering geometry, one might think of working with lines and circles on a flat plane as in Euclidean geometry. However, doing geometry in other spaces is possible, as the existence of spherical and hyperbolic geometry demonstrates.…
In this paper we investigate the evolution of the concept of area in Peano's works, taking into account the main role played by Grassmann's geometric-vector calculus and Peano's theory on derivative of measures. Geometric (1887) and…
In Euclidean space, one can use the dot product to give a formula for the area of a triangle in terms of the coordinates of each vertex. Since this formula involves only addition, subtraction, and multiplication, it can be used as a…
The aim of this work is to use Napoleon's Theorem in different regular polygons, and decide whether we can prove Napoleon's Theorem is only limited with triangles or it could be done in other regular polygons that can create regular…
This is an innovative treatise on triangles, resting upon 1) 3-body problem techniques including mass-weighted relative Jacobi coordinates. 2) Part I's detailed layer-by-layer topological and geometrical study of Kendall-type shape spaces -…
Richard Guy asked the following question: can we find a triangle with rational sides, medians, and area? Such a triangle is called a \emph{perfect triangle} and no example has been found to date. It is widely believed that such a triangle…
For three-body problem, R.Montgomery [3] proved a reconstruction formula which calculates the overall rotation relating two similar triangle configurations if the initial triangular configuration is similar to the configuration formed at…
The so-called Riemann sums have their origin in the efforts of Greek mathematicians to find the center of gravity or the volume of a solid body. These researches led to the method of exhaustion, discovered by Archimedes and described using…
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines…
Plutarchus, circa 100 AD, in his early book on "astrophysics" --in which he exposed, in a sense, a general theory of gravitation-- wrote the noticeable passage: <<The Moon gets the guarantee of not falling down just from its motion and from…
The goal of this text is to present the computation by Salmon, in the second half of the XIXth century, of various numbers enumerating planes with a prescribed tangency pattern with a sufficiently general surface $S$ in $\mathbf{P}^3$ (or,…
We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…