2013/03/23 by Mieczkowski, Edward
#05B45 #05E99 #51M20 #52C20 #Combinatorics (math.CO) #F.2.2 #FOS: Mathematics #G.2.0 #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1303.5865
In this paper, we consider a set of similar triangles with parallel sides, along with a set of points in the plane. It turns out that the set ℝ2= \± =± (x2,x,1); x∈ℝ \ describes this set of triangles quite well. The set ℝ2 is a subset of the ring ℝ3=ℝ×ℝ×ℝ= \ (x,y,z) ; x,y,z∈ℝ \ with addition and multiplication defined coordinate-wise. The set ℝ2 is equipped with two operations. Multiplication is inherited from the ring ℝ3, while addition is a ternary operation that represents homothety and translation of elements in ℝ2. However, the defined addition has its limitations. It turns out that, within this framework, the reduction of terms with different signs is not always possible. This leads to the distinction between an equation that is true in the arithmetic sense and one that is true in the geometric sense. A novel form of addition in ℝ2 leads to intriguing properties of multiplication in ℝ2, which are examined in a dedicated chapter. In the next section we use the construction of adding to describe the dissection of the triangle into 15 triangles of different sides. In the final two sections, we consider a set of two kinds of vectors, along with a set of points on the line. The set ℝ1= \± =± (x,1); x∈ℝ \ describes this set vectors quite well and it is a one-dimensional reduction of the set ℝ2.