Monochromatic equilateral triangles in the unit distance graph

Monochromatic equilateral right triangles on the integer grid

For any coloring of the N×N grid using less than log log n colors, one can always find a monochromatic equilateral right triangle, a triangle with vertex coordinates (x, y), (x + d, y), and (x, y + d). AMS Mathematics Subject Classification: 05D10

Packing and Covering a Unit Equilateral Triangle with Equilateral Triangles

Packing and covering are elementary but very important in combinatorial geometry , they have great practical and theoretical significance. In this paper, we discuss a problem on packing and covering a unit equilateral triangle with smaller triangles which is originated from one of Erd˝ os' favorite problems.

Equilateral Triangles in Finite

Translative Packing of Unit Squares into Equilateral Triangles

Every collection of n (arbitrary-oriented) unit squares can be packed translatively into any equilateral triangle of side length 2.3755 ̈ ? n. Let the coordinate system in the Euclidean plane be given. For 0 ≤ αi ă π{2, denote by Spαiq a square in the plane with sides of unit length and with an angle between the x-axis and a side of Spαiq equal to αi. Furthermore, let T psq be an equilateral tr...

Counting All Equilateral Triangles In

We describe a procedure of counting all equilateral triangles in the three dimensional space whose coordinates are allowed only in the set {0, 1, ..., n}. This sequence is denoted here by ET (n) and it has the entry A102698 in “The On-Line Encyclopedia of Integer Sequences”. The procedure is implemented in Maple and its main idea is based on the results in [3]. Using this we calculated the valu...