On a Covering Problem for Equilateral Triangles

On a Covering Problem for Equilateral Triangles

Let T be a unit equilateral triangle, and T1, . . . , Tn be n equilateral triangles that cover T and satisfy the following two conditions: (i) Ti has side length ti (0 < ti < 1); (ii) Ti is placed with each side parallel to a side of T . We prove a conjecture of Zhang and Fan asserting that any covering that meets the above two conditions (i) and (ii) satisfies ∑n i=1 ti ≥ 2. We also show that ...

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.

Quantization for uniform distributions on equilateral triangles

We approximate the uniform measure on an equilateral triangle by a measure supported on n points. We find the optimal sets of points (n-means) and corresponding approximation (quantization) error for n ≤ 4, give numerical optimization results for n ≤ 21, and a bound on the quantization error for n → ∞. The equilateral triangle has particularly efficient quantizations due to its connection with ...

Equilateral Triangles: A Challenge for Connectionist Vision

In this paper we explore the problem of dynamically computing visual relations in a connectionist system. The task. of detecting equilateral triangles from clusters of points is used to test our architecture. We argue that this is a difficult task for traditional feed-forward architectures although it is a simple task for people. Our solution implements a biologically inspired networlc which us...

Equilateral Triangles in Finite