Chorded keyboard (a writing system for the blind) uses either 6 or 8 tactile 'points' from which all letters and numbers are formed. When Louis Braille invented it, it was produced with a needle holing successively all needed points in a cardboard sheet. In 1892, Frank Haven Hall, superintendent of the Illinois Institute for the Education of the Blind, created the Hall Braille Writer, which was like a typewriter with 6 keys, one for each dot in a braille cell. The Perkins Brailler, first manufactured in 1951, uses a 6-key chord keyboard (plus a spacebar) to produce braille output, and has been
Dot product representation of a graph Dot product representation of a graph A dot product representation of a simple graph is a method of representing a graph using vector spaces and the dot product from linear algebra. Every graph has a dot product representation. Let "G" be a graph with vertex set "V". Let "F" be a field, and "f" a function from "V" to "F" such that "xy" is an edge of "G" if and only if "f"("x")·"f"("y") ≥ "t". This is the dot product representation of "G". The number "t" is called the dot product threshold, and the smallest possible value of "k" is