recursively, to higher dimensions.

In 2D, every node has 2 components, namely $x$ and $y$. If we include one one component, $z$, and assign $1$ to all third components of the sensors, Lemma \[lem:hard2d\] remains true.

Every component, in the context of this paper, adds an extra dimension to the nodes. Thus, for higher dimensions, we are able to set the new component’s value to $1$. The new copies of the sensor nodes will not be neighbors of the older ones. Also, the flags will be preserved. Therefore, the problem remains -hard in higher dimensions.

Localization process
====================

In this section, we discuss the details of utilizing the grouping function. First, we describe an algorithm to localize two collinear groups with respect to each other. Then, we use the same idea to localize two coplanar groups with respect to each other. Finally, we show that our algorithm is able to perform localization easily where the quadrilateration algorithm fails.

For the sake of completeness, we give the quadrilateration algorithm in Algorithm \[alg:quadrialteration\]. We would like to point out that algorithms given in this section do not use any GPS, and therefore they can only produce a point formation isometric (i.e. up 